Content-Type: multipart/mixed; boundary="-------------0606080500408" This is a multi-part message in MIME format. ---------------0606080500408 Content-Type: text/plain; name="06-178.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-178.keywords" Navier-Stokes equations, random force, stationary measure, exponentiial mixing, random attractor, Strong Law of Large Numbers, Eulerian limit, Central Limit Theorem, balance relation, turbulence ---------------0606080500408 Content-Type: application/postscript; name="Paper.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Paper.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: main.dvi %%Pages: 50 0 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMMI10 CMR10 CMSY10 MSBM10 CMR7 EUFM10 CMMI7 CMEX10 %%+ CMMI12 CMSY7 MSBM7 CMMI5 CMR5 MSAM10 CMSY5 TeX-cmex7 CMBX10 CMMI8 %%+ CMSY8 CMR8 CMMI6 CMBX8 CMR6 CMSY6 %%DocumentPaperSizes: a4 %%EndComments %%BeginProcSet: PStoPS 1 15 userdict begin [/showpage/erasepage/copypage]{dup where{pop dup load type/operatortype eq{1 array cvx dup 0 3 index cvx put bind def}{pop}ifelse}{pop}ifelse}forall [/letter/legal/executivepage/a4/a4small/b5/com10envelope /monarchenvelope/c5envelope/dlenvelope/lettersmall/note /folio/quarto/a5]{dup where{dup wcheck{exch{}put} {pop{}def}ifelse}{pop}ifelse}forall /setpagedevice {pop}bind 1 index where{dup wcheck{3 1 roll put} {pop def}ifelse}{def}ifelse /PStoPSmatrix matrix currentmatrix def /PStoPSxform matrix def/PStoPSclip{clippath}def /defaultmatrix{PStoPSmatrix exch PStoPSxform exch concatmatrix}bind def /initmatrix{matrix defaultmatrix setmatrix}bind def /initclip[{matrix currentmatrix PStoPSmatrix setmatrix [{currentpoint}stopped{$error/newerror false put{newpath}} {/newpath cvx 3 1 roll/moveto cvx 4 array astore cvx}ifelse] {[/newpath cvx{/moveto cvx}{/lineto cvx} {/curveto cvx}{/closepath cvx}pathforall]cvx exch pop} stopped{$error/errorname get/invalidaccess eq{cleartomark $error/newerror false put cvx exec}{stop}ifelse}if}bind aload pop /initclip dup load dup type dup/operatortype eq{pop exch pop} {dup/arraytype eq exch/packedarraytype eq or {dup xcheck{exch pop aload pop}{pop cvx}ifelse} {pop cvx}ifelse}ifelse {newpath PStoPSclip clip newpath exec setmatrix} bind aload pop]cvx def /initgraphics{initmatrix newpath initclip 1 setlinewidth 0 setlinecap 0 setlinejoin []0 setdash 0 setgray 10 setmiterlimit}bind def end %%EndProcSet %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o paper.ps main %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2006.06.08:1049 %%BeginProcSet: tex.pro 0 0 %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 50 /element put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 54 /six put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put readonly def /FontBBox{-59 -250 1235 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 42 /arrowrighttophalf put dup 66 /B put dup 116 /t put dup 120 /x put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 24 /xi put dup 33 /omega put dup 66 /B put dup 78 /N put dup 84 /T put dup 110 /n put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 6 /Sigma put dup 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 61 /equal put dup 99 /c put dup 100 /d put dup 105 /i put dup 108 /l put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 17 /equivalence put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 55 /mapsto put dup 66 /B put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /gamma put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 33 /omega put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 66 /B put dup 67 /C put dup 69 /E put dup 72 /H put dup 75 /K put dup 78 /N put dup 81 /Q put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 110 /n put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 69 /E put dup 80 /P put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: TeX-cmex7 %!PS-AdobeFont-1.0: TeX-cmex7 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex7) def /FullName (TeX\040cmex7\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox {-14 -2954 1627 771} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 63 /perpendicular put readonly def /FontBBox{21 -944 1448 791}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 46 /lessorsimilar put dup 98 /subsetdbl put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 20 /kappa put dup 23 /nu put dup 33 /omega put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 78 /N put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 116 /t put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 63 /perpendicular put dup 67 /C put dup 72 /H put dup 76 /L put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 98 /hatwide put dup 101 /tildewide put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 15 /epsilon1 put dup 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z put dup 96 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 6 /Sigma put dup 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 76 /L put dup 80 /P put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 118 /v put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 47AEDE4191CCDB8B61345FD070FD30C4F375D8418DDD454729A251B3F61DAE7C 8882384282FDD6102AE8EEFEDE6447576AFA181F27A48216A9CAD730561469E4 78B286F22328F2AE84EF183DE4119C402771A249AAC1FA5435690A28D1B47486 1060C8000D3FE1BF45133CF847A24B4F8464A63CEA01EC84AA22FD005E74847E 01426B6890951A7DD1F50A5F3285E1F958F11FC7F00EE26FEE7C63998EA1328B C9841C57C80946D2C2FC81346249A664ECFB08A2CE075036CEA7359FCA1E90C0 F686C3BB27EEFA45D548F7BD074CE60E626A4F83C69FE93A5324133A78362F30 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MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 28 /lessmuch put dup 29 /greatermuch put dup 33 /arrowright put dup 37 /arrownortheast put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 8 /Phi put dup 10 /Omega put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 76 /L put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 42 /arrowrighttophalf put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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003FC000000007FC00003FC000000003FE00003FC000000003FE00003FC000000003FE00 003FC000000001FF00003FC000000001FF00003FC000000001FF00003FC000000001FF00 003FC000000001FF80003FC000000000FF80003FC000000000FF80003FC000000000FF80 003FC000000000FF80003FC000000000FF80003FC000000000FF80003FC000000000FF80 003FC000000000FF80003FC000000000FF80003FC000000000FF80003FC000000000FF80 003FC000000000FF80003FC000000001FF00003FC000000001FF00003FC000000001FF00 003FC000000001FF00003FC000000001FF00003FC000000003FE00003FC000000003FE00 003FC000000003FC00003FC000000007FC00003FC000000007FC00003FC00000000FF800 003FE00000000FF000003FE00000001FF000003FF00000003FE000003FF80000003FC000 003FF80000007FC000003FFC000000FF8000003FDE000001FF0000003FCF000003FE0000 003FC7800007FC0000003FC3C0000FF00000003FC1F0003FE00000003FC07E01FF800000 003FC03FFFFE000000003FC00FFFF8000000003FC000FF80000000003FC0000000000000 003FC0000000000000003FC0000000000000003FC0000000000000003FC0000000000000 003FC0000000000000003FC0000000000000003FC0000000000000003FC0000000000000 003FC0000000000000003FC0000000000000003FC0000000000000003FC0000000000000 003FC0000000000000003FC0000000000000003FC0000000000000003FC0000000000000 003FC0000000000000003FC0000000000000003FC0000000000000003FC0000000000000 007FE000000000000000FFF0000000000000FFFFFFF00000000000FFFFFFF00000000000 FFFFFFF00000000000FFFFFFF00000000000415B7CBE4B>I<000F0007F8000FFF001FFE 00FFFF007FFF80FFFF00F81FC0FFFF01E03FC0FFFF03C07FE003FF07807FE000FF0F007F E000FF0E007FE0007F1C007FE0007F18003FC0007F38001F80007F30000600007F700000 00007F70000000007F60000000007FE0000000007FE0000000007FC0000000007FC00000 00007FC0000000007FC0000000007FC0000000007F80000000007F80000000007F800000 00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000 00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000 00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000 00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000 00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000 00007F8000000000FFC000000001FFF0000000FFFFFFF80000FFFFFFF80000FFFFFFF800 00FFFFFFF800002B3F7BBE34>114 D<0000FFC00300000FFFF80700007FFFFE0F0001FF 003F9F0003F00007DF0007E00003FF000F800000FF001F8000007F003F0000007F003E00 00003F007E0000001F007C0000001F00FC0000000F00FC0000000F00FC0000000F00FC00 00000F00FC0000000700FE0000000700FE0000000700FF0000000700FF80000007007FC0 000007007FE0000000007FF8000000003FFE000000001FFFE00000000FFFFF0000000FFF FFF0000003FFFFFE000001FFFFFF8000007FFFFFE000001FFFFFF8000007FFFFFC000000 FFFFFE00000007FFFF000000003FFF8000000007FF8000000001FFC0000000007FC0E000 00003FE0E00000001FE0E00000001FF0F00000000FF0F000000007F0F000000007F0F000 000007F0F800000003F0F800000003F0F800000003F0FC00000003F0FC00000003F0FE00 000003E0FE00000007E0FF00000007E0FF00000007C0FF8000000FC0FFC000000F80FFE0 00001F80FDF000003F00F8F800007E00F87C0001FC00F01F801FF000F00FFFFFC000E003 FFFF0000C0007FF000002C417CBF35>I<00007000000000007000000000007000000000 00700000000000700000000000700000000000700000000000F00000000000F000000000 00F00000000000F00000000000F00000000001F00000000001F00000000001F000000000 03F00000000003F00000000007F00000000007F0000000000FF0000000000FF000000000 1FF0000000003FF0000000007FF000000000FFF000000003FFF00000000FFFFFFFFF00FF FFFFFFFF00FFFFFFFFFF00FFFFFFFFFF00000FF0000000000FF0000000000FF000000000 0FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000000000 0FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000000000 0FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000000000 0FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000000000 0FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00001C000 0FF00001C0000FF00001C0000FF00001C0000FF00001C0000FF00001C0000FF00001C000 0FF00001C0000FF00001C0000FF00001C0000FF00001C0000FF00001C0000FF00001C000 0FF00001C00007F80003800007F80003800007F80003800007F80007800003FC00070000 03FC000F000001FE000E000000FF001E0000007F803C0000003FE0F80000001FFFF00000 0007FFC000000000FF00002A597ED734>I121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end userdict/PStoPSxform PStoPSmatrix matrix currentmatrix matrix invertmatrix matrix concatmatrix matrix invertmatrix put %%EndSetup %%Page: (0,1) 1 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 1 0 bop 165 872 a FM(Randomly)46 b(forced)f(nonlinear)i (PDEs)d(and)h(statistical)489 1054 y(h)l(ydro)t(dynamics)g(in)h(2)f (space)g(dimensions)1300 1295 y FL(Sergei)32 b(Kuksin)992 1411 y(Departmen)m(t)h(of)f(Mathematics)843 1528 y(Heriot\025W)-8 b(att)31 b(Univ)m(ersit)m(y)-8 b(,)33 b(Edin)m(burgh)1516 1644 y(and)892 1760 y(Steklo)m(v)g(Institute)g(of)f(Mathematics)1097 1876 y(8)g(Gubkina)g(St.,)h(Mosco)m(w)1004 1992 y(email:)41 b(kuksin@ma.h)m(w.ac.uk)0 2326 y FK(Con)l(ten)l(ts)0 2509 y FJ(0)77 b(In)m(tro)s(duction)2490 b(3)0 2691 y(1)77 b(F)-8 b(unction)31 b(spaces)2359 b(6)125 2791 y FI(1.1)83 b(F)-7 b(unction)28 b(spaces)f(for)g(functions)h(of)f FH(x)g FI(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)133 b(6)125 2891 y(1.2)83 b(F)-7 b(unctions)28 b(of)g FH(t)f FI(and)h FH(x)62 b FI(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)133 b(8)0 3073 y FJ(2)77 b(The)31 b(deterministic)f(2D)h(Na)m(vier-Stok)m (es)h(Equation)1058 b(10)125 3173 y FI(2.1)83 b(Lera)n(y)26 b(decomp)r(osition)51 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)92 b(10)125 3273 y(2.2)83 b(Prop)r(erties)27 b(of)h(the)g (nonlinearit)n(y)22 b FH(B)1413 3271 y(B)1415 3273 y(B)28 b FI(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(13)125 3372 y(2.3)83 b(The)28 b(existence)f(and)h(uniqueness)f(theorem)49 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)f(.)h(.)92 b(15)125 3472 y(2.4)83 b(Impro)n(ving)27 b(the)g(smo)r(othness)g(of)h(solutions)50 b(.)42 b(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(19)125 3572 y(2.5)83 b(The)28 b(NS)g(semigroup)61 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(23)125 3671 y(2.6)83 b(Singular)27 b(forces)75 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(24)125 3771 y(2.7)83 b(Some)28 b(h)n(ydro)r(dynamical)e(terminology) 56 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)92 b(27)0 3953 y FJ(3)77 b(Random)30 b(kic)m(k-forces)2148 b(29)125 4053 y FI(3.1)83 b(Ingredien)n(ts)27 b(for)g(the)h(constructions)73 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(29)125 4153 y(3.2)83 b(The)28 b(kic)n(k)n(ed)f(NSE)85 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(30)125 4252 y(3.3)83 b(Stationary)27 b(measures)64 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(32)125 4352 y(3.4)83 b(More)27 b(estimates)57 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(33)0 4535 y FJ(4)77 b(White-forced)31 b(equations)1995 b(35)125 4634 y FI(4.1)83 b(White)29 b(in)f(time)g(forces)58 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(35)125 4734 y(4.2)83 b(The)28 b(white-forced)f(2D)g(NSE)57 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(36)1574 5059 y(1)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 2 1 bop 125 523 a FI(4.3)83 b(Estimates)28 b(for)f(solutions)36 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.) 92 b(38)125 623 y(4.4)83 b(Stationary)27 b(measures)64 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(40)125 722 y(4.5)83 b(High-frequency)27 b(random)g(kic)n(ks)65 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(41)0 905 y FJ(5)77 b(Preliminaries)28 b(from)j(the)h(measure)e(theory)1344 b(44)125 1005 y FI(5.1)83 b(W)-7 b(eak)28 b(con)n(v)n(ergence)d(of)i (measures)f(and)i(Lipsc)n(hitz-dual)f(distance)d(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(44)125 1104 y(5.2)83 b(V)-7 b(ariational)26 b(distance)78 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.) g(.)f(.)h(.)f(.)h(.)92 b(45)125 1204 y(5.3)83 b(Coupling)23 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)92 b(46)125 1303 y(5.4)83 b(Kan)n(toro)n(vic)n(h)25 b(functionals)67 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(47)0 1486 y FJ(6)77 b(Uniqueness)31 b(of)g(a)h(stationary)h(measure:) 40 b(kic)m(k-forces)975 b(48)125 1586 y FI(6.1)83 b(The)28 b(main)g(lemma)55 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) f(.)h(.)92 b(48)125 1685 y(6.2)83 b(W)-7 b(eak)28 b(solution)f(of)g (\(6.1\))60 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(50)125 1785 y(6.3)83 b(The)28 b(theorem)78 b(.)42 b(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(51)125 1885 y(6.4)83 b(Corollaries)25 b(from)j(the)f(theorem)75 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(55)125 1984 y(6.5)83 b(3D)28 b(NSE)g(with)h(small)e(random)f(kic)n(ks)40 b(.)i(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)92 b(56)125 2084 y(6.6)83 b(Stationary)27 b(measures)f(and)h (random)g(attractors)54 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)92 b(57)125 2183 y(6.7)83 b(App)r(endix:)38 b(Summary)27 b(of)h(the)g(pro)r(of)f(of)g(Theorem)g(6.4)40 b(.)i(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(58)0 2366 y FJ(7)77 b(Uniqueness)31 b(of)g(a)h(stationary)h(measure:) 40 b(white-forces)911 b(61)125 2466 y FI(7.1)83 b(The)28 b(main)g(theorem)64 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.) g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)92 b(61)125 2565 y(7.2)83 b(Stationary)27 b(measures)f(for)h (equation,)g(p)r(erturb)r(ed)h(b)n(y)f(high)g(frequency)h(kic)n(ks)50 b(.)41 b(.)h(.)92 b(63)0 2748 y FJ(8)77 b(Ergo)s(dicit)m(y)31 b(and)h(the)g(strong)f(la)m(w)h(of)g(large)f(n)m(um)m(b)s(ers)987 b(65)0 2931 y(9)77 b(The)31 b(martingale)g(appro)m(ximation)g(and)h(CL) -8 b(T)1292 b(68)0 3113 y(10)29 b(The)i(Eulerian)h(limit)2192 b(71)125 3213 y FI(10.1)41 b(White-forces,)27 b(prop)r(ortional)f(to)h (the)h(square-ro)r(ot)d(of)j(the)g(viscosit)n(y)72 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)92 b(71)125 3312 y(10.2)41 b(One)28 b(negativ)n(e)e(result)32 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)f(.)h(.)92 b(77)125 3412 y(10.3)41 b(Other)27 b(scalings)f(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(79)125 3512 y(10.4)41 b(Discussion)c(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(80)125 3611 y(10.5)41 b(Kic)n(k)n(ed)27 b(equations)54 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(80)0 3794 y FJ(11)29 b(Balance)j(relations)f(for)h(the)f (white\025forced)h(NSE)1152 b(83)125 3894 y FI(11.1)41 b(The)28 b(balance)f(relations)21 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)92 b(83)125 3993 y(11.2)41 b(The)28 b(co-area)d(form)i (of)h(the)g(balance)f(relations)76 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)92 b(86)0 4176 y FJ(12)29 b(Commen)m(ts)2521 b(89)1574 5059 y FI(2)p eop end PStoPSsaved restore %%Page: (2,3) 2 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 3 2 bop 0 523 a FK(0)131 b(In)l(tro)t(duction)0 705 y FI(This)33 b(short)f(b)r(o)r(ok)h(is)g(based)f(on)h(a)g (lecture\025course)e(on)h(the)i(randomly)e(forced)g(t)n(w)n (o\025dimensional)0 804 y(Na)n(vier\025Stok)n(es)21 b(equation)i(\(2D)h (NSE\))g(and)g(t)n(w)n(o\025dimensional)d(statistical)i(h)n(ydro)r (dynamics)f(whic)n(h)0 904 y(the)g(author)e(taugh)n(t)h(at)g (ETH-Z\374ric)n(h)f(during)h(the)h(win)n(ter)e(term)h(of)h(the)f(y)n (ear)f(2004/2005.)30 b(The)21 b(goal)0 1004 y(of)d(the)h(course)e(w)n (as)h(to)g(review)g(the)h(recen)n(t)e(progress)f(in)j(the)g(qualitativ) n(e)e(theory)h(of)g(randomly)g(forced)0 1103 y(nonlinear)29 b(PDE)j(\(esp)r(ecially)-7 b(,)31 b(the)f(2D)h(NSE\),)g(and)f(discuss)g (applications)f(of)h(the)h(corresp)r(onding)0 1203 y(results)36 b(to)g(the)h(2D)g(statistical)f(h)n(ydro)r(dynamics,)i(including)e(the) h(2D)g(turbulence.)64 b(The)36 b(b)r(o)r(ok,)0 1303 y(as)c(w)n(ell)h (as)f(the)h(lecture-course,)g(is)f(aimed)h(on)g(p)r(eople)g(with)g (some)f(bac)n(kground)f(in)i(PDE,)i(or)d(in)0 1402 y(probabilit)n(y)-7 b(,)29 b(or)g(in)h(ph)n(ysics.)43 b(F)-7 b(or)29 b(the)h(b)r(ene\034t)g (of)g(the)g(last)f(t)n(w)n(o)g(groups)g(of)g(readers)f(w)n(e)i (included)0 1502 y(in)e(the)g(b)r(o)r(ok)f(a)g(section)g(on)h(the)g (deterministic)f(2D)h(NSE.)125 1601 y(Due)20 b(to)h(the)g(luc)n(k)f(of) g(time,)i(the)f(lectures)f(and)g(this)h(b)r(o)r(ok)f(do)g(not)g (include)h(all)f(relev)-5 b(an)n(t)20 b(material.)0 1701 y(The)38 b(author)f(restricted)g(himself)h(to)g(the)g(results,)i (related)e(to)f(his)h(curren)n(t)f(scien)n(ti\034c)h(in)n(terests)0 1801 y(\026)d(the)g(statistical)g(h)n(ydro)r(dynamics)e(of)i(randomly)f (forced)h(t)n(w)n(o\025dimensional)e(\035uids.)59 b(So)35 b(some)0 1900 y(imp)r(ortan)n(t)d(relev)-5 b(an)n(t)32 b(topics)g(are)g(not)g(represen)n(ted)g(in)h(the)f(b)r(o)r(ok.)52 b(Probably)-7 b(,)33 b(the)g(most)f(serious)0 2000 y(omissions)g(are)g (the)i(results)f(on)g(the)g(free)g(NSE)i(with)e(random)g(initial)g (data.)54 b(Concerning)32 b(them)0 2100 y(w)n(e)38 b(refer)g(the)g (reader)f(to)i(the)f(b)r(o)r(oks)g([VF88])g(and)g([FMR)-7 b(T01].)69 b(Some)39 b(imp)r(ortan)n(t)f(results)f(on)0 2199 y(randomly)26 b(forced)g(2D)h(\035uids)h(also)e(are)g(not)h(co)n (v)n(ered)e(b)n(y)i(the)g(b)r(o)r(ok.)36 b(With)28 b(the)g(exception)e (of)h(v)n(ery)0 2299 y(short)g(Section)g(6.5)g(w)n(e)g(a)n(v)n(oid)f (the)h(randomly)g(forced)g(3D)g(NSE)h(since)g(not)f(m)n(uc)n(h)g(is)h (kno)n(wn)e(there,)0 2399 y(and)h(what)f(is)h(kno)n(wn)f(di\033ers)g (in)h(spirit)g(from)f(the)h(2D)g(results)f(w)n(e)h(are)e(in)n(terested) i(in.)36 b(See)27 b([Fla05)o(].)125 2498 y(The)c(b)r(o)r(ok)g(con)n (tains)f(only)h(rigorously)e(pro)n(v)n(en)g(theorems.)35 b(Connections)22 b(with)i(the)g(\(heuristic\))0 2598 y(theory)f(of)h(turbulence)g(are)f(reduced)g(to)h(short)f(discussions)g (on)g(relev)-5 b(ance)23 b(of)h(the)g(obtained)g(results)0 2697 y(to)j(the)g(theory)g(of)g(turbulence,)g(made)g(at)g(the)h(ends)f (of)g(the)g(main)g(sections.)36 b(There)27 b(w)n(e)g(sho)n(w)f(that)0 2797 y(the)19 b(theorems)g(form)f(a)h(rigorous)d(mathematical)j (foundation)f(for)h(the)g(theory)g(of)f(2D)h(space-p)r(erio)r(dic)0 2897 y(turbulence.)37 b(In)28 b(particular,)e(the)i(results)f(obtained) g(imply)h(that)125 2996 y(i\))h(when)f(time)h(gro)n(ws,)e(statistical)h (c)n(haracteristics)e(of)j(a)f(turbulen)n(t)h(\035o)n(w)f(stabilise)g (to)g(c)n(harac-)0 3096 y(teristics,)f(indep)r(enden)n(t)i(from)e(the)h (initial)f(v)n(elo)r(cit)n(y)g(\034eld)h(\(Sections)f(6,)h(7\);)125 3196 y(ii\))35 b(for)f(an)n(y)g(c)n(haracteristic)e(of)j(a)f(turbulen)n (t)h(\035o)n(w,)h(its)f(time-a)n(v)n(erage)d(equals)i(the)h(ensem)n (ble-)0 3295 y(a)n(v)n(erage)25 b(\(Section)i(8\);)125 3395 y(iii\))h(in)f(large)g(time-scale)f(the)i(turbulen)n(t)g(\035o)n (w)f(is)h(a)f(Gaussian)f(pro)r(cess)h(\(Section)g(9\).)125 3494 y(In)h(the)h(last)f(t)n(w)n(o)g(sections)g(of)g(the)h(b)r(o)r(ok)f (w)n(e)g(pro)n(v)n(e)f(and)h(discuss)g(some)g(recen)n(t)g(results,)g (whic)n(h)0 3594 y(seem)f(to)h(b)r(e)g(unkno)n(wn)f(to)g(exp)r(erts)g (in)h(turbulence.)37 b(Namely)-7 b(,)28 b(w)n(e)f(sho)n(w)g(that)125 3694 y(iv\))j(when)g(the)h(co)r(e\036cien)n(t)f(of)g(kinematic)h (viscosit)n(y)d(deca)n(ys)h(to)i(zero)e(and)h(the)g(random)g(force,)0 3793 y(applied)i(to)f(the)h(\035uid,)i(is)d(scaled)g(to)h(k)n(eep)f (the)h(energy)e(of)i(the)g(\035uid)g(of)g(order)e(one,)j(the)f (solution)0 3893 y(of)c(the)g(2D)g(NSE)h(con)n(v)n(erges)d(in)i (distribution)g(to)g(a)f(random)g(\034eld)i(suc)n(h)e(that)h(eac)n(h)g (its)g(realisation)0 3993 y(satis\034es)f(the)h(free)f(2D)h(Euler)f (equation)g(\(Section)h(10\),)125 4092 y(v\))22 b(stationary)f(in)h (space)g(and)g(time)h(solutions)f(of)g(randomly)f(forced)h(2D)g(NSE)i (satisfy)d(in\034nitely)0 4192 y(man)n(y)29 b(explicit)h(algebraic)e (relations)h(\(i.e.)44 b(\020space-p)r(erio)r(dic)28 b(2D)i(turbulence)g(is)g(in)n(tegrable\021;)f(Sec-)0 4291 y(tion)f(11\).)125 4391 y(The)f(results)g(i\)\025v\))h(follo)n(w)f (from)g(rigorous)e(analysis)h(of)i(the)f(randomly)g(forced)g(2D)g(NSE) 736 4574 y FG(_)-37 b FH(u)18 b FF(\000)g FH(\027)5 b FG(\001)p FH(u)18 b FG(+)g(\()p FH(u)g FF(\001)h(r)p FG(\))p FH(u)f FG(+)g FF(r)p FH(p)23 b FG(=)g FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)98 b FI(div)14 b FH(u)22 b FG(=)h(0)14 b FH(;)550 b FI(\(0.1\))0 4756 y(where)19 b FH(\021)k FI(is)d(a)f(random)g(\034eld.)34 b(Usually)20 b(the)g(equation)f(is)h(supplemen)n(ted)g(b)n(y)f(the)h(p) r(erio)r(dic)g(b)r(oundary)1574 5059 y(3)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 4 3 bop 0 523 a FI(conditions)1238 623 y FH(x)24 b FF(2)f FE(T)1442 588 y FD(2)1503 623 y FG(=)g FE(R)1651 588 y FD(2)1688 623 y FH(=)p FG(2)p FH(\031)s FE(Z)1877 588 y FD(2)1928 623 y FH(:)0 766 y FI(Eq.)51 b(\(0.1\))31 b(is)h(the)h(main)f(ob)5 b(ject,)33 b(studied)f(in)g(this) h(b)r(o)r(ok.)50 b(Sections)31 b(1\0255)g(con)n(tain)g(preliminaries,)0 866 y(and)f(the)g(rest)g(of)g(the)g(b)r(o)r(ok)f(\025)h(the)g(new)g (results)g(on)f(the)i(equation)e(\(whic)n(h)h(imply)h(the)f(assertions) 0 965 y(i\)\025v\))e(ab)r(o)n(v)n(e\).)125 1065 y(Most)e(of)h(the)h (results)e(in)i(Sections)f(6\0259)e(hold)i(true)g(for)g(eq.)36 b(\(0.1\))27 b(in)h(a)e(b)r(ounded)i(domain)f(with)0 1165 y(suitable)f(b)r(oundary)e(conditions)h(\(sa)n(y)-7 b(,)26 b(Diric)n(hlet\),)g(or)f(in)h(a)f(t)n(w)n(o-dimensional)f (compact)h(Riemann)0 1264 y(surface,)32 b(e.g.)50 b(in)32 b(a)g(sphere)f(\(if)i(the)g(action)e(of)h(the)g(Laplacian)f(on)h(v)n (ector-\034elds)e FH(u)p FG(\()p FH(x)p FG(\))j FI(is)f(de\034ned)0 1364 y(accordingly\).)i(More)24 b(generally)-7 b(,)24 b(the)h(results)f(hold)h(for)f(solutions)g(of)g(man)n(y)g(nonlinear)g (dissipativ)n(e)0 1463 y(equations)19 b(in)g(b)r(ounded)h(domains)f (\(or)g(in)g(a)g(torus\),)i(p)r(erturb)r(ed)f(b)n(y)f(a)g(random)f (force.)34 b(In)19 b(particular,)0 1563 y(for)27 b(the)h (reaction-di\033usion)e(equation)1175 1736 y FG(_)-38 b FH(u)18 b FF(\000)g FH(\027)5 b FG(\001)p FH(u)18 b FG(+)g FH(u)1621 1702 y FD(3)1681 1736 y FG(=)23 b FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))g(;)990 b FI(\(0.2\))0 1908 y(or)27 b(for)g(the)h(Ginsburg\025Landau)d(equation)1120 2081 y FG(_)-37 b FH(u)18 b FF(\000)g FH(\027)5 b FG(\001)p FH(u)18 b FG(+)g FH(i)p FF(j)p FH(u)p FF(j)1642 2047 y FD(2)1679 2081 y FH(u)k FG(=)h FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))p FH(;)936 b FI(\(0.3\))0 2254 y(where)27 b FH(u)p FG(\()p FH(t;)h(x)p FG(\))23 b FF(2)h FE(C)j FI(and)h(dim)23 b FH(x)h FF(\024)e FG(3)p FI(;)28 b(or)e(for)h(the)h (equation)1023 2426 y FG(_)-38 b FH(u)18 b FF(\000)g FG(\()p FH(\027)25 b FG(+)18 b FH(i)p FG(\)\001)p FH(u)g FG(+)g FH(i)p FF(j)p FH(u)p FF(j)1740 2392 y FD(2)1776 2426 y FH(u)23 b FG(=)f FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))p FH(;)839 b FI(\(0.4\))0 2599 y(where)27 b FH(u)p FG(\()p FH(t;)h(x)p FG(\))c FF(2)f FE(C)p FI(,)28 b(dim)c FH(x)g FF(\024)f FG(4)p FI(.)37 b(F)-7 b(rom)27 b(time)h(to)g(time)g(w) n(e)f(brie\035y)g(discuss)h(these)f(equations)g(and)0 2699 y(prop)r(erties)g(of)g(their)h(solutions,)e(similar)h(to)h(those)f (of)g(2D)h(NSE.)125 2798 y(In)c(con)n(trast,)g(the)h(results)f(of)g (Section)h(10)e(only)h(hold)g(for)g(eq.)36 b(\(0.1\))24 b(with)h FC(some)g FI(b)r(oundary)f(con-)0 2898 y(ditions.)36 b(F)-7 b(or)24 b(example,)h(they)h(do)e(not)h(apply)g(to)g(\(0.1\))g (with)g(the)g(Diric)n(hlet)g(b)r(oundary)g(conditions,)0 2998 y(but)h(they)g(hold)g(for)f(the)i(equation)e(on)g(a)h(Riemann)g (surface.)35 b(Moreo)n(v)n(er,)23 b(the)k(results)e(are)g(v)-5 b(alid)26 b(for)0 3097 y(some)h(other)g(equations.)36 b(In)28 b(particular)e(\026)h(for)h(eq.)36 b(\(0.4\).)125 3197 y(The)d(results)g(of)h(Section)g(11)e(are)h(the)h(most)g(rigid:)48 b(they)34 b(only)f(hold)h(for)f(the)h(2D)g(NSE)g(\(0.1\))0 3296 y(under)22 b(the)h(p)r(erio)r(dic)f(b)r(oundary)f(conditions)h (\(so)g(only)g(the)g(p)r(erio)r(dic)h(2D)f(turbulence)g(is)g(in)n (tegrable,)0 3396 y(cf.)37 b(v\))28 b(ab)r(o)n(v)n(e\).)125 3496 y(In)39 b(this)g(b)r(o)r(ok)f(w)n(e)h(do)f(not)h(discuss)g(prop)r (erties)e(of)i(equations)f(\(0.2\)\025\(0.4\))g(whic)n(h)h(ha)n(v)n(e)e (no)0 3595 y(pro)n(v)n(en)c(analogies)g(for)i(the)g(2D)g(NSE)h(\(e.g.,) h(see)e([Kuk97)n(,)h(Kuk99)n(])f(for)g(a)g(study)g(of)41 b(\(0.3\))35 b(when)0 3695 y FH(\027)e FF(!)28 b FG(0)p FI(\).)46 b(Similar,)31 b(w)n(e)f(do)h(not)f(touc)n(h)g(the)h(problem)f (of)h(Burgers)e(turbulence,)i(describ)r(ed)f(b)n(y)h(the)0 3795 y(randomly)26 b(forced)h(Burgers)f(equation)h(\(see)g([EKMS00]\).) 125 3894 y(W)-7 b(e)29 b(consider)f(t)n(w)n(o)h(classes)f(of)h(ransom)f (forces)g FH(\021)s FI(:)41 b(they)29 b(are)f(either)h(Gaussian)g (random)f(\034elds,)0 3994 y(smo)r(oth)35 b(in)g FH(x)h FI(and)f(white)g(as)f(functions)i(of)f FH(t)p FI(,)i(or)d(they)h(are)f (kic)n(k\025pro)r(cesses)e(as)j(functions)g(of)g FH(t)p FI(,)0 4093 y(smo)r(oth)21 b(in)h FH(x)p FI(.)35 b(In)21 b(the)h(former)e(case)h(the)g(equations)g(de\034ne)g(sto)r(c)n(hastic)g (\(in)g(Ito's)g(sense\))g(di\033eren)n(tial)0 4193 y(equations)d(in)i (function)f(spaces,)h(while)g(in)f(the)h(latter)f(case)f(they)h (de\034ne)h(Mark)n(o)n(v)d(c)n(hains)h(in)i(function)0 4293 y(spaces.)34 b(All)24 b(our)f(results,)g(apart)g(from)g(those)g (in)g(Section)h(11,)f(hold)g(for)g(b)r(oth)h(classes)e(of)h(the)h (forces.)0 4392 y(W)-7 b(e)21 b(think)f(that)h(this)f(is)g(imp)r(ortan) n(t)g(since)g(it)g(indicates)g(that)h(the)f(results)g(obtained)g(for)f (the)i(2D)f(NSE)0 4492 y(\(0.1\))27 b(are)g(not)g(prop)r(erties)g(of)h (a)f(sp)r(eci\034c)g(mo)r(del,)h(but)g(of)g(the)g(2D)f(statistical)g(h) n(ydro)r(dynamics.)125 4637 y FJ(Notations.)35 b FI(W)-7 b(e)28 b(de\034ne)g FE(Z)1034 4607 y FD(2)1034 4658 y(0)1094 4637 y FG(=)23 b FE(Z)1237 4607 y FD(2)1275 4637 y FF(nf)p FG(0)p FF(g)p FI(.)35 b(F)-7 b(or)26 b(a)i(Banac)n(h)e(space)h FH(X)34 b FI(w)n(e)27 b(set)1045 4810 y FB(B)1118 4822 y FA(r)1155 4810 y FG(\()p FH(X)7 b FG(\))23 b(=)f FF(f)p FH(x)h FF(2)h FH(X)c FF(j)14 b(k)p FH(x)p FF(k)1853 4822 y FA(X)1938 4810 y FF(\024)23 b FH(r)r FF(g)14 b FH(:)1574 5059 y FI(4)p eop end PStoPSsaved restore %%Page: (4,5) 3 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 5 4 bop 0 523 a FI(By)41 b FF(D)r FG(\()p FH(\030)t FG(\))h FI(w)n(e)f(denote)g(the)h(distribution)f(of)g(a)g (random)f(v)-5 b(ariable)41 b FH(\030)t FI(.)78 b(Eac)n(h)41 b(metric)g(space)f FH(M)0 623 y FI(is)c(pro)n(vided)f(with)h(the)h FH(\033)s FI(-algebra)d(of)i(its)g(Borel)f(sets)g FF(B)s FG(\()p FF(M)p FG(\))h FI(\(so)f(`measurable')g(means)h(`Borel-)0 722 y(measurable'\).)57 b(W)-7 b(e)35 b(denote)g(b)n(y)f FH(C)1149 734 y FA(b)1183 722 y FG(\()p FH(M)9 b FG(\))35 b FI(the)g(space)f(of)h(b)r(ounded)g(con)n(tin)n(uous)f(function)h(on)f FH(M)9 b FI(,)0 822 y(b)n(y)32 b FF(M)p FG(\()p FH(M)9 b FG(\))33 b FI(\026)f(the)h(set)g(of)f(\034nite)h(signed)f(Borel)f (measures,)i(and)f(b)n(y)g FF(P)7 b FG(\()p FH(M)i FG(\))33 b FI(\026)f(the)h(probabilit)n(y)0 922 y(Borel)h(measures)g(on)h FH(M)9 b FI(.)60 b(F)-7 b(or)35 b FH(f)45 b FF(2)36 b FH(C)1281 934 y FA(b)1315 922 y FG(\()p FH(M)9 b FG(\))36 b FI(and)f FH(\026)h FF(2)g(M)p FG(\()p FH(M)9 b FG(\))36 b FI(w)n(e)f(de\034ne)71 b FG(\()p FH(f)t(;)14 b(\026)p FG(\))37 b(=)e(\()p FH(\026;)14 b(f)9 b FG(\))36 b(=)0 951 y Fz(Z)46 1139 y FA(M)134 1064 y FH(f)9 b FG(\()p FH(u)p FG(\))14 b FH(\026)p FG(\()p FH(du)p FG(\))g FH(:)27 b FI(By)g FH(I)745 1076 y FA(Q)830 1064 y FI(w)n(e)g(denote)h(the)g(c)n (haracteristic)d(function)j(of)g(a)f(set)g FH(Q)p FI(.)125 1255 y(W)-7 b(e)27 b(adopt)h(the)g(Einstein)g(rule)f(of)h(summation)f (o)n(v)n(er)f(rep)r(eated)h(indexes.)125 1404 y FJ(A)m(c)m(kno)m (wledgemen)m(t.)50 b FI(I)33 b(wish)f(to)h(express)e(m)n(y)h(gratitude) g(to)h(the)g(Mathematical)f(Depart-)0 1504 y(men)n(t)26 b(of)g(ETH-Z\374ric)n(h)f(for)g(the)i(in)n(vitation)e(to)h(teac)n(h)f (the)h(Nac)n(hdiplom)f(V)-7 b(orlesungen,)25 b(and)h(to)g(the)0 1603 y(F)-7 b(orc)n(h)n(ungsinstitut)24 b(\025)g(for)g(the)i (hospitalit)n(y)e(during)g(m)n(y)g(sta)n(ying)g(in)h(Z\374ric)n(h,)f (and)h(for)f(the)h(excellen)n(t)0 1703 y(w)n(orking)31 b(conditions.)52 b(I)32 b(am)h(obliged)f(to)g(the)h(listeners)f(of)h (the)g(course)e(for)i(their)f(questions)g(and)0 1802 y(commen)n(ts.)45 b(In)31 b(particular)e(\025)h(to)g(Alan)g(Sznitman)h (for)f(man)n(y)g(his)g(critical)g(remarks)f(and)h(sugges-)0 1902 y(tions)24 b(made)g(during)f(the)i(lectures,)f(as)f(w)n(ell)h(as)g (for)f(his)h(help)h(to)f(organise)e(them)i(on)g(high)g(lev)n(el.)35 b(My)0 2002 y(sp)r(ecial)26 b(thanks)g(\025)f(to)i(Alan's)f(PhD)h (studen)n(ts)f(Loren)n(t)f(Go)r(eregen)g(and)h(T)-7 b(om)26 b(Sc)n(hmitz)h(who)e(help)r(ed)0 2101 y(to)f(prepare)e(a)h(T)505 2119 y(E)551 2101 y(X-v)n(ersion)f(of)i(the)g(lecture)f(notes.)35 b(Finally)-7 b(,)25 b(I)f(wish)f(to)h(thank)f(Gregory)f(F)-7 b(alk)n(o)n(vic)n(h)0 2201 y(for)27 b(his)h(remarks)d(on)j(a)f(preprin) n(t)g(v)n(ersion)f(of)h(this)h(b)r(o)r(ok.)125 2301 y(My)f(researc)n(h) e(is)j(supp)r(orted)f(b)n(y)h(EPSR)n(C,)g(gran)n(t)e(S68712/01.)1574 5059 y(5)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 6 5 bop 0 523 a FK(1)131 b(F)-11 b(unction)45 b(spaces)0 721 y Fy(1.1)112 b(F)-9 b(unction)37 b(spaces)h(for)g (functions)f(of)g Fx(x)0 875 y FI(Let)28 b FH(Q)f FI(b)r(e)h(an)f(op)r (en)h(domain)f(of)h FE(R)1121 845 y FA(d)1187 875 y FI(or)f(the)g (torus)g FE(T)1699 845 y FA(d)1761 875 y FG(=)c FE(R)1909 845 y FA(d)1948 875 y FH(=)p FG(2)p FH(\031)s FE(Z)2137 845 y FA(d)2175 875 y FI(.)125 974 y FC(L)l(eb)l(esgue)29 b(sp)l(ac)l(es.)125 1074 y FI(W)-7 b(e)32 b(denote)g(with)h FH(L)795 1086 y FA(p)832 1074 y FG(\()p FH(Q)p FG(;)14 b FE(R)1027 1044 y FA(n)1072 1074 y FG(\))p FH(;)g FG(1)31 b FF(\024)f FH(p)g FF(\024)h(1)p FI(,)i(the)f(usual)g(Leb)r(esgue)g (spaces)f(of)h(v)n(ector-v)-5 b(alued)0 1174 y(functions,)28 b(abbreviate)e FH(L)846 1186 y FA(p)884 1174 y FG(\()p FH(Q)p FG(;)14 b FE(R)p FG(\))23 b(=)g FH(L)1279 1186 y FA(p)1317 1174 y FG(\()p FH(Q)p FG(\))p FI(,)28 b(and)f(denote)g(the) h FH(L)2126 1186 y FD(2)2191 1174 y FI(scalar-pro)r(duct)d(b)n(y)i FF(h\001)p FH(;)14 b FF(\001i)p FI(.)125 1273 y FC(Sob)l(olev)30 b(sp)l(ac)l(es)f FH(W)756 1243 y FA(m;p)872 1273 y FG(\()p FH(Q)p FG(;)14 b FE(R)1067 1243 y FA(n)1112 1273 y FG(\))p FI(.)125 1373 y(Let)19 b FH(C)330 1343 y Fw(1)324 1393 y FA(c)400 1373 y FG(\()p FH(Q)p FG(;)14 b FE(R)595 1343 y FA(n)640 1373 y FG(\))19 b FI(b)r(e)h(the)f(space)f(of)h (in\034nitely)h(di\033eren)n(tiable)e(maps)h FH(\036)k FG(:)g FH(Q)g FF(!)g FE(R)2640 1343 y FA(n)2704 1373 y FI(with)d(compact)0 1472 y(supp)r(ort)26 b(in)h FH(Q)p FI(.)36 b(Supp)r(ose)26 b FH(u;)14 b(v)29 b FI(are)d(lo)r(cally)f(in)n (tegrable)g(functions)i(on)f FH(Q)g FI(and)g FH(\013)e FG(=)e(\()p FH(\013)2760 1484 y FD(1)2798 1472 y FH(;)14 b(:)g(:)g(:)g(;)g(\013)3036 1484 y FA(d)3074 1472 y FG(\))27 b FI(is)0 1572 y(a)k(m)n(ultiindex.)49 b(W)-7 b(e)32 b(sa)n(y)f(that)g FH(v)k FI(is)d(the)g FH(\013)1373 1542 y FA(th)1441 1572 y FI(-w)n(eak)e(partial)h(deriv)-5 b(ativ)n(e)31 b(of)g FH(u)p FI(,)h(written)g FH(D)2904 1542 y FA(\013)2951 1572 y FH(u)d FG(=)h FH(v)s(;)0 1672 y FI(pro)n(vided)397 1605 y Fz(R)436 1701 y FA(Q)506 1672 y FH(uD)625 1642 y FA(\013)672 1672 y FH(\036)14 b(dx)47 b FG(=)f(\()p FF(\000)p FG(1\))1154 1642 y Fw(j)p FA(\013)p Fw(j)1254 1605 y Fz(R)1293 1701 y FA(Q)1364 1672 y FH(v)s(\036)14 b(dx)g(;)42 b FI(for)f(all)g(test)h(functions)f FH(\036)47 b FF(2)f FH(C)2718 1642 y Fw(1)2712 1692 y FA(c)2789 1672 y FG(\()p FH(Q)p FG(\))p FI(.)78 b(Here)0 1817 y FF(j)p FH(\013)p FF(j)123 1769 y FD(def)136 1817 y FG(=)36 b FH(\013)290 1829 y FD(1)346 1817 y FG(+)18 b FH(:)c(:)g(:)k FG(+)g FH(\013)680 1829 y FA(d)719 1817 y FI(.)125 1916 y(Let)28 b FH(m)23 b FF(2)h FE(N)k FI(and)g FG(1)23 b FF(\024)h FH(p)f FF(\024)h(1)p FI(.)38 b(The)28 b(space)f FH(W)1634 1886 y FA(m;p)1751 1916 y FG(\()p FH(Q;)14 b FE(R)1946 1886 y FA(n)1991 1916 y FG(\))28 b FI(consists)f(of)h(all)g(lo)r(cally)f(in)n(tegrable)0 2016 y(functions)j FH(u)c FG(:)h FH(Q)g FF(!)g FE(R)747 1986 y FA(n)822 2016 y FI(suc)n(h)i(that)h(for)f(eac)n(h)g(m)n (ultiindex)i FH(\013)f FI(with)g FF(j)p FH(\013)p FF(j)e(\024)e FH(m)p FI(,)k FH(D)2618 1986 y FA(\013)2666 2016 y FH(u)f FI(exists)h(in)g(the)0 2115 y(w)n(eak)i(sense)h(and)f(b)r(elongs)h(to)g FH(L)1066 2127 y FA(p)1104 2115 y FG(\()p FH(Q)p FG(;)14 b FE(R)1299 2085 y FA(n)1344 2115 y FG(\))p FI(.)53 b(W)-7 b(e)34 b(shall)e(only)h(use)g(these)g(spaces)f(with)i FH(p)e FG(=)g(2)g FI(and)0 2215 y(adopt)27 b(the)h(notations:)618 2391 y FH(W)708 2357 y FA(m;)p FD(2)824 2391 y FG(\()p FH(Q)p FG(;)14 b FE(R)1019 2357 y FA(n)1064 2391 y FG(\))23 b(=)g FH(H)1283 2357 y FA(m)1345 2391 y FG(\()p FH(Q)p FG(;)14 b FE(R)1540 2357 y FA(n)1585 2391 y FG(\))g FH(;)97 b(H)1827 2357 y FA(m)1890 2391 y FG(\()p FH(Q)p FG(;)14 b FE(R)2085 2357 y FD(1)2122 2391 y FG(\))24 b(=)e FH(H)2341 2357 y FA(m)2404 2391 y FG(\()p FH(Q)p FG(\))14 b FH(:)0 2567 y FI(If)28 b FH(u)23 b FF(2)g FH(H)308 2537 y FA(m)371 2567 y FG(\()p FH(Q)p FG(;)14 b FE(R)566 2537 y FA(n)611 2567 y FG(\))p FI(,)28 b(w)n(e)f(de\034ne)h(its)g(norm)e(to)i(b)r(e:) 1094 2817 y FF(k)o FH(u)p FF(k)1224 2841 y FA(m)1311 2817 y FG(=)1398 2699 y Fz(\022)1510 2738 y(X)1473 2920 y Fw(j)p FA(\013)p Fw(j\024)p FA(m)1680 2817 y FF(j)p FH(D)1774 2782 y FA(\013)1822 2817 y FH(u)p FF(j)1893 2782 y FD(2)1893 2837 y(2)1930 2699 y Fz(\023)1991 2717 y FD(1)p FA(=)p FD(2)0 3090 y FI(When)d(analysing)e(eq.)36 b(\(0.1\),)25 b(w)n(e)f(will)g(mostly)h(restrict)e(ourselv)n(es)g(to)h (the)h(case)f(where)g(the)h(domain)0 3214 y FH(Q)i FI(is)h(the)g(torus) f FE(T)588 3184 y FD(2)648 3167 y(def)662 3214 y FG(=)36 b FE(R)823 3184 y FD(2)860 3214 y FH(=)p FE(Z)957 3184 y FD(2)994 3214 y FI(.)125 3314 y FJ(\(S1\))f FI(If)g FH(Q)g FG(=)h FE(T)682 3284 y FA(d)721 3314 y FI(,)h(then)e FH(u)g FF(2)h FH(L)1208 3326 y FD(2)1280 3314 y FI(can)f(b)r(e)g (written)h(as)e FH(u)p FG(\()p FH(x)p FG(\))i(=)2263 3252 y Fz(P)2351 3339 y FA(s)p Fw(2)p Fv(Z)2466 3322 y Fu(d)2519 3314 y FH(u)2567 3326 y FA(s)2602 3314 y FH(e)2641 3284 y FA(is)p Fw(\001)p FA(x)2757 3314 y FI(.)59 b(It)36 b(is)f(then)0 3422 y(p)r(ossible)24 b(to)g(de\034ne)g FH(H)719 3392 y FA(m)782 3422 y FG(\()p FE(T)869 3392 y FA(d)908 3422 y FG(;)14 b FE(R)1005 3392 y FA(n)1050 3422 y FG(\))25 b FI(ev)n(en)e(for)h FH(m)f FF(2)g FE(R)p FI(.)36 b(F)-7 b(or)23 b(this)i(purp)r(ose,)f(w)n(e)g(de\034ne)g(a)g (norm)f(whic)n(h)0 3522 y(is)k(equiv)-5 b(alen)n(t)28 b(to)f FF(k\001k)685 3547 y FA(m)776 3522 y FI(in)h(\(1.1\))f(if)h FH(m)23 b FF(2)h FE(N)p FI(:)898 3719 y FF(k)p FH(u)p FF(k)1029 3677 y FD(2)1029 3744 y FA(m)1115 3719 y FG(=)1217 3640 y Fz(X)1202 3824 y FA(s)p Fw(2)p Fv(Z)1317 3807 y Fu(d)1366 3651 y Fz(\000)1404 3719 y FG(1)18 b(+)g FF(j)p FH(s)p FF(j)1632 3684 y FD(2)1669 3651 y Fz(\001)1707 3669 y FA(m)1770 3719 y FF(j)p FH(u)1841 3731 y FA(s)1876 3719 y FF(j)1899 3684 y FD(2)1937 3719 y FH(;)97 b(m)23 b FF(2)g FE(R)0 4000 y FI(No)n(w)33 b(for)g FH(m)g FF(\025)g FG(0)g FI(w)n(e)g(set)h FH(H)947 3970 y FA(m)1010 4000 y FG(\()p FE(T)1097 3970 y FA(d)1136 4000 y FG(;)14 b FE(R)1233 3970 y FA(n)1278 4000 y FG(\))34 b(=)e FF(f)p FH(u)h FF(2)g FH(L)1709 3970 y FD(2)1746 4000 y FG(\()p FE(T)1833 3970 y FA(d)1872 4000 y FG(;)14 b FE(R)1969 3970 y FA(n)2014 4000 y FG(\))34 b(:)f FF(k)o FH(u)p FF(k)2266 3959 y FD(2)2266 4025 y FA(m)2363 4000 y FH(<)f FF(1g)p FI(,)j(and)f(for)f FH(m)g(<)f FG(0)0 4100 y FI(de\034ne)c(the)g (space)f FH(H)681 4070 y FA(m)743 4100 y FG(\()p FE(T)830 4070 y FA(d)870 4100 y FG(;)14 b FE(R)967 4070 y FA(n)1012 4100 y FG(\))28 b FI(as)f(the)h(closure)e(of)h FH(L)1745 4112 y FD(2)1782 4100 y FG(\()p FE(T)1869 4070 y FA(d)1909 4100 y FG(;)14 b FE(R)2006 4070 y FA(n)2051 4100 y FG(\))28 b FI(in)f(the)h FF(k\001k)2456 4125 y FA(m)2519 4100 y FI(-norm.)0 4261 y FJ(Lemma)c(1.1.)35 b FC(F)-6 b(or)25 b(any)h FH(r)f FF(2)f FE(R)h FC(and)g(any)h(multiindex)f FH(\013)g FC(the)g(line)l(ar)h(map)g FH(D)2453 4231 y FA(\013)2525 4261 y FC(is)f(c)l(ontinuous)f(fr)l(om)0 4361 y FH(H)76 4330 y FA(r)113 4361 y FG(\()p FE(T)200 4330 y FA(d)239 4361 y FG(;)14 b FE(R)336 4330 y FA(n)381 4361 y FG(\))34 b FC(to)g FH(H)627 4330 y FA(r)r Fw(\000j)p FA(\013)p Fw(j)798 4361 y FG(\()p FE(T)885 4330 y FA(d)925 4361 y FG(;)14 b FE(R)1022 4330 y FA(n)1067 4361 y FG(\))p FC(.)52 b(A)l(c)l(c)l(or)l(dingly,)36 b(the)e(map)h FG(\001)c(:)h FH(H)2209 4330 y FA(r)2245 4361 y FG(\()p FE(T)2332 4330 y FA(d)2371 4361 y FG(;)14 b FE(R)2468 4330 y FA(n)2513 4361 y FG(\))32 b FF(!)f FH(H)2767 4330 y FA(r)r Fw(\000)p FD(2)2888 4361 y FG(\()p FE(T)2975 4330 y FA(d)3015 4361 y FG(;)14 b FE(R)3112 4330 y FA(n)3157 4361 y FG(\))0 4460 y FC(is)30 b(c)l(ontinuous.)125 4621 y FJ(\(S2\))d FI(If)h FF(h)p FH(u)p FF(i)545 4574 y FD(def)559 4621 y FG(=)660 4554 y Fz(R)699 4651 y Fv(T)738 4634 y Fu(d)791 4621 y FH(u)p FG(\()p FH(x)p FG(\))14 b FH(dx)24 b FG(=)e(0)p FI(,)28 b(then)g FH(u)1495 4633 y FD(0)1554 4621 y FG(=)23 b(0)p FI(.)37 b(Therefore)26 b(in)i(the)g(space)817 4810 y FH(H)893 4776 y FA(m)886 4831 y FD(0)956 4810 y FG(\()p FE(T)1043 4776 y FA(d)1082 4810 y FG(;)14 b FE(R)1179 4776 y FA(n)1224 4810 y FG(\))23 b(=)g FF(f)p FH(u)f FF(2)i FH(H)1634 4776 y FA(m)1696 4810 y FG(\()p FE(T)1783 4776 y FA(d)1823 4810 y FG(;)14 b FE(R)1920 4776 y FA(n)1965 4810 y FG(\))23 b FF(j)g(h)p FH(u)p FF(i)g FG(=)g(0)p FF(g)1574 5059 y FI(6)p eop end PStoPSsaved restore %%Page: (6,7) 4 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 7 6 bop 0 523 a FI(the)28 b(norm)f(can)g(b)r(e)h(equiv)-5 b(alen)n(tly)27 b(de\034ned)h(b)n(y)f(the)h(relation)1183 706 y FF(k)p FH(u)p FF(k)1314 664 y FD(2)1314 731 y FA(m)1400 706 y FG(=)1488 627 y Fz(X)1490 806 y FA(s)p Fw(6)p FD(=0)1621 706 y FF(j)p FH(s)p FF(j)1706 672 y FD(2)p FA(m)1802 706 y FF(j)p FH(u)1873 718 y FA(s)1909 706 y FF(j)1932 672 y FD(2)1983 706 y FH(:)0 985 y FI(In)g(particular,)e FF(k)p FH(u)p FF(k)641 943 y FD(2)641 1010 y(1)701 985 y FG(=)c FF(jr)p FH(u)p FF(j)951 997 y FD(2)989 985 y FH(:)125 1084 y FJ(\(S3\))27 b FC(Sob)l(olev)k(Emb)l(e)l(ddings)0 1184 y FI(Let)d FH(Q)f FI(b)r(e)h(an)f(op)r(en)h(subset)f(of)h FE(R)1080 1154 y FA(d)1146 1184 y FI(with)g(a)f(Lipsc)n(hitz)h(b)r (oundary)-7 b(,)27 b(or)f(the)i(torus)f FE(T)2661 1154 y FA(d)2700 1184 y FI(.)101 1341 y(1.)42 b(If)28 b FH(m)23 b FF(\024)484 1308 y FA(d)p 484 1322 35 4 v 485 1370 a FD(2)556 1341 y FI(and)28 b FG(2)22 b FF(\024)h FH(q)j FF(\024)1086 1308 y FD(2)p FA(d)p 1031 1322 179 4 v 1031 1370 a(d)p Fw(\000)p FD(2)p FA(m)1219 1341 y FH(;)i(q)e(<)c FF(1)p FI(,)28 b(then)1255 1512 y FH(H)1331 1478 y FA(m)1393 1512 y FG(\()p FH(Q)p FG(;)14 b FE(R)1588 1478 y FA(n)1633 1512 y FG(\))24 b FF(\032)e FH(L)1833 1524 y FA(q)1870 1512 y FG(\()p FH(Q)p FG(;)14 b FE(R)2065 1478 y FA(n)2110 1512 y FG(\))876 b FI(\(1.1\))101 1714 y(2.)42 b(If)28 b FH(m)23 b(>)484 1681 y FA(d)p 484 1695 35 4 v 485 1743 a FD(2)547 1714 y FG(+)18 b FH(\013;)28 b FG(0)23 b FF(\024)f FH(\013)i(<)f FG(1)p FI(,)k(then)1245 1885 y FH(H)1321 1851 y FA(m)1384 1885 y FG(\()p FH(Q)p FG(;)14 b FE(R)1579 1851 y FA(n)1624 1885 y FG(\))23 b FF(\032)g FH(C)1832 1851 y FA(\013)1879 1885 y FG(\()p FH(Q)p FG(;)14 b FE(R)2074 1851 y FA(n)2119 1885 y FG(\))867 b FI(\(1.2\))208 2057 y FH(C)273 2026 y FA(\013)320 2057 y FG(\()p FH(Q)p FG(\))p FH(;)28 b(\013)i(>)f FG(0)p FI(,)j(denotes)g(the)f(space)g(of)h FH(\013)p FI(-H\366lder)f(con)n(tin)n(uous)f(functions)i(and)f FH(C)2917 2026 y FD(0)2986 2057 y FI(is)h(the)208 2156 y(space)26 b(of)i(con)n(tin)n(uous)e(functions.)101 2318 y(3.)42 b(If)27 b FH(Q)h FI(is)f(an)g(op)r(en)g(b)r(ounded)h(subset)g (of)f FE(R)1530 2288 y FA(d)1596 2318 y FI(with)h(Lipsc)n(hitz)f(b)r (oundary)-7 b(,)27 b(or)g(if)h FH(Q)22 b FG(=)h FE(T)2938 2288 y FA(d)2977 2318 y FI(,)28 b(then)208 2417 y(the)38 b(em)n(b)r(edding)h(\(1.2\))f(is)g(compact,)i(and)e(the)g(em)n(b)r (edding)i(\(1.1\))d(is)h(compact)g(as)f(far)h(as)208 2517 y FH(q)26 b(<)424 2484 y FD(2)p FA(d)p 368 2498 179 4 v 368 2546 a(d)p Fw(\000)p FD(2)p FA(m)557 2517 y FI(.)37 b(Besides,)27 b(in)g(this)h(case)934 2688 y FH(H)1010 2654 y FA(m)1069 2662 y Ft(1)1105 2688 y FG(\()p FH(Q)p FG(;)14 b FE(R)1300 2654 y FA(n)1345 2688 y FG(\))23 b Fs(b)g FH(H)1564 2654 y FA(m)1623 2662 y Ft(2)1659 2688 y FG(\()p FH(Q)p FG(;)14 b FE(R)1854 2654 y FA(n)1899 2688 y FG(\))83 b FI(if)47 b FH(m)2182 2700 y FD(1)2242 2688 y FH(>)23 b(m)2403 2700 y FD(2)2440 2688 y FH(:)555 b FI(\(1.3\))0 2890 y FJ(Exercise)31 b(1.2.)40 b FC(Pr)l(ove)d FI(\(1.2\))29 b FC(for)i FH(\013)23 b FG(=)g(0)29 b FC(and)h FH(Q)23 b FG(=)g FE(T)1771 2860 y FA(d)1810 2890 y FC(.)0 3047 y FJ(Solution:)95 b FI(W)-7 b(e)31 b(ha)n(v)n(e)g(to)g(sho)n(w)g (that)h FH(H)1385 3017 y FA(m)1447 3047 y FG(\()p FH(Q)p FG(;)14 b FE(R)1642 3017 y FA(n)1687 3047 y FG(\))30 b FF(\032)g FH(C)1909 3017 y FD(0)1946 3047 y FG(\()p FH(Q)p FG(;)14 b FE(R)2141 3017 y FA(n)2186 3047 y FG(\))32 b FI(if)g FH(m)e(>)2537 3014 y FA(d)p 2537 3028 35 4 v 2538 3076 a FD(2)2581 3047 y FI(.)49 b(It)32 b(is)f(clear)g(that)0 3147 y FH(u)23 b FG(=)158 3084 y Fz(P)246 3172 y FA(s)p Fw(2)p Fv(Z)361 3155 y Fu(d)414 3147 y FH(u)462 3159 y FA(s)497 3147 y FH(e)536 3117 y FA(is)p Fw(\001)p FA(x)675 3147 y FI(with)861 3084 y Fz(P)962 3147 y FF(j)p FH(u)1033 3159 y FA(s)1068 3147 y FF(j)g FH(<)g FF(1)h FI(is)f(con)n(tin)n(uous.) 35 b(So)23 b(it)i(su\036ce)e(to)h(c)n(hec)n(k)f(that)2765 3084 y Fz(P)2866 3147 y FF(j)p FH(u)2937 3159 y FA(s)2972 3147 y FF(j)g FH(<)g FF(1)0 3257 y FI(for)k FH(u)c FF(2)g FH(H)352 3227 y FA(m)415 3257 y FG(\()p FH(Q)p FG(;)14 b FE(R)610 3227 y FA(n)655 3257 y FG(\))28 b FI(with)g FH(m)23 b(>)1097 3225 y FA(d)p 1097 3239 V 1098 3286 a FD(2)1142 3257 y FI(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e:)485 3374 y Fz(X)470 3558 y FA(s)p Fw(2)p Fv(Z)585 3541 y Fu(d)634 3453 y FF(j)p FH(u)705 3465 y FA(s)740 3453 y FF(j)p FG(\(1)18 b(+)g FF(j)p FH(s)p FF(j)1023 3419 y FD(2)1060 3453 y FG(\))1092 3419 y Fw(\000)p FA(m=)p FD(2)1275 3453 y FG(\(1)g(+)g FF(j)p FH(s)p FF(j)1535 3419 y FD(2)1572 3453 y FG(\))1604 3419 y FA(m=)p FD(2)786 3797 y FF(\024)873 3630 y Fz(0)873 3780 y(@)961 3718 y(X)946 3902 y FA(s)p Fw(2)p Fv(Z)1061 3885 y Fu(d)1109 3797 y FF(j)p FH(u)p FF(j)1203 3763 y FD(2)1240 3797 y FG(\(1)h(+)f FF(j)p FH(s)p FF(j)1501 3763 y FD(2)1538 3797 y FG(\))1570 3763 y FA(m)1633 3630 y Fz(1)1633 3780 y(A)1706 3647 y FD(1)p FA(=)p FD(2)1824 3630 y Fz(0)1824 3780 y(@)1912 3718 y(X)1897 3902 y FA(s)p Fw(2)p Fv(Z)2012 3885 y Fu(d)2046 3797 y FG(\(1)h(+)f FF(j)p FH(s)p FF(j)2307 3763 y FD(2)2344 3797 y FG(\))2376 3763 y Fw(\000)p FA(m)2491 3630 y Fz(1)2491 3780 y(A)2564 3647 y FD(1)p FA(=)p FD(2)2696 3797 y FH(:)0 4068 y FI(The)31 b(\034rst)g(factor)f(on)g(the)h(r.h.s.) 47 b(is)31 b(\034nite)g(since)g FH(u)d FF(2)h FH(H)1827 4038 y FA(m)1889 4068 y FG(\()p FH(Q)p FG(;)14 b FE(R)2084 4038 y FA(n)2129 4068 y FG(\))p FI(.)48 b(The)31 b(second)f(factor)g (is)h(\034nite)0 4168 y(b)r(ecause)c FG(2)p FH(m)c(>)f(d)28 b FI(and)297 4285 y Fz(X)283 4469 y FA(s)p Fw(2)p Fv(Z)398 4452 y Fu(d)432 4364 y FG(\(1)18 b(+)h FF(j)p FH(s)p FF(j)693 4329 y FD(2)730 4364 y FG(\))762 4329 y Fw(\000)p FA(m)900 4364 y FF(\024)k FG(1)18 b(+)1213 4285 y Fz(X)1131 4469 y FA(s)p Fw(2)p Fv(Z)1246 4452 y Fu(d)1280 4469 y Fw(nf)p FD(0)p Fw(g)1429 4364 y FF(j)p FH(s)p FF(j)1514 4329 y Fw(\000)p FD(2)p FA(m)1686 4364 y FF(\024)k FH(c)d FG(+)f FH(c)1947 4329 y Fw(0)1984 4251 y Fz(Z)2030 4439 y Fw(j)p FA(x)p Fw(j)p FA(>)p FD(1)2210 4364 y FF(j)p FH(x)p FF(j)2303 4329 y Fw(\000)p FD(2)p FA(m)2451 4364 y FH(dx)24 b(<)f FF(1)p FH(:)p 2846 4364 4 57 v 2850 4311 50 4 v 2850 4364 V 2899 4364 4 57 v 0 4639 a FJ(Example:)77 b FI(If)28 b FH(Q)23 b FG(=)f FE(T)782 4609 y FD(2)820 4639 y FI(,)28 b(then)1120 4810 y FH(H)1196 4776 y FD(1)1233 4810 y FG(\()p FH(Q)p FG(\))23 b Fs(b)g FH(L)1531 4822 y FA(q)1567 4810 y FG(\()p FH(Q)p FG(\))69 b FF(8)p FH(q)26 b(<)d FF(1)p FH(;)949 b FI(\(1.4\))1574 5059 y(7)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 8 7 bop 0 523 a FI(and)1286 623 y FH(H)1372 566 y Ft(1)p 1372 575 29 3 v 1372 608 a(2)1415 623 y FG(\()p FH(Q)p FG(\))23 b FF(\032)f FH(L)1712 635 y FD(4)1749 623 y FG(\()p FH(Q)p FG(\))p FH(:)1116 b FI(\(1.5\))125 770 y FJ(\(S4\))27 b FI(The)h(spaces)e FH(H)828 740 y FA(m)891 770 y FG(\()p FE(T)978 740 y FA(d)1018 770 y FG(;)14 b FE(R)1115 740 y FA(n)1160 770 y FG(\))28 b FI(and)f FH(H)1457 740 y Fw(\000)p FA(m)1572 770 y FG(\()p FE(T)1659 740 y FA(d)1698 770 y FG(;)14 b FE(R)1795 740 y FA(n)1840 770 y FG(\))28 b FI(are)f(dual:)709 950 y FF(8)p FH(u)22 b FF(2)i FH(C)969 915 y Fw(1)1040 950 y FG(\()p FE(T)1127 915 y FA(d)1166 950 y FG(;)14 b FE(R)1263 915 y FA(n)1308 950 y FG(\))p FH(;)83 b FF(k)p FH(u)p FF(k)1577 974 y FA(m)1663 950 y FG(=)210 b(sup)1751 1023 y FA(v)r Fw(2)p FA(C)1883 1006 y Fr(1)1943 1023 y FA(;)p Fw(k)p FA(v)r Fw(k)2066 1042 y Fr(\000)p Fu(m)2166 1023 y Fw(\024)p FD(1)2251 950 y FF(h)p FH(u;)14 b(v)s FF(i)g FH(:)0 1202 y FJ(Exercise)31 b(1.3.)40 b FC(Pr)l(ove)31 b(this)f(r)l(elation.)125 1366 y FI(In)18 b(particular,)h(the)g(scalar) d(pro)r(duct)j(in)f FH(L)1425 1336 y FD(2)1480 1366 y FI(extends)h(to)f(a)g(con)n(tin)n(uous)f(bilinear)h(map)g FH(H)2874 1336 y FA(m)2937 1366 y FG(\()p FE(T)3024 1336 y FA(d)3063 1366 y FG(;)c FE(R)3160 1336 y FA(n)3205 1366 y FG(\))p FF(\002)0 1465 y FH(H)76 1435 y Fw(\000)p FA(m)191 1465 y FG(\()p FE(T)278 1435 y FA(d)317 1465 y FG(;)g FE(R)414 1435 y FA(n)459 1465 y FG(\))23 b FF(!)h FE(R)p FI(.)125 1565 y FJ(\(S5\))j FC(Interp)l(olation)k(ine)l (quality.)93 b FI(Let)28 b FH(a)23 b(<)f(b)28 b FI(and)f FG(0)c FF(\024)f FH(\022)k FF(\024)c FG(1)p FI(.)37 b(Then)1068 1756 y FF(k)o FH(u)p FF(k)1198 1781 y FA(\022)r(a)p FD(+\(1)p Fw(\000)p FA(\022)r FD(\))p FA(b)1545 1756 y FF(\024)23 b(k)o FH(u)p FF(k)1763 1714 y FA(\022)1763 1781 y(a)1831 1756 y FH(:)14 b FF(k)p FH(u)p FF(k)1999 1714 y FD(1)p Fw(\000)p FA(\022)1999 1781 y(b)0 1954 y FC(Pr)l(o)l(of)29 b FI(\(for)e(the)h(case)f FH(Q)22 b FG(=)h FE(T)930 1924 y FA(d)997 1954 y FI(and)k FF(h)p FH(u)p FF(i)c FG(=)g(0)p FI(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)605 2233 y FF(k)p FH(u)p FF(k)736 2191 y FD(2)736 2258 y FA(\022)r(a)p FD(+\(1)p Fw(\000)p FA(\022)r FD(\))p FA(b)1095 2233 y FG(=)1196 2154 y Fz(X)1198 2333 y FA(s)p Fw(6)p FD(=0)1330 2233 y FF(j)p FH(u)1401 2245 y FA(s)1436 2233 y FF(j)1459 2199 y FD(2)1496 2233 y FF(j)p FH(s)p FF(j)1581 2199 y FD(2\()p FA(\022)r(a)p FD(+\(1)p Fw(\000)p FA(\022)r FD(\))p FA(b)p FD(\))1095 2473 y FG(=)1196 2394 y Fz(X)1198 2573 y FA(s)p Fw(6)p FD(=0)1330 2473 y FF(j)p FH(u)1401 2485 y FA(s)1436 2473 y FF(j)1459 2439 y FD(2)p FA(\022)1529 2473 y FF(j)p FH(s)p FF(j)1614 2439 y FD(2)p FA(\022)r(a)1721 2473 y FF(j)p FH(u)1792 2485 y FA(s)1827 2473 y FF(j)1850 2439 y FD(2\(1)p Fw(\000)p FA(\022)r FD(\))2058 2473 y FF(j)p FH(s)p FF(j)2143 2439 y FD(2\(1)p Fw(\000)p FA(\022)r FD(\))p FA(b)1095 2815 y FF(\024)1196 2649 y Fz(0)1196 2798 y(@)1268 2737 y(X)1271 2915 y FA(s)p Fw(6)p FD(=0)1402 2815 y FF(j)p FH(u)1473 2827 y FA(s)1508 2815 y FF(j)1531 2781 y FD(2)1569 2815 y FF(j)p FH(s)p FF(j)1654 2781 y FD(2)p FA(a)1727 2649 y Fz(1)1727 2798 y(A)1800 2666 y FA(\022)1851 2649 y Fz(0)1851 2798 y(@)1924 2737 y(X)1926 2915 y FA(s)p Fw(6)p FD(=0)2057 2815 y FF(j)p FH(u)2128 2827 y FA(s)2163 2815 y FF(jj)p FH(s)p FF(j)2271 2781 y FD(2)p FA(b)2338 2649 y Fz(1)2338 2798 y(A)2411 2666 y FD(1)p Fw(\000)p FA(\022)2561 2815 y FH(;)0 3089 y FI(where)f(in)h(the)g(last)f(step)h(w)n(e)f(use)g(the)h (H\366lder)f(inequalit)n(y)-7 b(.)0 3238 y FJ(Example:)75 b FI(In)26 b(the)g(Example)f(in)h(\(S3\),)h(using)e(the)h(in)n(terp)r (olation)f(with)h FH(a)d FG(=)f(0)p FH(;)14 b(b)22 b FG(=)h(1)p FH(;)14 b(\022)25 b FG(=)3002 3205 y FD(1)p 3002 3219 34 4 v 3002 3266 a(2)3045 3238 y FI(,)h(w)n(e)0 3350 y(get)h(that)56 b FF(j)p FH(u)p FF(j)440 3362 y FD(4)500 3350 y FF(\024)22 b FH(c)14 b FF(k)p FH(u)p FF(k)778 3353 y Ft(1)p 778 3362 29 3 v 778 3395 a(2)843 3350 y FF(\024)23 b FH(c)967 3281 y Fz(p)p 1050 3281 313 4 v 69 x FF(j)p FH(u)p FF(j)1144 3362 y FD(2)1194 3350 y FF(k)p FH(u)p FF(k)1325 3375 y FD(1)1376 3350 y FH(:)28 b FI(This)f(is)h(the)g FC(L)l(adyzhenskaya)k(ine)l(quality)p FI(.)0 3582 y Fy(1.2)112 b(F)-9 b(unctions)37 b(of)h Fx(t)f Fy(and)i Fx(x)0 3735 y FI(The)30 b(solutions)g(of)g(the)g (equations,)g(men)n(tioned)h(in)f(the)h(in)n(tro)r(duction,)f(are)f (functions)i(dep)r(ending)0 3835 y(on)c(time)h FH(t)g FI(and)f(space)g FH(x)p FI(.)38 b(W)-7 b(e)28 b(\034x)f FH(T)34 b(>)23 b FG(0)k FI(and)g(view)h FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))28 b FI(with)g FG(0)23 b FF(\024)f FH(t)h FF(\024)g FH(T)39 b FI(as)26 b(a)i(map)673 4014 y FG([0)p FH(;)14 b(T)e FG(])21 b FF(\000)-14 b(!)23 b FI(\020space)k(of)g(functions)h(of)f FH(x)p FI(\021)7 b FH(;)180 b(t)23 b FF(7!)g FH(u)p FG(\()p FH(t;)14 b FF(\001)p FG(\))g FH(:)0 4219 y FI(A)n(ccordingly)-7 b(,)26 b(w)n(e)h(can)h(de\034ne)f(spaces)55 b FH(L)1329 4231 y FA(p)1366 4151 y Fz(\000)1404 4219 y FG(0)p FH(;)14 b(T)e FG(;)i FH(L)1638 4231 y FA(q)1673 4219 y FG(\()p FH(Q)p FG(\))1803 4151 y Fz(\001)1864 4172 y FD(def)1878 4219 y FG(=)36 b FH(L)2036 4231 y FA(p)2074 4151 y Fz(\000)2112 4219 y FG([0)p FH(;)14 b(T)e FG(])p FH(;)i(L)2392 4231 y FA(q)2426 4219 y FG(\()p FH(Q)p FG(\))2556 4151 y Fz(\001)2595 4219 y FI(,)30 4352 y FH(L)87 4364 y FA(p)124 4284 y Fz(\000)162 4352 y FG(0)p FH(;)g(T)e FG(;)i FH(H)415 4322 y FA(k)454 4352 y FG(\()p FH(Q)p FG(\))584 4284 y Fz(\001)649 4305 y FD(def)662 4352 y FG(=)39 b FH(L)823 4364 y FA(p)861 4284 y Fz(\000)899 4352 y FG([0)p FH(;)14 b(T)e FG(])p FH(;)i(H)1198 4322 y FA(k)1237 4352 y FG(\()p FH(Q)p FG(\))1367 4284 y Fz(\001)1406 4352 y FI(,)59 b FH(C)1553 4284 y Fz(\000)1592 4352 y FG(0)p FH(;)14 b(T)e FG(;)i FH(L)1826 4364 y FA(q)1860 4352 y FG(\()p FH(Q)p FG(\))1990 4284 y Fz(\001)2054 4305 y FD(def)2068 4352 y FG(=)39 b FH(C)2237 4284 y Fz(\000)2275 4352 y FG([0)p FH(;)14 b(T)e FG(])p FH(;)i(L)2555 4364 y FA(q)2590 4352 y FG(\()p FH(Q)p FG(\))2720 4284 y Fz(\001)2758 4352 y FI(,)30 b(and)g(so)e(on.)0 4451 y(F)-7 b(ubini's)28 b(theorem)f(implies)h(that)974 4631 y FH(L)1031 4643 y FA(p)1069 4563 y Fz(\000)1107 4631 y FG(0)p FH(;)14 b(T)e FG(;)i FH(L)1341 4643 y FA(p)1377 4631 y FG(\()p FH(Q)p FG(\))1507 4563 y Fz(\001)1568 4631 y FG(=)23 b FH(L)1713 4643 y FA(p)1751 4563 y Fz(\000)1789 4631 y FG([0)p FH(;)14 b(T)e FG(])17 b FF(\002)h FH(Q)2141 4563 y Fz(\001)2192 4631 y FH(;)0 4810 y FI(if)28 b FH(p)23 b(<)g FF(1)p FI(.)37 b(Discussion)27 b(of)g(these)h(spaces)e(see)h(in)h ([Lio69)o(].)1574 5059 y(8)p eop end PStoPSsaved restore %%Page: (8,9) 5 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 9 8 bop 125 523 a FI(W)-7 b(e)27 b(shall)h(denote)740 706 y FH(C)805 671 y Fw(1)899 706 y FG(=)23 b FF(f)p FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))23 b FF(2)g FH(C)1421 671 y Fw(1)1492 706 y FF(g)45 b FI(or)h FH(C)1765 671 y Fw(1)1858 706 y FG(=)23 b FF(f)p FH(u)p FG(\()p FH(x)p FG(\))g FF(2)g FH(C)2313 671 y Fw(1)2384 706 y FF(g)p FH(;)0 888 y FI(dep)r(ending)28 b(on)f(the)h(con)n(text.)0 1054 y FJ(Exercise)j(1.4.)40 b FC(L)l(et)30 b FH(Q)22 b FG(=)h FE(R)948 1024 y FA(d)1016 1054 y FC(and)30 b FH(T)35 b FG(=)22 b(1)p FC(.)38 b(Consider)32 b(the)d(he)l(at)h (kernel:)981 1286 y FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)f(\(2)1393 1214 y FF(p)p 1462 1214 81 4 v 72 x FH(\031)s(t)p FG(\))1574 1251 y Fw(\000)p FA(d)1679 1286 y FG(exp)1819 1169 y Fz(\022)1880 1286 y FF(\000)1955 1229 y(j)p FH(x)p FF(j)2048 1199 y FD(2)p 1955 1266 131 4 v 1985 1343 a FG(4)p FH(t)2096 1169 y Fz(\023)2184 1286 y FH(:)0 1513 y FC(Pr)l(ove)31 b(that)952 1642 y FF(j)p FH(u)p FF(j)1046 1657 y FA(L)1092 1665 y Fu(p)1126 1657 y FD(\(0)p FA(;)p FD(1;)p FA(L)1303 1665 y Ft(2)1335 1657 y FD(\))1388 1642 y FG(=)1476 1525 y Fz(\032)1556 1591 y FF(1)p FH(;)208 b(pd)23 b FF(\025)g FG(4)1556 1691 y FH(<)g FF(1)p FH(;)120 b FC(otherwise)1574 5059 y FI(9)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 10 9 bop 0 523 a FK(2)131 b(The)44 b(deterministic)i(2D)d (Na)l(vier-Stok)l(es)i(Equation)0 705 y FI(In)21 b(the)f(forthcoming,)h (w)n(e)f(write)g(\0202D)h(NSE\021)27 b(for)20 b(the)h(\020t)n(w)n (o-dimensional)d(Na)n(vier\025Stok)n(es)g(equation\021,)0 804 y(and)25 b(often)g(abbreviate)f(2D)h(NSE)h(to)f(NSE.)h(W)-7 b(e)26 b(will)f(consider)f(2D)h(NSE)h(with)g(p)r(erio)r(dic)f(b)r (oundary)0 925 y(conditions.)48 b(That)32 b(is,)h(w)n(e)e(assume)g (that)h(the)g(space-v)-5 b(ariable)29 b FH(x)j FI(is)f(a)h(p)r(oin)n(t) f(in)h(the)g(torus)f FE(T)3030 895 y FD(2)3098 878 y(def)3111 925 y FG(=)0 1024 y FE(R)60 994 y FD(2)97 1024 y FH(=)p FG(2)p FH(\031)s FE(Z)286 994 y FD(2)323 1024 y FI(.)37 b(The)27 b(Na)n(vier\025Stok)n(es)d(equation)i(in)i(a)e(b)r(ounded)h(t) n(w)n(o-dimensional)e(domain)i(under)g(the)0 1124 y(Diric)n(hlet)35 b(b)r(oundary)f(conditions)g(can)g(b)r(e)i(studied)f(v)n(ery)e (similar.)58 b(In)35 b(an)f(un)n(b)r(ounded)h(domain,)0 1224 y(e.g.)42 b(in)29 b(the)h(whole)f(plane,)h(the)g(equation)e(b)r (ecomes)h(a)g(bit)h(more)f(di\036cult)h(since)f(in)h(this)f(case)g(the) 0 1323 y(Sob)r(olev)e(em)n(b)r(eddings)g(in)h(\(S3\))g(are)f(not)g (compact.)125 1423 y(The)34 b(2D)h(NSE)h(on)e(the)h(torus)f(is)h(the)g (follo)n(wing)f(system)g(of)h(three)f(equations)g(for)h(three)f(un-)0 1533 y(kno)n(wn)i(functions:)55 b(t)n(w)n(o)36 b(comp)r(onen)n(ts)g(of) h(the)g(v)n(ector-function)f FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))39 b(=)2519 1466 y Fz(\000)2557 1533 y FH(u)2605 1503 y FD(1)2642 1533 y FG(\()p FH(t;)14 b(x)p FG(\))p FH(;)g(u)2905 1503 y FD(2)2943 1533 y FG(\()p FH(t;)g(x)p FG(\))3121 1466 y Fz(\001)3160 1483 y FA(t)0 1633 y FI(\(the)28 b(v)n(elo)r(cit)n(y\))f(and)h(the)f(scalar)f(function)i FH(p)p FG(\()p FH(t;)14 b(x)p FG(\))29 b FI(\(the)f(pressure\),)f (where)g FH(x)c FF(2)g FE(T)2626 1603 y FD(2)2692 1633 y FI(and)k FH(t)c FF(2)h FE(R)p FI(:)440 1748 y Fz(\()522 1833 y FG(_)-38 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))19 b FF(\000)f FH(\027)5 b FG(\001)p FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))20 b(+)e(\()p FH(u)p FG(\()p FH(t;)c(x)p FG(\))19 b FF(\001)f(r)p FG(\))p FH(u)p FG(\()p FH(t;)c(x)p FG(\))20 b(+)e FF(r)p FH(p)p FG(\()p FH(t;)c(x)p FG(\))24 b(=)2445 1811 y(~)2427 1833 y FH(f)9 b FG(\()p FH(t;)14 b(x)p FG(\))507 1953 y FI(div)g FH(u)p FG(\()p FH(t;)g(x)p FG(\))24 b(=)e(0)14 b FH(:)3018 1890 y FI(\(2.1\))0 2142 y(Usually)29 b(w)n(e)f(study)h(the)h(equation)e(for)g FH(t)e FF(\025)f FG(0)j FI(and)h(supplemen)n(t)g(it)h(with)f(the)g (initial)h(condition)e(at)0 2242 y FH(t)23 b FG(=)g(0)p FI(:)1328 2342 y FH(u)p FG(\()p FF(\001)p FH(;)14 b FG(0\))22 b(=)h FH(u)1700 2354 y FD(0)1737 2342 y FG(\()p FF(\001)p FG(\))14 b FH(:)125 2491 y FI(Standard)26 b(references)h(are,)f(e.g.,)i ([Lio69)n(,)g(CF88)o(])g(and)f([BV92)o(].)0 2723 y Fy(2.1)112 b(Lera)m(y)38 b(decomp)s(osition)0 2877 y FI(Let)33 b FH(u)e FF(2)g FH(L)376 2889 y FD(2)413 2877 y FG(\()p FE(T)500 2846 y FD(2)538 2877 y FG(;)14 b FE(R)635 2846 y FD(2)672 2877 y FG(\))p FI(,)34 b(then)f FH(u)f FI(can)h(b)r(e)f (written)h(as)f(a)g(F)-7 b(ourier)32 b(series:)45 b FH(u)p FG(\()p FH(x)p FG(\))32 b(=)2631 2814 y Fz(P)2733 2877 y FH(u)2781 2889 y FA(s)2816 2877 y FH(e)2855 2846 y FA(is)p Fw(\001)p FA(x)2971 2877 y FI(,)i(with)0 2976 y FH(u)48 2988 y FA(s)106 2976 y FF(2)23 b FE(C)244 2946 y FD(2)307 2976 y FI(and)i FH(u)514 2988 y Fw(\000)p FA(s)624 2976 y FG(=)i(\026)-47 b FH(u)759 2988 y FA(s)794 2976 y FI(.)36 b(If)26 b FH(u)p FG(\()p FH(x)p FG(\))d FF(2)h FH(C)1260 2946 y Fw(1)1330 2976 y FG(\()p FE(T)1417 2946 y FD(2)1455 2976 y FG(;)14 b FE(R)1552 2946 y FD(2)1589 2976 y FG(\))p FI(,)26 b(then)g(div)f FH(u)p FG(\()p FH(x)p FG(\))e(=)2265 2914 y Fz(P)2353 3001 y FA(s)p Fw(2)p Fv(Z)2468 2985 y Ft(2)2518 2976 y FH(is)13 b FF(\001)g FH(u)2683 2988 y FA(s)2718 2976 y FH(e)2757 2946 y FA(is)p Fw(\001)p FA(x)2873 2976 y FI(.)36 b(Denote)0 3076 y(with)28 b FH(H)34 b FI(the)28 b(space)241 3258 y FH(H)35 b FI(=)27 b(the)h(closure)e(in)i FH(L)1011 3270 y FD(2)1048 3258 y FG(\()p FE(T)1135 3224 y FD(2)1173 3258 y FG(;)14 b FE(R)1270 3224 y FD(2)1307 3258 y FG(\))28 b FI(of)f FF(f)p FH(u)p FG(\()p FH(x)p FG(\))c FF(2)h FH(C)1829 3224 y Fw(1)1899 3258 y FG(\()p FE(T)1986 3224 y FD(2)2024 3258 y FG(;)14 b FE(R)2121 3224 y FD(2)2158 3258 y FG(\))p FF(j)28 b FI(div)g FH(u)23 b FG(=)f(0)p FH(;)14 b FF(h)p FH(u)p FF(i)23 b FG(=)g(0)p FF(g)o FH(:)0 3441 y FI(Then)28 b(it)g(holds)f(that)472 3624 y FH(H)j FG(=)23 b FF(f)p FH(u)p FG(\()p FH(x)p FG(\))g(=)984 3545 y Fz(X)970 3730 y FA(s)p Fw(2)p Fv(Z)1085 3710 y Ft(2)1085 3748 y(0)1131 3624 y FH(u)1179 3636 y FA(s)1214 3624 y FH(e)1253 3590 y FA(is)p Fw(\001)p FA(x)1392 3624 y FF(2)h FH(L)1528 3636 y FD(2)1565 3624 y FG(\()p FE(T)1652 3590 y FD(2)1689 3624 y FG(;)14 b FE(R)1786 3590 y FD(2)1824 3624 y FG(\))23 b FF(j)g FH(u)1973 3636 y Fw(\000)p FA(s)2083 3624 y FG(=)28 b(\026)-47 b FH(u)2219 3636 y FA(s)2254 3624 y FH(;)27 b(s)19 b FF(\001)f FH(u)2451 3636 y FA(s)2509 3624 y FG(=)23 b(0)p FF(g)14 b FH(;)300 b FI(\(2.2\))0 3915 y(where)30 b FE(Z)298 3885 y FD(2)298 3936 y(0)364 3915 y FG(=)e FE(Z)512 3885 y FD(2)571 3915 y FF(n)20 b(f)p FG(0)p FF(g)p FI(.)45 b(The)31 b(norm)f(in)h FH(H)38 b FI(will)31 b(b)r(e)g(denoted)g(with)g FF(j)21 b(\001)g(j)p FI(,)32 b(and)e(the)i(inner)e(pro)r(duct)0 4015 y(with)e FF(h\001)p FH(;)14 b FF(\001i)p FI(.)0 4181 y FJ(Exercise)47 b(2.1.)h FH(H)g FC(c)l(an)43 b(b)l(e)f(de\034ne)l(d)g(as)g FH(H)52 b FG(=)45 b FF(f)p FH(u)g FF(2)g FH(L)1922 4193 y FD(2)1959 4181 y FF(jh)p FH(u)p FF(i)h FG(=)f(0)p FH(;)27 b FC(div)k FH(u)45 b FG(=)g(0)p FF(g)p FC(,)f(wher)l(e)f(the)0 4281 y(derivatives)32 b(ar)l(e)e(viewe)l(d)h(in)f(the)f(sense)h(of)h (gener)l(alise)l(d)g(functions.)125 4447 y FI(W)-7 b(e)34 b(next)h(in)n(tro)r(duce)f(a)g(basis)g(of)g FH(H)7 b FI(.)58 b(Let)34 b(us)h(de\034ne)f FE(Z)1959 4417 y FD(2)1959 4467 y(+)2049 4447 y FG(=)h FF(f)p FG(\()p FH(s)2262 4459 y FD(1)2299 4447 y FH(;)14 b(s)2375 4459 y FD(2)2412 4447 y FG(\))34 b FF(j)h FG(\()p FH(s)2607 4459 y FD(1)2679 4447 y FH(>)f FG(0\))28 b FI(or)e FG(\()p FH(s)3052 4459 y FD(1)3124 4447 y FG(=)0 4546 y(0)p FH(;)14 b(s)118 4558 y FD(2)178 4546 y FH(>)22 b FG(0\))p FF(g)14 b FH(:)27 b FI(Then)921 4646 y FE(Z)976 4612 y FD(2)976 4666 y(0)1037 4646 y FG(=)c FE(Z)1180 4612 y FD(2)1180 4666 y(+)1254 4646 y FF([)18 b FG(\()p FF(\000)p FE(Z)1479 4612 y FD(2)1479 4666 y(+)1535 4646 y FG(\))c FH(;)97 b FE(Z)1756 4612 y FD(2)1756 4666 y(+)1830 4646 y FF(\\)18 b(\000)p FE(Z)2023 4612 y FD(2)2023 4666 y(+)2102 4646 y FG(=)k FF(;)14 b FH(;)1553 5059 y FI(10)p eop end PStoPSsaved restore %%Page: (10,11) 6 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 11 10 bop 0 523 a FI(and)27 b(w)n(e)h(de\034ne)f(the)h (follo)n(wing)f(set)g(of)h(v)n(ectors)e FF(f)p FH(e)1604 535 y FA(s)1662 523 y FF(j)d FH(s)g FF(2)g FE(Z)1903 493 y FD(2)1903 544 y(0)1941 523 y FF(g)p FI(:)935 780 y FH(e)974 792 y FA(s)1032 780 y FG(=)1120 638 y Fz(\()1187 724 y FH(c)1223 736 y FA(s)1258 724 y FH(s)1297 694 y Fw(?)1367 724 y FG(sin\()p FH(s)18 b FF(\001)h FH(x)p FG(\))p FH(;)98 b(s)23 b FF(2)g FE(Z)1995 694 y FD(2)1995 744 y(+)2064 724 y FH(;)1187 843 y(c)1223 855 y FA(s)1258 843 y FH(s)1297 813 y Fw(?)1367 843 y FG(cos)o(\()p FH(s)c FF(\001)f FH(x)p FG(\))p FH(;)98 b(s)23 b FF(2)g(\000)p FE(Z)2069 813 y FD(2)2069 864 y(+)2138 843 y FH(;)0 1046 y FI(where)30 b FH(c)279 1058 y FA(s)343 1046 y FG(=)529 1013 y FD(1)p 446 1027 200 4 v 446 1036 a Fw(p)p 500 1036 34 3 v 500 1084 a FD(2)p FA(\031)r Fw(j)p FA(s)p Fw(j)655 1046 y FI(,)i(and)e(if)i FH(s)c FG(=)g(\()p FH(s)1185 1058 y FD(1)1222 1046 y FH(;)14 b(s)1298 1058 y FD(2)1335 1046 y FG(\))1367 1016 y FA(t)1397 1046 y FI(,)32 b(then)f FH(s)1683 1016 y Fw(?)1767 1046 y FG(=)d(\()p FF(\000)p FH(s)1996 1058 y FD(2)2033 1046 y FH(;)14 b(s)2109 1058 y FD(1)2146 1046 y FG(\))2178 1016 y FA(t)2208 1046 y FI(.)46 b(The)31 b(set)g FF(f)p FH(e)2665 1058 y FA(s)2699 1046 y FF(g)g FI(is)f(a)h(Hilb)r(ert)0 1164 y(basis)c(of)g FH(H)7 b FI(,)28 b(formed)f(b)n(y)g(eigen\025v)n(ectors)e(of)j FF(\000)p FG(\001)p FI(:)1228 1347 y FF(\000)p FG(\001)p FH(e)1401 1359 y FA(s)1459 1347 y FG(=)22 b FF(j)p FH(s)p FF(j)1631 1312 y FD(2)1668 1347 y FH(e)1707 1359 y FA(s)1826 1347 y FF(8)14 b FH(s)g(:)125 1529 y FI(W)-7 b(e)27 b(further)h(in)n (tro)r(duce)f(the)h(space)1037 1732 y FF(r)p FH(H)1182 1698 y FD(1)1242 1685 y(def)1256 1732 y FG(=)36 b FF(fr)p FH(f)9 b FG(\()p FH(x)p FG(\))p FF(j)p FH(f)32 b FF(2)23 b FH(H)1879 1698 y FD(1)1916 1732 y FG(\()p FE(T)2003 1698 y FD(2)2041 1732 y FG(\))p FF(g)14 b FH(:)0 1915 y FI(Equiv)-5 b(alen)n(tly)e(,)928 2027 y FF(r)p FH(H)1073 1992 y FD(1)1133 2027 y FG(=)23 b FF(f)1277 1948 y Fz(X)1263 2133 y FA(s)p Fw(2)p Fv(Z)1378 2113 y Ft(2)1378 2151 y(0)1423 2027 y FH(sa)1506 2039 y FA(s)1541 2027 y FH(e)1580 1992 y FA(is)p Fw(\001)p FA(x)1719 2027 y FF(2)h FH(L)1855 2039 y FD(2)1892 2027 y FG(\()p FE(T)1979 1992 y FD(2)2017 2027 y FG(;)14 b FE(R)2114 1992 y FD(2)2151 2027 y FG(\))p FF(g)g FH(;)756 b FI(\(2.3\))0 2285 y(so)27 b FF(r)p FH(H)247 2255 y FD(1)312 2285 y FI(is)g(a)g(closed)g(subspace)g(of)h FH(H)7 b FI(.)125 2385 y(The)29 b(relations)f(\(2.2\))g(and)h(\(2.3\))g (immediately)g(imply)h(the)f(follo)n(wing)f(classical)g(result)h(due)g (to)0 2484 y(Helmholtz,)d(whic)n(h)f(is)h(a)f(common)g(to)r(ol)g(to)g (study)h(the)f(Na)n(vier\025Stok)n(es)e(equation)i(since)g(the)h(w)n (orks)0 2584 y(of)i(Lera)n(y:)0 2750 y FJ(Lemma)k(2.2.)41 b FC(The)32 b(sp)l(ac)l(e)g FH(L)978 2762 y FD(2)1015 2750 y FG(\()p FE(T)1102 2720 y FD(2)1140 2750 y FG(;)14 b FE(R)1237 2720 y FD(2)1274 2750 y FG(\))31 b FC(admits)h(the)g(fol)t (lowing)i(de)l(c)l(omp)l(osition)f(in)e(a)h(dir)l(e)l(ct)g(sum)0 2850 y(of)f(thr)l(e)l(e)e(close)l(d)i(ortho)l(gonal)g(subsp)l(ac)l(es) 1049 3032 y FH(L)1106 3044 y FD(2)1143 3032 y FG(\()p FE(T)1230 2998 y FD(2)1268 3032 y FG(;)14 b FE(R)1365 2998 y FD(2)1402 3032 y FG(\))23 b(=)g FH(H)i FF(\010)18 b(r)p FH(H)1867 2998 y FD(1)1923 3032 y FF(\010)g FE(R)2066 2998 y FD(2)2117 3032 y FH(;)0 3215 y FC(wher)l(e)30 b FE(R)294 3185 y FD(2)361 3215 y FC(stands)g(for)h(the)e(sp)l(ac)l(e)i (of)f(c)l(onstant)f(ve)l(ctor)h(\034elds.)125 3381 y FI(The)g(orthogonal)f(pro)5 b(jection)29 b FG(\005)g(:)g FH(L)1314 3393 y FD(2)1350 3381 y FG(\()p FE(T)1437 3351 y FD(2)1475 3381 y FG(;)14 b FE(R)1572 3351 y FD(2)1609 3381 y FG(\))29 b FF(!)f FH(H)38 b FI(is)31 b(called)f(the)h FC(L)l(er)l(ay)i(pr)l(oje)l(ction)p FI(.)48 b(Note)0 3481 y(that)1022 3580 y FG(\005\()p FF(r)p FH(p)p FG(\))24 b(=)f(0)p FH(;)96 b FG(\005\()p FH(constant)p FG(\))24 b(=)e(0)14 b FH(:)0 3730 y FI(Let)35 b FG(\()p FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))p FH(;)g(p)p FG(\()p FH(t;)g(x)p FG(\)\))36 b FI(b)r(e)f(a)f(smo)r(oth)g(solution)g(of)g(\(2.1\).)58 b(Let)34 b(us)h(denote)f FH(u)p FG(\(0)p FH(;)14 b(x)p FG(\))35 b(=)f FH(u)2872 3742 y FD(0)2909 3730 y FG(\()p FH(x)p FG(\))h FI(and)0 3829 y(assume)27 b(that)888 3929 y FF(h)p FH(u)968 3941 y FD(0)1005 3929 y FG(\()p FH(x)p FG(\))p FF(i)e FG(=)d(0)p FH(;)41 b FF(h)1417 3907 y FG(~)1398 3929 y FH(f)9 b FG(\()p FH(t;)14 b(x)p FG(\))p FF(i)25 b FG(=)d(0)27 b FI(for)g(all)h FH(t)23 b FF(\025)f FG(0)14 b FH(:)0 4078 y FI(Then,)28 b(in)n(tegrating)e(the)i(\034rst)f (equation)g(of)h(\(2.1\))f(o)n(v)n(er)f(space,)h(w)n(e)g(obtain)879 4248 y FH(d)p 864 4285 74 4 v 864 4361 a(dt)947 4304 y FF(h)p FH(u)p FF(i)18 b(\000)g(h)p FG(\001)p FH(u)p FF(i)h FG(+)f FF(h)p FG(\()p FH(u)h FF(\001)f(r)p FG(\))p FH(u)p FF(i)h FG(+)f FF(hr)p FH(p)p FF(i)24 b FG(=)e FF(h)2234 4282 y FG(~)2216 4304 y FH(f)9 b FF(i)14 b FH(:)0 4530 y FI(Since)28 b FF(h)p FG(\001)p FH(u)p FF(i)23 b FG(=)g FF(hr)p FH(p)p FF(i)g FG(=)g FF(h)845 4508 y FG(~)827 4530 y FH(f)9 b FF(i)23 b FG(=)g(0)k FI(and)g(for)g FH(l)e FG(=)d(1)p FH(;)14 b FG(2)27 b FI(w)n(e)g(ha)n(v)n(e)595 4756 y FF(h)p FG(\(\()p FH(u)19 b FF(\001)f(r)p FG(\))p FH(u)p FG(\))980 4721 y FA(l)1006 4756 y FF(i)23 b FG(=)1149 4643 y Fz(Z)1246 4756 y FH(u)1294 4721 y FA(j)1328 4756 y FG(\()1412 4700 y FH(@)p 1370 4737 132 4 v 1370 4813 a(@)5 b(x)1466 4789 y FA(j)1511 4756 y FH(u)1559 4721 y FA(l)1584 4756 y FG(\))p FH(dx)24 b FG(=)f FF(\000)1897 4643 y Fz(Z)1993 4756 y FI(div)28 b FH(u)18 b FF(\001)h FH(u)2290 4721 y FA(l)2315 4756 y FH(dx)k FG(=)g(0)14 b FH(;)1553 5059 y FI(11)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 12 11 bop 0 523 a FI(then)212 490 y FA(d)p 199 504 60 4 v 199 552 a(dt)269 523 y FF(h)p FH(u)p FF(i)23 b FG(=)f(0)14 b FH(:)27 b FI(Hence,)h(if)g FH(u)991 535 y FD(0)1051 523 y FF(2)c FH(H)7 b FI(,)27 b(then)h FH(u)p FG(\()p FH(t)p FG(\))23 b FF(2)h FH(H)34 b FI(for)27 b(all)g FH(t)c FF(\025)g FG(0)p FI(.)125 623 y(No)n(w,)k(apply)g(the)h (pro)5 b(jection)26 b FG(\005)i FI(to)g(\(2.1\).)36 b(Since)28 b FG(\005)p FH(u)23 b FG(=)f FH(u)p FI(,)28 b(w)n(e)f(\034nd)h(that) 1066 805 y FG(_)-38 b FH(u)18 b FF(\000)g FH(\027)5 b FG(\005\001)p FH(u)19 b FG(+)f(\005\()p FH(u)h FF(\001)f(r)p FG(\))p FH(u)23 b FG(=)g(\005)2069 783 y(~)2051 805 y FH(f)g(:)0 988 y FI(With)28 b(the)g(notation)323 1171 y FH(Lu)450 1124 y FD(def)464 1171 y FG(=)36 b FF(\000)p FG(\005\001)p FH(u)22 b FG(=)h FF(\000)p FG(\001)p FH(u)k FI(and)g FH(B)t FG(\()p FH(u)p FG(\))1491 1124 y FD(def)1505 1171 y FG(=)36 b(\005\()p FH(u)18 b FF(\001)h(r)p FG(\))p FH(u;)42 b FI(for)27 b FH(u)22 b FF(2)i FH(H)7 b FI(;)133 b FH(f)2603 1124 y FD(def)2616 1171 y FG(=)36 b(\005)2797 1149 y(~)2779 1171 y FH(f)23 b(;)0 1353 y FI(w)n(e)k(are)g(lead)g(to)g (the)h(equation)1079 1464 y Fz(\()1160 1549 y FG(_)-37 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)17 b FG(+)h FH(B)t FG(\()p FH(u)p FG(\))24 b(=)e FH(f)9 b FG(\()p FH(t)p FG(\))14 b FH(;)1146 1669 y(u)p FG(\()p FH(t)p FG(\))23 b FF(2)g FH(H)e(:)3018 1606 y FI(\(2.4\))0 1869 y FJ(Lemma)j(2.3.)36 b FC(If)25 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))24 b FF(2)f FH(C)994 1839 y Fw(1)1065 1869 y FG(\()p FE(T)1152 1839 y FD(2)1190 1869 y FG(\))i FC(satis\034es)h(\(2.4\),)h(then)e (ther)l(e)h(exists)f FH(p)p FG(\()p FH(t;)14 b(x)p FG(\))23 b FF(2)h FH(C)2777 1839 y Fw(1)2847 1869 y FG(\()p FE(T)2934 1839 y FD(2)2972 1869 y FG(\))i FC(such)0 1968 y(that)k(\(2.1\))h (holds.)0 2134 y(Pr)l(o)l(of.)43 b FI(Denote)e FG(_)-38 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))i FF(\000)e FH(\027)5 b FG(\001)p FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))i(+)f FH(B)t FG(\()p FH(u)p FG(\()p FH(t;)f(x)p FG(\)\))i FF(\000)1771 2112 y FG(~)1753 2134 y FH(f)9 b FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)e FF(\000)p FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\))p FI(.)37 b(Then)26 b FG(\005)p FH(\030)i FG(=)23 b(0)p FI(.)35 b(So,)26 b(b)n(y)0 2234 y(Lemma)h(2.2,)g FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)f FF(r)p FH(p)p FG(\()p FH(t;)14 b(x)p FG(\))19 b(+)f FH(C)6 b FG(\()p FH(t)p FG(\))p FI(.)38 b(F)-7 b(or)27 b(the)h(same)f(reasons)e (as)i(ab)r(o)n(v)n(e,)g FH(C)6 b FG(\()p FH(t)p FG(\))23 b FF(\021)g FG(0)p FI(.)p 3128 2234 4 57 v 3132 2181 50 4 v 3132 2234 V 3182 2234 4 57 v 125 2400 a(Belo)n(w)j(w)n(e)h (study)h(eq.)f(\(2.4\))h(instead)f(of)h(\(2.1\).)0 2549 y FJ(Notations:)86 b FI(W)-7 b(e)30 b(denote)f FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))25 b(=)g(\005\()p FH(u)20 b FF(\001)f(r)p FG(\))p FH(v)33 b FI(\(so)28 b FH(B)t FG(\()p FH(u)p FG(\))e(=)f FH(B)t FG(\()p FH(u;)14 b(u)p FG(\))p FI(\).)41 b(F)-7 b(or)28 b FH(r)g FF(\025)d FG(0)j FI(w)n(e)h(de\034ne)0 2649 y(the)38 b(space)f FH(H)461 2619 y FA(r)535 2649 y FI(as)g FH(H)723 2619 y FA(r)799 2649 y FG(:=)i FH(H)32 b FF(\\)26 b FH(H)1184 2619 y FA(r)1220 2649 y FG(\()p FE(T)1307 2619 y FD(2)1345 2649 y FH(;)14 b FE(R)1442 2619 y FD(2)1479 2649 y FG(\))p FI(,)41 b(and)c(for)g FH(r)43 b(<)c FG(0)e FI(\025)g(as)g FH(H)2413 2619 y FA(r)2489 2649 y FG(:=)h FI(closure)e(of)h FH(H)45 b FI(in)0 2749 y FH(H)76 2719 y FA(r)113 2749 y FG(\()p FE(T)200 2719 y FD(2)237 2749 y FH(;)14 b FE(R)334 2719 y FD(2)372 2749 y FG(\))p FI(.)37 b(Then)605 2956 y FH(H)681 2922 y FA(r)741 2956 y FG(=)828 2864 y Fz(n)884 2956 y FH(u)22 b FG(=)1056 2877 y Fz(X)1042 3063 y FA(s)p Fw(2)p Fv(Z)1157 3043 y Ft(2)1157 3081 y(0)1217 2956 y FH(u)1265 2968 y FA(s)1314 2956 y FH(e)1353 2968 y FA(s)1388 2956 y FG(\()p FH(x)p FG(\))14 b FF(j)g(k)p FH(u)p FF(k)1682 2922 y FD(2)1682 2977 y FA(r)1741 2956 y FG(=)1829 2877 y Fz(X)1977 2956 y FF(j)p FH(s)p FF(j)2062 2922 y FD(2)p FA(r)2132 2956 y FF(j)p FH(u)2203 2968 y FA(s)2238 2956 y FF(j)2261 2922 y FD(2)2321 2956 y FH(<)23 b FF(1)2492 2864 y Fz(o)2561 2956 y FH(:)0 3237 y FI(F)-7 b(or)27 b FH(u)c FF(2)g FH(H)374 3207 y FA(r)438 3237 y FI(w)n(e)k(ha)n(v)n(e)1340 3437 y FF(k)p FH(u)p FF(k)1472 3402 y FD(2)1472 3457 y FA(r)1531 3437 y FG(=)c FF(h)p FH(L)1708 3402 y FA(r)1744 3437 y FH(u;)14 b(u)p FF(i)g FH(;)0 3586 y FI(since)30 b FH(u)d FF(2)g FH(H)439 3556 y FA(r)476 3586 y FG(\()p FE(T)563 3556 y FD(2)600 3586 y FH(;)14 b(R)701 3556 y FD(2)738 3586 y FG(\))31 b FI(with)f FF(h)p FH(u)p FF(i)d FG(=)g(0)j FI(satis\034es)f FF(k)p FH(u)p FF(k)1735 3556 y FD(2)1735 3607 y FA(r)1798 3586 y FG(=)e FF(h)p FG(\()p FF(\000)p FG(\001\))2120 3556 y FA(r)2171 3586 y FH(u;)14 b(u)p FF(i)60 b FI(\(see)29 b(\(S2\)\),)j(and)e FH(Le)3063 3598 y FA(s)3124 3586 y FG(=)0 3686 y FF(j)p FH(s)p FF(j)85 3656 y FD(2)122 3686 y FH(e)161 3698 y FA(s)196 3686 y FI(.)37 b(In)28 b(particular,)739 3868 y FF(k)p FH(u)p FF(k)871 3834 y FD(2)871 3889 y(1)930 3868 y FG(=)23 b FF(h)p FH(Lu;)14 b(u)p FF(i)22 b FG(=)g FF(jr)p FH(u)p FF(j)1544 3880 y FA(L)1590 3888 y Ft(2)1640 3868 y FH(;)97 b FF(h)p FH(Lu;)14 b(v)s FF(i)23 b FG(=)f FF(hr)p FH(u;)14 b FF(r)p FH(v)s FF(i)569 b FI(\(2.5\))0 4051 y(for)27 b FH(u;)14 b(v)26 b FF(2)d FH(H)432 4021 y FD(1)469 4051 y FI(.)125 4151 y(W)-7 b(e)27 b(note)h(that)g FH(H)708 4120 y FA(r)739 4128 y Ft(1)798 4151 y FF(\032)23 b FH(H)962 4120 y FA(r)993 4128 y Ft(2)1056 4151 y FI(if)28 b FH(r)1169 4163 y FD(1)1230 4151 y FF(\025)23 b FH(r)1355 4163 y FD(2)1393 4151 y FI(,)k(that)1278 4266 y Fz(\\)1308 4441 y FA(r)1384 4345 y FH(H)1460 4311 y FA(r)1520 4345 y FG(=)22 b FH(H)k FF(\\)18 b FH(C)1840 4311 y Fw(1)0 4603 y FI(\(this)28 b(follo)n(ws)f(from)g(\(1.2\))o(\),)h(and)g(that)g(the)g(linear)e (space)h FF(\\)1908 4615 y FA(r)1945 4603 y FH(H)2021 4572 y FA(r)2086 4603 y FI(is)g(dense)g(in)h(eac)n(h)f(space)g FH(H)2977 4572 y FA(s)3012 4603 y FI(.)1553 5059 y(12)p eop end PStoPSsaved restore %%Page: (12,13) 7 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 13 12 bop 0 523 a FJ(Lemma)29 b(2.4.)40 b FC(\(\020Bounde)l(d)31 b(p)l(oly-line)l(ar)g(map)g(is)f(c)l (ontinuous\021\))0 623 y(If)f FH(X)155 635 y FD(1)192 623 y FH(;)14 b(X)298 635 y FD(2)364 623 y FC(ar)l(e)29 b(Banach)h(sp)l(ac)l(es)g(and)f FH(F)35 b FG(:)23 b FH(X)1416 635 y FD(1)1470 623 y FF(\002)16 b(\001)e(\001)g(\001)i(\002)g FH(X)1814 635 y FD(1)1874 623 y FF(!)23 b FH(X)2049 635 y FD(2)2115 623 y FC(is)29 b(a)g(p)l(oly-line)l(ar)i(map)e(such)g(that) 839 800 y FF(k)p FH(F)12 b FG(\()p FH(u)1026 812 y FD(1)1062 800 y FH(;)i(:)g(:)g(:)g(;)g(u)1295 812 y FA(r)1331 800 y FG(\))p FF(k)1405 812 y FA(X)1459 820 y Ft(2)1519 800 y FF(\024)22 b FH(C)6 b FF(k)p FH(u)1761 812 y FD(1)1798 800 y FF(k)1840 812 y FA(X)1894 820 y Ft(1)1944 800 y FH(:)14 b(:)g(:)g FF(k)p FH(u)2145 812 y FA(r)2181 800 y FF(k)2223 812 y FA(X)2277 820 y Ft(1)2327 800 y FH(;)0 978 y FC(then)27 b FH(F)39 b FC(is)27 b(c)l(ontinuous.)37 b(Mor)l(e)l(over,)29 b(if)f FH(V)1325 990 y FD(1)1385 978 y FF(\032)23 b FH(X)1542 990 y FD(1)1606 978 y FC(is)k(a)h(dense)f (line)l(ar)h(subsp)l(ac)l(e)f(and)g(the)g(ine)l(quality)0 1078 y(ab)l(ove)g(holds)h(for)e FH(u)606 1090 y FA(j)664 1078 y FF(2)d FH(V)790 1090 y FD(1)828 1078 y FC(,)k(then)f FH(F)38 b FC(extends)25 b(to)h(a)h(p)l(oly-line)l(ar)g(c)l(ontinuous)f (map)g FH(X)2672 1090 y FD(1)2720 1078 y FF(\002)10 b(\001)k(\001)g (\001)c(\002)g FH(X)3046 1090 y FD(1)3106 1078 y FF(!)0 1177 y FH(X)69 1189 y FD(2)106 1177 y FC(.)125 1339 y FI(The)27 b(pro)r(of)g(of)h(this)f(result)h(is)f(straigh)n(t-forw)n (ard.)0 1571 y Fy(2.2)112 b(Prop)s(erties)36 b(of)i(the)f(nonlinearit)m (y)31 b Fx(B)1747 1568 y(B)1750 1571 y(B)0 1724 y FJ(\(B1\))d FI(If)g FH(u)p FI(,)f FH(v)s FI(,)h FH(w)d FF(2)f FH(C)721 1694 y Fw(1)810 1724 y FF(\\)19 b FH(H)7 b FI(,)27 b(then)111 1886 y(i\))42 b FF(h)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FH(;)g(v)s FF(i)24 b FG(=)e(0)p FI(,)88 2050 y(ii\))42 b FF(h)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FH(;)g(w)r FF(i)24 b FG(=)f FF(\000h)p FH(B)t FG(\()p FH(u;)14 b(w)r FG(\))p FH(;)g(v)s FF(i)p FI(.)0 2212 y FC(Pr)l(o)l(of.)43 b FI(i\))28 b(In)n(tegrating)e(b)n(y)h(parts)g(w)n (e)g(ha)n(v)n(e:)435 2433 y FF(h)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FH(;)g(v)s FF(i)24 b FG(=)949 2320 y Fz(Z)995 2509 y Fv(T)1034 2492 y Ft(2)1085 2433 y FH(u)1133 2399 y FA(j)1167 2341 y Fz(\020)1268 2377 y FH(@)p 1227 2414 132 4 v 1227 2490 a(@)5 b(x)1323 2466 y FA(j)1391 2433 y FH(v)1434 2399 y FA(l)1459 2341 y Fz(\021)1523 2433 y FH(v)1566 2399 y FA(l)1605 2433 y FH(dx)24 b FG(=)1794 2377 y(1)p 1794 2414 42 4 v 1794 2490 a(2)1882 2320 y Fz(Z)1928 2509 y Fv(T)1967 2492 y Ft(2)2017 2433 y FH(u)2065 2399 y FA(j)2174 2377 y FH(@)p 2133 2414 132 4 v 2133 2490 a(@)5 b(x)2229 2466 y FA(j)2288 2433 y FF(j)p FH(v)s FF(j)2377 2399 y FD(2)2428 2433 y FH(dx)1719 2655 y FG(=)18 b FF(\000)1895 2599 y FG(1)p 1895 2636 42 4 v 1895 2712 a(2)1983 2542 y Fz(Z)2030 2731 y Fv(T)2069 2714 y Ft(2)2105 2655 y FG(\()p FI(div)c FH(u)p FG(\))p FF(j)p FH(u)p FF(j)2438 2621 y FD(2)2475 2655 y FH(dx)23 b FG(=)g(0)14 b FH(:)125 2879 y FI(ii\))28 b(Apply)g(i\))g(with)g FH(v)e FG(:=)d FH(v)e FG(+)d FH(w)r FI(.)p 3128 2879 4 57 v 3132 2827 50 4 v 3132 2879 V 3182 2879 4 57 v 125 3045 a FJ(\(B2\))27 b FI(If)h FH(u)p FI(,)f FH(v)s FI(,)h FH(w)e FF(2)d FH(C)845 3014 y Fw(1)934 3045 y FF(\\)c FH(H)7 b FI(,)28 b(then)111 3207 y(i\))42 b FF(jh)p FH(B)t FG(\()p FH(u;)14 b(w)r FG(\))p FH(;)g(v)s FF(ij)24 b FG(=)f FF(jh)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FH(;)g(w)r FF(ij)25 b(\024)d FH(C)6 b FF(k)p FH(u)p FF(k)1563 3222 y FD(1)p FA(=)p FD(2)1666 3207 y FF(k)p FH(v)s FF(k)1793 3222 y FD(1)p FA(=)p FD(2)1897 3207 y FF(k)p FH(w)r FF(k)2042 3219 y FD(1)2079 3207 y FI(,)88 3371 y(ii\))42 b FF(k)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FF(k)551 3383 y Fw(\000)p FD(1)662 3371 y FF(\024)23 b FH(C)6 b FF(k)p FH(u)p FF(k)947 3386 y FD(1)p FA(=)p FD(2)1064 3371 y FF(k)p FH(v)s FF(k)1191 3386 y FD(1)p FA(=)p FD(2)1322 3371 y FI(.)0 3533 y FC(Pr)l(o)l(of.)43 b FI(i\))28 b(implies)g(ii\))g (b)n(y)f(the)h(dualit)n(y)-7 b(,)28 b(see)f(\(S4\).)37 b(No)n(w)27 b(w)n(e)g(pro)n(v)n(e)f(i\):)347 3763 y FF(jh)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FH(;)g(w)r FF(ij)838 3704 y Fq(\(B1\))882 3763 y FG(=)67 b FF(j)18 b(\000)g(h)p FH(B)t FG(\()p FH(u;)c(w)r FG(\))p FH(;)g(v)s FF(ij)25 b(\024)e FH(C)1774 3650 y Fz(Z)1871 3763 y FF(j)p FH(u)p FF(j)f(jr)p FH(w)r FF(j)i(j)p FH(v)s FF(j)p FH(dx)838 3943 y Fq(H\366lder)918 4012 y FF(\024)102 b FH(C)1144 4024 y FD(1)1196 4012 y FF(jr)p FH(w)r FF(j)1372 4024 y FA(L)1418 4032 y Ft(2)1469 4012 y FF(j)p FH(u)p FF(j)1563 4024 y FA(L)1609 4032 y Ft(4)1658 4012 y FF(j)p FH(v)s FF(j)1747 4024 y FA(L)1793 4032 y Ft(4)1853 3932 y Fq(\(1.5\))1900 4012 y FF(\024)69 b FH(C)2093 4024 y FD(2)2130 4012 y FF(k)p FH(w)r FF(k)2275 4024 y FD(1)2326 4012 y FF(k)p FH(u)p FF(k)2458 4027 y FD(1)p FA(=)p FD(2)2575 4012 y FF(k)p FH(v)s FF(k)2702 4027 y FD(1)p FA(=)p FD(2)2819 4012 y FH(:)p 3128 4190 V 3132 4137 50 4 v 3132 4190 V 3182 4190 4 57 v 0 4355 a FI(So)27 b(b)n(y)h(Lemma)f(2.4)g(and)g (\(B2)14 b(ii\),)27 b FH(B)32 b FI(extends)c(to)f(a)g(bilinear)g(con)n (tin)n(uous)g(map,)1130 4533 y FH(B)g FG(:)c FH(H)1342 4498 y FD(1)p FA(=)p FD(2)1465 4533 y FF(\002)18 b FH(H)1624 4498 y FD(1)p FA(=)p FD(2)1751 4533 y FF(!)23 b FH(H)1933 4498 y Fw(\000)p FD(1)2036 4533 y FH(:)125 4710 y FJ(\(B3\))i FI(If)h FH(u;)14 b(v)26 b FF(2)d FH(H)e FF(\\)15 b FH(C)875 4680 y Fw(1)946 4710 y FI(,)26 b(then)g FF(k)p FH(B)t FG(\()p FH(u;)14 b(v)s FG(\))p FF(k)1525 4722 y Fw(\000)p FD(3)1637 4710 y FF(\024)22 b FH(C)6 b FF(j)p FH(u)p FF(j)14 b(j)p FH(v)s FF(j)p FI(.)37 b(So)25 b(b)n(y)g(Lemma)h(2.4,)f FH(B)30 b FI(extends)25 b(to)0 4810 y(a)i(con)n(tin)n(uous)g(map)g FH(B)g FG(:)c FH(H)j FF(\002)18 b FH(H)29 b FF(!)24 b FH(H)1264 4780 y Fw(\000)p FD(3)1352 4810 y FI(.)1553 5059 y(13)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 14 13 bop 125 523 a FI(The)27 b(pro)r(of)g(is)h(left)g (as)f(an)g(exercise.)125 623 y(Let)g(us)h(de\034ne)g(the)f(space)804 805 y FF(H)d FG(=)e FF(f)p FH(u)g FF(2)i FH(L)1233 817 y FD(2)1270 805 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1555 771 y FD(1)1590 805 y FG(\))g FF(j)29 b FG(_)-38 b FH(u)23 b FF(2)g FH(L)1879 817 y FD(2)1916 805 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2201 771 y Fw(\000)p FD(1)2288 805 y FG(\))p FF(g)p FH(;)930 1050 y FF(k)p FH(u)p FF(k)1062 1015 y FD(2)1062 1070 y Fw(H)1144 1050 y FG(=)1232 937 y Fz(Z)1315 957 y FA(T)1278 1125 y FD(0)1381 982 y Fz(\000)1419 1050 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)1645 1015 y FD(2)1645 1070 y(1)1700 1050 y FG(+)k FF(k)c FG(_)-37 b FH(u)o FG(\()p FH(t)p FG(\))p FF(k)2008 1015 y FD(2)2008 1070 y Fw(\000)p FD(1)2097 982 y Fz(\001)2149 1050 y FH(dt)14 b(:)0 1251 y FI(Here)38 b(for)g FH(u)i FF(2)i FH(L)587 1263 y FD(2)623 1251 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)908 1221 y FD(1)944 1251 y FG(\))39 b FI(and)f FH(\030)45 b FF(2)c FH(L)1421 1263 y FD(2)1458 1251 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1743 1221 y Fw(\000)p FD(1)1831 1251 y FG(\))38 b FI(w)n(e)g(sa)n(y)g(that)53 b FG(_)-38 b FH(u)41 b FG(=)f FH(\030)t FI(,)i(if)c(there)g(exists)0 1362 y FH(\021)d FF(2)c FH(H)238 1332 y Fw(\000)p FD(1)360 1362 y FI(suc)n(h)h(that)h FH(u)p FG(\()p FH(t)p FG(\))e(=)g FH(\021)25 b FG(+)1159 1295 y Fz(R)1214 1316 y FA(t)1198 1392 y FD(0)1257 1362 y FH(\030)t FG(\()p FH(s)p FG(\))14 b FH(ds)p FI(,)34 b(for)f(a.a.)50 b FH(t)32 b FF(2)g FG([0)p FH(;)14 b(T)e FG(])p FI(.)50 b(Note)33 b(that)f FF(H)i FI(is)e(a)h(Hilb)r(ert)0 1462 y(space.)0 1628 y FJ(Lemma)i(2.5.)42 b FF(H)35 b FC(is)f(c)l(ontinuously)g(and)g(c)l (omp)l(actly)h(emb)l(e)l(dde)l(d)g(in)f FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))p FC(.)50 b(In)34 b(p)l(articular,)0 1727 y(for)d(any)f FH(u)22 b FF(2)i(H)30 b FC(and)g(e)l(ach)h FG(0)23 b FF(\024)f FH(t)h FF(\024)g FH(T)12 b FC(,)29 b FH(u)p FG(\()p FH(t)p FG(\))h FC(is)g(a)g(wel)t(l)h (de\034ne)l(d)f(ve)l(ctor)g(in)g FH(H)7 b FC(.)0 1893 y(Pr)l(o)l(of.)43 b FI(The)19 b(em)n(b)r(edding)f FF(H)24 b(\032)f FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))17 b FI(is)h(an)h(exercise.)32 b(F)-7 b(or)18 b(its)g(compactness)g(see)g([Lio69)n(].)p 3128 1893 4 57 v 3132 1841 50 4 v 3132 1893 V 3182 1893 4 57 v 0 2109 a FJ(\(B4\))28 b FI(F)-7 b(or)26 b(an)n(y)h FH(u;)14 b(v)26 b FF(2)d(H)29 b FI(and)e(an)n(y)g FG(0)c FF(\024)f FH(t)h FF(\024)g FH(T)39 b FI(w)n(e)27 b(ha)n(v)n(e)817 2241 y Fz(Z)900 2261 y FA(T)863 2429 y FD(0)953 2354 y FF(h)p FH(Lu)p FG(\()p FH(t)p FG(\))p FH(;)g(v)s FG(\()p FH(t)p FG(\))p FF(i)14 b FH(dt)24 b FG(=)1602 2241 y Fz(Z)1685 2261 y FA(t)1648 2429 y FD(0)1714 2354 y FF(hr)p FH(u)p FG(\()p FH(t)p FG(\))p FH(;)29 b FF(r)p FH(v)s FG(\()p FH(t)p FG(\))p FF(i)14 b FH(dt)g(;)817 2470 y Fz(Z)900 2491 y FA(t)863 2659 y FD(0)930 2583 y FF(h)g FG(_)-37 b FH(u)o(;)14 b(u)p FF(i)g FH(ds)23 b FG(=)1343 2527 y(1)p 1343 2564 42 4 v 1343 2640 a(2)1394 2583 y(\()p FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1614 2549 y FD(2)1670 2583 y FF(\000)18 b(j)p FH(u)p FG(\(0\))p FF(j)1953 2549 y FD(2)1990 2583 y FG(\))c FH(:)0 2812 y FC(Pr)l(o)l(of.)43 b FI(The)28 b(\034rst)g(relation)f(follo)n(ws)g(from)h(\(2.5\))o(.)39 b(T)-7 b(o)28 b(establish)f(the)i(second)e(w)n(e)h(\034rst)g(pro)n(v)n (e)e(it)j(for)0 2912 y(smo)r(oth)e FH(u)g FI(and)h FH(v)s FI(,)g(and)f(then)h(argue)e(b)n(y)i(con)n(tin)n(uit)n(y)-7 b(.)p 3128 2912 4 57 v 3132 2859 50 4 v 3132 2912 V 3182 2912 4 57 v 125 3078 a(Noting)23 b(that)i FH(t)e FF(7!)g(h)15 b FG(_)-38 b FH(u;)14 b(u)p FF(i)22 b(2)i FH(L)1086 3090 y FD(1)1123 3078 y FG(\(0)p FH(;)14 b(T)e FG(\))p FI(,)23 b(w)n(e)h(see)g(that)g(the)h(second)e(relation)g(in)h(\(B4\))g(means)g (that)0 3178 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)188 3148 y FD(2)253 3178 y FI(is)k(an)f(absolutely)g(con)n(tin)n(uous)f (function)i(and)1063 3347 y FH(d)p 1048 3384 74 4 v 1048 3460 a(dt)1154 3404 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1342 3369 y FD(2)1403 3404 y FG(=)22 b(2)p FF(h)15 b FG(_)-38 b FH(u;)14 b(u)p FF(i)p FH(;)96 b FF(8)p FH(u)23 b FF(2)g(H)15 b FH(:)0 3614 y FJ(\(B5\))28 b FI(i\))g FH(B)j FI(de\034nes)d(a)f(three-linear)f(con)n(tin)n(uous)h(map)360 3797 y FF(H)19 b(\002)f(H)i(\002)e(H)24 b(!)f FH(L)962 3809 y FD(1)999 3797 y FG([0)p FH(;)14 b(T)e FG(])i FH(;)178 b FG(\()p FH(u)1480 3809 y FD(1)1517 3797 y FH(;)14 b(u)1602 3809 y FD(2)1639 3797 y FH(;)g(u)1724 3809 y FD(3)1760 3797 y FG(\))24 b FF(7\000)-15 b(!)23 b(h)p FH(B)2071 3729 y Fz(\000)2110 3797 y FH(u)2158 3809 y FD(1)2195 3797 y FG(\()p FH(t)p FG(\))p FH(;)28 b(u)2388 3809 y FD(2)2425 3797 y FG(\()p FH(t)p FG(\))2519 3729 y Fz(\001)2557 3797 y FH(;)14 b(u)2642 3809 y FD(3)2679 3797 y FG(\()p FH(t)p FG(\))p FF(i)p FH(:)0 3979 y FI(ii\))28 b(If)g FH(u)237 3991 y FD(1)274 3979 y FH(;)14 b(u)359 3991 y FD(2)396 3979 y FH(;)g(u)481 3991 y FD(3)540 3979 y FF(2)24 b(H)k FI(and)g FH(u)927 3991 y FD(2)986 3979 y FG(=)23 b FH(u)1122 3991 y FD(3)1159 3979 y FI(,)28 b(then)g FF(h)p FH(B)t FG(\()p FH(u)1578 3991 y FD(1)1615 3979 y FH(;)14 b(u)1700 3991 y FD(2)1737 3979 y FG(\))p FH(;)g(u)1854 3991 y FD(2)1891 3979 y FF(i)23 b FG(=)g(0)k FI(in)h FH(L)2257 3991 y FD(1)2293 3979 y FG([0)p FH(;)14 b(T)e FG(])p FI(.)0 4145 y FC(Pr)l(o)l(of.)43 b FI(i\))28 b(Note)g(\034rst)f(that)h FF(H)c(\032)e FH(L)1133 4157 y FD(4)1170 4145 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1455 4115 y FD(1)p FA(=)p FD(2)1558 4145 y FG(\))p FI(.)37 b(Indeed,)28 b(since)f FF(k)p FH(u)p FF(k)2278 4115 y FD(4)2278 4172 y(1)p FA(=)p FD(2)2404 4145 y FF(\024)c(j)p FH(u)p FF(j)2586 4115 y FD(2)2637 4145 y FF(k)p FH(u)p FF(k)2769 4115 y FD(2)2769 4166 y(1)2804 4145 y FI(,)28 b(then)299 4410 y FF(k)p FH(u)p FF(k)431 4376 y FD(4)431 4437 y FA(L)477 4445 y Ft(4)507 4437 y FD(\(0)p FA(;T)9 b FD(;)p FA(H)711 4420 y Ft(1)p Fu(=)p Ft(2)803 4437 y FD(\))856 4410 y FG(=)944 4297 y Fz(Z)1027 4318 y FA(T)990 4486 y FD(0)1093 4410 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1328 4376 y FD(4)1328 4433 y(1)p FA(=)p FD(2)1445 4410 y FH(ds)23 b FF(\024)g FG(sup)1688 4476 y FA(t)1800 4410 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1988 4376 y FD(2)2053 4297 y Fz(Z)2136 4318 y FA(T)2099 4486 y FD(0)2202 4410 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)2428 4376 y FD(2)2428 4431 y(1)2478 4410 y FH(dt)g FF(\024)g(k)p FH(u)p FF(k)2794 4376 y FD(4)2794 4431 y Fw(H)2867 4410 y FH(:)1553 5059 y FI(14)p eop end PStoPSsaved restore %%Page: (14,15) 8 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 15 14 bop 0 523 a FI(Hence,)28 b(for)f FH(u)445 535 y FD(1)482 523 y FH(;)14 b(u)567 535 y FD(2)603 523 y FH(;)g(u)688 535 y FD(3)748 523 y FF(2)23 b(H)d(\\)f FH(C)1055 493 y Fw(1)1153 523 y FI(w)n(e)27 b(ha)n(v)n(e)399 661 y Fz(Z)482 681 y FA(T)445 850 y FD(0)548 774 y FF(jh)p FH(B)t FG(\()p FH(u)750 786 y FD(1)788 774 y FH(;)14 b(u)873 786 y FD(2)910 774 y FG(\))p FH(;)g(u)1027 786 y FD(3)1064 774 y FF(ij)g FH(ds)1238 693 y Fq(\(B2\))1282 774 y FF(\024)67 b FH(C)1507 661 y Fz(Z)1590 681 y FA(T)1553 850 y FD(0)1670 774 y FF(k)p FH(u)1760 786 y FD(3)1796 774 y FF(k)1838 786 y FD(1)1888 774 y FF(k)p FH(u)1978 786 y FD(1)2015 774 y FF(k)2057 789 y FD(1)p FA(=)p FD(2)2174 774 y FF(k)p FH(u)2264 786 y FD(2)2301 774 y FF(k)2343 789 y FD(1)p FA(=)p FD(2)2460 774 y FH(ds)2566 705 y Fq(H\366lder)2645 774 y FF(\024)399 1007 y FH(C)464 915 y Fz(\020)528 894 y(Z)611 915 y FA(T)574 1083 y FD(0)691 1007 y FF(k)p FH(u)781 1019 y FD(3)817 1007 y FF(k)859 973 y FD(2)859 1028 y(1)910 1007 y FH(ds)992 915 y Fz(\021)1042 933 y FD(1)p FA(=)p FD(2)1169 915 y Fz(\020)1232 894 y(Z)1315 915 y FA(T)1279 1083 y FD(0)1395 1007 y FF(k)p FH(u)1485 1019 y FD(1)1522 1007 y FF(k)1564 973 y FD(4)1564 1030 y(1)p FA(=)p FD(2)1681 1007 y FH(ds)1763 915 y Fz(\021)1813 933 y FD(1)p FA(=)p FD(4)1941 915 y Fz(\020)2004 894 y(Z)2087 915 y FA(T)2050 1083 y FD(0)2167 1007 y FF(k)p FH(u)2257 1019 y FD(2)2293 1007 y FF(k)2335 973 y FD(4)2335 1030 y(1)p FA(=)p FD(2)2453 1007 y FH(ds)2535 915 y Fz(\021)2585 933 y FD(1)p FA(=)p FD(4)1727 1153 y FF(\024)23 b FH(C)d FF(k)p FH(u)1984 1165 y FD(3)2020 1153 y FF(k)2062 1165 y Fw(H)2123 1153 y FF(k)p FH(u)2213 1165 y FD(1)2249 1153 y FF(k)2291 1165 y Fw(H)2352 1153 y FF(k)p FH(u)2442 1165 y FD(2)2478 1153 y FF(k)2520 1165 y Fw(H)2594 1153 y FH(:)0 1335 y FI(No)n(w)27 b(i\))h(follo)n(ws)f(from)g(Lemma)g(2.4.) 125 1434 y(If)k FH(u)259 1446 y FD(1)296 1434 y FG(\()p FH(t)p FG(\))p FH(;)14 b(u)475 1446 y FD(2)512 1434 y FG(\()p FH(t)p FG(\))p FH(;)g(u)691 1446 y FD(3)729 1434 y FG(\()p FH(t)p FG(\))30 b FF(2)f FH(C)1002 1404 y Fw(1)1094 1434 y FF(\\)21 b FH(H)68 b FI(for)31 b(all)g FH(t)p FI(,)i(then)f(ii\))g(holds)f(b)n(y)g(\(B1\).)48 b(No)n(w)31 b(ii\))h(follo)n(ws)e(b)n(y)0 1534 y(con)n(tin)n(uit)n(y)-7 b(.)p 3128 1534 4 57 v 3132 1481 50 4 v 3132 1534 V 3182 1534 4 57 v 0 1766 a Fy(2.3)112 b(The)38 b(existence)f(and)h (uniqueness)g(theorem)0 1919 y FI(T)-7 b(o)27 b(simplify)h(notations,)f (w)n(e)g(assume)g(that)h FH(\027)g FG(=)23 b(1)p FI(.)0 2084 y FJ(Theorem)30 b(2.6.)40 b FC(L)l(et)30 b FH(f)h FF(2)24 b FH(L)947 2096 y FD(2)983 2084 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1268 2054 y Fw(\000)p FD(1)1356 2084 y FG(\))30 b FC(and)g FH(u)1627 2096 y FD(0)1687 2084 y FF(2)23 b FH(H)7 b FC(.)39 b(Then)0 2219 y FI(a\))29 b FC(\(2.4\))i(has)g(a)f(unique)f(solution)h FH(u)23 b FF(2)g(H)31 b FC(such)f(that)f FH(u)p FG(\(0\))23 b(=)g FH(u)2034 2231 y FD(0)2070 2219 y FC(.)0 2355 y FI(b\))30 b FC(This)h(solution)f(satis\034es)581 2597 y FG(sup)538 2667 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)762 2505 y Fz(\020)811 2597 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)999 2563 y FD(2)1055 2597 y FG(+)1138 2484 y Fz(Z)1221 2505 y FA(t)1185 2673 y FD(0)1278 2597 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1513 2563 y FD(2)1513 2618 y(1)1563 2597 y FH(ds)1645 2505 y Fz(\021)1718 2597 y FF(\024)23 b(j)p FH(u)1877 2609 y FD(0)1914 2597 y FF(j)1937 2563 y FD(2)1992 2597 y FG(+)2075 2484 y Fz(Z)2158 2505 y FA(T)2122 2673 y FD(0)2225 2597 y FF(k)p FH(f)9 b FG(\()p FH(t)p FG(\))p FF(k)2453 2563 y FD(2)2453 2618 y Fw(\000)p FD(1)2555 2597 y FH(dt:)367 b FI(\(2.6\))0 2833 y FC(Pr)l(o)l(of.)0 2933 y FJ(Uniqueness.)34 b FI(Let)24 b FH(u)727 2945 y FD(1)764 2933 y FH(;)14 b(u)849 2945 y FD(2)909 2933 y FF(2)23 b(H)i FI(b)r(e)f(t)n(w)n(o)f(solutions,)h(equal)f(to)h FH(u)2072 2945 y FD(0)2132 2933 y FI(at)g FH(t)f FG(=)g(0)p FI(.)35 b(Denote)24 b FH(u)e FG(=)h FH(u)2958 2945 y FD(2)3006 2933 y FF(\000)11 b FH(u)3130 2945 y FD(1)3166 2933 y FI(.)0 3032 y(Then)509 3132 y FG(_)-38 b FH(u)18 b FG(+)g FH(Lu)g FG(+)g FH(B)t FG(\()p FH(u)996 3144 y FD(2)1033 3132 y FH(;)c(u)1118 3144 y FD(2)1155 3132 y FG(\))k FF(\000)g FH(B)t FG(\()p FH(u)1435 3144 y FD(2)1472 3132 y FH(;)c(u)1557 3144 y FD(1)1594 3132 y FG(\))19 b(+)f FH(B)t FG(\()p FH(u)1875 3144 y FD(2)1912 3132 y FH(;)c(u)1997 3144 y FD(1)2034 3132 y FG(\))19 b FF(\000)f FH(B)t FG(\()p FH(u)2315 3144 y FD(1)2352 3132 y FH(;)c(u)2437 3144 y FD(1)2473 3132 y FG(\))24 b(=)e(0)14 b FH(;)0 3281 y FI(or)986 3380 y FG(_)-38 b FH(u)18 b FG(+)g FH(Lu)g FG(+)g FH(B)t FG(\()p FH(u)1473 3392 y FD(2)1510 3380 y FH(;)c(u)p FG(\))k(+)g FH(B)t FG(\()p FH(u;)c(u)1960 3392 y FD(1)1997 3380 y FG(\))23 b(=)g(0)14 b FH(:)0 3529 y FI(By)30 b(\(B5)14 b(i\))30 b(w)n(e)f(can)h(m)n(ultiply)h(this)f (relation)f(in)h FH(H)37 b FI(b)n(y)30 b FH(u)p FG(\()p FH(s)p FG(\))g FI(and)g(in)n(tegrate)f(o)n(v)n(er)f FH(ds)j FI(from)e FG(0)h FI(to)g FH(t)p FI(.)0 3628 y(Using)d(\(B5)14 b(ii\))28 b(w)n(e)f(ha)n(v)n(e)f(for)h(all)h FH(t)23 b FF(\025)f FG(0)p FI(:)807 3754 y Fz(Z)890 3774 y FA(t)853 3943 y FD(0)933 3800 y Fz(\000)971 3867 y FF(h)15 b FG(_)-38 b FH(u;)14 b(u)p FF(i)k FG(+)g FF(h)p FH(Lu;)c(u)p FF(i)j FG(+)h FF(h)p FH(B)t FG(\()p FH(u;)c(u)1887 3879 y FD(1)1924 3867 y FG(\))p FH(;)g(u)p FF(i)2073 3800 y Fz(\001)2125 3867 y FH(ds)23 b FG(=)g(0)p FH(:)0 4100 y FI(Since)28 b FF(h)p FH(Lu;)14 b(u)p FF(i)22 b FG(=)g FF(k)p FH(u)p FF(k)712 4070 y FD(2)712 4120 y(1)748 4100 y FI(,)28 b(then)g(using)f(\(B4\))h(w)n(e)f(obtain)g(the)h(follo)n(wing)f (equalit)n(y)f(in)i FH(L)2741 4112 y FD(1)2778 4100 y FG(\(0)p FH(;)14 b(T)e FG(\))p FI(:)941 4268 y FH(d)p 926 4305 74 4 v 926 4381 a(dt)1033 4324 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1221 4290 y FD(2)1277 4324 y FG(+)18 b(2)p FF(k)p FH(u)p FF(k)1534 4290 y FD(2)1534 4345 y(1)1592 4324 y FG(=)23 b FF(\000)p FG(2)p FF(h)p FH(B)t FG(\()p FH(u;)14 b(u)2051 4336 y FD(1)2087 4324 y FG(\))p FH(;)g(u)p FF(i)g FH(:)0 4533 y FI(By)27 b(\(B2)14 b(i\))28 b(and)f(the)h(in)n(terp)r(olation)f(inequalit)n(y)-7 b(,)205 4753 y FF(jh)p FH(B)t FG(\()p FH(u;)14 b(u)492 4765 y FD(1)529 4753 y FG(\))p FH(;)g(u)p FF(ij)23 b(\024)g FH(C)d FF(k)p FH(u)981 4765 y FD(1)1017 4753 y FF(k)1059 4765 y FD(1)1133 4753 y FF(k)p FH(u)p FF(k)1265 4719 y FD(2)1273 4759 y Ft(1)p 1273 4768 29 3 v 1273 4801 a(2)1352 4753 y FF(\024)j FH(C)d FF(k)p FH(u)1609 4765 y FD(1)1645 4753 y FF(k)1687 4765 y FD(1)1747 4753 y FF(j)p FH(u)p FF(j)14 b(k)p FH(u)p FF(k)1987 4765 y FD(1)2046 4753 y FF(\024)2143 4697 y FG(1)p 2143 4734 42 4 v 2143 4810 a(2)2218 4753 y FF(k)p FH(u)p FF(k)2350 4719 y FD(2)2350 4774 y(1)2404 4753 y FG(+)k FH(C)2546 4765 y FD(1)2597 4753 y FF(k)p FH(u)2687 4765 y FD(1)2724 4753 y FF(k)2766 4719 y FD(2)2766 4774 y(1)2816 4753 y FF(j)p FH(u)p FF(j)2910 4719 y FD(2)2961 4753 y FH(:)1553 5059 y FI(15)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 16 15 bop 0 523 a FI(Therefore,)1203 605 y FH(d)p 1188 642 74 4 v 1188 718 a(dt)1294 661 y FF(j)p FH(u)p FF(j)1388 627 y FD(2)1448 661 y FF(\024)23 b FH(C)d FF(j)p FH(u)p FF(j)1709 627 y FD(2)1769 661 y FF(k)p FH(u)1859 673 y FD(1)1895 661 y FF(k)1937 627 y FD(2)1937 682 y(1)1988 661 y FH(:)0 834 y FI(By)27 b(Gron)n(w)n(all's)f(lemma,) 1028 944 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1216 910 y FD(2)1277 944 y FF(\024)d FH(e)1404 863 y Fp(R)1448 884 y Fu(t)1436 932 y Ft(0)1498 910 y FA(C)t Fw(k)p FA(u)1623 918 y Ft(1)1656 910 y FD(\()p FA(s)p FD(\))p Fw(k)1773 885 y Ft(2)1773 926 y(1)1817 910 y FA(ds)1887 944 y FF(j)p FH(u)p FG(\(0\))p FF(j)2087 910 y FD(2)2137 944 y FH(:)858 b FI(\(2.7\))0 1094 y(So)27 b FH(u)p FG(\()p FH(t)p FG(\))c FF(\021)g FG(0)k FI(since)h FH(u)p FG(\(0\))22 b(=)h(0)p FI(.)0 1193 y FJ(Existence:)36 b(Step)31 b(1)h(\(a-priori)g(estimate\)) p FI(.)125 1293 y(Let)g FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))33 b FI(b)r(e)g(a)f FC(smo)l(oth)h FI(solution)f(of)g(\(2.4\).)51 b(Let)32 b(us)h(consider)e(the)i(function)g FH(u)d FF(7\000)-14 b(!)3001 1260 y FD(1)p 3001 1274 34 4 v 3001 1321 a(2)3058 1293 y FF(j)p FH(u)p FF(j)3152 1263 y FD(2)0 1392 y FI(and)27 b(calculate)g(its)h(deriv)-5 b(ativ)n(e)27 b(along)f(tra)5 b(jectories)26 b(of)h(\(2.4\):)205 1571 y FG(1)p 205 1608 42 4 v 205 1684 a(2)305 1571 y FH(d)p 290 1608 74 4 v 290 1684 a(dt)396 1627 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)584 1593 y FD(2)644 1627 y FG(=)c FF(h)p FH(u)p FG(\()p FH(t)p FG(\))p FH(;)43 b FG(_)-38 b FH(u)p FG(\()p FH(t)p FG(\))p FF(i)24 b FG(=)e FF(h)p FH(u;)14 b FF(\000)p FH(Lu)j FF(\000)h FH(B)t FG(\()p FH(u)p FG(\))h(+)f FH(f)9 b FG(\()p FH(t)p FG(\))p FF(i)2109 1569 y Fq(\(B1\))2153 1627 y FG(=)644 1827 y(=)23 b FF(\000k)p FH(u)p FF(k)929 1793 y FD(2)929 1848 y(1)983 1827 y FG(+)18 b FF(h)p FH(u;)c(f)9 b FF(i)23 b(\024)f(\000k)p FH(u)p FF(k)1572 1793 y FD(2)1572 1848 y(1)1626 1827 y FG(+)c FF(k)p FH(u)p FF(k)1841 1839 y FD(1)1900 1827 y FF(k)p FH(f)9 b FF(k)2034 1839 y Fw(\000)p FD(1)2145 1827 y FF(\024)22 b(\000)2307 1771 y FG(1)p 2307 1808 42 4 v 2307 1884 a(2)2381 1827 y FF(k)p FH(u)p FF(k)2513 1793 y FD(2)2513 1848 y(1)2568 1827 y FG(+)2661 1771 y(1)p 2661 1808 V 2661 1884 a(2)2735 1827 y FF(k)p FH(f)9 b FF(k)2869 1793 y FD(2)2869 1848 y Fw(\000)p FD(1)2971 1827 y FH(:)0 2032 y FI(So,)1129 2115 y FH(d)p 1114 2152 74 4 v 1114 2228 a(dt)1221 2171 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1409 2136 y FD(2)1464 2171 y FG(+)18 b FF(k)p FH(u)p FF(k)1679 2136 y FD(2)1679 2191 y(1)1738 2171 y FF(\024)23 b(k)p FH(f)9 b FF(k)1960 2136 y FD(2)1960 2191 y Fw(\000)p FD(1)2062 2171 y FH(;)0 2343 y FI(and)27 b(for)g FG(0)c FF(\024)g FH(t)g FF(\024)f FH(T)39 b FI(w)n(e)27 b(ha)n(v)n(e)688 2574 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)876 2540 y FD(2)932 2574 y FG(+)1015 2461 y Fz(Z)1098 2481 y FA(t)1061 2649 y FD(0)1155 2574 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1390 2540 y FD(2)1390 2594 y(1)1440 2574 y FH(ds)c FF(\024)g(j)p FH(u)1704 2586 y FD(0)1741 2574 y FF(j)1764 2540 y FD(2)1820 2574 y FG(+)1903 2461 y Fz(Z)1986 2481 y FA(t)1949 2649 y FD(0)2043 2574 y FF(k)p FH(f)9 b FG(\()p FH(s)p FG(\))p FF(k)2280 2540 y FD(2)2280 2594 y Fw(\000)p FD(1)2382 2574 y FH(ds)14 b(:)517 b FI(\(2.8\))0 2798 y(Hence,)459 2898 y FF(j)p FH(u)p FF(j)553 2863 y FD(2)553 2920 y FA(L)599 2928 y Fr(1)659 2920 y FD(\(0)p FA(;T)9 b FD(;)p FA(H)t FD(\))894 2898 y FH(;)37 b FF(j)p FH(u)p FF(j)1048 2863 y FD(2)1048 2920 y FA(L)1094 2904 y Ft(2)1125 2920 y FD(\(0)p FA(;T)9 b FD(;)p FA(H)1329 2904 y Ft(1)1362 2920 y FD(\))1415 2898 y FF(\024)23 b(j)p FH(u)1574 2910 y FD(0)1611 2898 y FF(j)1634 2863 y FD(2)1689 2898 y FG(+)18 b FF(j)p FH(f)9 b FF(j)1868 2863 y FD(2)1868 2921 y FA(L)1914 2929 y Ft(2)1946 2921 y FD(\(0)p FA(;T)g FD(;)p FA(H)2150 2904 y Fr(\000)p Ft(1)2228 2921 y FD(\))2281 2898 y FG(=:)23 b FH(C)6 b FG(\()p FH(u)2537 2910 y FD(0)2574 2898 y FH(;)14 b(f)9 b FG(\))14 b FH(:)288 b FI(\(2.9\))0 3056 y(Due)28 b(to)g(\(B2\))f(and)g(the)h(in)n(terp)r(olation)f(inequalit)n (y)-7 b(,)951 3238 y FF(k)p FH(B)t FG(\()p FH(u)p FG(\))p FF(k)1214 3250 y Fw(\000)p FD(1)1325 3238 y FF(\024)23 b FH(C)d FF(k)p FH(u)p FF(k)1624 3204 y FD(2)1624 3261 y(1)p FA(=)p FD(2)1750 3238 y FF(\024)j FH(C)d FF(j)p FH(u)p FF(j)i(k)p FH(u)p FF(k)2165 3250 y FD(1)2215 3238 y FH(:)0 3421 y FI(Expressing)41 b FG(_)-37 b FH(u)27 b FI(in)h(terms)f(of)g FH(u)h FI(from)f(\(2.4\))g(and)h(using)f (\(2.9\),)g(w)n(e)g(get)h(that)826 3604 y FF(j)15 b FG(_)-38 b FH(u)p FF(j)920 3619 y FA(L)966 3603 y Ft(2)998 3619 y FD(\(0)p FA(;T)9 b FD(:)p FA(H)1202 3603 y Fr(\000)p Ft(1)1280 3619 y FD(\))1333 3604 y FF(\024)22 b FH(C)1485 3569 y Fw(0)1509 3536 y Fz(\000)1547 3604 y FH(C)6 b FG(\()p FH(u)1692 3616 y FD(0)1729 3604 y FH(;)14 b(f)9 b FG(\))19 b(+)f FH(C)2015 3569 y FD(2)2052 3604 y FG(\()p FH(u)2132 3616 y FD(0)2169 3604 y FH(;)c(f)9 b FG(\))2288 3536 y Fz(\001)2340 3604 y FH(:)0 3786 y FI(So,)1277 3886 y FF(j)p FH(u)p FF(j)1371 3898 y Fw(H)1455 3886 y FF(\024)23 b FH(C)1602 3898 y FD(1)1639 3886 y FG(\()p FH(u)1719 3898 y FD(0)1756 3886 y FH(;)14 b(f)9 b FG(\))14 b FH(:)1065 b FI(\(2.10\))0 4035 y FJ(Step)32 b(2:)41 b(Galerkin)32 b(appro)m(ximations)125 4135 y FI(F)-7 b(or)26 b(the)i(basis)f FF(f)p FH(e)701 4147 y FA(s)736 4135 y FH(;)14 b(s)23 b FF(2)g FE(Z)968 4105 y FD(2)968 4156 y(0)1006 4135 y FF(g)p FI(,)k(w)n(e)g(set)1106 4318 y FH(H)1175 4333 y FD(\()p FA(N)6 b FD(\))1313 4318 y FG(=)22 b(span)p FF(f)p FH(e)1648 4330 y FA(s)1683 4318 y FH(;)27 b FF(j)p FH(s)p FF(j)c(\024)g FH(N)9 b FF(g)14 b FH(:)0 4500 y FI(Let)28 b FH(P)202 4512 y FA(N)288 4500 y FG(:)23 b FH(H)30 b FF(!)23 b FH(H)608 4515 y FD(\()p FA(N)6 b FD(\))751 4500 y FI(b)r(e)28 b(the)g(orthogonal)d(pro) 5 b(jection.)36 b(Clearly)-7 b(,)551 4683 y FH(H)620 4698 y FD(\()p FA(N)6 b FD(\))758 4683 y FF(\032)22 b FH(C)910 4649 y Fw(1)1000 4683 y FF(\\)c FH(H)r(;)37 b FG(dim)15 b FH(H)1426 4698 y FD(\()p FA(N)6 b FD(\))1564 4683 y FH(<)22 b FF(1)p FH(;)51 b FI(and)37 b FH(L)22 b FG(:)h FH(H)2173 4698 y FD(\()p FA(N)6 b FD(\))2311 4683 y FF(!)23 b FH(H)2486 4698 y FD(\()p FA(N)6 b FD(\))2615 4683 y FH(:)1553 5059 y FI(16)p eop end PStoPSsaved restore %%Page: (16,17) 9 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 17 16 bop 0 523 a FI(Let)28 b(us)f(apply)g FH(P)536 535 y FA(N)627 523 y FI(to)h(\(2.4\):)976 706 y FH(P)1029 718 y FA(N)1107 706 y FG(_)-38 b FH(u)18 b FG(+)g FH(P)1294 718 y FA(N)1371 706 y FH(Lu)g FG(+)g FH(P)1630 718 y FA(N)1707 706 y FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(P)2050 718 y FA(N)2127 706 y FH(f)f(:)0 888 y FI(A)28 b(curv)n(e)e FH(u)p FG(\()p FH(t)p FG(\))e FF(2)f FH(H)624 903 y FD(\()p FA(N)6 b FD(\))767 888 y FI(satis\034es)26 b(this)i(relation)f(i\033)1113 1071 y FG(_)-37 b FH(u)18 b FG(+)g FH(Lu)f FG(+)h FH(P)1506 1083 y FA(N)1584 1071 y FH(B)t FG(\()p FH(u)p FG(\))23 b(=)f FH(P)1926 1083 y FA(N)2004 1071 y FH(f)g(:)887 b FI(\(2.11\))0 1254 y(This)36 b(is)g(an)g(ODE)g(in)h FH(H)809 1269 y FD(\()p FA(N)6 b FD(\))923 1254 y FI(,)39 b(de\034ned)d(b)n(y)g(a)g(smo)r(oth)f(v)n(ector)g(\034eld,)k(with)d(an) g FH(L)2632 1266 y FD(2)2669 1254 y FI(\025r.h.s.)61 b(Let)36 b(us)0 1353 y(supplemen)n(t)28 b(it)g(with)g(the)g(initial)g (condition:)1336 1536 y FH(u)p FG(\(0\))23 b(=)g FH(P)1654 1548 y FA(N)1731 1536 y FH(u)1779 1548 y FD(0)1829 1536 y FH(:)0 1719 y FI(Then)28 b(the)g(equation)g(has)f(a)h(unique)g (solution)f FH(u)c FG(=)g FH(u)1709 1731 y FA(N)1772 1719 y FI(,)28 b(whic)n(h)g(exists)g(till)g FH(t)c FG(=)f FH(T)12 b FI(,)27 b(or)g(till)i(it)f(blo)n(ws)0 1818 y(up)g(at)f(time)h FH(T)459 1830 y FA(N)545 1818 y FF(\024)23 b FH(T)12 b FI(:)733 2001 y FH(u)781 2013 y FA(N)867 2001 y FF(2)23 b FH(C)6 b FG(\([0)p FH(;)14 b(T)1193 2013 y FA(N)1256 2001 y FG(])p FH(;)g(H)1385 2016 y FD(\()p FA(N)6 b FD(\))1500 2001 y FG(\))p FH(;)51 b FG(_)-37 b FH(u)1640 2013 y FA(N)1725 2001 y FF(2)24 b FH(L)1861 2013 y FD(2)1897 2001 y FG(\([0)p FH(;)14 b(T)2080 2013 y FA(N)2143 2001 y FG(])p FH(;)g(H)2272 2016 y FD(\()p FA(N)6 b FD(\))2387 2001 y FG(\))14 b FH(;)793 2183 y FG(0)22 b FH(<)h(T)994 2195 y FA(N)1079 2183 y FF(\024)g FH(T)7 b(;)96 b(T)1391 2195 y FA(N)1477 2183 y FH(<)23 b(T)61 b FG(=)-14 b FF(\))23 b(j)p FH(u)1903 2195 y FA(N)1966 2183 y FG(\()p FH(t)p FG(\))p FF(j)48 b(\000)-14 b(!)2106 2237 y FA(t)p Fw(!)p FA(T)2236 2245 y Fu(N)2313 2183 y FF(1)581 b FI(\(2.12\))0 2369 y FJ(A-priori)31 b(estimate)f(for)e FH(u)944 2381 y FA(N)898 2367 y FH(u)946 2379 y FA(N)900 2369 y FH(u)948 2381 y FA(N)1010 2369 y FI(.)0 2469 y(Consider)f(the)g (deriv)-5 b(ativ)n(e)27 b(of)975 2436 y FD(1)p 975 2450 34 4 v 975 2497 a(2)1041 2469 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1229 2439 y FD(2)1294 2469 y FI(along)f(tra)5 b(jectories)26 b(of)h(\(2.11\):)512 2646 y FG(1)p 512 2683 42 4 v 512 2759 a(2)611 2646 y FH(d)p 596 2683 74 4 v 596 2759 a(dt)702 2702 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)890 2668 y FD(2)951 2702 y FG(=)c FF(h)p FH(u;)28 b FG(_)-37 b FH(u)o FF(i)23 b FG(=)g FF(h)p FH(u;)14 b FF(\000)p FH(Lu)p FF(\000)n FH(P)1749 2714 y FA(N)1826 2702 y FH(B)t FG(\()p FH(u)p FG(\))k(+)h FH(P)2160 2714 y FA(N)2237 2702 y FH(f)9 b FF(i)1719 2862 y FG(=)22 b FF(h)p FH(u;)14 b FF(\000)p FH(Lu)j FF(\000)h FH(B)t FG(\()p FH(u)p FG(\))h(+)f FH(f)9 b FG(\()p FH(t)p FG(\))p FF(i)14 b FH(:)0 3041 y FI(Note)28 b(that)p 322 3103 2546 4 v 322 3236 4 133 v 350 3190 a(This)g(is)f(the)h(same)f(relation)g(as)g(for)g(solutions)f(of)i (\(2.4\))f(obtained)g(in)h(Step)g(1.)p 2864 3236 V 322 3239 2546 4 v 0 3354 a(Hence,)501 3454 y FH(u)549 3466 y FA(N)611 3454 y FG(\()p FH(t)p FG(\))56 b FI(satis\034es)27 b(\(2.8\),)g(\(2.10\))g(uniformly)g(in)h FH(N)36 b FI(\(with)29 b FH(T)34 b FG(:=)23 b FH(T)2571 3466 y FA(N)2633 3454 y FI(\).)125 3604 y(Due)d(to)f(\(2.12\),)i(it)f(means)f(that)h FH(T)1183 3616 y FA(N)1268 3604 y FG(=)j FH(T)31 b FI(for)19 b(all)g FH(N)9 b FI(.)34 b(Therefore)19 b(the)h(Galerkin)e(appro)n (ximations)0 3703 y(satisfy)1075 3738 y Fz(8)1075 3813 y(>)1075 3838 y(<)1075 3988 y(>)1075 4012 y(:)1167 3797 y FF(f)p FH(u)1257 3809 y FA(N)1319 3797 y FG(\()p FH(t)p FG(\))p FH(;)37 b FG(0)23 b FF(\024)f FH(t)h FF(\024)g FH(T)12 b FF(g)22 b(\032)g(H)15 b FH(;)1167 3933 y FF(k)p FH(u)1257 3945 y FA(N)1319 3933 y FF(k)1361 3945 y Fw(H)1444 3933 y FF(\024)23 b FH(C)1591 3945 y FD(1)1642 3933 y FH(;)1204 4068 y(u)1252 4080 y FA(N)1314 4068 y FG(\(0\))g(=)g FH(P)1584 4080 y FA(N)1661 4068 y FH(u)1709 4080 y FD(0)1760 4068 y FH(:)0 4213 y FJ(Step)32 b(3:)41 b(T)-8 b(ransition)32 b(to)g(the)f(limit)0 4313 y FI(By)j(the)h(Banac)n(h-Alaoglu)d(Theorem)i (a)g(closed)g(ball)h(in)g FF(H)g FI(is)g(compact)f(in)h(the)g(w)n(eak)e (top)r(ology)-7 b(.)0 4413 y(Hence,)31 b(there)g(is)f(a)g(subsequence)g FF(f)p FH(N)1228 4425 y FA(j)1262 4413 y FF(g)g FI(suc)n(h)h(that)f FH(u)1755 4425 y FA(N)1808 4433 y Fu(j)1843 4413 y FH(*u)d FF(2)h(H)k FI(as)e FH(j)j FF(!)28 b(1)p FI(.)45 b(Since)31 b(the)g(linear)0 4512 y(maps)1050 4606 y FF(H)24 b(!)f FH(L)1307 4618 y FD(2)1344 4606 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1629 4576 y Fw(\000)p FD(1)1716 4606 y FG(\))p FH(;)37 b(u)22 b FF(7\000)-14 b(!)37 b FG(_)-37 b FH(u)1050 4742 y FF(H)24 b(!)f FH(L)1307 4754 y FD(2)1344 4742 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1629 4712 y Fw(\000)p FD(1)1716 4742 y FG(\))p FH(;)37 b(u)22 b FF(7\000)-14 b(!)23 b FH(Lu)1553 5059 y FI(17)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 18 17 bop 0 523 a FI(are)27 b(con)n(tin)n(uous,)f(then) 1115 704 y FG(_)-38 b FH(u)1148 716 y FA(N)1201 724 y Fu(j)1259 704 y FH(*)55 b FG(_)-37 b FH(u)82 b FI(in)41 b FH(L)1680 716 y FD(2)1717 704 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2002 670 y Fw(\000)p FD(1)2090 704 y FG(\))855 b FI(\(2.13\))1030 874 y FH(L)14 b(u)1149 886 y FA(N)1202 894 y Fu(j)1259 874 y FH(*)36 b(Lu)f FI(in)h FH(L)1680 886 y FD(2)1717 874 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2002 839 y Fw(\000)p FD(1)2090 874 y FG(\))g FH(:)818 b FI(\(2.14\))125 1055 y(No)n(w)27 b(consider)f(the)i(term)g FF(f)p FH(B)t FG(\()p FH(u)1168 1067 y FA(N)1221 1075 y Fu(j)1255 1055 y FG(\))p FF(g)p FI(.)36 b(By)28 b(Lemma)f(2.5,)1158 1236 y FH(u)1206 1248 y FA(N)1259 1256 y Fu(j)1316 1236 y FF(!)d FH(u)36 b FI(in)h FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))14 b FH(:)945 b FI(\(2.15\))0 1417 y(Noting)27 b(that)h(b)n(y)f(\(B3\))h(the)g(map)751 1598 y FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))22 b FF(\000)-14 b(!)23 b FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1662 1563 y Fw(\000)p FD(3)1750 1598 y FG(\))p FH(;)37 b(u)p FG(\()p FH(t)p FG(\))23 b FF(7\000)-14 b(!)23 b FH(B)t FG(\()p FH(u)p FG(\()p FH(t)p FG(\)\))0 1779 y FI(is)k(con)n(tin)n(uous,)g(w)n(e)g(ha)n(v)n(e)933 1960 y FH(B)t FG(\()p FH(u)1080 1972 y FA(N)1133 1980 y Fu(j)1168 1960 y FG(\))c FF(\000)-14 b(!)23 b FH(B)t FG(\()p FH(u)p FG(\))84 b FI(in)37 b FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2099 1925 y Fw(\000)p FD(3)2186 1960 y FG(\))g FH(:)722 b FI(\(2.16\))0 2141 y(Finally)-7 b(,)28 b(due)f(to)h(\(2.15\),)1184 2240 y FH(u)1232 2252 y FA(N)1285 2260 y Fu(j)1319 2240 y FG(\(0\))23 b FF(\000)-14 b(!)23 b FH(u)p FG(\(0\))37 b FI(in)27 b FH(H)21 b(:)972 b FI(\(2.17\))0 2389 y(Let)25 b(us)f(tak)n(e)g(an)n(y)g FH(m)h FI(and)f(apply)g FH(P)1114 2401 y FA(m)1203 2389 y FI(to)g(\(2.11\))g(with)h FH(N)1791 2401 y FA(j)1849 2389 y FF(\025)d FH(m)j FI(in)g(place)f(of)h FH(N)9 b FI(.)35 b(Since)25 b FH(P)2830 2401 y FA(m)2906 2389 y FF(\016)13 b FH(P)3014 2401 y FA(N)3067 2409 y Fu(j)3124 2389 y FG(=)0 2488 y FH(P)53 2500 y FA(m)116 2488 y FI(,)28 b(w)n(e)f(ha)n(v)n(e)844 2588 y FH(P)897 2600 y FA(m)989 2588 y FG(_)-38 b FH(u)1022 2600 y FA(N)1075 2608 y Fu(j)1128 2588 y FG(+)18 b FH(P)1264 2600 y FA(m)1341 2588 y FH(Lu)1446 2600 y FA(N)1499 2608 y Fu(j)1551 2588 y FG(+)g FH(P)1687 2600 y FA(m)1765 2588 y FH(B)t FG(\()p FH(u)1912 2600 y FA(N)1965 2608 y Fu(j)1999 2588 y FG(\))24 b(=)e FH(P)2195 2600 y FA(m)2259 2588 y FH(f)g(:)0 2736 y FI(Send)28 b FH(N)270 2748 y FA(j)332 2736 y FI(to)g FF(1)p FI(.)37 b(Due)28 b(to)f(\(2.13\)\025\(2.16\),)976 2917 y FH(P)1029 2929 y FA(m)1120 2917 y FG(_)-37 b FH(u)18 b FG(+)g FH(P)1308 2929 y FA(m)1385 2917 y FH(Lu)f FG(+)h FH(P)1643 2929 y FA(m)1721 2917 y FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(P)2064 2929 y FA(m)2127 2917 y FH(f)f(:)0 3098 y FI(\(This)28 b(equalit)n(y)f(holds)g(in)h(the)g(space)e FH(L)1273 3110 y FD(2)1310 3098 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1595 3068 y Fw(\000)p FD(3)1683 3098 y FG(\))p FI(\).)37 b(Hence,)28 b(since)f FH(m)h FI(is)f(arbitrary)-7 b(,)26 b(w)n(e)h(ha)n(v)n(e)1253 3279 y FG(_)-38 b FH(u)18 b FG(+)g FH(Lu)g FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(f)t(:)0 3460 y FI(Due)28 b(to)g(\(2.17\),)928 3560 y FH(u)p FG(\(0\))23 b(=)f(lim)15 b FH(u)1370 3572 y FA(N)1423 3580 y Fu(j)1457 3560 y FG(\(0\))23 b(=)g(lim)14 b FH(P)1856 3572 y FA(N)1909 3580 y Fu(j)1944 3560 y FH(u)1992 3572 y FD(0)2052 3560 y FG(=)22 b FH(u)2187 3572 y FD(0)2238 3560 y FH(:)0 3709 y FI(No)n(w)27 b(the)h(assertion)e (a\))h(is)h(pro)n(v)n(ed.)125 3857 y(T)-7 b(o)27 b(pro)n(v)n(e)f(b\))i (w)n(e)f(note)g(that)h(\(2.9\))f(and)h(\(2.15\))f(imply)g(the)h (estimate)1153 4038 y FF(j)p FH(u)p FF(j)1247 4004 y FD(2)1247 4061 y FA(L)1293 4044 y Fr(1)1353 4061 y FD(\(0)p FA(;T)9 b FD(;)p FA(H)t FD(\))1610 4038 y FF(\024)23 b FH(C)6 b FG(\()p FH(u)1843 4050 y FD(0)1880 4038 y FH(;)14 b(f)9 b FG(\))14 b FH(:)0 4233 y FI(Since)27 b FH(u)264 4245 y FA(N)317 4253 y Fu(j)374 4233 y FH(*)c(u)k FI(in)g FF(H)q FI(,)g(then)h FH(u)1009 4245 y FA(N)1062 4253 y Fu(j)1119 4233 y FH(*)23 b(u)j FI(in)i FH(L)1453 4245 y FD(2)1489 4233 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1774 4203 y FD(1)1810 4233 y FG(\))p FI(.)37 b(As)27 b(in)g(a)g(Hilb)r(ert)g(space)f(a)h(norm)f(of)h(the)0 4332 y(w)n(eak)g(limit)h(is)f(no)h(bigger)e(than)h(the)h(liminf)h(of)e (the)h(norms,)f(then)1151 4513 y FF(j)p FH(u)p FF(j)1245 4479 y FD(2)1245 4536 y FA(L)1291 4519 y Ft(2)1322 4536 y FD(\(0)p FA(;T)9 b FD(;)p FA(H)1526 4519 y Ft(1)1559 4536 y FD(\))1613 4513 y FF(\024)22 b FH(C)6 b FG(\()p FH(u)1845 4525 y FD(0)1882 4513 y FH(;)14 b(f)9 b FG(\))14 b FH(:)0 4694 y FI(This)30 b(pro)n(v)n(es)e(\(2.6\))i(with)h(the)g (extra)e(factor)h(2)f(in)i(the)g(r.h.s.)44 b(T)-7 b(o)30 b(get)g(rid)g(of)g(this)h(factor,)f(one)g(has)0 4794 y(to)d(rep)r(eat)h(the)g(argumen)n(ts,)e(used)h(in)h(the)g(pro)r(of)f (of)h(the)g(Prop)r(osition)e(2.11)h(b)r(elo)n(w.)p 3128 4794 4 57 v 3132 4741 50 4 v 3132 4794 V 3182 4794 4 57 v 1553 5059 a(18)p eop end PStoPSsaved restore %%Page: (18,19) 10 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 19 18 bop 0 523 a FJ(Remark)41 b(2.7.)46 b FC(Our)37 b(pr)l(o)l(of)i(shows)g(that)g(any)f(se)l(quenc)l(e)g(of)h (Galerkin)g(solutions)f FH(u)2764 535 y FA(N)2817 543 y Fu(j)2890 523 y FC(c)l(ontains)0 623 y(a)d(subse)l(quenc)l(e)f(that)h (c)l(onver)l(ges)g(to)g(a)g(solution.)54 b(By)36 b(the)f(uniqueness,)h (this)f(solution)g(is)g FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))p FC(.)0 722 y(Henc)l(e,)30 b(the)g(whole)h(se)l(quenc)l(e)e(c)l (onver)l(ges)h(to)g FH(u)p FC(:)1231 891 y FH(u)1279 903 y FA(N)1465 891 y FH(*)1411 944 y FA(N)6 b Fw(!1)1672 891 y FH(u)45 b FC(i)q FH(n)h FF(H)1020 b FI(\(2.18\))0 1092 y FC(\()p FH(u)82 1104 y FA(N)174 1092 y FC(is)30 b(a)g(solution)g(of)48 b FI(\(2.11\))p FC(\).)38 b(So,)30 b FH(u)1266 1104 y FA(N)1352 1092 y FF(!)23 b FH(u)29 b FC(in)h FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))29 b FC(by)h(L)l(emma)g(2.5.)0 1247 y FJ(Prop)s(osition)20 b(2.8.)31 b FC(L)l(et)21 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))22 b FC(b)l(e)g(a)f(solution)h(as)g(in)f(The)l(or)l(em)i(2.6.)37 b(Then)22 b(one)f(c)l(an)h(\034nd)f FH(p)p FG(\()p FH(t;)14 b(x)p FG(\))24 b FF(2)0 1346 y FH(L)57 1358 y FD(2)94 1346 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(L)360 1358 y FD(2)395 1346 y FG(\()p FE(T)482 1316 y FD(2)520 1346 y FG(\)\))24 b(=)e FH(L)752 1358 y FD(2)789 1346 y FG(\(\(0)p FH(;)14 b(T)e FG(\))18 b FF(\002)g FE(T)1181 1316 y FD(2)1218 1346 y FG(\))30 b FC(such)g(that)g FG(\()p FH(u;)14 b(p)p FG(\))29 b FC(satis\034es)h(\(2.1\).)0 1501 y FJ(Exercise)h(2.9.)40 b FC(Pr)l(ove)31 b(the)f(Pr)l(op)l(osition.)125 1700 y FI(The)40 b(presen)n(ted)f(pro)r(of)h(of)g(the)g(existence)g(and)g (uniqueness)g(of)g(a)f(solution)h(applies)g(to)g(the)0 1799 y(2D)21 b(NSE)h(in)f(a)f(b)r(ounded)h(domain)g(with)g(the)g(Diric) n(hlet)g(b)r(oundary)f(conditions.)34 b(Similar)20 b(argumen)n(ts)0 1899 y(apply)i(to)g(\(0.4\),)h(but)g(not)f(to)g(\(0.3\).)35 b(T)-7 b(o)22 b(handle)g(the)h(latter)f(equation)f(\(with)i FH(d)h FF(\025)e FG(2)p FI(\))h(one)e(has)h(to)g(use)0 1998 y(the)28 b(Maxim)n(um)g(Principle,)g(whic)n(h)g(holds)f(for)g (solutions)g(of)h(the)g(equation,)f(but)i(not)f(for)f(solutions)0 2098 y(of)h(its)f(Galerkin)g(appro)n(ximations.)35 b(So)27 b(in)h(this)g(case)e(one)h(should)h(argue)e(di\033eren)n(tly)-7 b(.)0 2328 y Fy(2.4)112 b(Impro)m(ving)36 b(the)h(smo)s(othness)h(of)g (solutions)0 2481 y FJ(Lemma)29 b(2.10.)40 b FC(If)30 b FH(u)23 b FF(2)g FH(H)j FF(\\)18 b FH(C)1045 2451 y Fw(1)1116 2481 y FC(,)30 b(then)g FF(h)p FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b FG(\001)p FH(u)p FF(i)23 b FG(=)g(0)p FC(.)0 2636 y(Pr)l(o)l(of.)43 b FI(W)-7 b(e)28 b(ha)n(v)n(e)560 2854 y FF(h)p FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b FG(\001)p FH(u)p FF(i)24 b FG(=)1068 2741 y Fz(Z)1165 2854 y FH(u)1213 2820 y FA(i)1264 2798 y FH(@)5 b(u)1361 2768 y FA(j)p 1264 2835 132 4 v 1268 2911 a FH(@)g(x)1364 2887 y FA(i)1482 2798 y FH(@)1531 2768 y FD(2)1568 2798 y FH(u)1616 2768 y FA(j)p 1429 2835 274 4 v 1429 2911 a FH(@)g(x)1525 2887 y FA(k)1566 2911 y FH(@)g(x)1662 2887 y FA(k)1727 2854 y FH(dx)981 3083 y FG(=)1068 2970 y Fz(Z)1165 3083 y FH(u)1213 3049 y FA(i)1310 3027 y FH(@)1359 2997 y FD(2)1396 3027 y FH(u)1444 2997 y FA(j)p 1264 3064 261 4 v 1264 3140 a FH(@)g(x)1360 3116 y FA(i)1388 3140 y FH(@)g(x)1484 3116 y FA(k)1562 3027 y FH(@)g(u)1659 2997 y FA(j)p 1559 3064 137 4 v 1559 3140 a FH(@)g(x)1655 3116 y FA(k)1720 3083 y FH(dx)19 b FG(+)1912 2970 y Fz(Z)2025 3027 y FH(@)5 b(u)2122 2997 y FA(i)p 2018 3064 V 2018 3140 a FH(@)g(x)2114 3116 y FA(k)2189 3027 y FH(@)g(u)2286 2997 y FA(j)p 2189 3064 132 4 v 2193 3140 a FH(@)g(x)2289 3116 y FA(i)2357 3027 y FH(@)g(u)2454 2997 y FA(j)p 2354 3064 137 4 v 2354 3140 a FH(@)g(x)2450 3116 y FA(k)2515 3083 y FH(dx:)0 3327 y FI(The)28 b(\034rst)f(in)n(tegral)g(in)h(the)g (r.h.s.)38 b(v)-5 b(anishes)27 b(since)h(its)g(in)n(tegrand)e(equals)h FH(u)p FG(\()p FH(x)p FG(\))19 b FF(\001)g(r)2662 3235 y Fz(\020)2724 3295 y FA(@)t(u)2802 3270 y Fu(j)p 2722 3309 113 4 v 2722 3357 a FA(@)t(x)2799 3340 y Fu(k)2857 3295 y FA(@)t(u)2935 3270 y Fu(j)p 2855 3309 V 2855 3357 a FA(@)t(x)2932 3340 y Fu(k)2978 3235 y Fz(\021)3055 3327 y FI(and)0 3460 y(div)14 b FH(u)23 b FG(=)g(0)p FI(.)38 b(The)28 b(in)n(tegrand)f(in)h(the)g(second)f(in)n(tegral)g (equals)g(tr)2060 3393 y Fz(\000)2098 3460 y FH(U)2164 3430 y FD(2)2201 3460 y FH(U)2267 3430 y FA(t)2296 3393 y Fz(\001)2334 3460 y FI(,)h(where)g FH(U)36 b FI(is)28 b(the)h(matrix)0 3560 y(of)g FH(du)g FI(\(i.e.,)h(of)f(the)g (linearisation)f(of)h(the)g(mapping)g FH(x)d FF(7!)f FH(u)p FG(\()p FH(x)p FG(\))p FI(\).)43 b(As)29 b(tr)14 b FH(U)34 b FG(=)p FI(div)14 b FH(u)24 b FG(=)h(0)p FI(,)k(then)h(the)0 3659 y(matrix)g FH(U)339 3629 y FD(2)407 3659 y FI(is)h(prop)r (ortional)e(to)i(the)h(iden)n(tit)n(y)-7 b(.)47 b(Therefore)30 b(the)h(in)n(tegrand)f(v)-5 b(anishes)30 b(iden)n(tically)-7 b(,)0 3759 y(and)27 b(the)h(lemma)g(is)f(pro)n(v)n(ed.)p 3128 3759 4 57 v 3132 3706 50 4 v 3132 3759 V 3182 3759 4 57 v 125 3922 a(W)-7 b(e)19 b(note)f(that)i(the)f(lemma's)f (assertion)g(do)r(es)g(not)h(hold)g(for)f(smo)r(oth)g(div)n (ergence\025free)f(functions)0 4022 y(in)34 b(a)g(domain,)h(v)-5 b(anishing)34 b(on)g(the)g(domain's)g(b)r(oundary)-7 b(.)55 b(So)34 b(Lemma)g(2.10)f(do)r(es)g(not)h(apply)g(to)0 4122 y(study)28 b(the)g(2D)f(NSE)i(under)e(the)h(Diric)n(hlet)g(b)r (oundary)e(conditions.)125 4256 y(Let)c FH(u)316 4268 y FD(0)376 4256 y FF(2)i FH(H)531 4226 y FD(1)591 4256 y FI(and)e FH(f)32 b FF(2)23 b FH(L)955 4268 y FD(2)992 4256 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))p FI(.)34 b(Consider)22 b(the)h(functional:)35 b FH(u)22 b FF(7!)2452 4199 y FG(1)p 2452 4237 42 4 v 2452 4313 a(2)2527 4256 y FF(k)p FH(u)p FF(k)2659 4221 y FD(2)2659 4276 y(1)2718 4256 y FG(=)2838 4199 y(1)p 2838 4237 V 2838 4313 a(2)2913 4256 y FF(h)p FH(u;)14 b(Lu)p FF(i)p FI(.)0 4383 y(If)28 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))28 b FI(is)g(a)f(smo)r(oth)g(solution,)g(then)h(b)n(y)f(the)h(lemma)486 4539 y FG(1)p 486 4576 V 486 4652 a(2)585 4539 y FH(d)p 570 4576 74 4 v 570 4652 a(dt)676 4595 y FF(k)p FH(u)p FF(k)808 4561 y FD(2)808 4615 y(1)867 4595 y FG(=)23 b FF(h)p FH(Lu;)28 b FG(_)-37 b FH(u)o FF(i)23 b FG(=)g FF(\000k)p FH(u)p FF(k)1516 4561 y FD(2)1516 4615 y(2)1569 4595 y FG(+)18 b FF(h)p FH(Lu;)c(f)9 b FF(i)1231 4795 y(\024)23 b(\000k)p FH(u)p FF(k)1516 4760 y FD(2)1516 4815 y(2)1569 4795 y FG(+)18 b FF(k)p FH(u)p FF(k)1784 4807 y FD(2)1834 4795 y FF(j)p FH(f)9 b FF(j)23 b(\024)g(\000)2116 4738 y FG(1)p 2116 4776 42 4 v 2116 4852 a(2)2190 4795 y FF(k)p FH(u)p FF(k)2322 4760 y FD(2)2322 4815 y(2)2376 4795 y FG(+)2469 4738 y(1)p 2469 4776 V 2469 4852 a(2)2543 4795 y FF(j)p FH(f)9 b FF(j)2639 4760 y FD(2)2690 4795 y FH(:)1553 5059 y FI(19)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 20 19 bop 0 523 a FI(In)n(tegrating)26 b(o)n(v)n(er)g(time)i(w)n(e)f(get)g(that)h(for)f(an)n(y)g FH(t)c FF(\024)g FH(T)12 b FI(,)132 759 y FH(')p FG(\()p FH(u)p FG(\))24 b(:=)e FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)658 724 y FD(2)658 779 y(1)713 759 y FG(+)796 646 y Fz(Z)879 666 y FA(t)842 834 y FD(0)936 759 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1171 724 y FD(2)1171 779 y(2)1221 759 y FH(ds)h FF(\024)g(k)p FH(u)1504 771 y FD(0)1540 759 y FF(k)1582 724 y FD(2)1582 779 y(1)1638 759 y FG(+)1721 646 y Fz(Z)1804 666 y FA(t)1767 834 y FD(0)1861 759 y FF(j)p FH(f)9 b FG(\()p FH(s)p FG(\))p FF(j)2060 724 y FD(2)2097 759 y FH(ds)23 b FG(=:)2329 738 y Fz(e)2313 759 y FH(C)6 b FG(\()p FH(u)2458 771 y FD(0)2495 759 y FH(;)14 b(f)9 b FG(\))23 b FF(\024)g(1)14 b FH(:)132 b FI(\(2.19\))0 983 y(As)28 b(in)f(Step)i(2,)e(w)n(e)g(see)g (that)515 1166 y(this)h(estimate)f(also)g(holds)g(for)g(Galerkin)g (appro)n(ximations)e FH(u)2494 1178 y FA(N)2556 1166 y FG(\()p FH(t)p FG(\))q FH(:)303 b FI(\(2.20\))0 1348 y FJ(Prop)s(osition)30 b(2.11.)40 b FC(The)30 b(estimate)g(\(2.19\))i (holds)f(for)f(the)g(solution)g FH(u)f FC(as)h(in)g(The)l(or)l(em)h (2.6.)0 1514 y(Pr)l(o)l(of.)43 b FI(By)c(\(2.18\))o(,)78 b FH(u)763 1526 y FA(N)933 1514 y FF(\000)-14 b(!)904 1568 y FA(N)6 b Fw(!1)1135 1514 y FH(u)75 b FI(in)38 b FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))p FI(.)66 b(So)75 b FH(')p FG(\()p FH(P)2138 1526 y FA(m)2202 1514 y FH(u)2250 1526 y FA(N)2312 1514 y FG(\))109 b FF(\000)-14 b(!)2424 1568 y FA(N)6 b Fw(!1)2655 1514 y FH(')p FG(\()p FH(P)2794 1526 y FA(m)2858 1514 y FH(u)p FG(\))75 b FI(since)0 1665 y FH(')21 b FF(\016)f FH(P)190 1677 y FA(m)283 1665 y FI(is)31 b(a)f(con)n(tin)n(uous)f(functional)h (on)h FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))p FI(.)44 b(Since)31 b FH(')p FG(\()p FH(P)2179 1677 y FA(m)2243 1665 y FH(u)2291 1677 y FA(N)2353 1665 y FG(\))d FF(\024)2522 1644 y Fz(e)2506 1665 y FH(C)36 b FI(for)30 b(all)g FH(N)40 b FI(due)30 b(to)0 1775 y(\(2.20\))o(,)j (then)f(for)f(eac)n(h)g FH(m)h FI(w)n(e)f(ha)n(v)n(e)g FH(')p FG(\()p FH(P)1349 1787 y FA(m)1413 1775 y FH(u)p FG(\))e FF(\024)1633 1754 y Fz(e)1617 1775 y FH(C)6 b FI(.)50 b(As)31 b FH(')p FG(\()p FH(P)2020 1787 y FA(m)2084 1775 y FH(u)p FG(\))f FF(\045)g FH(')p FG(\()p FH(u)p FG(\))p FI(,)j(then)g(the)f(assertion)0 1875 y(follo)n(ws.)p 3128 1875 4 57 v 3132 1822 50 4 v 3132 1875 V 3182 1875 4 57 v 125 2041 a(The)20 b(argumen)n(ts,)h(used)g(to)f(pro)n(v)n(e)f (the)i(Prop)r(osition)f(ab)r(o)n(v)n(e,)h(are)f(general.)33 b(They)20 b(ma)n(y)g(b)r(e)h(stated)0 2141 y(as)27 b(a)g FJ(Principle:)p 541 2251 2108 4 v 541 2500 4 250 v 587 2346 a FI(If)h(an)f(estimate)g(holds)h(for)f(all)g(Galerkin)g(appro)n (ximations,)587 2446 y(then)h(it)g(also)e(holds)h(for)g(the)h (solutions)f(as)g(in)h(Theorem)f(2.6.)p 2645 2500 V 541 2503 2108 4 v 2977 2397 a(\(2.21\))0 2677 y(Often)h(it)g(means)f(that:) p 560 2772 2069 4 v 560 3021 4 250 v 606 2866 a(If)h(an)f(estimate)h (holds)f(for)g(smo)r(oth)g(solutions,)g(then)h(it)g(also)606 2966 y(holds)f(for)g(solutions)g(as)g(in)h(Theorem)f(2.6)o FH(:)p 2626 3021 V 560 3024 2069 4 v 2977 2917 a FI(\(2.22\))0 3178 y(In)h(particular,)e(the)i(a-priori)e(estimate)h(\(2.8\))g (implies)h(\(2.6\).)0 3344 y FJ(Exercise)47 b(2.12.)h FI(Let)42 b FH(')f FI(b)r(e)h(a)f(smo)r(oth)g(functional,)k(de\034ned)c (on)g(some)g(Sob)r(olev)g(space)f FH(H)3129 3314 y FA(r)3166 3344 y FI(,)0 3443 y FH(r)35 b FF(\025)e FG(0)p FI(.)54 b(Find)33 b(under)h(what)f(assumptions)f(on)h FH(')h FI(the)g(equation)f(obtained)g(b)n(y)g(di\033eren)n(tiating)g FH(')0 3543 y FI(along)h(\(2.11\))e(is)i(the)g(same)f(as)g(for)g (di\033eren)n(tiating)g FH(')i FI(along)d(\(2.4\).)55 b([F)-7 b(or)33 b(suc)n(h)g(functionals)h(the)0 3643 y(a-priori)29 b(estimate)j(whic)n(h)f FH(')g FI(implies)h(for)f (\(2.11\))f(is)h(the)h(same)f(as)g(the)g(one)g(whic)n(h)g(it)h(implies) g(for)0 3742 y(\(2.4\);)27 b(so)g(\(2.22\))g(follo)n(ws)g(from)g (\(2.21\)].)0 3908 y FJ(The)32 b(smo)s(othing)d(prop)s(ert)m(y)-8 b(.)0 4008 y FI(The)30 b FC(smo)l(othing)i(pr)l(op)l(erty)e FI(states)f(that)h(if)g(in)g(\(2.4\))f FH(u)1736 4020 y FD(0)1800 4008 y FF(2)d FH(H)7 b FI(,)30 b(but)g FH(f)9 b FG(\()p FH(t;)14 b(x)p FG(\))27 b FF(2)g FH(C)2566 3978 y Fw(1)2637 4008 y FI(,)j(then)g FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))27 b FF(2)0 4108 y FH(C)65 4078 y Fw(1)163 4108 y FI(for)g FH(t)c(>)g FG(0)p FI(.)36 b(W)-7 b(e)28 b(start)f(with)0 4274 y FJ(Theorem)j(2.13.)39 b FC(If)29 b FH(u)776 4286 y FD(0)836 4274 y FF(2)24 b FH(H)35 b FC(and)30 b FH(f)i FF(2)23 b FH(L)1388 4286 y FD(2)1425 4274 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))p FC(,)28 b(then)h FH(u)p FG(\()p FH(s)p FG(\))23 b FF(2)h FH(H)2308 4244 y FD(2)2374 4274 y FC(for)30 b(a.a.)40 b FH(s)23 b(>)f FG(0)p FC(,)29 b(and)h(for)0 4373 y(e)l(ach)h FG(0)22 b FH(<)h(t)g FF(\024)g FH(T)40 b FC(we)30 b(have)209 4605 y FH(t)14 b FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)479 4570 y FD(2)479 4625 y(1)534 4605 y FG(+)617 4491 y Fz(Z)700 4512 y FA(t)663 4680 y FD(0)757 4605 y FH(s)p FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1031 4570 y FD(2)1031 4625 y(2)1081 4605 y FH(ds)23 b FF(\024)g(j)p FH(u)1345 4617 y FD(0)1381 4605 y FF(j)1404 4570 y FD(2)1460 4605 y FG(+)1543 4491 y Fz(Z)1626 4512 y FA(t)1589 4680 y FD(0)1683 4605 y FH(s)14 b FF(j)p FH(f)9 b FG(\()p FH(s)p FG(\))p FF(j)1935 4570 y FD(2)1986 4605 y FH(ds)19 b FG(+)2170 4491 y Fz(Z)2253 4512 y FA(t)2216 4680 y FD(0)2310 4605 y FF(k)p FH(f)9 b FG(\()p FH(s)p FG(\))p FF(k)2547 4570 y FD(2)2547 4625 y Fw(\000)p FD(1)2649 4605 y FH(ds)14 b(:)2977 4599 y FI(\(2.23\))1553 5059 y(20)p eop end PStoPSsaved restore %%Page: (20,21) 11 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 21 20 bop 0 523 a FC(Sketch)31 b(of)g(the)g(pr)l(o)l(of)p FI(.)41 b(T)-7 b(ak)n(e)27 b(the)i(functional)g FH(')p FG(\()p FH(t;)14 b(u)p FG(\))25 b(=)f FH(t)14 b FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)2094 493 y FD(2)2094 544 y(1)2130 523 y FI(.)40 b(Then)28 b(for)g(a)g(smo)r(oth)h(solution)0 623 y FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))p FI(,)28 b(w)n(e)f(ha)n(v)n (e)449 792 y FH(d)p 434 829 74 4 v 434 906 a(dt)541 849 y FG(\()p FH(t)p FF(k)p FH(u)p FF(k)735 814 y FD(2)735 869 y(1)771 849 y FG(\))c(=)g FF(k)p FH(u)p FF(k)1046 814 y FD(2)1046 869 y(1)1100 849 y FG(+)18 b(2)p FH(t)p FF(h)p FH(Lu;)27 b FG(_)-36 b FH(u)n FF(i)23 b FG(=)g FF(k)p FH(u)p FF(k)1750 814 y FD(2)1750 869 y(1)1804 849 y FG(+)18 b(2)p FH(t)p FF(h)p FH(Lu;)c FF(\000)p FH(Lu)i FF(\000)i FH(B)t FG(\()p FH(u)p FG(\))h(+)f FH(f)9 b FF(i)447 987 y FD(Lemma)18 b(2)p FA(:)p FD(10)590 1034 y FG(=)166 b FF(k)p FH(u)p FF(k)953 1000 y FD(2)953 1054 y(1)1007 1034 y FF(\000)18 b FG(2)p FH(t)c FF(k)p FH(u)p FF(k)1308 1000 y FD(2)1308 1054 y(2)1362 1034 y FG(+)k(2)p FH(t)p FF(h)p FH(Lu;)c(f)9 b FG(\()p FH(t)p FG(\))p FF(i)0 1216 y FI(With)1112 1316 y FG(2)p FH(t)p FF(h)p FH(Lu;)14 b(f)9 b FF(i)22 b(\024)h FH(t)14 b FF(k)p FH(u)p FF(k)1726 1282 y FD(2)1726 1337 y(2)1780 1316 y FG(+)k FH(t)c FF(j)p FH(f)9 b FF(j)2003 1282 y FD(2)2053 1316 y FH(;)0 1466 y FI(w)n(e)27 b(obtain)g(that)977 1531 y FH(d)p 962 1569 V 962 1645 a(dt)1068 1588 y FG(\()p FH(t)p FF(k)p FH(u)p FF(k)1262 1553 y FD(2)1262 1608 y(1)1298 1588 y FG(\))c FF(\024)g(k)p FH(u)p FF(k)1573 1553 y FD(2)1573 1608 y(1)1627 1588 y FF(\000)18 b FH(t)c FF(k)p FH(u)p FF(k)1886 1553 y FD(2)1886 1608 y(2)1940 1588 y FG(+)k FH(t)c FF(j)p FH(f)9 b FF(j)2163 1553 y FD(2)2214 1588 y FH(:)0 1760 y FI(In)n(tegrating)26 b(from)h FG(0)g FI(to)h FH(t)g FI(w)n(e)f(get)477 1996 y FH(t)14 b FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)747 1961 y FD(2)747 2016 y(1)802 1996 y FG(+)885 1883 y Fz(Z)968 1903 y FA(t)931 2071 y FD(0)1025 1996 y FH(s)p FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1299 1961 y FD(2)1299 2016 y(2)1349 1996 y FH(ds)23 b FF(\024)1541 1883 y Fz(Z)1624 1903 y FA(t)1588 2071 y FD(0)1681 1996 y FH(s)14 b FF(j)p FH(f)9 b FG(\()p FH(s)p FG(\))p FF(j)1933 1961 y FD(2)1984 1996 y FH(ds)19 b FG(+)2168 1883 y Fz(Z)2251 1903 y FA(t)2214 2071 y FD(0)2308 1996 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)2543 1961 y FD(2)2543 2016 y(1)2593 1996 y FH(ds)14 b(:)0 2225 y FI(Using)29 b(\(2.6\))f(w)n(e)g(see)h(that)g FH(u)f FI(satis\034es)g(\(2.23\).)40 b(This)28 b(estimate)h(also)f (holds)g(for)g(Galerkin)g(appro)n(x-)0 2324 y(imations)i FH(u)388 2336 y FA(N)450 2324 y FG(\()p FH(t;)14 b(x)p FG(\))p FI(.)45 b(As)30 b(in)h(the)f(pro)r(of)g(of)g(the)g(prop)r (osition)f(it)i(implies)f(that)g(the)h(estimate)e(holds)0 2424 y(for)e(solutions)g(of)g(\(2.4\))h(\(cf.)37 b(\(2.22\))o(\).)1909 b Fs(\003)0 2640 y FJ(Remark)31 b(2.14.)40 b FI(W)-7 b(e)28 b(used)g(that)g FF(h)p FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b(Lu)p FF(i)23 b FG(=)g(0)p FI(,)28 b(but)g(this)g(relation)f FJ(is)k(not)d FI(really)f(needed)h(for)0 2740 y(the)g(pro)r(of)f (\(instead)h(w)n(e)f(ma)n(y)g(estimate)g(this)h(term,)f(using)h(Lemma)f (2.16\))f(b)r(elo)n(w.)125 2906 y(No)n(w)h(assume)f(\(only)i(for)f (simplicit)n(y!\))37 b(that)28 b FH(f)j FG(=)23 b(0)p FI(.)37 b(Then)1032 3088 y FG(_)-38 b FH(u)18 b FG(+)g FH(Lu)f FG(+)i FH(B)t FG(\()p FH(u)p FG(\))k(=)f(0)p FH(;)60 b(u)p FG(\(0\))23 b(=)f FH(u)2098 3100 y FD(0)2149 3088 y FH(:)805 b FI(\(2.24\))0 3271 y FJ(Theorem)30 b(2.15.)40 b FC(If)30 b FH(u)778 3283 y FD(0)838 3271 y FF(2)23 b FH(H)37 b FC(and)30 b FH(u)f FC(is)h(a)h(solution)e(of)i (\(2.24\),)h(then)588 3511 y FH(t)618 3477 y FA(m)681 3511 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)907 3477 y FD(2)907 3532 y FA(m)987 3511 y FG(+)1070 3398 y Fz(Z)1153 3419 y FA(t)1117 3587 y FD(0)1210 3511 y FH(s)1249 3477 y FA(m)1312 3511 y FF(k)p FH(u)p FF(k)1444 3477 y FD(2)1444 3532 y FA(m)p FD(+1)1604 3511 y FH(ds)23 b FF(\024)g FH(C)1856 3523 y FA(m)1919 3511 y FG(\()p FF(j)p FH(u)2022 3523 y FD(0)2059 3511 y FF(j)2082 3477 y FD(2)2138 3511 y FG(+)18 b FF(j)p FH(u)2292 3523 y FD(0)2329 3511 y FF(j)2352 3477 y FD(4)p FA(m)p FD(+2)2532 3511 y FG(\))c FH(:)376 b FI(\(2.25\))0 3746 y FC(Pr)l(o)l(of.)43 b FI(T)-7 b(ak)n(e)27 b FH(')515 3758 y FA(m)578 3746 y FG(\()p FH(t;)14 b(u)p FG(\))23 b(=)g FH(t)898 3716 y FA(m)961 3746 y FF(k)p FH(u)p FF(k)1093 3716 y FD(2)1093 3766 y FA(m)1177 3746 y FG(=)g FH(t)1295 3716 y FA(m)1358 3746 y FF(h)p FH(L)1447 3716 y FA(m)1510 3746 y FH(u;)14 b(u)p FF(i)p FI(.)36 b(Then)28 b(for)f(a)g(smo)r(oth)g(solution)g FH(u)h FI(w)n(e)f(ha)n(v)n(e)626 3915 y FH(d)p 611 3953 V 611 4029 a(dt)718 3972 y(')772 3984 y FA(m)858 3972 y FG(=)c FH(m)14 b(t)1063 3937 y FA(m)p Fw(\000)p FD(1)1210 3972 y FF(k)p FH(u)p FF(k)1342 3937 y FD(2)1342 3992 y FA(m)1422 3972 y FG(+)k(2)p FH(t)1577 3937 y FA(m)1640 3972 y FF(h)p FH(L)1729 3937 y FA(m)1792 3972 y FH(u;)28 b FG(_)-37 b FH(u)o FF(i)858 4133 y FG(=)23 b FH(m)14 b(t)1063 4099 y FA(m)p Fw(\000)p FD(1)1210 4133 y FF(k)p FH(u)p FF(k)1342 4099 y FD(2)1342 4154 y FA(m)1422 4133 y FF(\000)k FG(2)c FH(t)1591 4099 y FA(m)1654 4133 y FF(k)p FH(u)p FF(k)1786 4099 y FD(2)1786 4154 y FA(m)p FD(+1)1950 4133 y FF(\000)k FG(2)p FH(t)2105 4099 y FA(m)2168 4133 y FF(h)p FH(L)2257 4099 y FA(m)2319 4133 y FH(u;)c(B)t(u)p FF(i)g FH(:)2977 4032 y FI(\(2.26\))0 4312 y FJ(Lemma)29 b(2.16.)40 b FC(F)-6 b(or)30 b FH(u)23 b FF(2)g FH(H)i FF(\\)19 b FH(C)1113 4282 y Fw(1)1213 4312 y FC(the)30 b(fol)t(lowing)i(ine)l(quality)f(holds:)741 4529 y FF(jh)p FH(L)853 4495 y FA(m)916 4529 y FH(u;)14 b(B)t(u)p FF(ij)23 b(\024)1291 4473 y FG(1)p 1291 4510 42 4 v 1291 4586 a(2)1357 4529 y FF(k)p FH(u)p FF(k)1489 4495 y FD(2)1489 4550 y FA(m)p FD(+1)1653 4529 y FG(+)18 b FH(C)1795 4541 y FA(m)1872 4529 y FF(k)p FH(u)p FF(k)2004 4486 y FD(2\()p FA(m)p FD(+1\))2004 4551 y(1)2235 4529 y FF(j)p FH(u)p FF(j)2329 4495 y FD(2)p FA(m)2425 4529 y FH(:)1553 5059 y FI(21)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 22 21 bop 125 523 a FI(P)n(ostp)r(oning)26 b(the)i(lemma's)f(pro)r(of,)g(w)n(e)g(con)n(tin)n(ue)f(to)h(pro)n(v)n (e)f(the)i(theorem.)36 b(Due)28 b(to)f(\(2.26\))f(and)0 623 y(the)i(lemma,)f(w)n(e)h(ha)n(v)n(e:)417 785 y FH(d)p 402 822 74 4 v 402 898 a(dt)508 841 y(t)538 806 y FA(m)601 841 y FF(k)p FH(u)p FF(k)733 806 y FD(2)733 861 y FA(m)813 841 y FG(+)18 b FH(t)926 806 y FA(m)989 841 y FF(k)p FH(u)p FF(k)1121 806 y FD(2)1121 861 y FA(m)p FD(+1)1290 841 y FF(\024)k FH(m)14 b(t)1494 806 y FA(m)p Fw(\000)p FD(1)1642 841 y FF(k)p FH(u)p FF(k)1774 806 y FD(2)1774 861 y FA(m)1854 841 y FG(+)k(2)p FH(t)2009 806 y FA(m)2086 841 y FH(C)2145 853 y FA(m)2208 841 y FF(k)p FH(u)p FF(k)2340 798 y FD(2\()p FA(m)p FD(+1\))2340 863 y(1)2571 841 y FF(j)p FH(u)p FF(j)2665 806 y FD(2)p FA(m)2774 841 y FH(:)0 1073 y FI(By)27 b(Theorems)g(2.6)f(and)i(2.13)e(with)i FH(f)k FG(=)22 b(0)p FI(,)28 b FH(t)1456 1042 y FA(m)1519 1073 y FF(k)p FH(u)p FF(k)1651 1029 y FD(2\()p FA(m)p FD(+1\))1651 1095 y(1)1881 1073 y FF(j)p FH(u)p FF(j)1975 1042 y FD(2)p FA(m)2094 1073 y FF(\024)23 b(k)p FH(u)p FF(k)2314 1042 y FD(2)2314 1093 y(1)2363 1073 y FF(j)p FH(u)2434 1085 y FD(0)2471 1073 y FF(j)2494 1042 y FD(2)p FA(m)2590 1073 y FF(j)p FH(u)2661 1085 y FD(0)2698 1073 y FF(j)2721 1042 y FD(2)p FA(m)2818 1073 y FI(.)125 1172 y(So,)k(in)n(tegrating)f(\(2.26\))h(w)n(e)g(get)g(that)487 1409 y FH(t)517 1375 y FA(m)594 1409 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)820 1375 y FD(2)820 1430 y FA(m)900 1409 y FG(+)983 1296 y Fz(Z)1066 1317 y FA(t)1030 1485 y FD(0)1123 1409 y FH(s)1162 1375 y FA(m)1225 1409 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1460 1375 y FD(2)1460 1430 y FA(m)p FD(+1)905 1646 y FF(\024)993 1533 y Fz(Z)1076 1554 y FA(t)1039 1722 y FD(0)1119 1646 y FH(s)1158 1612 y FA(m)p Fw(\000)p FD(1)1306 1646 y FH(m)p FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1614 1612 y FD(2)1614 1667 y FA(m)1676 1646 y FH(ds)18 b FG(+)g FH(C)1924 1612 y Fw(0)1918 1667 y FA(m)1982 1646 y FF(j)p FH(u)2053 1658 y FD(0)2090 1646 y FF(j)2113 1612 y FD(4)p FA(m)2223 1533 y Fz(Z)2306 1554 y FA(t)2269 1722 y FD(0)2349 1646 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)2584 1612 y FD(2)2584 1667 y(1)2620 1646 y FH(ds)905 1919 y FF(\024)993 1806 y Fz(Z)1076 1827 y FA(t)1039 1995 y FD(0)1119 1919 y FH(s)1158 1885 y FA(m)p Fw(\000)p FD(1)1306 1919 y FH(m)p FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1614 1885 y FD(2)1614 1940 y FA(m)1676 1919 y FH(ds)g FG(+)g FH(C)1924 1885 y Fw(0)1918 1940 y FA(m)1982 1919 y FF(j)p FH(u)2053 1931 y FD(0)2090 1919 y FF(j)2113 1885 y FD(4)p FA(m)p FD(+2)2307 1919 y FH(:)0 2145 y FI(No)n(w)35 b(the)h(estimate)f(\(2.25\))f(for)h(smo)r(oth)g(solutions) g(follo)n(ws)f(b)n(y)h(induction,)j(where)d(the)g(base)g(of)0 2245 y(induction)29 b(with)h FH(m)c FG(=)f(0)j FI(is)h(pro)n(vided)f(b) n(y)h(Theorem)f(2.6.)41 b(T)-7 b(o)29 b(complete)g(the)g(pro)r(of)g(w)n (e)f(again)g(use)0 2344 y(Principle)g(\(2.22\))f(\(justifying)h(this)g (as)f(b)r(efore\).)125 2444 y(It)g(remains)g(to)h(pro)n(v)n(e)d(the)j (lemma.)0 2544 y FJ(Pro)s(of)k(of)f(Lemma)f(2.16.)35 b FI(W)-7 b(e)28 b(\034rst)f(sho)n(w)g(that)830 2734 y FF(jh)p FH(L)942 2699 y FA(m)1004 2734 y FH(u;)14 b(B)t FG(\()p FH(u)p FG(\))p FF(ij)24 b(\024)e FH(C)e FF(k)p FH(u)p FF(k)1655 2659 y Ft(4)p Fu(m)p Fr(\000)p Ft(1)p 1654 2672 153 3 v 1690 2705 a(2)p Fu(m)1645 2756 y FA(m)p FD(+1)1834 2734 y FF(k)p FH(u)p FF(k)1976 2659 y Fu(m)p Ft(+1)p 1975 2672 122 3 v 1996 2705 a(2)p Fu(m)1966 2756 y FD(1)2125 2734 y FF(j)p FH(u)p FF(j)2219 2699 y FD(1)p FA(=)p FD(2)2336 2734 y FH(:)618 b FI(\(2.27\))0 2913 y(T)-7 b(o)27 b(pro)n(v)n(e)f(the)i(inequalit)n(y)f(w)n(e)g(note)h (that)858 3105 y FF(h)p FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b(L)1163 3070 y FA(m)1226 3105 y FH(u)p FF(i)23 b FG(=)1453 3026 y Fz(X)1416 3208 y Fw(j)p FA(\013)p Fw(j)p FD(=)p FA(m)1623 3105 y FH(C)1682 3117 y FA(\013)1743 3105 y FF(h)p FH(D)1846 3070 y FA(\013)1894 3105 y FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b(D)2181 3070 y FA(\013)2228 3105 y FH(u)p FF(i)p FH(;)0 3376 y FI(and)622 3504 y FF(h)p FH(D)725 3470 y FA(\013)773 3504 y FH(B)t FG(\()p FH(u)p FG(\))p FH(;)g(D)1060 3470 y FA(\013)1107 3504 y FH(u)p FF(i)23 b FG(=)1305 3426 y Fz(X)1298 3604 y FA(\014)s Fw(\024)p FA(\013)1447 3387 y Fz(\022)1526 3454 y FH(\013)1527 3553 y(\014)1597 3387 y Fz(\023)1672 3504 y FF(h)p FH(B)t FG(\()p FH(D)1874 3470 y FA(\013)p Fw(\000)p FA(\014)2015 3504 y FH(u;)14 b(D)2171 3470 y FA(\014)2215 3504 y FH(u)p FG(\))p FH(;)g(D)2403 3470 y FA(\013)2450 3504 y FH(u)p FF(i)g FH(:)0 3742 y FI(Since)28 b FF(h)p FH(B)t FG(\()p FH(u;)14 b(D)504 3712 y FA(\013)551 3742 y FH(u)p FG(\))p FH(;)g(D)739 3712 y FA(\013)786 3742 y FH(u)p FF(i)23 b FG(=)g(0)p FI(,)k(then)h(using)f(\(B2\))h(w)n (e)f(get)g(that)594 3938 y FF(h)p FH(D)697 3904 y FA(\013)759 3938 y FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b(D)1046 3904 y FA(\013)1093 3938 y FH(u)p FF(i)23 b(\024)g FH(C)1371 3860 y Fz(X)1363 4038 y FA(\014)s(<\013)1512 3938 y FF(k)p FH(D)1625 3904 y FA(\013)p Fw(\000)p FA(\014)1765 3938 y FH(u)p FF(k)1855 3953 y FD(1)p FA(=)p FD(2)1958 3938 y FF(k)p FH(D)2071 3904 y FA(\014)2115 3938 y FH(u)p FF(k)2205 3950 y FD(1)2242 3938 y FF(k)p FH(D)2355 3904 y FA(\013)2401 3938 y FH(u)p FF(k)2491 3953 y FD(1)p FA(=)p FD(2)1196 4214 y FF(\024)g FH(C)1371 4135 y Fz(X)1363 4314 y FA(\014)s(<\013)1512 4214 y FF(k)p FH(u)p FF(k)1644 4229 y FD(1)p FA(=)p FD(2+)p FA(m)p Fw(\000j)p FA(\014)s Fw(j)2003 4214 y FF(k)p FH(u)p FF(k)2135 4229 y FD(1+)p Fw(j)p FA(\014)s Fw(j)2302 4214 y FF(k)p FH(u)p FF(k)2434 4235 y FA(m)p FD(+)2552 4212 y Ft(1)p 2552 4221 29 3 v 2552 4255 a(2)2977 4114 y FI(\(2.28\))0 4492 y(where)h FG(0)f FF(\024)f(j)p FH(\014)t FF(j)i FH(<)e(m)p FI(.)36 b(Note)25 b(that)f(all)h(the)g(n)n(um)n(b)r(ers)1697 4459 y FD(1)p 1697 4473 34 4 v 1697 4520 a(2)1753 4492 y FG(+)12 b FH(m)g FF(\000)g(j)p FH(\014)t FF(j)p FI(,)26 b FG(1)12 b(+)g FF(j)p FH(\014)t FF(j)24 b FI(and)h FH(m)12 b FG(+)2721 4459 y FD(1)p 2721 4473 V 2721 4520 a(2)2789 4492 y FI(lie)24 b(b)r(et)n(w)n(een)0 4591 y(1)j(and)h FH(m)18 b FG(+)415 4559 y FD(1)p 415 4573 V 415 4620 a(2)458 4591 y FI(.)37 b(F)-7 b(or)27 b(an)n(y)g FH(a)c FF(2)g FG([1)p FH(;)14 b(m)k FG(+)1255 4559 y FD(1)p 1255 4573 V 1255 4620 a(2)1298 4591 y FG(])28 b FI(w)n(e)f(ha)n(v)n(e) 318 4810 y FF(k)p FH(u)p FF(k)450 4822 y FA(a)512 4810 y FF(\024)c(j)p FH(u)p FF(j)694 4774 y FD(1)p Fw(\000)833 4752 y Fu(a)p 788 4761 122 3 v 788 4794 a(m)p Ft(+1)947 4810 y FF(k)p FH(u)p FF(k)1133 4732 y Fu(a)p 1088 4741 V 1088 4775 a(m)p Ft(+1)1079 4832 y FA(m)p FD(+1)1248 4810 y FG(:=)g FH(X)75 b FI(and)69 b FF(k)p FH(u)p FF(k)1838 4822 y FA(a)1900 4810 y FF(\024)23 b(k)p FH(u)p FF(k)2120 4761 y FD(1)p Fw(\000)2213 4736 y Fu(a)p Fr(\000)p Ft(1)p 2213 4749 106 3 v 2240 4781 a Fu(m)2120 4832 y FD(1)2356 4810 y FF(k)p FH(u)p FF(k)2498 4736 y Fu(a)p Fr(\000)p Ft(1)p 2496 4749 V 2523 4781 a Fu(m)2488 4832 y FA(m)p FD(+1)2656 4810 y FG(:=)g FH(Y)33 b(:)1553 5059 y FI(22)p eop end PStoPSsaved restore %%Page: (22,23) 12 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 23 22 bop 0 523 a FI(T)-7 b(ak)n(e)27 b(an)n(y)g(term)h (in)h(the)f(sum)g(in)h(the)f(r.h.s.)38 b(of)28 b(\(2.28\).)38 b(Estimating)28 b(eac)n(h)g(its)g(factor)f(b)n(y)h FH(X)2963 493 y FA(\022)3000 523 y FH(Y)3066 493 y FD(1)p Fw(\000)p FA(\022)0 623 y FI(with)f(a)g(suitable)g FH(\022)r FI(,)g(dep)r(ending) h(on)e(the)i(factor)e(\(and)h(using)g(that)g FF(k)p FH(u)p FF(k)2271 635 y FD(1)2330 623 y FF(\024)22 b(k)p FH(u)p FF(k)2549 635 y FA(m)p FD(+1)2695 623 y FI(\),)27 b(w)n(e)g(get)g(that) 771 836 y FF(jh)p FH(D)897 802 y FA(\013)959 836 y FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b(D)1246 802 y FA(\013)1293 836 y FH(u)p FF(ij)23 b(\024)g FH(C)d FF(k)p FH(u)p FF(k)1728 762 y Ft(4)p Fu(m)p Fr(\000)p Ft(1)p 1726 775 153 3 v 1763 807 a(2)p Fu(m)1718 858 y FA(m)p FD(+1)1907 836 y FF(k)p FH(u)p FF(k)2049 762 y Fu(m)p Ft(+1)p 2047 775 122 3 v 2068 807 a(2)p Fu(m)2039 858 y FD(1)2183 836 y FF(j)p FH(u)p FF(j)2277 802 y FD(1)p FA(=)p FD(2)2395 836 y FH(:)0 1018 y FI(Therefore,)26 b(\(2.27\))h(follo)n(ws.)125 1117 y(No)n(w,)g(applying)g(the)h(Y)-7 b(oung)27 b(inequalit)n(y)589 1334 y FH(AB)g FF(\024)c FH(p)871 1299 y Fw(\000)p FD(1)960 1334 y FH(A)1022 1299 y FA(p)1079 1334 y FG(+)18 b FH(q)1202 1299 y Fw(\000)p FD(1)1305 1334 y FH(B)1372 1299 y FA(q)1478 1334 y FI(if)47 b FG(1)22 b FH(<)h(p)g(<)f FF(1)46 b FI(and)2196 1278 y FG(1)p 2196 1315 42 4 v 2196 1391 a FH(p)2266 1334 y FG(+)2359 1278 y(1)p 2359 1315 V 2360 1391 a FH(q)2434 1334 y FG(=)23 b(1)14 b FH(;)376 b FI(\(2.29\))0 1599 y(where)32 b FH(p)f FG(=)466 1567 y FD(4)p FA(m)p 424 1581 177 4 v 424 1628 a FD(4)p FA(m)p Fw(\000)p FD(1)611 1599 y FI(,)i FH(A)f FG(=)f FH(\016)s FF(k)p FH(u)p FF(k)1039 1525 y Ft(4)p Fu(m)p Fr(\000)p Ft(1)p 1037 1538 153 3 v 1074 1571 a(2)p Fu(m)1029 1621 y FA(m)p FD(+1)1236 1599 y FI(and)h FH(B)k FG(=)31 b FH(\016)1637 1569 y Fw(\000)p FD(1)1726 1599 y FH(C)20 b FF(k)p FH(u)p FF(k)1947 1525 y Fu(m)p Ft(+1)p 1945 1538 122 3 v 1967 1571 a(2)p Fu(m)1937 1621 y FD(1)2095 1599 y FF(j)p FH(u)p FF(j)2189 1569 y FD(1)p FA(=)p FD(2)2326 1599 y FI(with)33 b FH(\016)h FF(\034)d FG(1)p FI(,)i(w)n(e)f(get)h(the)0 1699 y(lemma's)27 b(assertion.)p 3128 1699 4 57 v 3132 1646 50 4 v 3132 1699 V 3182 1699 4 57 v 125 1914 a(A)g(v)n(ersion)e(of)i(the)g (estimate)g(\(2.25\))f(\(with)h(another)f(r.h.s.)37 b(and)26 b(with)i FG(0)22 b FF(\024)h FH(t)g FF(\024)f FH(T)12 b FG(\))27 b FI(holds)f(for)g(a)0 2014 y(solution)h(of)h(\(2.4\),)f (where)g FH(f)32 b FF(2)23 b FH(L)1078 2026 y FD(2)1115 2014 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1400 1984 y FA(s)1434 2014 y FG(\))51 b FI(for)27 b(all)g FH(s)p FI(.)125 2114 y(Due)c(to)g(Theorem)f(2.15,)g(solutions)g(of)h(\(2.24\)) f(with)h FH(u)1830 2126 y FD(0)1890 2114 y FF(2)g FH(H)30 b FI(are)22 b(smo)r(oth)g(in)i FH(t)e FI(and)h FH(x)g FI(for)g FH(t)g(>)f FG(0)p FI(.)0 2213 y(In)28 b(fact,)g(they)f(are)g (analytic.)36 b(See)28 b([DG95)o(].)0 2445 y Fy(2.5)112 b(The)38 b(NS)f(semigroup)0 2599 y FI(F)-7 b(or)27 b FH(\034)33 b(>)22 b FG(0)p FI(,)28 b(w)n(e)f(set)1061 2698 y FH(S)1112 2710 y FA(\034)1176 2698 y FG(:)c FH(H)30 b FF(\000)-14 b(!)23 b FH(H)r(;)97 b(u)1717 2710 y FD(0)1777 2698 y FF(7\000)-14 b(!)23 b FH(u)p FG(\()p FH(\034)9 b FG(\))14 b FH(;)849 b FI(\(2.30\))0 2847 y(where)26 b FH(u)p FG(\()p FH(t)p FG(\))h FI(is)g(a)g(solution)f(of)h(\(2.24\),)f (equal)h(to)f FH(u)1596 2859 y FD(0)1660 2847 y FI(at)h FH(t)c FG(=)g(0)p FI(.)36 b(Since)27 b FH(u)p FG(\()p FH(t)p FG(\))c FF(2)g FH(H)34 b FI(is)27 b(con)n(tin)n(uous)f(in)h FH(t)0 2947 y FI(\(see)g(Lemma)h(2.5\),)f(then)h(these)f(are)g(w)n(ell) g(de\034ned)h(maps.)37 b(Ob)n(viously)-7 b(,)1053 3129 y FH(S)1104 3141 y FD(0)1164 3129 y FG(=)23 b FI(id)14 b FH(;)97 b(S)1506 3141 y FA(t)1531 3149 y Ft(1)1586 3129 y FF(\016)18 b FH(S)1697 3141 y FA(t)1722 3149 y Ft(2)1781 3129 y FG(=)23 b FH(S)1920 3141 y FA(t)1945 3149 y Ft(1)1977 3141 y FD(+)p FA(t)2053 3149 y Ft(2)2113 3129 y FH(:)0 3311 y FI(So)34 b FF(f)p FH(S)215 3323 y FA(t)243 3311 y FF(g)g FI(is)f(a)h(semi-group)e(of)i(non-linear)e (transformations)g(of)i FH(H)7 b FI(,)35 b(called)f(the)68 b FC(Navier\025Stokes)0 3410 y(semigr)l(oup)28 b FI(or,)f(for)g(short,) g(the)h FC(NS)h(semigr)l(oup)p FI(.)125 3510 y(Prop)r(erties)d(of)i (the)g(NS)g(semigroup:)0 3609 y FJ(\(NS1\))1151 b FF(j)p FH(S)1475 3621 y FA(t)1504 3609 y FH(u)1552 3621 y FD(0)1589 3609 y FF(j)23 b(\024)g FH(e)1762 3579 y Fw(\000)p FA(t)1842 3609 y FF(j)p FH(u)1913 3621 y FD(0)1950 3609 y FF(j)14 b FH(:)0 3775 y FC(Pr)l(o)l(of.)43 b FI(Multiplying)28 b(\(2.24\))f(b)n(y)g FH(u)g FI(in)h FH(H)7 b FI(,)28 b(w)n(e)f(get:)834 3944 y FG(1)p 834 3981 42 4 v 834 4057 a(2)934 3944 y FH(d)p 919 3981 74 4 v 919 4057 a(dt)1025 4000 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1213 3966 y FD(2)1273 4000 y FG(=)c FF(\000h)p FH(Lu;)14 b(u)p FF(i)22 b FG(=)g FF(\000k)p FH(u)p FF(k)1986 3966 y FD(2)1986 4021 y(1)2045 4000 y FF(\024)g(\000j)p FH(u)p FF(j)2291 3966 y FD(2)2342 4000 y FH(:)0 4221 y FI(That)30 b(is,)343 4188 y FA(d)p 330 4202 60 4 v 330 4250 a(dt)414 4221 y FF(j)p FH(u)p FF(j)508 4191 y FD(2)572 4221 y FF(\024)d(\000)p FG(2)p FF(j)p FH(u)p FF(j)865 4191 y FD(2)901 4221 y FI(.)45 b(By)30 b(Gron)n(w)n(all's)f(lemma,)i FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1999 4191 y FD(2)2064 4221 y FF(\024)c(j)p FH(u)2227 4233 y FD(0)2264 4221 y FF(j)2287 4191 y FD(2)2338 4221 y FH(e)2377 4191 y Fw(\000)p FD(2)p FA(t)2505 4221 y FH(:)j FI(So)g(\(NS1\))h(follo)n(ws.) p 3128 4321 4 57 v 3132 4268 50 4 v 3132 4321 V 3182 4321 4 57 v 125 4486 a(Similar,)c(using)g(Lemma)g(2.10)f(w)n(e)i(get)f (that)0 4586 y FJ(\(NS)p FG(1)207 4556 y Fw(0)230 4586 y FJ(\))1041 b FF(k)p FH(S)1401 4598 y FA(t)1429 4586 y FH(u)1477 4598 y FD(0)1514 4586 y FF(k)1556 4598 y FD(1)1616 4586 y FF(\024)22 b FH(e)1742 4556 y Fw(\000)p FA(t)1823 4586 y FF(k)p FH(u)1913 4598 y FD(0)1950 4586 y FF(k)1992 4598 y FD(1)2042 4586 y FH(:)125 4710 y FI(The)27 b(maps)g FH(S)563 4722 y FA(t)620 4710 y FI(are)g(uniformly)g(Lipsc)n (hitz)g(on)h(b)r(ounded)g(sets:)0 4810 y FJ(\(NS2\))723 b FF(j)p FH(S)1047 4822 y FA(t)1076 4810 y FG(\()p FH(u)1156 4822 y FD(10)1226 4810 y FG(\))19 b FF(\000)f FH(S)1411 4822 y FA(t)1440 4810 y FG(\()p FH(u)1520 4822 y FD(20)1590 4810 y FG(\))p FF(j)24 b(\024)e FH(e)1795 4780 y FA(C)t Fw(j)p FA(u)1906 4788 y Ft(10)1966 4780 y Fw(j)1986 4755 y Ft(2)2023 4810 y FF(j)p FH(u)2094 4822 y FD(10)2182 4810 y FF(\000)c FH(u)2313 4822 y FD(20)2383 4810 y FF(j)p FI(.)1553 5059 y(23)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 24 23 bop 0 523 a FC(Pr)l(o)l(of.)43 b FI(Let)30 b FH(u)459 535 y FD(1)496 523 y FG(\()p FH(t)p FG(\))g FI(and)g FH(u)832 535 y FD(2)869 523 y FG(\()p FH(t)p FG(\))g FI(b)r(e)h(solutions)e(suc)n(h)g(that)h FH(u)1877 535 y FD(1)1914 523 y FG(\(0\))d(=)g FH(u)2187 535 y FD(10)2256 523 y FI(,)k FH(u)2358 535 y FD(2)2395 523 y FG(\(0\))c(=)f FH(u)2667 535 y FD(20)2737 523 y FI(.)44 b(W)-7 b(e)30 b(denote)0 623 y FH(w)59 635 y FD(0)120 623 y FG(=)22 b FH(u)255 635 y FD(20)342 623 y FF(\000)15 b FH(u)470 635 y FD(10)540 623 y FI(,)27 b FH(w)r FG(\()p FH(t)p FG(\))d(=)f FH(u)905 635 y FD(2)942 623 y FG(\()p FH(t)p FG(\))17 b FF(\000)f FH(u)1182 635 y FD(1)1218 623 y FG(\()p FH(t)p FG(\))p FI(.)38 b(Subtracting)26 b(from)g(the)h(equation)e(for)h FH(u)2672 635 y FD(2)2736 623 y FI(the)h(equation)0 722 y(for)g FH(u)175 734 y FD(1)212 722 y FI(,)h(w)n(e)f(get:)969 822 y FG(_)-49 b FH(w)21 b FG(+)d FH(Lw)j FG(+)d FH(B)t FG(\()p FH(u)1473 834 y FD(2)1510 822 y FH(;)c(w)r FG(\))19 b(+)f FH(B)t FG(\()p FH(w)r(;)c(u)1987 834 y FD(1)2025 822 y FG(\))23 b(=)g(0)14 b FH(:)0 957 y FI(Calculating)34 b FG(\()p FH(d=dt)p FG(\))p FF(j)p FH(w)r FG(\()p FH(t)p FG(\))p FF(j)873 926 y FD(2)947 957 y FI(and)h(transforming)e(the)j(result)e (as)g(in)i(the)f(pro)r(of)f(of)h(uniqueness)g(in)0 1056 y(Theorem)27 b(2.6,)f(w)n(e)i(get)f(that)h FH(w)r FG(\()p FH(t)p FG(\))h FI(satis\034es)d(the)i(estimate)g(\(2.7\))o(.)37 b(So)790 1230 y FF(j)p FH(w)r FG(\()p FH(t)p FG(\))p FF(j)991 1196 y FD(2)1052 1230 y FF(\024)23 b FH(e)1179 1196 y FA(C)1242 1149 y Fp(R)1287 1169 y Fu(t)1274 1217 y Ft(0)1325 1196 y Fw(k)p FA(u)1398 1204 y Ft(1)1431 1196 y FD(\()p FA(s)p FD(\))p Fw(k)1548 1170 y Ft(2)1548 1212 y(1)1592 1196 y FA(ds)1662 1230 y FF(j)p FH(w)1744 1242 y FD(0)1782 1230 y FF(j)1805 1196 y FD(2)1865 1230 y FF(\024)g FH(e)1992 1196 y FA(C)t Fw(j)p FA(u)2103 1204 y Ft(10)2163 1196 y Fw(j)2183 1170 y Ft(2)2219 1230 y FF(j)p FH(w)2301 1242 y FD(0)2339 1230 y FF(j)2362 1196 y FD(2)0 1388 y FI(\(w)n(e)k(use)h(\(2.6\))f(with)h FH(f)k FG(=)23 b(0)p FI(\),)k(and)g(w)n(e)h(ha)n(v)n(e)e(got)h (\(NS2\).)p 3128 1388 4 57 v 3132 1335 50 4 v 3132 1388 V 3182 1388 4 57 v 0 1549 a FJ(\(NS3\))g FI(The)h(maps)f FH(S)716 1561 y FA(t)768 1549 y FG(:)c FH(H)30 b FF(!)23 b FH(H)1095 1519 y FD(1)1160 1549 y FI(are)k(Lipsc)n(hitz)g(uniformly)g (on)h(b)r(ounded)g(sets,)f(and)456 1722 y FF(k)p FH(S)549 1734 y FA(t)577 1722 y FG(\()p FH(u)657 1734 y FD(10)727 1722 y FG(\))19 b FF(\000)f FH(S)912 1734 y FA(t)941 1722 y FG(\()p FH(u)1021 1734 y FD(20)1091 1722 y FG(\))p FF(k)1165 1734 y FD(1)1226 1722 y FF(\024)k FH(C)1372 1734 y FD(1)1410 1722 y FG(\()p FH(t)1472 1688 y Fw(\000)p FA(a)1583 1722 y FF(_)c FG(1\))c FH(e)1783 1688 y FA(C)1831 1696 y Ft(2)1863 1688 y FD(\()p Fw(j)p FA(u)1948 1696 y Ft(10)2008 1688 y Fw(j)2028 1663 y Ft(2)2060 1688 y FD(+)p Fw(j)p FA(u)2170 1696 y Ft(20)2231 1688 y Fw(j)2251 1663 y Ft(2)2283 1688 y FD(\))2313 1722 y FF(j)p FH(u)2384 1734 y FD(10)2473 1722 y FF(\000)k FH(u)2604 1734 y FD(20)2673 1722 y FF(j)c FH(;)0 1881 y FI(for)27 b(suitable)g(constan)n(ts)g FH(a;)14 b(C)944 1893 y FD(1)981 1881 y 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y(rem)27 b(2.6.)p 3128 988 4 57 v 3132 935 50 4 v 3132 988 V 3182 988 4 57 v 125 1154 a(De\034ne)485 1337 y FH(L)542 1302 y FD(0)542 1357 y Fw(1)612 1337 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)897 1302 y FD(1)932 1337 y FG(\))24 b(=)e FF(f)p FH(\020)29 b FF(2)24 b FH(L)1318 1349 y Fw(1)1388 1337 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1673 1302 y FD(1)1708 1337 y FG(\))24 b FF(j)f FH(\020)34 b FI(is)27 b(con)n(tin)n(uous)g(at)g FH(t)c FG(=)g(0)o FF(g)0 1519 y FJ(Prop)s(osition)30 b(2.18.)40 b FC(The)30 b(map)650 1702 y FH(H)25 b FF(\002)18 b FH(L)884 1668 y FD(0)884 1722 y Fw(1)954 1702 y FG(\(0)p FH(;)c(T)e FG(;)i FH(H)1239 1668 y FD(1)1275 1702 y FG(\))23 b FF(!)g FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))14 b FH(;)98 b FG(\()p FH(u)2033 1714 y FD(0)2070 1702 y FH(;)14 b(\020)6 b FG(\))23 b FF(7!)h FH(u)18 b FF(\000)g FH(\020)i(;)0 1885 y FC(wher)l(e)j FH(u)f FC(is)h(a)f(solution)h(of)g(\(2.31\))h (with)f(initial)g(data)h FH(u)1718 1897 y FD(0)1754 1885 y 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FD(2)1110 2961 y(1)1165 2936 y FG(+)1248 2869 y Fz(\000)1286 2936 y FH(B)t FG(\()p FH(w)r(;)c(\020)6 b FG(\))p FH(;)14 b(w)1655 2869 y Fz(\001)1248 3000 y(|)p 1285 3000 149 10 v 149 w({z)p 1508 3000 V 149 w(})1441 3078 y FA(I)1470 3086 y Ft(1)1719 2936 y FG(=)23 b FF(\000)14 b FG(\()p FH(L\020)6 b(;)14 b(w)r FG(\))1886 2992 y Fz(|)p 1923 2992 57 10 v 57 w({z)p 2054 2992 V 57 w(})1985 3070 y FA(I)2014 3078 y Ft(2)2161 2936 y FF(\000)2240 2869 y Fz(\000)2277 2936 y FH(B)t FG(\()p FH(\020)6 b FG(\))p FH(;)14 b(w)2548 2869 y Fz(\001)2240 3000 y(|)p 2277 3000 100 10 v 100 w({z)p 2451 3000 V 100 w(})2383 3078 y FA(I)2412 3086 y Ft(3)2602 2936 y FH(:)0 3249 y FI(W)-7 b(e)28 b(ha)n(v)n(e)e(from)i(\(B2\))f(that)783 3440 y FF(j)p FH(I)842 3452 y FD(1)880 3440 y FF(j)c(\024)g FH(C)d FF(k)p FH(\020)6 b FF(k)1219 3465 y FD(1)1270 3440 y FF(j)p FH(w)r FF(j)14 b(k)p FH(w)r FF(k)1536 3465 y FD(1)1596 3440 y FF(\024)1694 3407 y FD(1)p 1694 3421 34 4 v 1694 3469 a(4)1751 3440 y FF(k)o FH(w)r FF(k)1895 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FF(k)2170 3930 y FD(2)2170 3996 y(1)2221 3971 y FH(:)125 4121 y FI(Using)27 b(\(2.33\))g(w)n(e)g(obtain:)397 4290 y FH(d)p 382 4328 74 4 v 382 4404 a(dt)465 4347 y FF(j)p FH(w)r FF(j)572 4312 y FD(2)629 4347 y FG(+)722 4314 y FD(1)p 722 4328 34 4 v 722 4375 a(2)779 4347 y FF(k)o FH(w)r FF(k)923 4305 y FD(2)923 4372 y(1)979 4347 y FF(\000)18 b FH(c)c FF(k)o FH(\020)6 b FF(k)1237 4305 y FD(2)1237 4372 y(1)1288 4347 y FF(j)p FH(w)r FF(j)1395 4312 y FD(2)1457 4347 y FF(\024)22 b FH(C)1609 4279 y Fz(\002)1658 4347 y FF(k)o FH(\020)6 b FF(k)1783 4305 y FD(2)1783 4372 y(1)1839 4347 y FG(+)18 b FF(j)p FH(\020)6 b FF(j)2010 4312 y FD(2)2062 4347 y FF(k)o FH(\020)g FF(k)2187 4305 y FD(2)2187 4372 y(1)2238 4279 y Fz(\003)2296 4347 y FF(\024)23 b FH(C)6 b FG(\()p FH(R)2545 4312 y FD(2)2601 4347 y FG(+)18 b FH(R)2748 4312 y FD(4)2784 4347 y FG(\))0 4562 y FI(for)28 b(almost)g(all)g FH(t)c FF(\025)h FG(0)p FI(.)39 b(No)n(w)28 b(Gron)n(w)n(all's)e(lemma)j (implies)f(that)h FF(j)p FH(w)r FG(\()p 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FI(or)c(the)h FH(d)p FI(-torus)g FE(T)2816 2082 y FA(d)2855 2112 y FI(,)h FH(d)j FG(=)e(2)e FI(or)0 2211 y(3,)g(is)f(describ)r(ed)g(b)n(y) g(the)g FH(d)q FG(+)q(1)g FI(equations)f(\(2.1\),)j(supplemen)n(ted)e (b)n(y)g(suitable)f(b)r(oundary)h(conditions,)0 2311 y(if)25 b(necessary)-7 b(.)34 b(There)25 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))25 b FI(denotes)f(the)h(v)n(elo)r(cit)n(y)f(of)h (the)g(\035uid,)h FH(p)p FG(\()p FH(t;)14 b(x)p FG(\))25 b FI(the)h(pressure)d(and)i FH(\027)30 b FI(the)0 2453 y(kinematic)f(viscosit)n(y)-7 b(.)41 b(The)30 b(quan)n(tit)n(y)1264 2420 y FD(1)p 1264 2434 34 4 v 1264 2482 a(2)1334 2453 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1522 2423 y FD(2)1585 2453 y FG(=)1686 2420 y FD(1)p 1686 2434 V 1686 2482 a(2)1769 2340 y Fz(Z)1865 2453 y FF(j)p FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))p FF(j)2137 2419 y FD(2)2175 2453 y FH(dx)30 b FI(is)g(the)f FC(ener)l(gy)j(of)g(the)g(\035uid)0 2595 y FI(\(at)c(time)g FH(t)p FI(\).)125 2695 y(Assume)f FH(f)32 b FG(=)22 b(0)28 b FI(and)f(formally)g(m)n(ultiply)h(\(2.1\))f (with)h FH(u)p FG(\()p FH(t)p FG(\))g FI(in)g FH(L)2194 2707 y FD(2)2230 2695 y FI(:)1015 2862 y FG(1)p 1015 2899 42 4 v 1015 2975 a(2)1115 2862 y FH(d)p 1100 2899 74 4 v 1100 2975 a(dt)1206 2918 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1394 2883 y FD(2)1450 2918 y FG(+)18 b FH(\027)1607 2805 y Fz(Z)1718 2918 y FF(jr)p FH(u)p FF(j)1881 2883 y FD(2)1918 2918 y FH(dx)23 b FG(=)g(0)p FH(:)0 3191 y FI(A)n(ccordingly)-7 b(,)26 b(the)i(quan)n(tit)n(y)f FH(")c FG(:=)g FH(\027)1183 3078 y Fz(Z)1293 3191 y FF(jr)p FH(u)p FF(j)1456 3156 y FD(2)1494 3191 y FH(dx)28 b FI(is)f(called)g (the)h FC(r)l(ate)i(of)h(dissip)l(ation)g(of)g(ener)l(gy)p FI(.)0 3376 y(The)d FC(R)l(eynolds)i(numb)l(er)d FI(of)g(the)h(\035o)n (w)f(is)g(de\034ned)h(as)501 3599 y FH(R)23 b FG(=)685 3543 y FF(h)p FI(c)n(haracteristic)j(scale)g(for)h FH(x)p FF(i)20 b(\001)e(h)p FI(c)n(haracteristic)26 b(scale)h(for)g FH(u)p FF(i)p 685 3580 1957 4 v 1640 3656 a FH(\027)2665 3599 y(:)0 3802 y FI(The)40 b(terms)g(in)g(the)h(n)n(umerator)d(are)h (am)n(biguous.)73 b(F)-7 b(or)39 b(the)i(purp)r(oses)e(of)h(these)g (Lectures)f(w)n(e)0 3902 y(understand)27 b(them)h(as)f(follo)n(ws:)125 4026 y FF(h)p FI(c)n(haracteristic)e(scale)i(for)g FH(x)p FF(i)d FG(=)j FI(diameter)g(of)h(domain)f FH(O)j FI(\(or)d(the)h(p)r (erio)r(d)f(of)h(torus\),)125 4125 y FF(h)p FI(c)n(haracteristic)d (scale)i(for)55 b FH(u)p FF(i)23 b FG(=)1248 4093 y FD(1)p 1221 4107 88 4 v 1221 4115 a Fw(p)p 1276 4115 34 3 v 48 x FD(2)1333 4125 y FF(j)p FH(u)p FF(j)1427 4137 y FA(H)1512 4125 y FG(=)1600 4055 y FF(p)p 1669 4055 67 4 v 70 x FH(E)c FI(.)0 4236 y(If)28 b FH(u)f FI(dep)r(ends)h(on)f(a)h (random)e(parameter,)g(then)i(w)n(e)f(mo)r(dify)h(this)g(de\034nition)g (as)125 4354 y FF(h)p FI(c)n(haracteristic)d(scale)i(for)g FH(u)p FF(i)c FG(=)1184 4287 y Fz(\000)1222 4354 y Fk(E)14 b FH(E)1365 4287 y Fz(\001)1402 4304 y FD(1)p FA(=)p FD(2)1521 4354 y FI(,)0 4454 y(where)27 b Fk(E)g FI(stands)h(for)f(the) h(mathematical)f(exp)r(ectation.)125 4602 y(In)k(the)i FJ(2D)j(case)c FI(w)n(e)g(denote)f(with)i FH(\030)h FG(=)29 b(curl)44 b FH(u)29 b FG(=)1860 4546 y FH(@)5 b(u)1957 4516 y FD(2)p 1860 4583 134 4 v 1860 4659 a FH(@)g(x)1956 4635 y FD(1)2025 4602 y FF(\000)2120 4546 y FH(@)g(u)2217 4516 y FD(1)p 2120 4583 V 2120 4659 a FH(@)g(x)2216 4635 y FD(2)2296 4602 y FI(the)32 b FC(vorticity)h FI(of)f(the)g(\035o)n(w,) 0 4780 y(and)27 b(call)h FG(\012)23 b(=)494 4747 y FD(1)p 494 4761 34 4 v 494 4809 a(2)551 4667 y Fz(Z)648 4780 y FH(\030)688 4746 y FD(2)725 4780 y FH(dx)29 b FI(the)f FC(enstr)l(ophy)p FI(.)1553 5059 y(27)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 28 27 bop 0 553 a FJ(Exercise)31 b(2.21.)40 b FI(Sho)n(w)27 b(that)h FG(\012)23 b(=)1196 520 y FD(1)p 1196 534 34 4 v 1196 582 a(2)1275 440 y Fz(Z)1386 553 y FF(jr)p FH(u)p FF(j)1549 519 y FD(2)1586 553 y FH(dx)125 761 y FI(Applying)k(the)h(op)r(erator)e FC(curl)i FI(to)f(\(2.1\),)g(w) n(e)h(obtain)977 922 y FG(_)959 944 y FH(\030)23 b FF(\000)18 b FH(\027)5 b FG(\001)p FH(\030)23 b FG(+)18 b(curl)1511 877 y Fz(\000)1549 944 y FG(\()p FH(u)g FF(\001)g(r)p FG(\))p FH(u)1837 877 y Fz(\001)1898 944 y FG(=)23 b(curl)37 b FH(f)t(:)0 1135 y FI(It)28 b(is)f(easy)g(to)g(see)h(that)f FG(curl)924 1068 y Fz(\000)962 1135 y FG(\()p FH(u)18 b FF(\001)h(r)p FG(\))p FH(u)1251 1068 y Fz(\001)1312 1135 y FG(=)j(\()p FH(u)d FF(\001)f(r)p FG(\))p FH(\030)t FI(.)125 1235 y(If)28 b(w)n(e)f(assume)g(that)g FH(f)32 b FG(=)23 b(0)k FI(and)g(m)n(ultiply)h(the)g(previous)f(equation)g(b)n (y)g FH(\030)32 b FI(in)27 b FH(L)2668 1247 y FD(2)2705 1235 y FI(,)h(w)n(e)f(get:)1041 1400 y FG(1)p 1041 1437 42 4 v 1041 1513 a(2)1140 1400 y FH(d)p 1125 1437 74 4 v 1125 1513 a(dt)1231 1456 y FF(j)p FH(\030)t FF(j)1317 1422 y FD(2)1317 1477 y FA(L)1363 1485 y Ft(2)1418 1456 y FG(+)18 b FH(\027)1561 1343 y Fz(Z)1672 1456 y FF(jr)p FH(\030)t FF(j)1827 1422 y FD(2)1879 1456 y FH(dx)23 b FG(=)g(0)14 b FH(:)0 1732 y FI(W)-7 b(e)28 b(deduce)g(that)f FH(\027)660 1619 y Fz(Z)770 1732 y FF(jr)p FH(\030)t FF(j)925 1698 y FD(2)977 1732 y FH(dx)h FI(is)g(the)g FC(r)l(ate)h(of)i(dissip)l(ation)g(of)g(enstr)l(ophy)p FI(.)0 1991 y FJ(Exercise)g(2.22.)40 b FI(Sho)n(w)27 b(that)1015 1878 y Fz(Z)1112 1991 y FF(jr)p FH(\030)t FF(j)1267 1957 y FD(2)1304 1991 y FH(dx)d FG(=)f FF(j)p FG(\001)p FH(u)p FF(j)1669 1957 y FD(2)1669 2012 y FA(L)1715 2020 y Ft(2)1751 1991 y FI(.)1553 5059 y(28)p eop end PStoPSsaved restore %%Page: (28,29) 15 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 29 28 bop 0 523 a FK(3)131 b(Random)43 b(kic)l(k-forces)0 721 y Fy(3.1)112 b(Ingredien)m(ts)37 b(for)g(the)h(constructions)0 875 y FI(W)-7 b(e)28 b(consider)e(a)i (sequence)f FH(b)916 887 y FA(s)974 875 y FF(\025)22 b FG(0)p FI(,)28 b FH(s)23 b FF(2)g FE(Z)1349 845 y FD(2)1349 895 y(0)1387 875 y FI(,)k(of)h(real)e(n)n(um)n(b)r(ers)h(with)1270 1074 y FH(B)1333 1086 y FD(0)1394 1074 y FG(=)1481 995 y Fz(X)1615 1074 y FH(b)1651 1040 y FD(2)1651 1094 y FA(s)1711 1074 y FH(<)c FF(1)14 b FH(:)0 1278 y FI(Let)22 b FG(\(\012)p FH(;)14 b FF(F)8 b FH(;)14 b Fk(P)p FG(\))22 b FI(b)r(e)g(a)f(probabilit)n(y)g(space,)h(and)f FF(f)p FH(\030)1558 1290 y FA(sk)1644 1278 y FF(j)14 b FH(s)23 b FF(2)g FE(Z)1876 1248 y FD(2)1876 1299 y(0)1914 1278 y FH(;)37 b(k)26 b FF(2)d FE(Z)p FF(g)f FI(b)r(e)g(a)f(family)h(of)f (indep)r(enden)n(t)0 1378 y(random)27 b(v)-5 b(ariables,)26 b(suc)n(h)h(that)h FF(j)p FH(\030)1098 1390 y FA(sk)1170 1378 y FG(\()p FH(!)s FG(\))p FF(j)c(\024)e FG(1)28 b FI(for)f(all)g FH(s)c FF(2)g FE(Z)1930 1348 y FD(2)1930 1399 y(0)1968 1378 y FH(;)14 b(k)26 b FF(2)d FE(Z)p FH(;)14 b(!)26 b FF(2)e FG(\012)p FI(,)j(and)963 1561 y FF(D)r FG(\()p FH(\030)1097 1573 y FA(sk)1170 1561 y FG(\))d(=)e FH(p)1355 1573 y FA(s)1391 1561 y FG(\()p FH(r)r FG(\))14 b FH(dr)n(;)98 b(s)23 b FF(2)g FE(Z)1902 1531 y FD(2)1902 1581 y(0)1940 1561 y FH(;)37 b(k)26 b FF(2)d FE(Z)q FH(:)0 1743 y FI(Here)k FH(p)238 1755 y FA(s)273 1743 y FI(,)h FH(s)23 b FF(2)h FE(Z)520 1713 y FD(2)520 1764 y(0)557 1743 y FI(,)k(are)e(Lipsc)n(hitz)i(functions)g(with)g(supp)r(ort)f(in)h FG([)p FF(\000)p FG(1)p FH(;)14 b FG(1])p FI(,)26 b(and)i FH(p)2537 1755 y FA(s)2572 1743 y FG(\(0\))23 b FF(6)p FG(=)g(0)p FI(.)0 1909 y FJ(Remark)37 b(3.1.)43 b FI(In)34 b(fact,)h(it)e(su\036ces)g(that)g(eac)n(h)g(densit)n(y)f FH(p)1953 1921 y FA(s)1989 1909 y FG(\()p FH(r)r FG(\))i FI(has)f(\034nite)g(total)g(v)-5 b(ariation)32 b(and)0 1942 y Fz(R)55 1963 y FA(")39 2038 y Fw(\000)p FA(")140 2009 y FH(p)182 2021 y FA(s)232 2009 y FH(dr)26 b(>)c FG(0)p FI(,)27 b(for)g(eac)n(h)g FH(")c(>)g FG(0)p FI(.)0 2175 y(The)28 b FC(kick)i(numb)l(er)f FH(k)s FI(,)f FH(k)e FF(2)d FE(Z)p FI(,)28 b(is)g(the)g(random)e(v)-5 b(ariable)27 b FH(\021)1870 2187 y FA(k)1934 2175 y FG(:)c(\012)g FF(!)g FH(H)7 b FI(,)27 b(de\034ned)h(b)n(y)1258 2370 y FH(\021)1299 2382 y FA(k)1364 2370 y FG(=)1465 2291 y Fz(X)1451 2476 y FA(s)p Fw(2)p Fv(Z)1566 2456 y Ft(2)1566 2494 y(0)1612 2370 y FH(b)1648 2382 y FA(s)1697 2370 y FH(\030)1733 2382 y FA(sk)1820 2370 y FH(e)1859 2382 y FA(s)1908 2370 y FH(;)0 2656 y FI(where)e FF(f)p FH(e)320 2668 y FA(s)355 2656 y FF(g)g FI(is)h(the)g(orthonormal)e(basis)h(of)h FH(H)7 b FI(,)27 b(de\034ned)g(in)g(Section)g(2.1.)35 b(Note)27 b(that)g FF(f)p FH(\021)2844 2668 y FA(k)2885 2656 y FH(;)14 b(k)26 b FF(2)d FE(Z)p FF(g)p FI(,)0 2755 y(are)k(i.i.d.)37 b(random)26 b(v)-5 b(ariables)27 b(in)g FH(H)7 b FI(.)37 b(The)28 b(kic)n(ks)e FH(\021)1635 2767 y FA(k)1704 2755 y FI(p)r(ossess)h(the)h(follo)n(wing)e(ob)n(vious)g (prop)r(erties:)652 2950 y Fk(E)p FF(j)p FH(\021)779 2962 y FA(k)820 2950 y FF(j)843 2920 y FD(2)916 2950 y FG(=)1003 2871 y Fz(X)1048 3046 y FA(s)1151 2950 y FH(b)1187 2916 y FD(2)1187 2971 y FA(s)1224 2950 y FH(d)1267 2962 y FA(s)1326 2950 y FG(=:)c FH(D)1505 2962 y FD(0)1565 2950 y FH(<)h(B)1716 2962 y FD(0)1767 2950 y FH(;)14 b FI(where)27 b FH(d)2087 2962 y FA(s)2146 2950 y FG(=)22 b Fk(E)14 b FH(\030)2350 2920 y FD(2)2346 2974 y FA(sk)2441 2950 y FF(\024)23 b FG(1)p FI(;)601 3141 y Fk(E)14 b FF(k)p FH(\021)761 3153 y FA(k)801 3141 y FF(k)843 3111 y FD(2)843 3162 y(1)916 3141 y FG(=)1003 3062 y Fz(X)1048 3237 y FA(s)1151 3141 y FH(b)1187 3107 y FD(2)1187 3162 y FA(s)1238 3141 y FF(j)p FH(s)p FF(j)1323 3107 y FD(2)1374 3141 y FH(d)1417 3153 y FA(s)1476 3141 y FG(=:)22 b FH(D)1655 3153 y FD(1)1715 3141 y FF(\024)h(1)14 b FG(;)595 3332 y FF(j)p FH(\021)659 3344 y FA(k)700 3332 y FG(\()p FH(!)s FG(\))p FF(j)842 3302 y FD(2)916 3332 y FG(=)1003 3253 y Fz(X)1048 3427 y FA(s)1151 3332 y FH(b)1187 3298 y FD(2)1187 3353 y FA(s)1238 3332 y FH(\030)1278 3298 y FD(2)1274 3353 y FA(sk)1346 3332 y FG(\()p FH(!)s FG(\))23 b FF(\024)g FH(B)1639 3344 y FD(0)1676 3332 y FH(;)79 b FI(for)27 b(all)g FH(k)f FF(2)d FE(Z)p FI(,)28 b FH(!)e FF(2)e FG(\012)p FI(.)3018 3170 y(\(3.1\))0 3585 y(Since)k(the)g (densit)n(y)f FH(p)688 3597 y FA(s)751 3585 y FI(of)g FH(\030)881 3597 y FA(sk)982 3585 y FI(is)g(con)n(tin)n(uous)g(and)g (do)r(es)g(not)h(v)-5 b(anish)27 b(around)g(0,)g(w)n(e)g(ha)n(v)n(e) 1013 3767 y Fk(P)p FG(\()14 b FF(j)p FH(\021)1188 3779 y FA(k)1229 3767 y FF(j)23 b(\024)g FH(")p FG(\))g FH(>)g FG(0)p FH(;)96 b FI(for)27 b(all)g FH(")c(>)g FG(0)13 b FH(:)125 3950 y FI(Consider)26 b(an)n(y)h FH(\034)33 b(>)22 b FG(0)p FI(.)37 b(The)27 b FC(\(r)l(andom\))j(kick-for)l(c)l(e) g FH(\021)s FG(\()p FH(t)p FG(\))23 b FF(2)h FH(H)7 b FI(,)27 b FH(t)c FF(2)h FE(R)p FI(,)j(is)h(then)g(de\034ned)g(b)n(y) 1012 4149 y FH(\021)s FG(\()p FH(t)p FG(\))c(=)e FH(\021)1305 4115 y FA(!)1353 4149 y FG(\()p FH(t)p FG(\))i(=)1559 4070 y Fz(X)1559 4249 y FA(k)q Fw(2)p Fv(Z)1707 4149 y FH(\021)1751 4115 y FA(!)1748 4170 y(k)1813 4149 y FH(\016)s FG(\()p FH(t)18 b FF(\000)g FH(\034)9 b(k)s FG(\))14 b FH(:)842 b FI(\(3.2\))0 4410 y(As)26 b(a)f(function)h(of)f FH(t)p FI(,)h(this)g(is)g(a)f(random)f(v)n(ector\025v)-5 b(alued)24 b(measure)h(that)g(tak)n(es)g(v)-5 b(alues)25 b(in)h FH(H)7 b FI(.)36 b(Then)0 4510 y(\(cf.)g(Example)25 b(2.19\))e FH(\021)s FG(\()p FH(t)p FG(\))h(=)971 4477 y FA(@)p 959 4491 65 4 v 959 4539 a(@)t(t)1047 4510 y FH(\020)6 b FG(\()p FH(t)p FG(\))p FI(,)26 b(where)e(the)h(function)g FH(\020)6 b FG(\()p FH(t)p FG(\))24 b(=)e FH(\020)2220 4480 y FA(!)2269 4510 y FG(\()p FH(t)p FG(\))j FI(is)f(con)n(tin)n (uous)g(from)g(the)0 4610 y(righ)n(t,)j(constan)n(t)g(in)g FE(R)p FF(n)p FH(\034)9 b FE(Z)p FI(,)29 b(and)e(its)h(jump)g(at)g FH(t)23 b FG(=)f FH(j)5 b(\034)38 b FI(is)27 b FH(\021)1879 4622 y FA(j)1914 4610 y FI(.)1553 5059 y(29)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 30 29 bop 0 523 a Fy(3.2)112 b(The)38 b(kic)m(k)m(ed)f (NSE)0 676 y FI(W)-7 b(e)28 b(consider)e(the)i(equation)1172 776 y FG(_)-37 b FH(u)18 b FG(+)g FH(Lu)f FG(+)h FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(\021)1846 742 y FA(w)1900 776 y FG(\()p FH(t)p FG(\))14 b FH(;)987 b FI(\(3.3\))0 925 y(with)34 b FH(\021)s FG(\()p FH(t)p FG(\))h FI(as)f(in)g(\(3.2\).) 55 b(A)35 b(random)e(pro)r(cess)f FH(u)1580 895 y FA(!)1628 925 y FG(\()p FH(t)p FG(\))j FI(is)e(a)h(solution)f(if)i(it)f (satis\034es)f(\(3.3\))h(for)f(a.e.)0 1025 y FH(!)i FF(2)e FG(\012)h FI(in)f(the)h(sense)f(of)g(Section)g(2.6.)53 b(W)-7 b(e)34 b(c)n(ho)r(ose)e FH(\034)42 b FG(=)33 b(1)g FI(to)g(simplify)g(notations.)54 b(Then)33 b(\(see)0 1125 y(Example)28 b(2.19\),)972 1224 y FH(u)p FG(\()p FH(k)s FG(\))23 b(=)g FH(S)1297 1157 y Fz(\000)1335 1224 y FH(u)p FG(\()p FH(k)e FF(\000)d FG(1\))1636 1157 y Fz(\001)1692 1224 y FG(+)g FH(\021)1816 1236 y FA(k)1871 1224 y FH(;)97 b(k)26 b FF(2)d FE(Z)p FH(;)802 b FI(\(3.4\))0 1374 y(where)38 b FH(S)47 b FG(=)42 b FH(S)507 1386 y FD(1)583 1374 y FI(is)c(the)i(time-one)e(shift)i(along)e(tra)5 b(jectories)37 b(of)h(the)i(free)e(equation.)71 b(F)-7 b(or)38 b(an)n(y)0 1473 y FG(0)23 b FF(\024)f FH(t)h(<)g FG(1)p FI(,)1219 1573 y FH(u)p FG(\()p FH(k)e FG(+)d FH(t)p FG(\))23 b(=)g FH(S)1670 1585 y FA(t)1699 1506 y Fz(\000)1737 1573 y FH(u)p FG(\()p FH(k)s FG(\))1895 1506 y Fz(\001)1947 1573 y FH(:)1048 b FI(\(3.5\))0 1722 y(W)-7 b(e)28 b(will)g(b)r(e)g(mainly)f(in)n(terested)g(in)h FH(u)p FG(\()p FH(t)p FG(\))p FI(,)g FH(t)23 b FF(2)g FE(Z)p FI(.)38 b(By)1137 1905 y FH(u)p FG(\()p FH(k)s FG(;)14 b FH(u)1348 1917 y FD(0)1384 1905 y FG(\))g FH(;)37 b(k)26 b FF(\025)d FG(0)p FH(;)59 b(u)1819 1917 y FD(0)1879 1905 y FF(2)24 b FH(H)r(;)0 2088 y FI(w)n(e)j(denote)h(the)g(solution)f (of)34 b(\(3.4\))o(,)28 b(equal)f FH(u)1436 2100 y FD(0)1500 2088 y FI(at)h FH(t)23 b FG(=)g(0)p FI(.)0 2237 y FJ(Basic)32 b(estimate.)i FI(W)-7 b(e)28 b(ha)n(v)n(e)e(for)h(an)n(y)g FH(k)f FF(2)d FE(Z)28 b FI(and)g FH(m)23 b FF(\025)f FG(0)216 2473 y FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)h FG(=)531 2402 y Fz(\014)531 2452 y(\014)559 2473 y FH(S)615 2406 y Fz(\000)652 2473 y FH(u)p FG(\()p FH(k)f FF(\000)c FG(1\))954 2406 y Fz(\001)1010 2473 y FG(+)g FH(\021)1134 2485 y FA(k)1175 2402 y Fz(\014)1175 2452 y(\014)1226 2396 y FD(\()p FA(N)6 b(S)s FD(1\))1287 2473 y FF(\024)1436 2397 y Fz(p)p 1519 2397 101 4 v 76 x FH(B)1582 2485 y FD(0)1638 2473 y FG(+)18 b FH(e)1760 2439 y Fw(\000)p FD(1)1849 2473 y FF(j)p FH(u)p FG(\()p FH(k)j FF(\000)d FG(1\))p FF(j)1249 2619 y(\024)1336 2543 y Fz(p)p 1419 2543 V 76 x FH(B)1482 2631 y FD(0)1538 2619 y FG(+)g FH(e)1660 2585 y Fw(\000)p FD(1)1749 2552 y Fz(\000)1787 2543 y(p)p 1870 2543 V 76 x FH(B)1933 2631 y FD(0)1989 2619 y FG(+)g FH(e)2111 2585 y Fw(\000)p FD(1)2213 2619 y FF(j)p FH(u)p FG(\()p FH(k)j FF(\000)d FG(2\))p FF(j)2560 2552 y Fz(\001)1249 2765 y FH(<)k FF(\001)14 b(\001)g(\001)23 b FH(<)1544 2689 y Fz(p)p 1627 2689 V 76 x FH(B)1690 2777 y FD(0)1741 2765 y FG(\(1)18 b(+)g FH(e)1955 2731 y Fw(\000)p FD(1)2062 2765 y FG(+)h FH(:)14 b(:)g(:)f FG(\))19 b(+)f FH(e)2429 2731 y Fw(\000)p FA(m)2557 2765 y FF(j)p FH(u)p FG(\()p FH(k)k FF(\000)c FH(m)p FG(\))p FF(j)c FH(:)0 2948 y FI(So,)496 3064 y FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)23 b(\024)811 2988 y Fz(p)p 894 2988 V 76 x FH(B)957 3076 y FD(0)1099 3008 y FH(e)p 1027 3045 182 4 v 1027 3121 a(e)18 b FF(\000)g FG(1)1237 3064 y(+)g FH(e)1359 3030 y Fw(\000)p FA(m)1488 3064 y FF(j)p FH(u)p FG(\()p FH(k)j FF(\000)d FH(m)p FG(\))p FF(j)p FH(;)60 b FI(for)27 b(all)h FH(m)23 b FF(\025)f FG(0)p FH(;)14 b(!)25 b FF(2)f FG(\012)p FH(:)325 b FI(\(3.6\))0 3243 y(In)28 b(particular,)787 3360 y FF(j)p FH(u)p FG(\()p FH(k)s FG(;)14 b(0\))p FF(j)23 b(\024)1262 3304 y FH(e)p 1190 3341 V 1190 3417 a(e)c FF(\000)f FG(1)1405 3283 y Fz(p)p 1488 3283 101 4 v 77 x FH(B)1551 3372 y FD(0)1602 3360 y FH(;)60 b FI(for)27 b(all)h FH(k)d FF(\025)e FG(0)p FH(;)14 b(!)25 b FF(2)f FG(\012)p FH(:)616 b FI(\(3.7\))0 3539 y(F)-7 b(or)27 b FH(r)f FF(2)d FE(R)28 b FI(w)n(e)f(denote)1179 3639 y FH(B)1242 3651 y FA(r)1302 3639 y FG(=)1390 3560 y Fz(X)1537 3639 y FF(j)p FH(s)p FF(j)1622 3604 y FD(2)p FA(r)1706 3639 y FH(b)1742 3604 y FD(2)1742 3659 y FA(s)1802 3639 y FF(\024)c(1)14 b FH(:)0 3805 y FI(Rep)r(eating)30 b(the)g(argumen)n(ts)e(ab)r(o)n(v)n(e)h(and)g(using)h(the)g(prop)r(ert) n(y)f(\(NS)p FG(1)2239 3775 y Fw(0)2262 3805 y FI(\))h(instead)g(of)g (\(NS1\))g(w)n(e)f(get)0 3904 y(that)408 4004 y FF(k)p FH(u)p FG(\()p FH(k)s FG(\))p FF(k)650 4016 y FD(1)710 4004 y FF(\024)797 3928 y Fz(p)p 880 3928 V 76 x FH(B)943 4016 y FD(1)1085 3948 y FH(e)p 1013 3985 182 4 v 1013 4061 a(e)19 b FF(\000)f FG(1)1223 4004 y(+)h FH(e)1346 3970 y Fw(\000)p FA(m)1460 4004 y FF(k)p FH(u)p FG(\()p FH(k)i FF(\000)d FH(m)p FG(\))p FF(k)1876 4016 y FD(1)1913 4004 y FH(;)60 b FI(for)27 b(all)g FH(m)c FF(\025)g FG(0)p FH(;)41 b(!)25 b FF(2)f FG(\012)14 b FH(;)237 b FI(\(3.8\))0 4183 y(if)28 b FH(B)139 4195 y FD(1)199 4183 y FH(<)23 b FF(1)p FI(.)125 4283 y(W)-7 b(e)27 b(ma)n(y)g(write)h(\(3.4\))f(as) 1163 4382 y FH(u)p FG(\()p FH(k)s FG(\))c(=)f FH(F)1484 4394 y FA(k)1525 4382 y FG(\()p FH(u)p FG(\()p FH(k)g FF(\000)c FG(1\))p FH(;)c(w)r FG(\))g FH(;)0 4532 y FI(where)23 b(the)i(map)f FH(F)610 4544 y FA(k)674 4532 y FG(:)f FH(H)18 b FF(\002)11 b FG(\012)23 b FF(!)g FH(H)31 b FI(is)24 b(measurable)e(and)i(lo)r(cally)f(Lipsc)n(hitz)h(in)g FH(u)f FF(2)g FH(H)7 b FI(.)36 b(So)23 b(\(3.4\))h(is)0 4632 y(a)k(Random)g(Dynamical)f(System)i(\(RDS\).)g(See)f(the)h(b)r(o)r (oks)f([Kif86)o(,)g(Arn98)o(].)39 b(Ev)n(ery)27 b(RDS)i(de\034nes)0 4731 y(a)e(Mark)n(o)n(v)e(c)n(hain:)1553 5059 y(30)p eop end PStoPSsaved restore %%Page: (30,31) 16 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 31 30 bop 0 523 a FJ(The)32 b(Mark)m(o)m(v)h(c)m(hain.)k FI(F)-7 b(or)27 b FH(u)1025 535 y FD(0)1085 523 y FF(2)d FH(H)7 b FI(,)27 b FG(0)c FF(\024)f FH(k)k FF(2)e FE(Z)k FI(and)f FG(\000)c FF(2)g(B)s FG(\()p FH(H)7 b FG(\))27 b FI(w)n(e)h(de\034ne)1080 676 y FH(P)1133 688 y FA(k)1174 676 y FG(\()p FH(u)1254 688 y FD(0)1291 676 y FH(;)14 b FG(\000\))23 b(=)g Fk(P)p FG(\()p FH(u)p FG(\()p FH(k)s FG(;)14 b FH(u)1831 688 y FD(0)1868 676 y FG(\))23 b FF(2)h FG(\000\))p FH(:)0 830 y(P)31 b FI(is)20 b(called)f(the)h FC(tr)l(ansition)i(function)d FI(for)g(\(3.4\).)34 b(It)20 b(satis\034es)e(the)i FC(Chapman-Kolmo)l(gor)l(ov)25 b(r)l(elation)p FI(:)0 929 y(for)i FH(k)s(;)14 b(n)23 b FF(\025)f FG(0)p FI(,)905 1067 y FH(P)958 1079 y FA(k)q FD(+)p FA(n)1091 1067 y FG(\()p FH(u)1171 1079 y FD(0)1208 1067 y FH(;)14 b FG(\000\))23 b(=)1440 954 y Fz(Z)1486 1143 y FA(H)1577 1067 y FH(P)1630 1079 y FA(k)1671 1067 y FG(\()p FH(u)1751 1079 y FD(0)1788 1067 y FH(;)14 b(du)p FG(\))g FH(P)2015 1079 y FA(n)2060 1067 y FG(\()p FH(u;)g FG(\000\))p FH(:)0 1245 y FI(So)38 b(\(3.4\))g(de\034nes)g(a)g(Mark)n (o)n(v)e(c)n(hain)i FF(f)p FH(u)p FG(\()p FH(k)s FG(\))p FF(g)1483 1257 y FA(k)q Fw(\025)p FD(0)1646 1245 y FI(in)g FH(H)7 b FI(.)69 b(\(This)38 b(is)g(almost)g(ob)n(vious,)i(or)d(see)h (in)0 1345 y([Kif86)o(,)28 b(Arn98)o(]\).)37 b(Due)28 b(to)f(\(3.6\))h(with)g FH(k)e FG(=)c FH(m)p FI(,)645 1428 y Fz(Z)755 1541 y FF(j)p FH(u)p FF(j)14 b FH(P)916 1553 y FA(k)957 1541 y FG(\()p FH(u)1037 1553 y FD(0)1074 1541 y FH(;)g(du)p FG(\))23 b(=)g Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)22 b(\024)1736 1464 y Fz(p)p 1819 1464 101 4 v 77 x FH(B)1882 1553 y FD(0)2024 1485 y FH(e)p 1952 1522 182 4 v 1952 1598 a(e)c FF(\000)g FG(1)2162 1541 y(+)g FH(e)2284 1507 y Fw(\000)p FA(k)2377 1541 y FF(j)p FH(u)2448 1553 y FD(0)2484 1541 y FF(j)c FH(;)474 b FI(\(3.9\))0 1737 y(where)27 b FH(u)p FG(\()p FH(t)p FG(\))h FI(is)f(a)g(solution)g(suc)n(h)h(that)f FH(u)p FG(\(0\))c(=)g FH(u)1556 1749 y FD(0)1593 1737 y FI(.)0 1874 y FJ(The)42 b(Mark)m(o)m(v)i(semigroups.)59 b FI(W)-7 b(e)37 b(next)g(in)n(tro)r(duce)f(the)h(t)n(w)n(o)e(Mark)n(o) n(v)f(semigroups,)j(one)f(on)0 1974 y FH(C)59 1986 y FA(b)93 1974 y FG(\()p FH(H)7 b FG(\))p FI(,)28 b(another)e(on)i FF(P)7 b FG(\()p FH(H)g FG(\))p FI(:)111 2159 y(i\))42 b FF(B)263 2171 y FA(k)326 2159 y FG(:)23 b FH(C)431 2171 y FA(b)465 2159 y FG(\()p FH(H)7 b FG(\))23 b FF(!)g FH(C)793 2171 y FA(b)827 2159 y FG(\()p FH(H)7 b FG(\))p FH(;)97 b FG(\()p FF(B)1174 2171 y FA(k)1215 2159 y FH(f)9 b FG(\)\()p FH(v)s FG(\))24 b(=)1515 2046 y Fz(Z)1561 2235 y FA(H)1652 2159 y FH(P)1705 2171 y FA(k)1746 2159 y FG(\()p FH(v)s(;)14 b(dz)t FG(\))g FH(f)9 b FG(\()p FH(z)t FG(\))p FI(,)208 2333 y(for)27 b FG(0)22 b FF(\024)h FH(k)j FF(2)d FE(Z)14 b FH(;)37 b(v)27 b FF(2)c FH(H)62 b FI(\(see)27 b(Exercise)g(3.3)g(b)r(elo)n(w\).)88 2529 y(ii\))42 b FF(B)266 2499 y Fw(\003)263 2553 y FA(k)326 2529 y FG(:)23 b FF(P)7 b FG(\()p FH(H)g FG(\))23 b FF(!)g(P)7 b FG(\()p FH(H)g FG(\))p FH(;)97 b FG(\()p FF(B)1121 2499 y Fw(\003)1118 2553 y FA(k)1159 2529 y FH(\026)p FG(\)\(\000\))23 b(=)1468 2416 y Fz(Z)1514 2605 y FA(H)1605 2529 y FH(P)1658 2541 y FA(k)1699 2529 y FG(\()p FH(v)s(;)14 b FG(\000\))g FH(\026)p FG(\()p FH(dv)s FG(\))p FI(,)208 2703 y(for)27 b FG(0)22 b FF(\024)h FH(k)j FF(2)d FE(Z)14 b FH(;)37 b FG(\000)23 b FF(2)h(B)s FG(\()p FH(H)7 b FG(\))p FI(.)125 2846 y(Let)27 b FH(u)321 2858 y FD(0)381 2846 y FF(2)c FH(H)35 b FI(b)r(e)28 b(a)f(random)g(v)-5 b(ariable,)26 b(suc)n(h)h(that)h FF(D)r FG(\()p FH(u)1897 2858 y FD(0)1935 2846 y FG(\))23 b(=)g FH(\026)p FI(.)37 b(Then)1079 2999 y FF(D)16 b FH(u)p FG(\()p FH(k)s FG(;)e FH(u)1370 3011 y FD(0)1407 2999 y FG(\))23 b(=)g FF(B)1608 2965 y Fw(\003)1605 3019 y FA(k)1659 2999 y FH(\026;)97 b FF(8)p FH(k)26 b FF(\025)c FG(0)14 b FH(:)0 3152 y FI(In)28 b(particular,)e(taking)h FH(k)f FG(=)c(1)28 b FI(and)f FH(\026)c FG(=)g FH(\016)1351 3164 y FA(v)1390 3152 y FI(,)28 b FH(v)e FF(2)e FH(H)7 b FI(,)27 b(w)n(e)g(ha)n(v)n(e) 1020 3306 y FF(B)1078 3271 y Fw(\003)1075 3326 y FD(1)1115 3306 y FH(\016)1152 3318 y FA(v)1215 3306 y FG(=)22 b FH(P)1355 3318 y FD(1)1393 3306 y FG(\()p FH(v)s(;)14 b FF(\001)p FG(\))24 b(=)e FF(D)r FG(\()p FH(S)5 b FG(\()p FH(v)s FG(\))20 b(+)e FH(\021)2076 3318 y FD(1)2113 3306 y FG(\))p FH(:)809 b FI(\(3.10\))125 3459 y(It)34 b(is)h(immediate)g (that)g(eac)n(h)e(op)r(erator)g FF(B)1505 3429 y Fw(\003)1502 3483 y FA(k)1577 3459 y FI(extends)i(to)f(a)h FJ(linear)f FI(map)g FF(M)p FG(\()p FH(H)7 b FG(\))35 b FF(!)g(M)p FG(\()p FH(H)7 b FG(\))p FI(,)0 3559 y(de\034ned)28 b(b)n(y)f(the)h (same)f(relation)g(ii\).)0 3696 y FJ(The)32 b(dualit)m(y)-8 b(.)37 b FI(F)-7 b(or)27 b(the)h(t)n(w)n(o)e(semigroups)g(the)i(dualit) n(y)g(relations)e(hold:)593 3850 y FG(\()p FF(B)680 3862 y FA(k)721 3850 y FH(f)t(;)14 b(\026)p FG(\))23 b(=)f(\()p FH(f)t(;)14 b FF(B)1167 3816 y Fw(\003)1164 3870 y FA(k)1219 3850 y FH(\026)p FG(\))p FH(;)97 b FI(for)27 b(all)g FH(f)32 b FF(2)23 b FH(C)1873 3862 y FA(b)1907 3850 y FH(;)28 b(\026)23 b FF(2)g(M)28 b FI(and)f FH(k)f FF(\025)d FG(0)0 4003 y FI(\(the)32 b(relations)e(do)h(not)g(mean)g(that)h(these) f(are)f(dual)h(semigroups)f(since)h(the)h(spaces)e FH(C)2850 4015 y FA(b)2884 4003 y FG(\()p FH(H)7 b FG(\))31 b FI(and)0 4103 y FF(M)p FG(\()p FH(H)7 b FG(\))28 b FI(are)e(not)i(dual.\))0 4245 y FJ(Example)i(3.2.)40 b FI(F)-7 b(or)27 b FH(u)c FF(2)g FH(H)7 b FI(,)28 b(let)f FH(\016)1165 4257 y FA(u)1236 4245 y FI(b)r(e)h(the)g(delta-measure)e(at)i FH(u)p FI(.)36 b(Then)56 b FF(B)2535 4215 y Fw(\003)2532 4269 y FA(k)2572 4245 y FG(\()p FH(\016)2641 4257 y FA(u)2685 4245 y FG(\))23 b(=)g FH(P)2881 4257 y FA(k)2922 4245 y FG(\()p FH(u;)14 b FF(\001)p FG(\))g FH(:)0 4388 y FJ(Exercise)42 b(3.3.)k FI(a\))36 b(Pro)n(v)n(e)g(that)h(for)g(an)n(y)f FH(k)41 b FF(\025)d FG(1)f FI(the)g(map)g FH(H)45 b FF(!)39 b(P)7 b FG(\()p FH(H)g FG(\))p FI(,)39 b FH(u)f FF(7\000)-14 b(!)39 b FH(P)2858 4400 y FA(k)2899 4388 y FG(\()p FH(u;)14 b FF(\001)p FG(\))p FI(,)39 b(is)0 4488 y(con)n(tin)n(uous,)23 b(where)f FF(P)7 b FG(\()p FH(H)g FG(\))22 b FI(is)h(endo)n(w)n(ed)f (with)h(the)g(w)n(eak)f(top)r(ology)-7 b(.)34 b(Sho)n(w)22 b(that)h(this)g(is)f(equiv)-5 b(alen)n(t)0 4587 y(to)28 b(the)g(fact)g(that)g FG(\()p FF(B)676 4599 y FA(k)716 4587 y FH(f)9 b FG(\))24 b FF(2)f FH(C)959 4599 y FA(b)993 4587 y FG(\()p FH(H)7 b FG(\))28 b FI(if)h FH(f)i FF(2)24 b FH(C)1448 4599 y FA(b)1482 4587 y FG(\()p FH(H)7 b FG(\))p FI(.)38 b(This)27 b(prop)r(ert)n(y)g(of)h(a)f(Mark)n(o)n(v)f(c) n(hain)h(is)g(called)0 4687 y(the)h FC(F)-6 b(el)t(ler)30 b(pr)l(op)l(erty)p FI(.)125 4787 y(b\))e(Pro)n(v)n(e)e(that)i(the)g (maps)f FF(B)1063 4756 y Fw(\003)1060 4810 y FA(k)1123 4787 y FG(:)c FF(P)7 b FG(\()p FH(H)g FG(\))23 b FF(!)g(P)7 b FG(\()p FH(H)g FG(\))28 b FI(are)f(con)n(tin)n(uous)f(\(in)i(the)g(w) n(eak)f(top)r(ology\).)1553 5059 y(31)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 32 31 bop 0 523 a Fy(3.3)112 b(Stationary)37 b(measures)0 676 y FJ(De\034nition)h(3.4.)44 b FC(A)35 b(me)l(asur)l(e)g FH(\026)f FF(2)g(P)7 b FG(\()p FH(H)g FG(\))35 b FC(is)h(a)g(stationary)h(me)l(asur)l(e)e(for)h(\(3.4\))h(if) g FF(B)2881 646 y Fw(\003)2878 697 y FD(1)2932 676 y FH(\026)c FG(=)g FH(\026)p FC(.)0 776 y(\(Then)d FF(B)308 746 y Fw(\003)305 799 y FA(k)360 776 y FH(\026)23 b FG(=)f FH(\026)30 b FC(for)h(e)l(ach)f FH(k)c FF(\025)d FG(0)p FC(\).)0 923 y FJ(Theorem)30 b(3.5.)40 b FC(A)30 b(stationary)g(me)l (asur)l(e)g FH(\026)f FC(exists.)0 1071 y(Pr)l(o)l(of.)43 b FI(F)-7 b(or)30 b(simplicit)n(y)-7 b(,)31 b(assume)f(that)h FH(B)1346 1083 y FD(1)1411 1071 y FH(<)d FF(1)p FI(.)46 b(\(F)-7 b(or)30 b FH(B)1903 1083 y FD(1)1968 1071 y FG(=)e FF(1)p FI(,)j(the)g(pro)r(of)f(is)g(sligh)n(tly)g(longer,)0 1171 y(see)d([KS00)o(]\).)125 1270 y(Let)g(us)h(tak)n(e)f(in)g(\(3.4\)) h FH(u)p FG(\(0\))22 b(=)h(0)p FI(,)k(denote)h FH(\026)1529 1282 y FA(k)1593 1270 y FG(=)22 b FF(D)r FG(\()p FH(u)p FG(\()p FH(k)s FG(\)\))29 b FI(and)e(set)p 1291 1466 51 4 v 1291 1512 a FH(\026)1341 1532 y FA(k)1405 1512 y FG(=)1504 1456 y(1)p 1502 1493 46 4 v 1502 1569 a FH(k)1618 1408 y FA(k)q Fw(\000)p FD(1)1619 1433 y Fz(X)1621 1610 y FA(j)s FD(=0)1776 1512 y FH(\026)1826 1524 y FA(j)1875 1512 y FH(:)0 1768 y FI(Note)k(that)g FF(B)445 1738 y Fw(\003)442 1789 y FD(1)496 1768 y FH(\026)546 1780 y FA(j)610 1768 y FG(=)d FH(\026)753 1780 y FA(j)s FD(+1)872 1768 y FI(.)47 b(W)-7 b(e)32 b(de\034ne)f FB(B)1405 1780 y FA(r)1442 1768 y FG(\()p FH(H)1550 1738 y FA(s)1586 1768 y FG(\))e(=)f FF(f)p FH(u)14 b FF(j)g(k)p FH(u)p FF(k)2013 1780 y FA(s)2074 1768 y FF(\024)29 b FH(r)r FF(g)14 b FI(.)47 b(Let)31 b FH(r)2522 1780 y FD(1)2588 1768 y FG(=)2734 1736 y FA(e)p 2692 1750 117 4 v 2692 1797 a(e)p Fw(\000)p FD(1)2847 1704 y FF(p)p 2916 1704 101 4 v 64 x FH(B)2979 1780 y FD(1)3016 1768 y FI(.)47 b(By)0 1868 y(\(3.8\),)27 b(w)n(e)h(see)f(that)990 1968 y FH(\026)1040 1980 y FA(j)1075 1900 y Fz(\000)1114 1968 y FB(B)1187 1980 y FA(r)1218 1988 y Ft(1)1254 1968 y FG(\()p FH(H)1362 1933 y FD(1)1400 1968 y FG(\))1432 1900 y Fz(\001)1493 1968 y FG(=)c(1)p FH(;)96 b FI(for)27 b(all)g FH(j)h FF(\025)23 b FG(0)p FI(.)0 2111 y(Hence)p 254 2066 51 4 v 35 w FH(\026)304 2132 y FA(k)345 2044 y Fz(\000)383 2111 y FB(B)456 2123 y FA(r)487 2131 y Ft(1)524 2111 y FG(\()p FH(H)632 2081 y FD(1)669 2111 y FG(\))701 2044 y Fz(\001)776 2111 y FG(=)35 b(1)g FI(for)g(all)f FH(k)39 b FF(\025)d FG(0)p FI(.)59 b(But)36 b(from)e(\(1.3\))h(w)n(e)g (kno)n(w)g(that)g FH(H)2717 2081 y FD(1)2790 2111 y Fs(b)g FH(H)r(;)71 b FI(and)0 2211 y(th)n(us)37 b(the)g(set)f FB(B)555 2223 y FA(r)586 2231 y Ft(1)623 2211 y FG(\()p FH(H)731 2181 y FD(1)768 2211 y FG(\))h FI(is)f(compact)g(in)h FH(H)7 b FI(.)64 b(It)37 b(follo)n(ws)e(from)i(the)f(Prokhoro)n(v)f (Theorem)g(\(see)0 2311 y([Bil99)o(,)30 b(Shi96)o(,)g(Dud89)o(]\))g (that)g(the)g(set)g FF(f)p 1339 2265 V FH(\026)1388 2331 y FA(k)1429 2311 y FH(;)14 b(k)29 b FF(\025)d FG(1)p FF(g)j FI(is)g(compact)g(in)h FF(P)7 b FG(\()p FH(H)g FG(\))p FI(,)30 b(where)f(the)h(latter)f(is)0 2410 y(pro)n(vided)d (with)j(the)e(w)n(eak)g(top)r(ology)-7 b(.)36 b(So)27 b(there)g(exists)g(a)h(subsequence)e FF(f)p FH(k)2427 2422 y FA(m)2490 2410 y FF(g)h FI(suc)n(h)h(that)p 1261 2524 V 1261 2570 a FH(\026)1311 2590 y FA(k)1346 2598 y Fu(m)1429 2570 y FH(*)23 b(\026)g FF(2)g(P)7 b FG(\()p FH(H)g FG(\))14 b FH(:)0 2730 y FI(W)-7 b(e)28 b(claim)f(that)h FH(\026)g FI(is)f(a)g(stationary)f(measure.)36 b(Indeed,)28 b(for)f(an)n(y)g FH(f)k FF(2)24 b FH(C)2312 2742 y FA(b)2346 2730 y FG(\()p FH(H)7 b FG(\))p FI(,)28 b(w)n(e)f(ha)n(v)n(e)190 2972 y FG(\()p FH(f)t(;)14 b FF(B)362 2938 y Fw(\003)359 2992 y FD(1)413 2972 y FH(\026)p FG(\))519 2925 y Fj(Exerc.)26 b(3.3)614 2972 y FG(=)156 b(lim)797 3022 y FA(m)p Fw(!1)989 2972 y FG(\()p FH(f)t(;)14 b FF(B)1161 2938 y Fw(\003)1158 2992 y FD(1)p 1212 2926 V 1212 2972 a FH(\026)1262 2992 y FA(k)1297 3000 y Fu(m)1356 2972 y FG(\))24 b(=)60 b(lim)1499 3022 y FA(m)p Fw(!1)1793 2916 y FG(1)p 1761 2953 107 4 v 1761 3029 a FH(k)1804 3041 y FA(m)1914 2867 y(k)1949 2875 y Fu(m)2004 2867 y Fw(\000)p FD(1)1941 2893 y Fz(X)1944 3070 y FA(j)s FD(=0)2103 2972 y FG(\()p FH(f)t(;)14 b FF(B)2275 2938 y Fw(\003)2272 2992 y FD(1)2326 2972 y FH(\026)2376 2984 y FA(j)2411 2972 y FG(\))519 3276 y(=)60 b(lim)606 3326 y FA(m)p Fw(!1)854 3220 y FG(1)p 821 3257 V 821 3333 a FH(k)864 3345 y FA(m)966 3171 y(k)1001 3179 y Fu(m)952 3197 y Fz(X)954 3374 y FA(j)s FD(=1)1094 3276 y FG(\()p FH(f)t(;)14 b(\026)1258 3288 y FA(j)1294 3276 y FG(\))23 b(=)61 b(lim)1437 3326 y FA(m)p Fw(!1)1656 3184 y Fz(\020)1705 3276 y FG(\()p FH(f)t(;)p 1819 3230 51 4 v 14 w(\026)1869 3296 y FA(k)1904 3304 y Fu(m)1964 3276 y FG(\))19 b(+)2140 3220 y(1)p 2108 3257 107 4 v 2108 3333 a FH(k)2151 3345 y FA(m)2247 3209 y Fz(\000)2285 3276 y FG(\()p FH(f)t(;)14 b(\026)2449 3288 y FA(k)2484 3296 y Fu(m)2544 3276 y FG(\))k FF(\000)h FG(\()p FH(f)t(;)14 b(\026)2842 3288 y FD(0)2879 3276 y FG(\))2911 3209 y Fz(\001)2949 3184 y(\021)519 3497 y FG(=)22 b(\()p FH(f)t(;)14 b(\026)p FG(\))g FH(:)0 3652 y FI(As)28 b(a)f(consequence,)f FF(B)741 3622 y Fw(\003)738 3673 y FD(1)792 3652 y FH(\026)e FG(=)e FH(\026)p FI(.)p 3128 3652 4 57 v 3132 3599 50 4 v 3132 3652 V 3182 3652 4 57 v 0 3813 a FJ(Remark)43 b(3.6.)j FC(The)40 b(weighte)l(d)g(sum,)h(de\034ning)f(the)f(me)l(asur) l(e)p 2107 3768 51 4 v 39 w FH(\026)2157 3834 y FA(k)2198 3813 y FC(,)j(is)d(c)l(al)t(le)l(d)h(the)g(Bo)l(golyub)l(ov\025)0 3913 y(Krylov)35 b(ansatz.)53 b(If)35 b FH(B)735 3925 y FD(1)803 3913 y FG(=)c FF(1)p FC(,)36 b(but)e FH(B)1251 3925 y FD(0)1320 3913 y FH(<)d FF(1)p FC(,)36 b(then)e(stil)t(l)h(the)g (set)f(of)h(me)l(asur)l(es)f FF(f)p 2698 3867 V FH(\026)2747 3933 y FA(k)2788 3913 y FF(g)g FC(is)h(we)l(akly)0 4013 y(pr)l(e)l(c)l(omp)l(act)30 b(in)g FF(P)7 b FG(\()p FH(H)g FG(\))30 b FC(and)g(every)h(limiting)f(me)l(asur)l(e)f(is)i (stationary,)g(se)l(e)e([KS00)q(].)0 4160 y FJ(Exercise)e(3.7.)36 b FI(Use)24 b(\(3.7\))f(to)h(pro)n(v)n(e)e(that)i FH(\026)1454 4093 y Fz(\000)1492 4160 y FB(B)1565 4172 y FA(r)1596 4180 y Ft(0)1633 4160 y FG(\()p FH(H)7 b FG(\))1773 4093 y Fz(\001)1834 4160 y FG(=)23 b(1)p FI(,)h(with)48 b FH(r)2257 4172 y FD(0)2318 4160 y FG(=)2458 4127 y FA(e)p 2415 4141 117 4 v 2415 4189 a(e)p Fw(\000)p FD(1)2565 4096 y FF(p)p 2634 4096 101 4 v 64 x FH(B)2697 4172 y FD(0)2734 4160 y FI(.)36 b(If)24 b FH(B)2935 4172 y FD(1)2995 4160 y FH(<)f FF(1)p FI(,)0 4275 y(then)28 b(also)f FH(\026)406 4207 y Fz(\000)444 4275 y FB(B)517 4287 y FA(r)548 4295 y Ft(1)584 4275 y FG(\()p FH(H)692 4244 y FD(1)730 4275 y FG(\))762 4207 y Fz(\001)823 4275 y FG(=)c(1)p FI(.)125 4422 y(Let)28 b FH(u)p FG(\()p FH(t)p FG(\))p FI(,)i FH(t)25 b FF(\025)g FG(0)p FI(,)j(b)r(e)i(a)e(solution)g(of)h(\(3.3\),) g(suc)n(h)f(that)h FF(D)17 b FH(u)p FG(\(0\))24 b(=)h FH(\026)p FI(,)k(where)g FH(\026)g FI(is)f(a)h(stationary)0 4522 y(measure.)35 b(Then)27 b FF(D)16 b FH(u)p FG(\()p FH(k)s FG(\))23 b(=)f FH(\026)27 b FI(for)e(all)h(in)n(teger)f FH(k)h FF(\025)d FG(0)p FI(,)j(i.e.,)g FF(f)p FH(u)p FG(\()p FH(k)s FG(\))p FF(g)f FI(is)h(a)g(stationary)e(pro)r(cess.)36 b(Due)0 4621 y(to)27 b(\(3.5\),)h(the)g(pro)r(cess)e FF(f)p FH(u)p FG(\()p FH(t)p FG(\))p FH(;)14 b(t)23 b FF(\025)f FG(0)p FF(g)27 b FI(is)g(1\025p)r(erio)r(dic)g(in)h (distribution:)1009 4781 y FF(D)1075 4714 y Fz(\000)1113 4781 y FH(u)p FG(\()p FH(t)19 b FG(+)f(1\))1399 4714 y Fz(\001)1460 4781 y FG(=)k FF(D)1627 4714 y Fz(\000)1665 4781 y FH(u)p FG(\()p FH(t)p FG(\))1807 4714 y Fz(\001)1929 4781 y FF(8)p FH(t)h FF(\025)f FG(0)p FH(:)1553 5059 y FI(32)p eop end PStoPSsaved restore %%Page: (32,33) 17 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 33 32 bop 0 523 a Fy(3.4)112 b(More)38 b(estimates)0 676 y FI(Let)28 b(us)f(assume)g(that)1015 776 y Fk(E)14 b FH(\030)1128 788 y FA(sk)1224 776 y FG(=)22 b(0)83 b FI(for)27 b(all)g FH(s)c FF(2)g FE(Z)1873 788 y FD(0)1911 776 y FH(;)14 b(k)26 b FF(2)d FE(Z)p FI(.)0 911 y(Then)28 b Fk(E)14 b FH(\021)335 923 y FA(k)398 911 y FG(=)23 b(0)p FI(.)37 b(Consider)26 b(the)i(follo)n(wing)e (scaled)h(NSE:)876 1071 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)1552 1007 y FF(p)p 1621 1007 47 4 v 64 x FH(\027)c(\021)s FG(\()p FH(t)p FG(\))p FH(;)98 b FG(0)23 b FH(<)f(\027)29 b FF(\024)22 b FG(1)14 b FH(:)649 b FI(\(3.11\))0 1231 y(\(This)25 b(equation)f(coincides)g(with)h(\(3.3\))f(if)i FH(\027)i FG(=)23 b(1)p FI(\).)35 b(As)25 b(b)r(efore,)g FH(\034)33 b FG(=)22 b(1)p FI(.)36 b(The)25 b(n)n(um)n(b)r(ers)f FH(D)2863 1243 y FD(0)2925 1231 y FI(and)g FH(D)3152 1243 y FD(1)0 1330 y FI(are)j(de\034ned)g(as)g(in)h(\(3.1\).)0 1478 y FJ(Exercise)j(3.8.)40 b FI(Let)28 b FH(u)p FG(\()p FH(t)p FG(\))p FI(,)g FH(t)23 b FF(\025)f FG(0)p FI(,)28 b(b)r(e)g(a)f(solution)g(of)g(\(3.11\).)36 b(Then)28 b(for)f(all)h FH(k)d FF(\025)e FG(0)p FI(,)570 1699 y Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))p FF(j)994 1665 y FD(2)1049 1699 y FG(+)g(2)p FH(\027)1234 1586 y Fz(Z)1317 1607 y FA(k)q FD(+1)1280 1775 y FA(k)1456 1699 y Fk(E)c FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1768 1665 y FD(2)1768 1720 y(1)1818 1699 y FH(ds)23 b FG(=)f Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)2291 1665 y FD(2)2347 1699 y FG(+)k FH(\027)5 b(D)2545 1711 y FD(0)2596 1699 y FH(:)0 1901 y FI(If)28 b FH(D)152 1913 y FD(1)212 1901 y FH(<)23 b FF(1)p FI(,)k(then)h(also)533 2118 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(k)20 b FG(+)e(1\))p FF(k)994 2083 y FD(2)994 2138 y(1)1049 2118 y FG(+)g(2)p FH(\027)1234 2005 y Fz(Z)1317 2025 y FA(k)q FD(+1)1280 2193 y FA(k)1456 2118 y Fk(E)c FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1768 2083 y FD(2)1768 2138 y(2)1818 2118 y FH(ds)23 b FG(=)f Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(k)s FG(\))p FF(k)2329 2083 y FD(2)2329 2138 y(1)2384 2118 y FG(+)k FH(\027)5 b(D)2582 2130 y FD(1)2633 2118 y FH(:)0 2319 y FI(Note)28 b(that)f(if)i FH(u)e FI(is)g(a)g(stationary) f(solution,)h(then)h(these)g(relations)e(imply)i(that)578 2536 y FG(2)634 2423 y Fz(Z)717 2443 y FA(k)q FD(+1)680 2612 y FA(k)856 2536 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1168 2502 y FD(2)1168 2556 y(1)1217 2536 y FH(ds)23 b FG(=)g FH(D)1479 2548 y FD(0)1516 2536 y FH(;)97 b FG(2)1692 2423 y Fz(Z)1774 2443 y FA(k)q FD(+1)1737 2612 y FA(k)1913 2536 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)2225 2502 y FD(2)2225 2556 y(2)2275 2536 y FH(ds)23 b FG(=)g FH(D)2537 2548 y FD(1)2587 2536 y FH(:)367 b FI(\(3.12\))125 2742 y(If)28 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))28 b FI(is)f(a)g(solution)g(of)h (\(3.11\),)f(then)1011 2918 y FH(u)p FG(\()p FH(k)22 b FG(+)c(1\))k(=)h FH(S)1479 2875 y FD(\()p FA(\027)t FD(\))1474 2940 y(1)1586 2918 y FH(u)p FG(\()p FH(k)s FG(\))18 b(+)1845 2854 y FF(p)p 1914 2854 V 64 x FH(\027)i(\021)2016 2930 y FA(k)q FD(+1)2154 2918 y FH(;)800 b FI(\(3.13\))0 3095 y(where)30 b FH(S)299 3052 y FD(\()p FA(\027)t FD(\))294 3117 y(1)422 3095 y FI(is)h(the)g(time-one)f(shift)h(along)f(tra)5 b(jectories)28 b(of)j(the)g(free)f(equation)g(\(3.11\).)45 b(No)n(w)30 b(the)0 3194 y(prop)r(ert)n(y)c(\(NS1\))i(tak)n(es)f(the)h (form)1280 3310 y FF(j)p FH(S)1359 3267 y FD(\()p FA(\027)t FD(\))1354 3332 y(1)1452 3310 y FH(u)p FF(j)22 b(\024)h FH(e)1672 3276 y Fw(\000)p FA(\027)1779 3310 y FF(j)p FH(u)p FF(j)14 b FH(:)1067 b FI(\(3.14\))0 3458 y FJ(Lemma)40 b(3.9.)46 b FC(Ther)l(e)39 b(exist)f FH(\027)5 b FC(-indep)l(endent)39 b(c)l(onstants)e FH(C)1971 3470 y FA(m)2035 3458 y FC(,)j FH(m)f FF(2)g FE(N)p FC(,)i(such)d(that)g(for)h(any)g FH(m)0 3557 y FC(solutions)30 b(of)g(\(3.13\))i(satisfy)803 3717 y Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))p FF(j)1227 3683 y FD(2)p FA(m)1346 3717 y FF(\024)23 b FG(\(1)18 b FF(\000)g FH(\027)5 b FG(\))14 b Fk(E)g FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)1982 3683 y FD(2)p FA(m)2097 3717 y FG(+)k FH(C)2239 3729 y FA(m)2302 3717 y FH(\027)h(:)592 b FI(\(3.15\))0 3876 y FC(If)30 b FH(D)156 3888 y FD(1)216 3876 y FH(<)23 b FF(1)p FC(,)30 b(then)g(also)785 4036 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(k)20 b FG(+)e(1\))p FF(k)1246 4002 y FD(2)p FA(m)1246 4056 y FD(1)1365 4036 y FF(\024)k FG(\(1)d FF(\000)f FH(\027)5 b FG(\))14 b Fk(E)g FF(k)p FH(u)p FG(\()p FH(k)s FG(\))p FF(k)2039 4002 y FD(2)p FA(m)2039 4056 y FD(1)2152 4036 y FG(+)k FH(C)2300 4002 y Fw(0)2294 4056 y FA(m)2358 4036 y FH(\027)0 4195 y FC(for)31 b(some)f(c)l(onstants)f FH(C)775 4165 y Fw(0)769 4216 y FA(m)832 4195 y FC(.)39 b(In)29 b(p)l(articular,)i (if)g FH(u)p FG(\(0\))22 b(=)h(0)p FC(,)30 b(then)802 4355 y Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)1083 4321 y FD(2)p FA(m)1215 4355 y FF(\024)35 b FH(C)1374 4367 y FA(m)1484 4355 y FC(for)30 b(al)t(l)h FH(k)26 b FF(\025)d FG(0)p FC(,)114 b(and)773 b FI(\(3.16\))766 4551 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(k)s FG(\))p FF(k)1085 4517 y FD(2)p FA(m)1085 4571 y FD(1)1215 4551 y FF(\024)35 b FH(C)1380 4517 y Fw(0)1404 4563 y FA(m)1513 4551 y FC(for)c(al)t(l)f FH(k)c FF(\025)d FG(0)p FC(,)30 b(if)g FH(D)2167 4563 y FD(1)2227 4551 y FH(<)23 b FF(1)p FC(.)554 b FI(\(3.17\))0 4710 y FC(If)30 b FH(u)p FG(\()p FH(t)p FG(\))p FH(;)e(t)23 b FF(\025)g FG(0)29 b FC(is)h(a)g(solution)g (of)g(\(3.11\),)i(e)l(qual)e(zer)l(o)g(at)g FH(t)23 b FG(=)f(0)p FC(,)30 b(then)f(\(3.16\))j(and)e(\(3.17\))h(hold)g(for)0 4810 y(any)f(r)l(e)l(al)g FH(k)c FF(\025)d FG(0)p FC(.)1553 5059 y FI(33)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 34 33 bop 0 523 a FC(Pr)l(o)l(of.)43 b FI(Let)28 b FH(u)p FG(\()p FH(k)s FG(\))f FI(b)r(e)h(a)g(solution)f(of) g(\(3.13\).)36 b(Then)802 722 y Fk(E)p FF(j)p FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))p FF(j)1212 688 y FD(2)p FA(m)1331 722 y FG(=)k Fk(E)14 b FF(j)p FH(S)1574 679 y FD(\()p FA(\027)t FD(\))1569 744 y(1)1667 722 y FH(u)p FG(\()p FH(k)s FG(\))k(+)1926 658 y FF(p)p 1996 658 47 4 v 1996 722 a FH(\027)28 b(\021)2106 734 y FA(k)q FD(+1)2231 722 y FF(j)2254 688 y FD(2)p FA(m)2364 722 y FH(:)590 b FI(\(3.18\))0 922 y(W)-7 b(e)27 b(denote)f FH(u)d FG(=)f FH(S)622 879 y FD(\()p FA(\027)t FD(\))617 944 y(1)715 922 y FH(u)p FG(\()p FH(k)s FG(\))27 b FI(and)f FH(\021)g FG(=)d FH(\021)1256 934 y FA(k)q FD(+1)1381 922 y FI(.)37 b(W)-7 b(e)26 b(claim)h(that)f(for)g(an)n(y)g FH(c)d(>)f FG(0)k FI(w)n(e)g(can)h(\034nd)f FH(C)2973 934 y FA(m)3060 922 y FH(>)c FG(0)0 1022 y FI(suc)n(h)27 b(that)724 1121 y FH(I)760 1133 y FA(\027)825 1121 y FG(:=)c Fk(E)14 b FF(j)p FH(u)j FG(+)1184 1057 y FF(p)p 1254 1057 V 1254 1121 a FH(\027)i(\021)s FF(j)1381 1087 y FD(2)p FA(m)1500 1121 y FF(\024)k FG(\(1)18 b(+)g FH(c\027)5 b FG(\))14 b Fk(E)g FF(j)p FH(u)p FF(j)2062 1087 y FD(2)p FA(m)2176 1121 y FG(+)k FH(C)2318 1133 y FA(m)2382 1121 y FH(\027)h(:)512 b FI(\(3.19\))0 1271 y(Indeed,)28 b(w)n(e)f(ha)n(v)n(e)749 1453 y FH(I)785 1465 y FA(\027)850 1453 y FG(=)c Fk(E)p FG(\()p FF(j)p FH(u)p FF(j)1127 1419 y FD(2)1182 1453 y FG(+)18 b FH(\027)h FF(j)p FH(\021)s FF(j)1415 1419 y FD(2)1471 1453 y FG(+)f(2)1596 1389 y FF(p)p 1665 1389 V 64 x FH(\027)h FF(h)p FH(u;)14 b(\021)s FF(i)p FG(\))1950 1419 y FA(m)850 1588 y FG(=)23 b Fk(E)14 b FF(j)p FH(u)p FF(j)1109 1554 y FD(2)p FA(m)1223 1588 y FG(+)k(2)p FH(m)1421 1524 y FF(p)p 1489 1524 V 1489 1588 a FH(\027)42 b Fk(E)p FG(\()p FF(j)p FH(u)p FF(j)1761 1554 y FD(2)p FA(m)p Fw(\000)p FD(2)1956 1588 y FF(h)p FH(u;)14 b(\021)s FF(i)p FG(\))19 b(+)f FH(\027)5 b(J)2375 1600 y FA(\027)2417 1588 y FH(;)0 1771 y FI(where)23 b FH(J)282 1783 y FA(\027)346 1771 y FG(=)g FH(O)r FG(\(1\))h FI(as)f FH(\027)28 b FF(!)23 b FG(0)p FI(.)36 b(The)23 b(second)g(term)g(in)h(the)g(r.h.s.) 35 b(v)-5 b(anishes)23 b(since)g FH(u)g FI(is)g(indep)r(enden)n(t)0 1870 y(of)28 b FH(\021)i FI(and)e Fk(E)14 b FH(\021)26 b FG(=)c(0)p FI(.)37 b(Let)27 b(us)h(estimate)f(the)h(third)g(term.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)699 2065 y FF(j)p FH(J)768 2077 y FA(\027)809 2065 y FF(j)c(\024)e FH(\027)989 2031 y FA(m)p Fw(\000)p FD(1)1151 2065 y Fk(E)14 b FF(j)p FH(\021)s FF(j)1318 2031 y FD(2)p FA(m)1433 2065 y FG(+)k FH(C)1581 2031 y Fw(0)1575 2086 y FA(m)1770 1986 y Fz(X)1652 2164 y FD(2)p FA(m)p Fw(\000)p FD(2)p Fw(\025)p FA(n)p Fw(\025)p FD(2)2021 2065 y FF(j)p FH(u)p FF(j)2115 2031 y FA(n)2174 2065 y FF(j)p FH(\021)s FF(j)2264 2031 y FD(2)p FA(m)p Fw(\000)p FA(n)2467 2065 y FH(:)487 b FI(\(3.20\))0 2337 y(By)27 b(the)h(Y)-7 b(oung)27 b(inequalit)n(y)i(\(2.29\))o(,)e (for)h(an)n(y)e FH(\024)d(>)g FG(0)k FI(and)g(an)n(y)g FH(n)c FF(2)g FG([2)p FH(;)14 b FG(2)p FH(m)k FF(\000)g FG(2])27 b FI(w)n(e)g(ha)n(v)n(e)608 2559 y FF(j)p FH(u)p FF(j)702 2525 y FA(n)761 2559 y FF(j)p FH(\021)s FF(j)851 2525 y FD(2)p FA(m)p Fw(\000)p FA(n)1063 2559 y FF(\024)1193 2503 y FH(n)p 1161 2540 115 4 v 1161 2616 a FG(2)p FH(m)1308 2559 y(\024)1366 2502 y Ft(2)p Fu(m)p 1366 2511 80 3 v 1387 2545 a(n)1459 2559 y FF(j)p FH(u)p FF(j)1553 2525 y FD(2)p FA(m)1668 2559 y FG(+)1761 2503 y(2)p FH(m)17 b FF(\000)h FH(n)p 1761 2540 266 4 v 1836 2616 a FG(2)p FH(m)2059 2559 y(\024)2107 2523 y Fw(\000)2210 2501 y Ft(2)p Fu(m)p 2169 2510 161 3 v 2169 2543 a Ft(2)p Fu(m)p Fr(\000)p Fu(n)2358 2559 y FF(j)p FH(\021)s FF(j)2448 2525 y FD(2)p FA(m)2558 2559 y FH(:)0 2769 y FI(Using)28 b(in)h(\(3.20\))e(this)i(inequalit)n(y)f(with)h(v)-5 b(arious)27 b FH(\024)d FG(=)g FH(\024)1805 2781 y FA(n)1850 2769 y FI(,)29 b(w)n(e)f(get)g(that)h(for)f(an)n(y)g FH(c)c(>)g FG(0)k FI(there)g(is)g(a)0 2869 y FH(C)59 2881 y FA(m)150 2869 y FI(suc)n(h)f(that)1046 2969 y FF(j)p FH(J)1115 2981 y FA(\027)1156 2969 y FF(j)c(\024)g FH(c)14 b Fk(E)g FF(j)p FH(u)p FF(j)1511 2934 y FD(2)p FA(m)1624 2969 y FG(+)k FH(C)1766 2981 y FA(m)1844 2969 y Fk(E)c FF(j)p FH(\021)s FF(j)2011 2934 y FD(2)p FA(m)2120 2969 y FH(:)0 3118 y FI(No)n(w)27 b(\(3.19\))g(follo)n(ws)f(since)i Fk(E)14 b FF(j)p FH(\021)s FF(j)1072 3088 y FD(2)p FA(m)1191 3118 y FF(\024)22 b FH(B)1345 3088 y FA(m)1341 3139 y FD(0)1409 3118 y FI(.)125 3218 y(Due)28 b(to)f(\(3.19\))g(and)g (\(3.14\),)g(w)n(e)g(obtain)g(from)h(\(3.18\))389 3416 y Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))p FF(j)813 3382 y FD(2)p FA(m)932 3416 y FF(\024)23 b FG(\(1)18 b(+)g FH(c\027)5 b FG(\))14 b Fk(E)g FF(j)p FH(S)1479 3373 y FD(\()p FA(\027)t FD(\))1474 3439 y(1)1586 3416 y FH(u)p FG(\()p FH(k)s FG(\))p FF(j)1767 3382 y FD(2)p FA(m)1881 3416 y FG(+)19 b FH(C)2024 3428 y FA(m)2087 3416 y FH(\027)489 3587 y FF(\024)k FG(\(1)18 b(+)g FH(c\027)5 b FG(\))14 b FH(e)919 3553 y Fw(\000)p FD(2)p FA(m\027)1118 3587 y Fk(E)g FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)1399 3553 y FD(2)p FA(m)1513 3587 y FG(+)k FH(C)1655 3599 y FA(m)1719 3587 y FH(\027)28 b FF(\024)23 b FG(\(1)18 b FF(\000)g FH(\027)5 b FG(\))24 b Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)2434 3553 y FD(2)p FA(m)2548 3587 y FG(+)k FH(C)2690 3599 y FA(m)2753 3587 y FH(\027)0 3770 y FI(\(since)31 b(for)g FH(c)g FI(small)g(enough)g(w)n(e)g(ha)n(v) n(e)f FG(\(1)20 b(+)h FH(c\027)5 b FG(\))14 b FH(e)1616 3740 y Fw(\000)p FD(2)p FA(\027)1771 3770 y FF(\024)29 b FG(1)20 b FF(\000)h FH(\027)5 b FI(\).)48 b(No)n(w)32 b(\(3.15\))f(is)g(pro)n(v)n(en.)46 b(Since)0 3869 y(the)26 b(estimate)g(\(3.14\))f(also)f(holds)i(for)f(the)h(norm)f FF(k)14 b(\001)h(k)1707 3881 y FD(1)1744 3869 y FI(,)26 b(then)g(rep)r(eating)f(the)h(same)f(argumen)n(ts)f(w)n(e)0 3969 y(get)j(the)h(required)f(b)r(ound)h(for)f Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(k)20 b FG(+)e(1\))p FF(k)1451 3939 y FD(2)p FA(m)1451 3990 y FD(1)1546 3969 y FI(.)125 4069 y(The)36 b(last)g(t)n(w)n(o)f(estimates)h(of)g(the)h(lemma)f(are)f (straigh)n(tforw)n(ard)f(consequences)g(of)j(the)f(\034rst)0 4180 y(t)n(w)n(o.)f(These)25 b(estimates)g(and)h(\(3.5\))f(imply)h(the) f(last)g(assertion)f(since)i(the)f(maps)g FH(S)2629 4137 y FD(\()p FA(\027)t FD(\))2624 4200 y FA(t)2748 4180 y FI(decrease)f(the)0 4279 y(norms)j(in)h FH(H)34 b FI(and)27 b(in)h FH(H)784 4249 y FD(1)821 4279 y FI(.)p 3128 4279 4 57 v 3132 4227 50 4 v 3132 4279 V 3182 4279 4 57 v 1553 5059 a(34)p eop end PStoPSsaved restore %%Page: (34,35) 18 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 35 34 bop 0 523 a FK(4)131 b(White-forced)45 b(equations)0 721 y Fy(4.1)112 b(White)36 b(in)h(time)f(forces)0 897 y FI(F)-7 b(or)27 b FH(x)c FF(2)h FE(T)353 867 y FD(2)418 897 y FI(and)k FH(t)23 b FF(\025)f FG(0)p FI(,)27 b(w)n(e)h(consider)e FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)1617 841 y FH(d)p 1602 878 74 4 v 1602 954 a(dt)1699 897 y(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))p FH(;)29 b FI(where)907 1133 y FH(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)1252 1054 y Fz(X)1239 1239 y FA(s)p Fw(2)p Fv(Z)1354 1219 y Ft(2)1354 1257 y(0)1400 1133 y FH(b)1436 1145 y FA(s)1485 1133 y FH(\014)1532 1145 y FA(s)1567 1133 y FG(\()p FH(t)p FG(\))14 b FH(e)1714 1145 y FA(s)1750 1133 y FG(\()p FH(x)p FG(\))p FH(;)61 b(\020)6 b FG(\(0\))23 b(=)g(0)14 b FH(:)0 1424 y FI(Here)27 b FF(f)p FH(\014)285 1436 y FA(s)320 1424 y FH(;)14 b(s)23 b FF(2)g FE(Z)552 1394 y FD(2)552 1445 y(0)590 1424 y FF(g)k FI(are)g(standard)f(indep)r(enden)n(t)j(Wiener)e(pro)r(cesses,)f (and)1184 1623 y FH(B)1247 1635 y FD(1)1307 1623 y FG(:=)1418 1545 y Fz(X)1565 1623 y FF(j)p FH(s)p FF(j)1650 1589 y FD(2)1702 1623 y FH(b)1738 1589 y FD(2)1738 1644 y FA(s)1798 1623 y FH(<)c FF(1)14 b FH(:)0 1820 y FJ(Basic)32 b(inequalities)e(for)i(the)f(random)g(\034eld)d FH(\020)1627 1818 y(\020)1628 1820 y(\020)7 b FJ(.)36 b FI(The)28 b(follo)n(wing)e(relations)h(are)f(ob)n(vious:)714 2015 y Fk(E)14 b FF(j)p FH(\020)6 b FG(\()p FH(t)p FG(\))p FF(j)973 1981 y FD(2)1034 2015 y FG(=)1121 1936 y Fz(X)1165 2110 y FA(s)1255 2015 y FH(b)1291 1981 y FD(2)1291 2036 y FA(s)1342 2015 y Fk(E)14 b FH(\014)1470 1981 y FD(2)1466 2036 y FA(s)1507 2015 y FG(\()p FH(t)p FG(\))23 b(=)g FH(t)1770 1936 y Fz(X)1903 2015 y FH(b)1939 1981 y FD(2)1939 2036 y FA(s)1999 2015 y FG(=)g FH(tB)2180 2027 y FD(0)2231 2015 y FH(;)673 2239 y Fk(E)14 b FF(k)p FH(\020)6 b FG(\()p FH(t)p FG(\))p FF(k)970 2205 y FD(2)970 2260 y FA(k)1034 2239 y FG(=)1121 2160 y Fz(X)1165 2335 y FA(s)1255 2239 y FH(b)1291 2205 y FD(2)1291 2260 y FA(s)1342 2239 y Fk(E)14 b FH(\014)1470 2205 y FD(2)1466 2260 y FA(s)1507 2239 y FG(\()p FH(t)p FG(\))g FF(k)p FH(e)1696 2251 y FA(s)1731 2239 y FF(k)1773 2205 y FD(2)1773 2260 y FA(k)1836 2239 y FG(=)23 b FH(t)1982 2160 y Fz(X)2026 2335 y FA(s)2115 2239 y FH(b)2151 2205 y FD(2)2151 2260 y FA(s)2202 2239 y FF(j)p FH(s)p FF(j)2287 2205 y FD(2)p FA(k)2384 2239 y FG(=)g FH(t)14 b(B)2579 2251 y FA(k)2633 2239 y FH(;)533 2504 y Fk(E)g FF(j)p FH(\020)6 b FF(j)698 2470 y FD(2)698 2527 y FA(L)744 2535 y Ft(2)776 2527 y FD(\(0)p FA(;T)j FD(;)p FA(H)t FD(\))1034 2504 y FG(=)22 b Fk(E)1212 2391 y Fz(Z)1295 2412 y FA(t)1258 2580 y FD(0)1351 2504 y FF(j)p FH(\020)6 b FG(\()p FH(s)p FG(\))p FF(j)1542 2470 y FD(2)1581 2504 y FH(ds)23 b FG(=)1784 2448 y(1)p 1784 2485 42 4 v 1784 2561 a(2)1858 2504 y FH(t)1888 2470 y FD(2)1925 2504 y FH(B)1988 2516 y FD(0)2039 2504 y FH(;)0 2728 y FI(where)k(w)n(e)g(denote)1189 2828 y FH(B)1252 2840 y FA(k)1316 2828 y FG(=)1404 2749 y Fz(X)1448 2924 y FA(s)1537 2828 y FF(j)p FH(s)p FF(j)1622 2794 y FD(2)p FA(k)1710 2828 y FH(b)1746 2794 y FD(2)1746 2849 y FA(s)1806 2828 y FF(\024)c(1)p FH(:)0 3052 y FI(The)28 b(Wiener)f(pro)r(cesses)g FH(\014)866 3064 y FA(s)901 3052 y FG(\()p FH(t)p FG(\))h FI(are)f(a.s.)37 b(con)n(tin)n(uous.)f(The)28 b(pro)r(cess)e FH(\020)6 b FG(\()p FH(t)p FG(\))29 b FI(also)e(is)g(a.s.)37 b(con)n(tin)n(uous)0 3152 y(in)28 b(the)g(corresp)r(onding)d(spaces:)0 3318 y FJ(Lemma)i(4.1.)39 b FC(Assume)27 b(that)h FH(B)1064 3330 y FA(k)1128 3318 y FH(<)23 b FF(1)p FC(.)38 b(Then)29 b(the)f(pr)l(o)l(c)l(ess)g FH(\020)6 b FG(\()p FH(t)p FG(\))p FC(,)30 b FG(0)23 b FF(\024)g FH(t)g(<)f FF(1)p FC(,)29 b(is)g(c)l(ontinuous)e(in)0 3418 y FH(H)76 3387 y FA(k)146 3418 y FC(a.s.,)32 b(and)1040 3517 y Fk(E)47 b FG(max)1130 3571 y FD(0)p Fw(\024)p FA(s)p Fw(\024)p FA(t)1337 3517 y FF(k)p FH(\020)6 b FG(\()p FH(s)p FG(\))p FF(k)1566 3477 y FA(p)1566 3542 y(k)1630 3517 y FF(\024)23 b FH(t)1748 3483 y FA(p)1800 3517 y FH(C)6 b FG(\()p FH(p;)14 b(B)2039 3529 y FA(k)2080 3517 y FG(\))g FH(;)869 b FI(\(4.1\))0 3705 y FC(for)31 b(e)l(ach)f FH(p)23 b(>)g FG(1)p FC(.)0 3871 y(Pr)l(o)l(of.)43 b FI(Set)32 b FH(\020)443 3883 y FA(N)507 3871 y FG(\()p FH(t)p FG(\))e(=)726 3809 y Fz(P)813 3896 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1013 3871 y FH(b)1049 3883 y FA(s)1098 3871 y FH(\014)1145 3883 y FA(s)1180 3871 y FG(\()p FH(t)p FG(\))14 b FH(e)1327 3883 y FA(s)1363 3871 y FI(.)49 b(Clearly)-7 b(,)32 b FH(\020)1781 3883 y FA(N)1845 3871 y FG(\()p FF(\001)p FG(\))e FF(2)h FH(C)6 b FG(\([0)p FH(;)14 b(T)e FG(];)i FH(H)2444 3841 y FA(k)2483 3871 y FG(\))32 b FI(a.s.,)h(for)e(an)n(y)g (\034xed)0 3981 y(constan)n(t)c FH(T)12 b FI(.)36 b(By)27 b(the)h(Do)r(ob)f(inequalit)n(y)h([IW89)o(,)g(Kry03)m(,)g(\330k03)o(],) f(for)g FH(M)32 b FF(\025)23 b FH(N)36 b FI(w)n(e)27 b(ha)n(v)n(e)373 4165 y Fk(E)58 b FG(max)463 4219 y FD(0)p Fw(\024)p FA(s)p Fw(\024)p FA(T)693 4165 y FF(k)p FG(\()p FH(\020)803 4177 y FA(M)896 4165 y FF(\000)18 b FH(\020)1015 4177 y FA(N)1078 4165 y FG(\)\()p FH(s)p FG(\))p FF(k)1255 4131 y FD(2)1255 4186 y FA(k)1319 4165 y FF(\024)23 b FG(4)14 b Fk(E)g FF(k)p FG(\()p FH(\020)1650 4177 y FA(M)1741 4165 y FF(\000)k FH(\020)1860 4177 y FA(N)1923 4165 y FG(\)\()p FH(T)12 b FG(\))p FF(k)2122 4131 y FD(2)2122 4186 y FA(k)2186 4165 y FG(=)1407 4358 y(4)p FH(T)1627 4279 y Fz(X)1536 4461 y FA(M)6 b Fw(\025j)p FA(s)p Fw(j)p FA(>N)1853 4358 y FH(b)1889 4324 y FD(2)1889 4379 y FA(s)1939 4358 y FF(j)p FH(s)p FF(j)2024 4324 y FD(2)p FA(k)2121 4358 y FF(\024)23 b FG(4)p FH(T)2369 4279 y Fz(X)2338 4461 y Fw(j)p FA(s)p Fw(j)p FA(>N)2534 4358 y FH(b)2570 4324 y FD(2)2570 4379 y FA(s)2620 4358 y FF(j)p FH(s)p FF(j)2705 4324 y FD(2)p FA(k)2793 4358 y FH(:)0 4646 y FI(Since)j FH(B)278 4658 y FA(k)341 4646 y FH(<)d FF(1)p FI(,)j(then)g(a.s.)36 b FF(f)p FH(\020)983 4658 y FA(N)1045 4646 y FG(\()p FF(\001)p FG(\))p FF(g)26 b FI(is)f(a)g(Cauc)n(h)n (y-sequence)f(in)h FH(C)6 b FG(\([0)p FH(;)14 b(T)e FG(];)i FH(H)2478 4616 y FA(k)2518 4646 y FG(\))p FI(.)37 b(Hence,)26 b(the)f(limit)0 4746 y FH(\020)6 b FG(\()p FH(t)p FG(\))24 b(=)248 4683 y Fz(P)349 4746 y FH(b)385 4758 y FA(s)434 4746 y FH(\014)481 4758 y FA(s)517 4746 y FG(\()p FH(t)p FG(\))14 b FH(e)664 4758 y FA(s)727 4746 y FI(exists)27 b(and)g(b)r(elongs)g(to)h FH(C)6 b FG(\([0)p FH(;)14 b(T)e FG(];)i FH(H)1913 4715 y FA(k)1952 4746 y FG(\))p FI(.)1553 5059 y(35)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 36 35 bop 125 523 a FI(The)27 b(estimate)h(\(4.1\))f (follo)n(ws)f(from)i(the)g(Do)r(ob)f(inequalit)n(y)865 751 y Fk(E)46 b FG(max)955 804 y FD(0)p Fw(\024)p FA(s)p Fw(\024)p FA(t)1162 751 y FF(k)p FH(\020)6 b FG(\()p FH(s)p FG(\))p FF(k)1391 711 y FA(p)1391 776 y(k)1455 751 y FF(\024)1543 634 y Fz(\022)1685 695 y FH(p)p 1614 732 185 4 v 1614 808 a(p)18 b FF(\000)g FG(1)1808 634 y Fz(\023)1870 651 y FA(p)1936 751 y Fk(E)c FF(k)p FH(\020)6 b FG(\()p FH(t)p FG(\))p FF(k)2233 716 y FA(P)2233 771 y(k)2301 751 y FH(;)0 977 y FI(since)27 b Fk(E)p FF(k)p FH(\020)6 b FG(\()p FH(t)p FG(\))p FF(k)486 937 y FA(p)486 1002 y(k)550 977 y FF(\024)23 b FH(t)668 946 y FA(p)706 977 y FH(C)6 b FG(\()p FH(p;)14 b(B)945 989 y FA(k)986 977 y FG(\))28 b FI(\(sa)n(y)-7 b(,)27 b(b)n(y)g(the)h(F)-7 b(ernique)27 b(theorem,)g(see)h([Bog98)m(]\).)p 3128 977 4 57 v 3132 924 50 4 v 3132 977 V 3182 977 4 57 v 125 1141 a(It)f(is)h(con)n(v)n(enien)n(t)e(to)i(rede\034ne)f FH(\020)34 b FI(on)27 b(a)h(n)n(ull-set)f(to)g(ac)n(hiev)n(e)789 1314 y FH(\020)831 1280 y FA(!)879 1314 y FG(\()p FF(\001)p FG(\))d FF(2)f FH(C)6 b FG(\([0)p FH(;)14 b FF(1)p FG(\))p FH(;)g(H)1495 1280 y FA(k)1536 1314 y FG(\))46 b FI(for)28 b(all)f FH(!)s FI(,)g(if)h FH(B)2101 1326 y FA(k)2165 1314 y FH(<)23 b FF(1)41 b FH(:)618 b FI(\(4.2\))0 1633 y FJ(Homogeneit)m(y)-8 b(.)32 b FI(The)24 b(random)e(\034eld)i FH(\020)6 b FG(\()p FH(x;)14 b(t)p FG(\))25 b FI(is)e(called)g FC(homo)l(gene)l(ous)h FI(\(in)g(the)g(space-v)-5 b(ariable)21 b FH(x)p FI(\),)0 1733 y(if)28 b(for)f(eac)n(h)g FH(\030)g FF(2)c FE(T)586 1703 y FD(2)652 1733 y FI(the)28 b(\034eld)g FH(T)1024 1745 y FA(\030)1059 1733 y FH(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))25 b(=)d FH(\020)6 b FG(\()p FH(t;)14 b(x)20 b FG(+)e FH(\030)t FG(\))28 b FI(has)f(the)h(same)f (distribution)h(as)e FH(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))p FI(.)0 1892 y FJ(Exercise)31 b(4.2.)40 b FI(Sho)n(w)27 b(that)h FH(\020)6 b FG(\()p FH(t;)14 b(s)p FG(\))29 b FI(is)e(homogeneous)f(in)i FH(x)g FI(if)g(and)f(only)g(if)1170 2065 y FH(b)1206 2077 y FA(s)1264 2065 y FG(=)c FH(b)1388 2077 y Fw(\000)p FA(s)1521 2065 y FI(for)k(all)g FH(s)c FF(2)g FE(Z)1958 2035 y FD(2)1958 2086 y(0)1996 2065 y FH(:)0 2296 y Fy(4.2)112 b(The)38 b(white-forced)f(2D)g(NSE)0 2449 y FI(The)28 b FC(white-for)l(c)l(e)l(d)j(NSE)c FI(is)1104 2587 y FG(_)-37 b FH(u)18 b FG(+)g FH(Lu)f FG(+)h FH(B)t FG(\()p FH(u)p FG(\))23 b(=)1759 2531 y FH(d)p 1744 2568 74 4 v 1744 2644 a(dt)1841 2587 y(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)920 b FI(\(4.3\))0 2759 y(where)27 b FH(\020)34 b FI(is)28 b(the)g(pro)r(cess)e(as)h(in)h(Section)f(4.1.) 36 b(W)-7 b(e)28 b(supplemen)n(t)g(\(4.3\))f(with)h(an)g(initial)f (condition:)1313 2933 y FH(u)p FG(\(0\))c(=)f FH(u)1625 2945 y FD(0)1685 2933 y FF(2)h FH(H)e(:)1142 b FI(\(4.4\))0 3106 y(Due)28 b(to)g(\(4.2\),)f(Theorem)g(2.17)f(and)h(Example)h(2.20)e (applied)i(with)g(an)n(y)e FH(T)35 b(<)22 b FF(1)p FI(,)28 b(and)f(w)n(e)g(obtain:)0 3265 y FJ(Theorem)j(4.3.)40 b FC(The)31 b(pr)l(oblem)g(has)f(a)g(unique)g(solution)g(such)f(that) 729 3438 y FH(u)22 b FF(2)i FH(C)6 b FG(\([0)p FH(;)14 b FF(1)p FG(\))p FH(;)g(H)7 b FG(\))18 b FF(\\)h FH(L)1486 3450 y FD(2)p FA(;)11 b FD(lo)r(c)1637 3438 y FG(\([0)p FH(;)j FF(1)p FG(\))p FH(;)g(H)1999 3404 y FD(1)2036 3438 y FG(\))47 b FC(for)30 b(al)t(l)h FH(!)17 b FC(.)558 b FI(\(4.5\))0 3611 y FC(Mor)l(e)l(over)31 b(for)g(any)f FH(T)k(>)22 b FG(0)30 b FC(and)g(any)g FH(!)s FC(,)1082 3785 y FH(u)1130 3797 y FA(N)1244 3785 y FF(\000)-14 b(!)1215 3838 y FA(N)6 b Fw(!1)1453 3785 y FH(u)36 b FG(in)h FH(C)6 b FG(\([0)p FH(;)14 b(T)e FG(])p FH(;)i(H)7 b FG(\))14 b FH(;)910 b FI(\(4.6\))0 3991 y FC(wher)l(e)30 b FH(u)282 4003 y FA(N)345 3991 y FG(\()p FH(t)p FG(\))g FC(is)g(a)g(Galerkin)h(appr)l(oximation.)41 b(Besides)992 4164 y FH(u)p FG(\()p FH(t)p FG(\))23 b(=)g FH(F)12 b FG(\()p FH(t;)i(u)1457 4176 y FD(0)1493 4164 y FH(;)g FF(f)p FH(\020)6 b FG(\()p FH(s)p FG(\))p FH(;)14 b FG(0)23 b FF(\024)g FH(s)g FF(\024)f FH(t)p FF(g)p FG(\))14 b FH(:)821 b FI(\(4.7\))0 4338 y FJ(Ampli\034cation.)34 b FI(Same)28 b(is)f(true)h(if)g FH(u)1232 4350 y FD(0)1292 4338 y FF(2)23 b FH(H)34 b FI(is)28 b(a)f(random)g(v)-5 b(ariable.)125 4437 y(Belo)n(w)26 b(either)h FH(u)653 4407 y FD(0)717 4437 y FI(is)h(non-random,)d(or)i FH(u)1439 4407 y FD(0)1499 4437 y FF(2)c FH(H)34 b FI(is)28 b(a)f(random)f(v)-5 b(ariable,)26 b(indep)r(enden)n(t)j(of)e FH(\020)6 b FG(\()p FF(\001)p FG(\))p FI(.)0 4537 y FJ(Measurabilit)m(y)-8 b(,)33 b(measures)f(and)i(w)m(eak)g(solutions.)39 b FI(In)29 b FG(\(\012)p FH(;)14 b FF(F)8 b FH(;)14 b Fk(P)p FG(\))p FI(,)31 b(w)n(e)d(consider)g(the)i(\034ltra-)0 4637 y(tion)e FF(F)231 4649 y FA(t)283 4637 y FF(\032)22 b(F)8 b FH(;)14 b(t)23 b FF(\025)g FG(0)p FI(,)k(generated)f(b)n(y)i(all)f(zero-sets)f (and)h(b)n(y)g(the)h(random)f(v)-5 b(ariables)1312 4810 y FH(\020)6 b FG(\()p FH(s)p FG(\))p FH(;)28 b FG(0)23 b FF(\024)g FH(s)g FF(\024)f FH(t)14 b(:)1553 5059 y FI(36)p eop end PStoPSsaved restore %%Page: (36,37) 19 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 37 36 bop 0 523 a FI(A)32 b(random)e(pro)r(cess)g FH(y)s FG(\()p FH(t)p FG(\))g(=)f FH(y)1001 493 y FA(!)1048 523 y FG(\()p FH(t)p FG(\))p FI(,)k FH(t)d FF(\025)f FG(0)p FI(,)j(is)f(called)g FF(F)1835 535 y FA(t)1864 523 y FC(-adapte)l(d)h FI(if)g FH(y)s FG(\()p FH(t)p FG(\))g FI(is)f FF(F)2595 535 y FA(t)2624 523 y FI(-measurable)f(for)0 623 y(all)d FH(t)c FF(\025)g FG(0)p FI(.)125 722 y(Due)28 b(to)f(\(4.7\),)g(the)h(random)f(pro)r(cess)f FH(u)p FG(\()p FH(t)p FG(\))p FI(,)i(in)n(terpreted)f(as)g(a)g(random)g(v)-5 b(ariable)1068 867 y FG(\012)23 b FF(3)h FH(!)h FF(7\000)-14 b(!)23 b FH(u)1512 833 y FA(!)1583 867 y FF(2)g FH(C)6 b FG(\([0)p FH(;)14 b FF(1)p FG(\))p FH(;)g(H)7 b FG(\))0 1012 y FI(has)27 b(the)h(form)1216 1111 y FH(!)e FF(7\000)-14 b(!)23 b FH(\020)1493 1077 y FA(!)1564 1111 y FF(7\000)-14 b(!)23 b FG(\010\()p FH(\020)1855 1077 y FA(!)1904 1111 y FG(\))14 b FH(;)1045 b FI(\(4.8\))0 1238 y(where)655 1363 y FG(\010)24 b(:)f FH(C)6 b FG(\([0)p FH(;)14 b FF(1)p FG(\))p FH(;)g(H)1212 1328 y FD(1)1249 1363 y FG(\))23 b FF(!)g FH(C)6 b FG(\([0)p FH(;)14 b FF(1)p FG(\))p FH(;)g(H)7 b FG(\))14 b FH(;)655 1496 y FG(\010)715 1429 y Fz(\000)754 1496 y FH(y)s FG(\()p FF(\001)p FG(\))885 1429 y Fz(\001)946 1496 y FG(=)23 b FH(Y)18 b FG(\()p FF(\001)p FG(\))p FH(;)98 b(Y)18 b FG(\()p FH(t)p FG(\))24 b(=)f FH(F)12 b FG(\()p FH(t;)i(u)1792 1508 y FD(0)1829 1496 y FH(;)g FF(f)p FH(y)s FG(\()p FH(s)p FG(\))p FH(;)g FG(0)22 b FF(\024)g FH(s)h FF(\024)g FH(t)p FF(g)p FG(\))14 b FH(:)0 1642 y FI(If)38 b(w)n(e)f(consider)g(the)h(pro)r(cess)e FH(u)1059 1612 y FA(!)1145 1642 y FI(for)h FH(t)j FF(2)g FG([0)p FH(;)14 b(T)e FG(])p FI(,)39 b(then)f(the)g(corresp)r(onding)e (random)g(v)-5 b(ariable)0 1742 y FH(!)26 b FF(7\000)-15 b(!)24 b FH(u)283 1711 y FA(!)330 1742 y FF(j)353 1757 y FD([0)p FA(;T)9 b FD(])524 1742 y FI(has)27 b(the)h(form)1010 1886 y FH(!)e FF(7\000)-14 b(!)23 b FH(\020)1287 1852 y FA(!)1349 1886 y FF(j)1372 1901 y FD([0)p FA(;T)9 b FD(])1538 1886 y FF(7\000)-14 b(!)23 b FG(\010)1755 1898 y FA(T)1807 1886 y FG(\()p FH(\020)1881 1852 y FA(!)1944 1886 y FF(j)1967 1901 y FD([0)p FA(;T)9 b FD(])2109 1886 y FG(\))14 b FH(;)840 b FI(\(4.9\))0 2031 y(where)1021 2130 y FG(\010)1081 2142 y FA(T)1156 2130 y FG(:)24 b FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1553 2096 y FD(1)1589 2130 y FG(\))23 b FF(!)g FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))14 b FH(:)0 2257 y FI(The)28 b(map)f FG(\010)h FI(and)f(the)h(maps)f FG(\010)1024 2269 y FA(T)1104 2257 y FI(agree:)1148 2402 y FG(\010)1208 2414 y FA(T)1260 2402 y FG(\()p FH(\020)21 b FF(j)1372 2417 y FD([0)p FA(;T)9 b FD(])1515 2402 y FG(\))23 b(=)g(\010\()p FH(\020)6 b FG(\))14 b FF(j)1861 2417 y FD([0)p FA(;T)9 b FD(])2018 2402 y FH(:)0 2546 y FI(By)27 b(Prop)r(osition)f(2.18,)g (the)i(maps)f FG(\010)1194 2558 y FA(T)1273 2546 y FI(are)f(lo)r(cally) h(Lipsc)n(hitz.)36 b(In)27 b(particular,)f(they)i(are)e(con)n(tin)n(u-) 0 2646 y(ous.)36 b(A)n(ccordingly)-7 b(,)26 b(the)i(map)g FG(\010)g FI(is)f(also)f(con)n(tin)n(uous.)125 2745 y(By)c(Theorem)g (4.3,)h FH(u)797 2715 y FA(!)845 2745 y FG(\()p FH(t)p FG(\))g FI(is)g(a)f(random)g(pro)r(cess)g(with)h(con)n(tin)n(uous)f (tra)5 b(jectories.)33 b(Due)24 b(to)e(\(4.9\))1246 2890 y FH(u)p FG(\()p FH(t)p FG(\))28 b FI(is)f FF(F)1559 2902 y FA(t)1588 2890 y FI(-adapted)14 b(.)1033 b(\(4.10\))0 3035 y(F)-7 b(or)36 b(an)n(y)f FH(t)353 3047 y FD(0)428 3035 y FF(\025)i FG(0)p FI(,)h(let)e(us)g(denote)g FH(u)1200 3005 y FA(t)1225 3013 y Ft(0)1261 3035 y FG(\()p FH(t)p FG(\))i(=)f FH(u)p FG(\()p FH(t)1605 3047 y FD(0)1667 3035 y FG(+)23 b FH(t)p FG(\))p FI(.)63 b(Then)37 b FH(u)2177 3005 y FA(t)2202 3013 y Ft(0)2274 3035 y FI(satis\034es)f(\(4.3\),)i (\(4.4\))e(with)0 3134 y FH(u)48 3146 y FD(0)108 3134 y FG(=)22 b FH(u)p FG(\()p FH(t)305 3146 y FD(0)343 3134 y FG(\))27 b FI(and)h FH(\020)6 b FG(\()p FH(t)p FG(\))29 b FI(replaced)d(b)n(y)h FH(\020)6 b FG(\()p FH(t)20 b FG(+)e FH(t)1408 3146 y FD(0)1445 3134 y FG(\))h FF(\000)f FH(\020)6 b FG(\()p FH(t)1683 3146 y FD(0)1721 3134 y FG(\))p FI(.)37 b(Applying)27 b(\(4.9\))h(to)f FH(u)2520 3104 y FA(t)2545 3112 y Ft(0)2609 3134 y FI(w)n(e)g(see)g(that)470 3279 y FH(u)p FG(\()p FH(t)p FG(\))14 b FH(;)37 b(t)23 b FF(\025)g FH(t)857 3291 y FD(0)908 3279 y FH(;)37 b FI(is)27 b(a)g(function)h(of)g FH(u)1588 3291 y FA(t)1613 3299 y Ft(0)1677 3279 y FI(and)f FH(\020)6 b FG(\()p FH(s)p FG(\))20 b FF(\000)e FH(\020)6 b FG(\()p FH(t)2190 3291 y FD(0)2228 3279 y FG(\))p FI(,)41 b FH(t)2354 3291 y FD(0)2415 3279 y FF(\024)22 b FH(s)h FF(\024)g FH(t)14 b(:)258 b FI(\(4.11\))125 3424 y(The)27 b(prop)r(erties)g(\(4.10\),)g (\(4.11\))f(imply)0 3559 y FJ(Prop)s(osition)36 b(4.4.)42 b FC(The)35 b(solution)g FH(u)p FG(\()p FH(t)p FG(\))f FC(is)g(an)h FF(F)1660 3571 y FA(t)1689 3559 y FC(-adapte)l(d)g(Markov) h(pr)l(o)l(c)l(ess)e(with)h(c)l(ontinuous)0 3659 y(tr)l(aje)l(ctories.) 125 3794 y FI(W)-7 b(e)27 b(omit)h(details)f(of)h(the)g(pro)r(of.)125 3894 y(Let)i FH(dW)39 b FG(=)26 b FF(D)r FG(\()p FH(\020)6 b FG(\))32 b FI(b)r(e)e(the)h(distribution)f(of)g FH(\020)36 b FI(in)30 b FH(C)6 b FG(\([0)p FH(;)14 b FF(1)p FG(\))p FH(;)g(H)2142 3864 y FD(1)2180 3894 y FG(\))30 b FI(\(i.e.,)h(the)f (Wiener)g(measure,)0 3994 y(corresp)r(onding)c(to)h(the)h(pro)r(cess)e FH(\020)6 b FI(\).)38 b(Then,)28 b(b)n(y)f(\(4.8\),)1214 4138 y FF(D)1280 4071 y Fz(\000)1318 4138 y FH(u)p FG(\()p FF(\001)p FG(\))1453 4071 y Fz(\001)1515 4138 y FG(=)22 b(\010)d FF(\016)f FG(\()p FH(dW)12 b FG(\))i FH(:)1002 b FI(\(4.12\))0 4283 y(Here)27 b FG(\010)19 b FF(\016)f FG(\()p FH(dW)12 b FG(\))28 b FI(denotes)f(the)h(push-forw)n(ard)e(of)h FH(dW)40 b FI(b)n(y)27 b(the)h(map)g FG(\010)p FI(.)0 4418 y FJ(De\034nition)e(4.5.)36 b FC(A)26 b(pr)l(o)l(c)l(ess)j Fz(e)-49 b FH(u)p FG(\()p FH(t)p FG(\))26 b FC(as)g(in)33 b FI(\(4.5\))25 b FC(is)i(a)f(we)l(ak)h(solution)f(of)h(\(4.3\),)h (\(4.4\))f(if)g(it)f(satis\034es)0 4518 y(\(4.3\),)31 b(\(4.4\))g(with)f(the)g(pr)l(o)l(c)l(ess)g FH(\020)6 b FG(\()p FH(t)p FG(\))31 b FC(r)l(eplac)l(e)l(d)g(by)f(some)g(pr)l(o)l (c)l(ess)g FH(\020)2165 4488 y Fw(0)2189 4518 y FG(\()p FH(t)p FG(\))g FC(such)g(that)g FF(D)r FG(\()p FH(\020)2812 4488 y Fw(0)2836 4518 y FG(\))24 b(=)e FF(D)r FG(\()p FH(\020)6 b FG(\))p FC(.)0 4654 y FJ(Prop)s(osition)30 b(4.6.)40 b FC(The)31 b(distribution)f(of)h(a)f(we)l(ak)g(solution)g FF(D)r FG(\()s Fz(e)-49 b FH(u)p FG(\))30 b FC(is)g(uniquely)g (de\034ne)l(d.)0 4789 y(Pr)l(o)l(of.)43 b FI(The)28 b(measure)e FF(D)r FG(\()s Fz(e)-49 b FH(u)p FG(\))28 b FI(satis\034es)f(\(4.12\).) 36 b(So)27 b(it)h(is)g(indep)r(enden)n(t)g(of)g(the)g(c)n(hoice)e(of)31 b Fz(e)-49 b FH(u)o FI(.)p 3128 4789 4 57 v 3132 4737 50 4 v 3132 4789 V 3182 4789 4 57 v 1553 5059 a(37)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 38 37 bop 0 523 a Fy(4.3)112 b(Estimates)36 b(for)h(solutions)0 676 y FI(The)28 b FH(N)9 b FI(-th)27 b(Galerkin)g(appro)n(ximation)e(to)j(equation)f(\(4.3\))g(is)h(the)f (equation)542 977 y FG(_)-38 b FH(u)23 b FG(=)888 831 y FA(V)927 839 y Fu(N)980 831 y FD(\()p FA(u)p FD(\))685 890 y Fz(z)p 722 890 220 10 v 220 w(}|)p 1016 890 V 220 w({)685 977 y FF(\000)p FH(Lu)17 b FF(\000)h FG(\005)1017 989 y FA(N)1095 977 y FH(B)t FG(\()p FH(u)p FG(\))c(+)1377 921 y FH(d)p 1363 958 74 4 v 1363 1034 a(dt)1513 898 y Fz(X)1482 1080 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1691 977 y FH(b)1727 989 y FA(s)1776 977 y FH(\014)1823 989 y FA(s)1859 977 y FG(\()p FH(t)p FG(\))g FH(e)2006 989 y FA(s)2055 977 y FH(;)527 1174 y(u)23 b FF(2)g FH(H)745 1186 y FA(N)831 1174 y FG(=)g(span)o FF(f)p FH(e)1166 1186 y FA(s)1215 1103 y Fz(\014)1215 1153 y(\014)1256 1174 y FF(j)p FH(s)p FF(j)g(\024)g FH(N)9 b FF(g)p FG(;)36 b FH(u)p FG(\(0\))23 b(=)g FH(u)1942 1186 y FD(0)p FA(;N)2080 1174 y FG(=)g(\005)2230 1186 y FA(N)2307 1174 y FH(u)2355 1186 y FD(0)2415 1174 y FF(2)g FH(H)2562 1186 y FA(N)2639 1174 y FH(:)2977 1012 y FI(\(4.13\))0 1353 y(W)-7 b(e)28 b(assume)f(that)h FH(u)658 1365 y FD(0)717 1353 y FG(=)23 b FH(u)853 1323 y FA(!)853 1374 y FD(0)928 1353 y FI(is)28 b(indep)r(enden)n(t)g(of)g FH(\020)6 b FG(\()p FH(t)p FG(\))p FI(,)28 b FH(t)23 b FF(\025)g FG(0)p FI(,)k(and)h Fk(E)14 b FF(j)p FH(u)2307 1365 y FD(0)2343 1353 y FF(j)2366 1323 y FD(2)2426 1353 y FH(<)23 b FF(1)p FI(.)125 1453 y(F)-7 b(or)39 b(the)i(same)e(reasons)f(as)i(in)g(Section)g(4.2,)j (this)d(equation)g(has)f(a)h(solution)g FH(u)j FG(=)h FH(u)3009 1465 y FA(N)3071 1453 y FG(\()p FH(t)p FG(\))p FI(.)0 1553 y(This)30 b(random)g(pro)r(cess)f(also)h(is)g(a)g(solution) g(of)g(\(4.13\),)h(understo)r(o)r(d)f(as)g(a)g(sto)r(c)n(hastic)f (di\033eren)n(tial)0 1652 y(equation.)36 b(That)28 b(is,)1022 1764 y FH(du)23 b FG(=)g FH(V)1272 1776 y FA(N)1335 1764 y FG(\()p FH(u)p FG(\))14 b FH(dt)19 b FG(+)1667 1685 y Fz(X)1636 1867 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1831 1764 y FH(b)1867 1776 y FA(s)1916 1764 y FH(e)1955 1776 y FA(s)2004 1764 y FH(d\014)2094 1776 y FA(s)2144 1764 y FH(:)0 2005 y FI(The)29 b(solutions)f FH(u)568 2017 y FA(N)631 2005 y FG(\()p FH(t)p FG(\))h FI(exists)g(till)h(a)e(blo)n (w\025up)g(time)i FH(T)1769 1975 y FA(!)1841 2005 y FF(\024)25 b(1)p FI(.)42 b(It)29 b(is)g(a)f(classical)g(result)h(that)g(due)0 2104 y(to)d(a-priori)e(estimate)i(\(4.16\))g(b)r(elo)n(w)f FH(T)1258 2074 y FA(!)1328 2104 y FG(=)e FF(1)j FI(a.e.)36 b(\(this)27 b(happ)r(ens)f(b)n(y)g(a)g(reason,)e(similar)i(to)g(that)0 2204 y(in)i(Section)f(2.3\),)g(see)h([Kha80)n(,)g(IW89)o(,)g(Kry03)m(,) g(\330k03)o(].)125 2304 y(Let)d FH(f)9 b FG(\()p FH(t;)14 b(u)p FG(\))26 b FI(b)r(e)g(a)f FH(C)769 2273 y FD(2)807 2304 y FI(-smo)r(oth)g(function)h(on)g FE(R)14 b FF(\002)h FH(H)1786 2316 y FA(N)1849 2304 y FI(.)36 b(Then)26 b(the)g FC(Ito)i(formula)f FI(applies)e(to)h(the)0 2403 y(pro)r(cess)g FH(f)32 b FG(=)23 b FH(f)9 b FG(\()p FH(t;)14 b(u)p FG(\()p FH(t)p FG(\)\))27 b FI(\(see)h([Kha80)n(,)g(IW89)o(,)g(Kry03)n(,)f (\330k03)o(]\),)h(and)f(w)n(e)g(ha)n(v)n(e)629 2564 y FH(d)p 614 2601 V 614 2677 a(dt)721 2620 y(f)9 b FG(\()p FH(t;)14 b(u)p FG(\()p FH(t)p FG(\)\))23 b(=)f(\()p FH(@)1230 2632 y FA(t)1260 2620 y FH(f)9 b FG(\)\()p FH(t;)14 b(u)p FG(\()p FH(t)p FG(\)\))19 b(+)f FF(hr)1818 2632 y FA(u)1862 2620 y FH(f)t(;)28 b(V)2006 2632 y FA(N)2069 2620 y FG(\()p FH(u)p FG(\))p FF(i)1062 2834 y FG(+)1155 2778 y(1)p 1155 2815 42 4 v 1155 2891 a(2)1274 2755 y Fz(X)1244 2937 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1472 2778 y FH(@)1521 2748 y FD(2)1558 2778 y FH(f)p 1472 2815 136 4 v 1473 2891 a(@)5 b(u)1570 2867 y FD(2)1570 2912 y FA(s)1640 2834 y FH(b)1676 2800 y FD(2)1676 2855 y FA(s)1732 2834 y FG(+)1815 2742 y Fz(D)1866 2834 y FF(r)1935 2846 y FA(u)1978 2834 y FH(f)t(;)2091 2755 y Fz(X)2060 2937 y Fw(j)p FA(s)p Fw(j\024)p FA(N)2255 2834 y FH(b)2291 2846 y FA(s)2347 2812 y FG(_)2326 2834 y FH(\014)2373 2846 y FA(s)2423 2834 y FH(e)2462 2846 y FA(s)2497 2742 y Fz(E)2561 2834 y FH(:)2977 2754 y FI(\(4.14\))0 3110 y(Cho)r(ose)27 b FH(f)9 b FG(\()p FH(t;)14 b(u)p FG(\))22 b(=)h FF(j)p FH(u)p FF(j)722 3080 y FD(2)759 3110 y FI(.)37 b(Then)501 3271 y FH(d)p 486 3308 74 4 v 486 3384 a(dt)592 3327 y FF(j)p FH(u)p FF(j)686 3293 y FD(2)746 3327 y FG(=)23 b(2)14 b FF(h)p FH(u;)g FF(\000)p FH(Lu)i FF(\000)i FH(B)t FG(\()p FH(u)p FG(\))p FF(i)890 3382 y Fz(|)p 927 3382 225 10 v 225 w({z)p 1226 3382 V 225 w(})1013 3461 y Fj(B1)i(i\))1051 3500 y FD(=)37 b Fw(\000)p FD(2)p Fw(k)p FA(u)p Fw(k)1331 3477 y Ft(2)1331 3518 y(1)1502 3327 y FG(+)1581 3235 y Fz(\020)1674 3248 y(X)1643 3430 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1839 3327 y FH(b)1875 3293 y FD(2)1875 3347 y FA(s)1912 3235 y Fz(\021)1581 3488 y(|)p 1618 3488 116 10 v 116 w({z)p 1808 3488 V 116 w(})1705 3566 y FA(B)1755 3574 y Ft(0)p Fu(N)1998 3327 y FG(+)23 b(2)2128 3235 y Fz(D)2178 3327 y FH(u;)2293 3248 y Fz(X)2263 3430 y Fw(j)p FA(s)p Fw(j\024)p FA(N)2458 3327 y FH(b)2494 3339 y FA(s)2564 3305 y FG(_)2543 3327 y FH(\014)2590 3339 y FA(s)2625 3235 y Fz(E)2690 3327 y FH(:)264 b FI(\(4.15\))0 3722 y(By)27 b(taking)g(exp)r(ectations,)g (w)n(e)g(\034nd:)752 3878 y FH(d)p 737 3915 74 4 v 737 3991 a(dt)843 3935 y Fk(E)14 b FF(j)p FH(u)p FF(j)1014 3900 y FD(2)1073 3935 y FG(=)23 b FF(\000)p FG(2)14 b Fk(E)g FF(k)p FH(u)p FF(k)1491 3900 y FD(2)1491 3955 y(1)1544 3935 y FG(+)k FH(B)1690 3947 y FD(0)p FA(N)1809 3935 y FF(\024)k(\000)p FG(2)14 b Fk(E)g FF(j)p FH(u)p FF(j)2188 3900 y FD(2)2242 3935 y FG(+)k FH(B)2388 3947 y FD(0)2439 3935 y FH(:)0 4131 y FI(Therefore,)26 b(b)n(y)i(Gron)n(w)n (all's)d(lemma,)287 4339 y Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)552 4305 y FD(2)612 4339 y FF(\024)22 b(\000)774 4283 y FG(1)p 774 4320 42 4 v 774 4396 a(2)825 4339 y FH(e)864 4305 y Fw(\000)p FD(2)p FA(t)992 4339 y FH(B)1055 4351 y FD(0)1111 4339 y FG(+)1204 4283 y(1)p 1204 4320 V 1204 4396 a(2)1255 4339 y FH(B)1318 4351 y FD(0)1374 4339 y FG(+)c FH(e)1496 4305 y Fw(\000)p FD(2)p FA(t)1623 4339 y Fk(E)c FF(j)p FH(u)1771 4351 y FD(0)1808 4339 y FF(j)1831 4305 y FD(2)1891 4339 y FF(\024)1989 4283 y FG(1)p 1989 4320 V 1989 4396 a(2)2040 4339 y FH(B)2103 4351 y FD(0)2159 4339 y FG(+)k FH(e)2281 4305 y Fw(\000)p FD(2)p FA(t)2409 4339 y Fk(E)c FF(j)p FH(u)2557 4351 y FD(0)2593 4339 y FF(j)2616 4305 y FD(2)2667 4339 y FH(:)287 b FI(\(4.16\))125 4541 y(In)n(tegrating)26 b(\(4.15\))g(w)n(e)i(see)f(that)692 4776 y FF(j)p FH(u)p FG(\()p FH(T)12 b FG(\))p FF(j)911 4742 y FD(2)966 4776 y FG(+)18 b(2)1118 4663 y Fz(Z)1201 4683 y FA(T)1164 4852 y FD(0)1281 4776 y FF(k)p FH(u)p FF(k)1413 4742 y FD(2)1413 4796 y(1)1463 4776 y FH(ds)23 b FG(=)g FF(j)p FH(u)1727 4788 y FD(0)1763 4776 y FF(j)1786 4742 y FD(2)1842 4776 y FG(+)18 b FH(B)1988 4788 y FD(0)p FA(N)2098 4776 y FH(T)30 b FG(+)18 b FH(M)9 b FG(\()p FH(T)j FG(\))p FH(;)479 b FI(\(4.17\))1553 5059 y(38)p eop end PStoPSsaved restore %%Page: (38,39) 20 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 39 38 bop 0 568 a FI(where)30 b FH(M)9 b FG(\()p FH(t)p FG(\))27 b(=)g(2)602 455 y Fz(Z)684 476 y FA(t)648 644 y FD(0)741 476 y Fz(D)792 568 y FH(u)p FG(\()p FH(\034)9 b FG(\))p FH(;)986 489 y Fz(X)1031 664 y FA(s)1121 568 y FH(b)1157 580 y FA(s)1206 568 y FH(d)14 b(\014)1310 580 y FA(s)1345 568 y FG(\()p FH(\034)9 b FG(\))1454 476 y Fz(E)1550 568 y FI(is)30 b(a)g(martingale.)43 b(T)-7 b(aking)30 b(the)g(exp)r(ectations,)g(w)n(e)0 737 y(obtain)894 878 y Fk(E)985 765 y Fz(Z)1068 786 y FA(T)1031 954 y FD(0)1148 878 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)1383 844 y FD(2)1383 899 y(1)1419 878 y FH(ds)23 b FF(\024)1622 822 y FG(1)p 1622 859 42 4 v 1622 935 a(2)1673 878 y(\()p FH(T)i(B)1842 890 y FD(0)1898 878 y FG(+)18 b Fk(E)c FF(j)p FH(u)2129 890 y FD(0)2165 878 y FF(j)2188 844 y FD(2)2226 878 y FG(\))g FH(:)682 b FI(\(4.18\))0 1068 y(Due)28 b(to)g(\(4.17\))e(with)i FH(T)39 b FI(replaced)27 b(b)n(y)g FH(t)c FF(\024)g FH(T)12 b FI(,)862 1241 y FG(sup)819 1311 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)1057 1241 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1245 1207 y FD(2)1306 1241 y FF(\024)22 b(j)p FH(u)1464 1253 y FD(0)1501 1241 y FF(j)1524 1207 y FD(2)1580 1241 y FG(+)c FH(B)1726 1253 y FD(0)1763 1241 y FH(T)30 b FG(+)60 b(sup)1925 1311 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)2149 1241 y FH(M)9 b FG(\()p FH(t)p FG(\))14 b FH(:)607 b FI(\(4.19\))0 1474 y(The)28 b(Burkholder-Da)n (vis-Gundy)c(inequalit)n(y)k(\(with)g FH(p)23 b FG(=)f(1\))28 b FI(states)f(that:)752 1663 y Fk(E)70 b FG(sup)843 1732 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)1067 1663 y FF(j)p FH(M)9 b FG(\()p FH(t)p FG(\))p FF(j)23 b(\024)g FH(N)1475 1675 y FD(1)1525 1663 y Fk(E)p FF(h)p FH(M)9 b FF(i)1742 1619 y FD(1)p FA(=)p FD(2)1742 1687 y FA(T)1870 1663 y FF(\024)22 b FH(N)2024 1675 y FD(1)2061 1588 y Fz(p)p 2144 1588 270 4 v 75 x Fk(E)p FF(h)p FH(M)9 b FF(i)2361 1675 y FA(T)2414 1663 y FH(;)540 b FI(\(4.20\))0 1965 y(where)27 b FF(h)p FH(M)9 b FF(i)394 1977 y FA(T)470 1965 y FG(=)557 1852 y Fz(Z)640 1873 y FA(T)603 2041 y FD(0)706 1873 y Fz(\020)770 1886 y(X)904 1965 y FH(u)952 1931 y FD(2)952 1986 y FA(s)988 1965 y FG(\()p FH(\034)g FG(\))p FH(b)1133 1931 y FD(2)1133 1986 y FA(s)1171 1873 y Fz(\021)1235 1965 y FH(d\034)g FI(,)28 b(see)g([Kry03)n(].)37 b(W)-7 b(e)28 b(ha)n(v)n(e)391 2255 y Fk(E)p FF(h)p FH(M)567 2267 y FA(T)618 2255 y FF(i)c FG(=)761 2142 y Fz(Z)844 2162 y FA(T)807 2330 y FD(0)911 2255 y Fk(E)974 2162 y Fz(\020)1037 2176 y(X)1170 2255 y FH(u)1218 2220 y FD(2)1218 2275 y FA(s)1255 2255 y FG(\()p FH(\034)9 b FG(\))14 b FH(b)1414 2220 y FD(2)1414 2275 y FA(s)1452 2162 y Fz(\021)1516 2255 y FH(d\034)33 b FF(\024)22 b FH(b)1751 2220 y FD(2)1751 2275 y(max)1914 2142 y Fz(Z)1997 2162 y FA(T)1961 2330 y FD(0)2064 2255 y Fk(E)14 b FF(j)p FH(u)p FG(\()p FH(\034)9 b FG(\))p FF(j)2344 2220 y FD(2)2395 2255 y FH(d\034)1715 2415 y FD(\(4)p FA(:)p FD(18\))1768 2492 y FF(\024)1919 2436 y FG(1)p 1919 2473 42 4 v 1919 2549 a(2)1994 2492 y FH(b)2030 2458 y FD(2)2030 2512 y(max)2156 2400 y Fz(\020)2206 2492 y FH(B)2269 2504 y FD(0)2306 2492 y FH(T)29 b FG(+)18 b Fk(E)c FF(j)p FH(u)2615 2504 y FD(0)2652 2492 y FF(j)2675 2458 y FD(2)2712 2400 y Fz(\021)2775 2492 y FH(:)0 2687 y FI(So,)872 2751 y Fz(p)p 955 2751 261 4 v 75 x Fk(E)p FF(h)p FH(M)1131 2838 y FA(T)1183 2826 y FF(i)23 b(\024)1336 2770 y FH(b)1372 2782 y FD(max)p 1336 2807 163 4 v 1362 2823 a FF(p)p 1431 2823 42 4 v 69 x FG(2)1508 2734 y Fz(\020)1558 2749 y(p)p 1641 2749 161 4 v 77 x FH(B)1704 2838 y FD(0)1741 2826 y FH(T)30 b FG(+)1903 2751 y Fz(p)p 1986 2751 245 4 v 75 x Fk(E)14 b FF(j)p FH(u)2134 2838 y FD(0)2170 2826 y FF(j)2193 2802 y FD(2)2230 2734 y Fz(\021)2294 2826 y FH(:)0 3017 y FI(Assume)28 b(that)f FH(T)35 b FF(\025)22 b FG(1)p FI(.)37 b(Then)27 b(\(4.19\))g(and)h(\(4.20\))e (imply:)747 3201 y Fk(E)56 b FG(sup)824 3271 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)1061 3201 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1249 3167 y FD(2)1310 3201 y FF(\024)22 b FH(C)1456 3213 y FD(1)1494 3134 y Fz(\000)1532 3201 y FH(T)29 b FG(+)18 b Fk(E)c FF(j)p FH(u)1841 3213 y FD(0)1878 3201 y FF(j)1901 3167 y FD(2)1957 3201 y FG(+)2040 3126 y Fz(p)p 2123 3126 V 75 x Fk(E)g FF(j)p FH(u)2271 3213 y FD(0)2307 3201 y FF(j)2330 3177 y FD(2)2367 3134 y Fz(\001)2419 3201 y FH(:)535 b FI(\(4.21\))0 3434 y(W)-7 b(e)24 b(remind)f(that)g(the)h(estimates)f(\(4.16\)\025\(4.21\)) e(are)h(obtained)h(for)f FH(u)h FG(=)g FH(u)2380 3446 y FA(N)2442 3434 y FI(,)h(a)f(solution)g(of)g(\(4.13\).)125 3534 y(Due)28 b(to)f(\(4.6\),)g(for)g(an)n(y)g FH(M)36 b FI(and)28 b(ev)n(ery)e FH(!)s FI(,)i(w)n(e)f(ha)n(v)n(e:)938 3706 y FG(\005)1000 3718 y FA(M)1088 3706 y FH(u)1136 3718 y FA(N)1250 3706 y FF(\000)-14 b(!)1222 3760 y FA(N)6 b Fw(!1)1436 3706 y FG(\005)1498 3718 y FA(M)1586 3706 y FH(u)36 b FI(in)28 b FH(L)1824 3718 y FD(2)1861 3706 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2146 3676 y FD(1)2181 3706 y FG(\))h FH(:)0 3912 y FI(So,)27 b(b)n(y)h(\(4.18\))e(and)i(the)g (F)-7 b(atou)27 b(Lemma,)331 4147 y Fk(E)408 4034 y Fz(Z)491 4054 y FA(T)454 4222 y FD(0)557 4147 y FF(k)p FG(\005)661 4159 y FA(M)748 4147 y FH(u)p FG(\()p FH(s)p FG(\))p FF(k)941 4112 y FD(2)941 4167 y(1)978 4147 y FH(ds)c FF(\024)61 b FG(lim)1171 4200 y FA(N)6 b Fw(!1)1376 4147 y Fk(E)1453 4034 y Fz(Z)1535 4054 y FA(T)1498 4222 y FD(0)1601 4147 y FF(k)p FG(\005)1705 4159 y FA(M)1793 4147 y FH(u)1841 4159 y FA(N)1903 4147 y FF(k)1945 4112 y FD(2)1945 4167 y(1)1996 4147 y FH(ds)23 b FF(\024)2199 4090 y FG(1)p 2199 4127 42 4 v 2199 4203 a(2)2250 4147 y(\()p FH(T)12 b(B)2406 4159 y FD(0)2461 4147 y FG(+)18 b Fk(E)c FF(j)p FH(u)2692 4159 y FD(0)2728 4147 y FF(j)2751 4112 y FD(2)2789 4147 y FG(\))g FH(:)0 4361 y FI(No)n(w,)27 b(applying)g(F)-7 b(atou)27 b(again,)g(w)n(e)g(obtain)g(that)470 4591 y Fk(E)547 4478 y Fz(Z)630 4498 y FA(T)593 4666 y FD(0)696 4591 y FF(k)p FH(u)p FF(k)828 4556 y FD(2)828 4611 y(1)877 4591 y FH(ds)c FF(\024)66 b FG(lim)1070 4644 y FA(M)6 b Fw(!1)1300 4591 y Fk(E)1377 4478 y Fz(Z)1459 4498 y FA(T)1422 4666 y FD(0)1525 4591 y FF(k)p FG(\005)1629 4603 y FA(M)1703 4591 y FH(u)p FF(k)1793 4556 y FD(2)1843 4591 y FH(ds)23 b FF(\024)2046 4535 y FG(1)p 2046 4572 V 2046 4648 a(2)2097 4591 y(\()p FH(T)i(B)2266 4603 y FD(0)2322 4591 y FG(+)18 b Fk(E)c FF(j)p FH(u)2553 4603 y FD(0)2590 4591 y FF(j)2613 4556 y FD(2)2650 4591 y FG(\))g FH(:)0 4810 y FI(In)26 b(a)f(similar)g(w)n(a)n(y)-7 b(,)25 b(w)n(e)g(can)g(pass)g(to)h(the)g(limits)g(in)g(the)g(estimates) f(\(4.16\))g(and)g(\(4.21\))g(and)g(obtain:)1553 5059 y(39)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 40 39 bop 0 523 a FJ(Theorem)39 b(4.7.)44 b FC(L)l(et)36 b FH(u)806 535 y FD(0)878 523 y FF(2)f FH(H)43 b FC(b)l(e)37 b(a)g(r)l(andom)f(variable,)41 b(indep)l(endent)c(of)g FH(\020)6 b FG(\()p FF(\001)p FG(\))p FC(,)40 b(and)d(such)f(that)0 623 y Fk(E)14 b FF(j)p FH(u)148 635 y FD(0)184 623 y FF(j)207 593 y FD(2)274 623 y FH(<)29 b FF(1)p FC(.)49 b(Then)34 b(the)f(solution)g FH(u)p FC(,)h(c)l(onstructe)l(d)f(in)g(The)l(or)l(em)h(4.3,)i (satis\034es)d(\(4.16\),)j(\(4.18\))0 722 y(and)30 b(\(4.21\).)0 888 y FJ(Exercise)h(4.8.)40 b FI(Apply)28 b(Ito's)g(form)n(ula)e(to)i FH(')p FG(\()p FH(u)p FG(\))23 b(=)g FF(k)p FH(u)p FF(k)1820 858 y FD(2)1820 909 y(1)1883 888 y FI(to)28 b(get:)363 1071 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)666 1040 y FD(2)666 1091 y(1)725 1071 y FF(\024)822 1038 y FD(1)p 822 1052 34 4 v 822 1099 a(2)866 1071 y FH(B)929 1083 y FD(1)984 1071 y FG(+)k FH(e)1106 1040 y Fw(\000)p FD(2)p FA(t)1234 1071 y Fk(E)c FF(k)p FH(u)1401 1083 y FD(0)1437 1071 y FF(k)1479 1040 y FD(2)1479 1091 y(1)1516 1071 y FH(;)363 1304 y Fk(E)440 1191 y Fz(Z)523 1212 y FA(T)486 1380 y FD(0)602 1304 y FF(k)p FH(u)p FG(\()p FH(s)p FG(\))p FF(k)837 1270 y FD(2)837 1325 y(2)888 1304 y FH(ds)23 b FF(\024)1090 1248 y FG(1)p 1090 1285 42 4 v 1090 1361 a(2)1142 1304 y(\()p FH(T)i(B)1311 1316 y FD(1)1367 1304 y FG(+)18 b Fk(E)c FF(k)p FH(u)1617 1316 y FD(0)1653 1304 y FF(k)1695 1270 y FD(2)1695 1325 y(1)1731 1304 y FG(\))g FH(;)363 1501 y Fk(E)56 b FG(sup)440 1571 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)663 1501 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)889 1471 y FD(2)889 1522 y(1)949 1501 y FF(\024)22 b FH(C)1101 1409 y Fz(\020)1151 1501 y FH(T)30 b FG(+)18 b Fk(E)c FF(k)p FH(u)1480 1513 y FD(0)1516 1501 y FF(k)1558 1471 y FD(2)1558 1522 y(1)1613 1501 y FG(+)1696 1429 y Fz(p)p 1779 1429 282 4 v 72 x Fk(E)g FF(k)p FH(u)1946 1513 y FD(0)1982 1501 y FF(k)2024 1472 y FD(2)2024 1523 y(1)2061 1409 y Fz(\021)2124 1501 y FH(;)37 b(C)29 b FG(=)23 b FH(C)6 b FG(\()p FH(b)2493 1513 y FD(max)2620 1501 y FH(;)14 b(B)2720 1513 y FD(1)2757 1501 y FG(\))g FH(:)0 1752 y FJ(Exercise)31 b(4.9.)40 b FI(Apply)28 b(Ito's)g(form)n(ula)e (to)i FG(exp)1551 1684 y Fz(\000)1589 1752 y FH(\033)s FF(j)p FH(u)p FF(j)1733 1722 y FD(2)1771 1684 y Fz(\001)1809 1752 y FI(,)f FH(\033)g(<)22 b FG(\(2)p FH(b)2130 1722 y FD(2)2130 1772 y(max)2257 1752 y FG(\))2289 1722 y Fw(\000)p FD(1)2378 1752 y FI(,)28 b(to)f(get)1025 1958 y Fk(E)14 b FH(e)1141 1924 y FA(\033)r Fw(j)p FA(u)p FD(\()p FA(t)p FD(\))p Fw(j)1337 1899 y Ft(2)1396 1958 y FF(\024)22 b FH(e)1522 1924 y Fw(\000)p FA(\033)r(t)1657 1958 y Fk(E)14 b FH(e)1773 1924 y FA(\033)r Fw(j)p FA(u)1872 1932 y Ft(0)1905 1924 y Fw(j)1925 1899 y Ft(2)1979 1958 y FG(+)k FH(C)i(:)0 2141 y FI(Similar,)27 b(apply)g(Ito's)h(form)n(ula) e(to)i FG(exp\()p FH(\033)s FF(k)p FH(u)p FF(k)1472 2111 y FD(2)1472 2161 y(1)1508 2141 y FG(\))g FI(to)f(get)h(that)996 2339 y Fk(E)14 b FH(e)1112 2305 y FA(\033)r Fw(k)p FA(u)p FD(\()p FA(t)p FD(\))p Fw(k)1336 2280 y Ft(2)1336 2321 y(1)1396 2339 y FF(\024)22 b FH(e)1522 2305 y Fw(\000)p FA(\033)r(t)1657 2339 y Fk(E)14 b FH(e)1773 2305 y FA(\033)r Fw(k)p FA(u)1886 2313 y Ft(0)1919 2305 y Fw(k)1953 2280 y Ft(2)1953 2321 y(1)2008 2339 y FG(+)k FH(C)i(:)0 2522 y FJ(Exercise)25 b(4.10.)604 2492 y Fw(\003)k Fo(2)728 2522 y FI(If)22 b FH(B)868 2534 y FA(k)932 2522 y FG(=)1020 2459 y Fz(P)1121 2522 y FF(j)p FH(s)p FF(j)1206 2492 y FD(2)p FA(k)1280 2522 y FH(b)1316 2492 y FD(2)1316 2542 y FA(s)1376 2522 y FH(<)h FF(1)p FI(,)g(apply)e(Ito's)h(form)n (ula)f(to)g(functionals)h FH(t)2838 2492 y FA(m)2901 2522 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)3127 2492 y FD(2)3127 2542 y FA(m)0 2621 y FI(and)27 b(argue)g(b)n(y)g (induction)h(to)f(see)g(that)503 2815 y Fk(E)566 2748 y Fz(\000)660 2815 y FG(sup)618 2885 y FD(0)p Fw(\024)p FA(t)p Fw(\024)p FA(T)842 2815 y FH(t)872 2781 y FA(k)927 2815 y FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)1153 2781 y FD(2)1153 2835 y FA(k)1192 2748 y Fz(\001)1231 2765 y FA(p)1292 2815 y FF(\024)c FH(C)d Fk(E)14 b FH(e)1575 2781 y FA(\033)r Fw(j)p FA(u)1674 2789 y Ft(0)1706 2781 y Fw(j)1726 2756 y Ft(2)1762 2815 y FH(;)65 b FI(with)28 b FH(C)h FG(=)23 b FH(C)6 b FG(\()p FH(k)s(;)14 b(p;)g(\033)n(;)g(T)e FG(\))i FH(;)0 3057 y FI(for)27 b(an)n(y)g FH(k)16 b(;)e(p)23 b FF(2)h FE(N)j FI(\(see)h([KS03)n(]\).)0 3290 y Fy(4.4)112 b(Stationary)37 b(measures)0 3443 y FI(The)28 b(semigroups)1058 3537 y FF(B)1113 3549 y FA(t)1165 3537 y FG(:)47 b FH(C)1294 3549 y FA(b)1328 3537 y FG(\()p FH(H)7 b FG(\))23 b FF(!)g FH(C)1656 3549 y FA(b)1690 3537 y FG(\()p FH(H)7 b FG(\))p FH(;)59 b(t)23 b FF(\025)g FG(0)14 b FH(;)1058 3672 y FF(B)1116 3642 y Fw(\003)1113 3693 y FA(t)1176 3672 y FG(:)36 b FF(P)7 b FG(\()p FH(H)g FG(\))23 b FF(!)g(P)7 b FG(\()p FH(H)g FG(\))p FH(;)115 b(t)23 b FF(\025)g FG(0)14 b FH(;)0 3818 y FI(are)27 b(de\034ned)g(as)g(in)h(Section)g (3.2.)36 b(That)27 b(is,)h(for)f FH(t)c FF(\025)f FG(0)p FH(;)14 b(v)26 b FF(2)e FH(H)34 b FI(and)27 b FG(\000)h FI(a)f(measurable)f(subset)i(of)f FH(H)7 b FI(:)795 4038 y FG(\()p FF(B)882 4050 y FA(t)911 4038 y FH(f)i FG(\)\()p FH(v)s FG(\))24 b(=)1212 3925 y Fz(Z)1258 4114 y FA(H)1349 4038 y FH(P)1402 4050 y FA(t)1431 4038 y FG(\()p FH(v)s(;)14 b(du)p FG(\))g FH(f)9 b FG(\()p FH(u)p FG(\))23 b(=)g Fk(E)14 b FH(f)2080 3971 y Fz(\000)2117 4038 y FH(u)p FG(\()p FH(t)p FG(;)g FH(v)s FG(\))2339 3971 y Fz(\001)2391 4038 y FH(;)563 b FI(\(4.22\))775 4260 y FG(\()p FF(B)865 4226 y Fw(\003)862 4280 y FA(t)902 4260 y FH(\026)p FG(\)\(\000\))24 b(=)1212 4147 y Fz(Z)1258 4336 y FA(H)1349 4260 y FH(P)1402 4272 y FA(t)1431 4260 y FG(\()p FH(v)s(;)14 b FG(\000\))g FH(\026)p FG(\()p FH(dv)s FG(\))g FH(;)891 4439 y FG(\()p FF(B)981 4405 y Fw(\003)978 4460 y FA(t)1019 4439 y FH(\026)p FG(\))23 b(=)g FF(D)1278 4372 y Fz(\000)1330 4439 y FH(u)p FG(\()p FH(t)p FG(;)14 b FH(u)1525 4451 y FD(0)1562 4439 y FG(\))1594 4372 y Fz(\001)1632 4439 y FH(;)41 b FI(if)29 b FF(D)r FG(\()p FH(u)1919 4451 y FD(0)1956 4439 y FG(\))23 b(=)g FH(\026)14 b(:)791 b FI(\(4.23\))p 0 4522 1276 4 v 92 4576 a Fn(2)127 4599 y Fm(As)23 b(usual,)h(the)h(asterisk)g (indicates)f(a)g(more)g(di\036cult)g(exercise)h(\(in)g(the)f(D.)g(Kn)n (uth's)g(b)r(o)r(ok)g(\020The)g(Art)h(of)e(Com-)0 4678 y(puter)g(Programming\021)j(this)c(notion)g(is)f(illustrated)h(b)n(y)g (the)g(follo)n(wing)f(example:)29 b Fl(Pr)l(ove)c(that)e(for)h(an)g (inte)l(ger)f Fi(n)c Fh(\025)h Fg(3)0 4757 y Fl(the)25 b(e)l(quation)i Fi(x)443 4734 y Ff(n)500 4757 y Fg(+)16 b Fi(y)608 4734 y Ff(n)670 4757 y Fg(=)k Fi(z)781 4734 y Ff(n)848 4757 y Fl(has)27 b(no)f(inte)l(ger)f(solutions)p Fm(.\))1553 5059 y FI(40)p eop end PStoPSsaved restore %%Page: (40,41) 21 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 41 40 bop 0 523 a FI(Here)25 b FH(P)247 535 y FA(t)276 523 y FG(\()p FH(v)s(;)14 b(du)p FG(\))26 b FI(is)f(the)g(Mark)n(o)n(v)e(transition)h(function,)i(and)f FH(u)p FG(\()p FH(t)p FG(;)14 b FH(u)2134 535 y FD(0)2170 523 y FG(\))26 b FI(is)e(a)h(solution)f(of)h(\(4.3\),)h(equal)0 623 y FH(u)48 635 y FD(0)112 623 y FI(at)i FH(t)23 b FG(=)g(0)k FI(\(in)h(\(4.23\))f FH(u)841 635 y FD(0)905 623 y FI(is)h(a)f(random)f(v)-5 b(ariable\).)125 722 y(W)e(e)26 b(note)g(that)h(b)n(y)f(Prop)r(osition)g(4.6)f(relations)g (\(4.22\))h(and)g(\(4.23\))f(remain)h(v)-5 b(alid)26 b(if)h FH(u)p FG(\()p FH(t)p FG(;)14 b FH(u)3038 734 y FD(0)3074 722 y FG(\))27 b FI(is)0 822 y(a)g(w)n(eak)g(solution.)125 922 y(A)g(measure)g FH(\026)c FF(2)g(P)7 b FG(\()p FH(H)g FG(\))28 b FI(is)f(called)h(a)f FC(stationary)j(me)l(asur)l(e)e FI(if)g FF(B)2157 891 y Fw(\003)2154 942 y FA(t)2208 922 y FH(\026)23 b FG(=)f FH(\026)28 b FI(for)f(all)g FH(t)d FF(\025)e FG(0)p FI(.)0 1088 y FJ(Theorem)30 b(4.11.)40 b FC(A)29 b(stationary)i(me)l(asur)l(e)e(exists.)0 1254 y(Pr)l(o)l(of.)43 b FI(Let)29 b FH(u)p FG(\()p FH(t)p FG(\))f FI(b)r(e)h(a)g(solution)f(suc)n(h)g(that)h FH(u)p FG(\(0\))24 b(=)g(0)p FI(.)39 b(W)-7 b(e)29 b(set)g FH(\026)2145 1266 y FA(t)2199 1254 y FG(=)24 b FF(D)r FG(\()p FH(u)p FG(\()p FH(t)p FG(\)\))p FI(.)41 b(Due)29 b(to)f(the)h(\034rst)0 1353 y(estimate)e(in)h(Exercise)f(4.8)g(and)g(the)h(Cheb)n(yc)n(hev)f (inequalit)n(y)g(w)n(e)g(ha)n(v)n(e)760 1536 y FH(\026)810 1548 y FA(t)839 1469 y Fz(\000)877 1536 y FH(H)7 b FF(n)p FB(B)1068 1548 y FA(R)1122 1536 y FG(\()p FH(H)1230 1502 y FD(1)1267 1536 y FG(\))1299 1469 y Fz(\001)1361 1536 y FG(=)22 b Fk(P)p FG(\()p FF(k)p FH(u)p FG(\()p FH(t)p FG(\))p FF(k)1771 1548 y FD(1)1831 1536 y FH(>)h(R)q FG(\))g FF(\024)g FH(B)2189 1548 y FD(1)2240 1536 y FH(R)2304 1502 y Fw(\000)p FD(2)2406 1536 y FH(:)0 1730 y FI(F)-7 b(or)27 b FH(")c(>)f FG(0)p FI(,)28 b(c)n(ho)r(osing)e FH(R)791 1742 y FA(")849 1730 y FG(=)d(\()p FH(B)1032 1742 y FD(1)1069 1730 y FH(=")p FG(\))1182 1700 y FD(1)p FA(=)p FD(2)1286 1730 y FI(,)28 b(w)n(e)f(see)g(that)959 1913 y FH(\026)1009 1925 y FA(t)1038 1845 y Fz(\000)1076 1913 y FH(H)7 b FF(n)p FB(B)1267 1925 y FA(R")1353 1913 y FG(\()p FH(H)1461 1878 y FD(1)1498 1913 y FG(\))1530 1845 y Fz(\001)1591 1913 y FH(<)23 b(";)41 b FI(for)27 b(all)h FH(t)23 b FF(\025)f FG(0)p FH(:)0 2115 y FI(Setting)p 297 2069 51 4 v 39 w FH(\026)347 2135 y FA(t)418 2115 y FG(=)534 2082 y FD(1)p 534 2096 34 4 v 538 2144 a FA(t)591 2048 y Fz(R)646 2069 y FA(t)630 2145 y FD(0)690 2115 y FH(\026)740 2127 y FA(s)789 2115 y FH(ds)p FI(,)42 b FH(t)f FF(\025)h FG(1)p FI(,)f(w)n(e)d(see)h(that)p 1690 2069 51 4 v 39 w FH(\026)1740 2135 y FA(t)1808 2115 y FI(also)e(satis\034es)h(this)h(estimate.)70 b(Since)39 b(b)n(y)0 2215 y(\(1.3\),)c FB(B)302 2227 y FA(R)352 2235 y Ft(1)389 2215 y FG(\()p FH(H)497 2185 y FD(1)534 2215 y FG(\))f Fs(b)f FH(H)7 b FI(,)35 b(then)f(the)g(Prokhoro)n(v)d (theorem)i(applies,)i(and)f(the)g(set)f FF(f)p 2718 2169 V FH(\026)2768 2235 y FA(t)2797 2215 y FI(,)i FH(t)f FF(\025)e FG(1)p FF(g)h FI(is)0 2314 y(precompact)d(in)h FF(P)7 b FG(\()p FH(H)g FG(\))p FI(.)47 b(Cho)r(osing)30 b(a)h(con)n(v)n(erging)d(sequence)p 2029 2269 V 31 w FH(\026)2079 2335 y FA(t)2104 2343 y Fu(j)2168 2314 y FH(*)g(\026)p FI(,)k(w)n(e)f(see)g(as)f(in)h(the)h(pro)r(of)0 2414 y(of)c(Theorem)e(3.5)h(that)h FH(\026)f FI(is)h(a)f(stationary)f (measure.)p 3128 2414 4 57 v 3132 2361 50 4 v 3132 2414 V 3182 2414 4 57 v 125 2580 a(The)h(constructed)g(measure)g FH(\026)g FI(has)g(\034nite)h(second)f(momen)n(t:)1247 2688 y Fz(Z)1344 2801 y FF(j)p FH(u)p FF(j)1438 2766 y FD(2)1489 2801 y FH(\026)p FG(\()p FH(du)p FG(\))c FF(\024)g FH(B)1868 2813 y FD(0)1919 2801 y FH(:)0 3031 y FI(Indeed,)31 b(b)n(y)f(Theorem)f(4.7,)927 2964 y Fz(R)1010 3031 y FF(j)p FH(u)p FF(j)1104 3001 y FD(2)1155 3031 y FH(\026)1205 3043 y FA(t)1234 3031 y FG(\()p FH(du)p FG(\))f FF(\024)f FH(B)1572 3043 y FD(0)1640 3031 y FI(for)i(eac)n(h)h FH(t)p FI(.)45 b(Hence,)31 b(the)f(measure)p 2801 2985 51 4 v 29 w FH(\026)2851 3051 y FA(t)2911 3031 y FI(satis\034es)0 3131 y(the)e(same)f(inequalit)n(y)-7 b(.)36 b(F)-7 b(or)27 b(ev)n(ery)g FH(M)k(<)23 b FF(1)k FI(w)n(e)h(ha)n(v)n(e)650 3238 y Fz(Z)746 3351 y FG(\()p FF(j)p FH(u)p FF(j)872 3317 y FD(2)928 3351 y FF(^)19 b FH(M)9 b FG(\))14 b FH(\026)p FG(\()p FH(du)p FG(\))23 b(=)g(lim)1583 3238 y Fz(Z)1666 3351 y FG(\()p FF(j)p FH(u)p FF(j)1792 3317 y FD(2)1847 3351 y FF(^)c FH(M)9 b FG(\))p 2057 3305 V 14 w FH(\026)2107 3371 y FA(t)2136 3351 y FG(\()p FH(du)p FG(\))24 b FF(\024)e FH(B)2465 3363 y FD(0)2516 3351 y FH(:)0 3571 y FI(No)n(w)27 b(the)h(F)-7 b(atou)27 b(lemma)h(implies)g (the)f(desired)g(estimate.)0 3737 y FJ(Exercise)36 b(4.12.)42 b FI(Use)32 b(Exercise)f(4.10)f(to)i(pro)n(v)n(e)e(that)i(if)g FH(B)1958 3749 y FA(k)2029 3737 y FH(<)d FF(1)p FI(,)k(then)g(the)f (constructed)f(sta-)0 3837 y(tionary)26 b(measure)h FH(\026)h FI(satis\034es)1032 3944 y Fz(Z)1129 4057 y FF(k)p FH(u)p FF(k)1261 4017 y FA(p)1261 4082 y(k)1315 4057 y FH(\026)p FG(\()p FH(du)p FG(\))23 b FH(<)g FF(1)166 b(8)14 b FH(p)22 b FF(\025)h FG(0)p FH(:)0 4322 y Fy(4.5)112 b(High-frequency)38 b(random)f(kic)m(ks)0 4475 y FI(Consider)27 b(the)g(kic)n(k-force)691 4658 y FH(\021)732 4670 y FA(")768 4658 y FG(\()p FH(t)p FG(\))c(=)973 4594 y FF(p)p 1042 4594 39 4 v 64 x FH(")1118 4579 y Fz(X)1159 4758 y FA(k)1265 4658 y FH(\021)1309 4623 y FA(!)1306 4678 y(k)1371 4658 y FH(\016)s FG(\()p FH(t)c FF(\000)f FH("k)s FG(\))p FH(;)97 b(\021)1853 4670 y FA(k)1917 4658 y FG(=)2018 4579 y Fz(X)2005 4764 y FA(s)p Fw(2)p Fv(Z)2120 4744 y Ft(2)2120 4782 y(0)2180 4658 y FH(b)2216 4670 y FA(s)2265 4658 y FH(\030)2301 4670 y FA(sk)2387 4658 y FH(e)2426 4670 y FA(s)2475 4658 y FH(:)1553 5059 y FI(41)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 42 41 bop 0 523 a FI(The)28 b(random)e(v)-5 b(ariables)26 b(are)h(as)g(in)h(Section)f(3,)g(and)h(in)g(addition)953 706 y Fk(E)14 b FH(\030)1066 718 y FA(sk)1161 706 y FG(=)23 b(0)p FH(;)96 b(B)1473 718 y FD(1)1534 706 y FG(=)1635 627 y Fz(X)1621 812 y FA(s)p Fw(2)p Fv(Z)1736 792 y Ft(2)1736 830 y(0)1796 706 y FF(j)p FH(s)p FF(j)1881 672 y FD(2)1932 706 y FH(b)1968 672 y FD(2)1968 727 y FA(s)2028 706 y FH(<)23 b FF(1)14 b FH(:)0 975 y FI(W)-7 b(e)28 b(set)1212 1112 y FH(\020)1248 1124 y FA(")1283 1112 y FG(\()p FH(t)p FG(\))c(=)1489 999 y Fz(Z)1572 1019 y FA(t)1535 1188 y FD(0+)1651 1112 y FH(\021)1692 1124 y FA(")1727 1112 y FG(\()p FH(\034)9 b FG(\))14 b FH(d\034)25 b(;)0 1303 y FI(and)i(normalise)g FH(\020)572 1315 y FA(")635 1303 y FI(to)h(b)r(e)g(con)n(tin)n(uous)e(from)h(the)h(righ)n(t.)125 1403 y(W)-7 b(e)27 b(w)n(an)n(t)g(to)h(compare)e(the)i(force)f FH(\021)1289 1415 y FA(")1352 1403 y FI(with)h(the)g(white-force)f FH(\021)s FI(,)964 1612 y FH(\021)g FG(=)1144 1556 y FH(d)p 1129 1593 74 4 v 1129 1669 a(dt)1235 1612 y(\020)6 b(;)98 b(\020)29 b FG(=)1565 1534 y Fz(X)1551 1719 y FA(s)p Fw(2)p Fv(Z)1666 1699 y Ft(2)1666 1737 y(0)1726 1612 y FH(b)1762 1624 y FA(s)1811 1612 y FH(\014)1858 1624 y FA(s)1893 1612 y FG(\()p FH(t)p FG(\))14 b FH(e)2040 1624 y FA(s)2076 1612 y FG(\()p FH(x)p FG(\))g FH(:)0 1887 y FI(Fixing)30 b FH(u)310 1899 y FD(0)374 1887 y FF(2)d FH(H)7 b FI(,)31 b(w)n(e)e(denote)h(b)n(y)g FH(u)1146 1899 y FA(")1181 1887 y FG(\()p FH(t)p FG(;)14 b FH(u)1328 1899 y FD(0)1365 1887 y FG(\))30 b FI(the)h(solution)e(of)h(\(3.3\))g (with)g FH(\020)k FG(=)27 b FH(\020)2576 1899 y FA(")2612 1887 y FH(;)14 b(u)2697 1899 y FA(")2732 1887 y FG(\(0)p FH(;)g(u)2891 1899 y FD(0)2927 1887 y FG(\))30 b(=)d FH(u)3129 1899 y FD(0)3166 1887 y FI(,)0 1987 y(and)g(b)n(y)h FH(u)p FG(\()p FH(t)p FG(;)14 b FH(u)472 1999 y FD(0)508 1987 y FG(\))28 b FI(\025)f(the)h(solution)f(of)h(\(4.3\))f(with)h FH(u)p FG(\(0;)14 b FH(u)1783 1999 y FD(0)1819 1987 y FG(\))23 b(=)g FH(u)2010 1999 y FD(0)2047 1987 y FI(.)0 2143 y FJ(Theorem)30 b(4.13.)40 b FC(F)-6 b(or)30 b(any)g(\034nite)f (time-interval)h FG([0)p FH(;)14 b(T)e FG(])p FC(,)29 b(we)h(have)738 2314 y FF(D)804 2247 y Fz(\000)842 2314 y FH(u)890 2326 y FA(")926 2314 y FG(\()p FF(\001)14 b FG(;)g FH(u)1080 2326 y FD(0)1116 2314 y FG(\))1148 2247 y Fz(\001)1210 2314 y FH(*)23 b FF(D)1382 2247 y Fz(\000)1420 2314 y FH(u)p FG(\()p FF(\001)14 b FG(;)g FH(u)1622 2326 y FD(0)1659 2314 y FG(\))1691 2247 y Fz(\001)1766 2314 y FC(in)30 b FF(P)1933 2247 y Fz(\000)1970 2314 y FH(L)2027 2326 y Fw(1)2097 2314 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))2414 2247 y Fz(\001)0 2485 y FC(and)916 2585 y FF(D)r FG(\()p FH(u)1062 2597 y FA(")1097 2585 y FG(\()p FH(T)12 b FG(;)i FH(u)1275 2597 y FD(0)1311 2585 y FG(\)\))24 b FH(*)f FF(D)r FG(\()p FH(u)p FG(\()p FH(T)12 b FG(;)i FH(u)1829 2597 y FD(0)1865 2585 y FG(\)\))38 b FC(in)29 b FF(P)7 b FG(\()p FH(H)g FG(\))0 2727 y FC(as)30 b FH(")23 b FF(!)g FG(0)p FC(.)125 2884 y(Sketch)k(of)h(the)g(pr)l(o)l (of.)38 b FI(Let)25 b FH(\020)1043 2854 y Fw(0)1037 2904 y FA(")1073 2884 y FG(\()p FH(t)p FG(\))h FI(b)r(e)f(a)g (piecewise-linear)e(con)n(tin)n(uous)h(pro)r(cess,)h(equal)f FH(\020)2934 2896 y FA(")2970 2884 y FG(\()p FH(t)p FG(\))i FI(for)0 2984 y FG(0)d FF(\024)f FH(t)h FF(2)h FH(")14 b FE(Z)p FI(.)37 b(Then)953 3083 y FF(D)r FG(\()p FH(\020)1093 3049 y Fw(0)1087 3104 y FA(")1124 3083 y FG(\))30 b Fe(*)1179 3134 y FA(")p Fw(!)p FD(0)1333 3083 y FF(D)r FG(\()p FH(\020)6 b FG(\))38 b FI(in)28 b FF(P)1705 3016 y Fz(\000)1743 3083 y FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2093 3053 y FD(1)2128 3083 y FG(\))2160 3016 y Fz(\001)2213 3083 y FH(;)0 3256 y FI(b)n(y)38 b(the)g(Donsk)n(er)f(principle,)k(see) c([Bil99)o(,)h(Dur91)o(].)69 b(Let)38 b(us)g(\034x)g(an)n(y)f(sequence) h FH(")2711 3268 y FA(j)2786 3256 y FF(!)j FG(0)p FI(.)68 b(Due)0 3356 y(to)35 b(the)h(Sk)n(orokho)r(d)e(em)n(b)r(edding)i (theorem)f(\(see)g([Shi96],)j(Section)d(I)r(I)r(I.8)h(or)f([IW89)o(],)j (Section)d(1,)0 3467 y(Theorem)d(2.7\),)i(there)f(exist)g(pro)r(cesses) 1346 3445 y Fz(e)1336 3467 y FH(\020)1378 3443 y Fw(0)1402 3487 y FA(")1433 3495 y Fu(j)1501 3467 y FI(and)1673 3445 y Fz(e)1668 3467 y FH(\020)7 b FI(,)34 b(de\034ned)f(on)g(the)h (same)e(probabilit)n(y)g(space,)0 3566 y(suc)n(h)27 b(that)174 3737 y FF(D)r FG(\()282 3715 y Fz(e)272 3737 y FH(\020)314 3713 y Fw(0)338 3758 y FA(")369 3766 y Fu(j)405 3737 y FG(\))c(=)g FF(D)r FG(\()p FH(\020)688 3703 y Fw(0)682 3758 y FA(")713 3766 y Fu(j)749 3737 y FG(\))p FH(;)61 b FF(D)r FG(\()968 3715 y Fz(e)963 3737 y FH(\020)7 b FG(\))23 b(=)g FF(D)r FG(\()p FH(\020)6 b FG(\))84 b FI(and)1632 3715 y Fz(e)1622 3737 y FH(\020)1664 3713 y Fw(0)1688 3758 y FA(")1719 3766 y Fu(j)1815 3737 y FF(\000)-14 b(!)1800 3790 y FA(j)s Fw(!1)2015 3715 y Fz(e)2010 3737 y FH(\020)52 b FI(a.s.)36 b(in)24 b FH(C)6 b FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)2698 3703 y FD(1)2734 3737 y FG(\))g FH(:)174 b FI(\(4.24\))0 3971 y(Let)148 3949 y Fz(e)143 3971 y FH(\020)179 3983 y FA(")210 3991 y Fu(j)267 3971 y FI(b)r(e)22 b(a)f(pro)r(cess)f(whic)n(h)i(is)f(con)n (tin)n(uous)g(from)g(the)h(righ)n(t,)g(constan)n(t)e(on)i(the)g(in)n (terv)-5 b(als)20 b FG(\()p FH(k)s(";)14 b FG(\()p FH(k)9 b FG(+)0 4090 y(1\))p FH(")p FG(\))p FH(;)14 b(k)26 b FF(\025)c FG(0)27 b FI(and)h(equal)f(to)899 4068 y Fz(e)889 4090 y FH(\020)931 4066 y Fw(0)955 4110 y FA(")986 4118 y Fu(j)1049 4090 y FI(for)g FH(t)c FF(2)h FH(")14 b FE(Z)p FI(.)37 b(Then)1287 4288 y FF(D)r FG(\()1390 4266 y Fz(e)1385 4288 y FH(\020)1421 4300 y FA(")1452 4308 y Fu(j)1488 4288 y FG(\))24 b(=)e FF(D)r FG(\()p FH(\020)1765 4300 y FA(")1796 4308 y Fu(j)1833 4288 y FG(\))14 b FH(:)0 4459 y FI(De\034ne)28 b(the)g(map)f FG(\010)644 4471 y FA(T)724 4459 y FI(as)g(in)h(\(4.9\).)37 b(Then)1031 4641 y FF(D)1097 4574 y Fz(\000)1135 4641 y FH(u)1183 4653 y FA(")1214 4661 y Fu(j)1249 4641 y FG(\()p FF(\001)14 b FG(;)g FH(u)1403 4653 y FD(0)1440 4641 y FG(\))1472 4574 y Fz(\001)1533 4641 y FG(=)23 b(\010)1681 4653 y FA(T)1752 4641 y FF(\016)1812 4574 y Fz(\000)1850 4641 y FF(D)r FG(\()1953 4619 y Fz(e)1948 4641 y FH(\020)1984 4653 y FA(")2015 4661 y Fu(j)2051 4641 y FG(\))2083 4574 y Fz(\001)2135 4641 y FH(;)1097 4822 y FF(D)1163 4754 y Fz(\000)1202 4822 y FH(u)p FG(\()p FF(\001)14 b FG(;)g FH(u)1404 4834 y FD(0)1440 4822 y FG(\))1472 4754 y Fz(\001)1533 4822 y FG(=)23 b(\010)1681 4834 y FA(T)1752 4822 y FF(\016)1812 4754 y Fz(\000)1850 4822 y FF(D)r FG(\()1953 4800 y Fz(e)1948 4822 y FH(\020)7 b FG(\))2023 4754 y Fz(\001)2075 4822 y FH(;)2977 4727 y FI(\(4.25\))1553 5059 y(42)p eop end PStoPSsaved restore %%Page: (42,43) 22 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 43 42 bop 0 523 a FI(in)28 b FF(P)7 b FG(\()p FH(L)251 535 y Fw(1)321 523 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\)\))p FI(.)125 623 y(Since)27 b(the)h(Bro)n(wnian)e (motion)h(is)h(H\366lder-con)n(tin)n(uous,)d(w)n(e)j(ha)n(v)n(e)e(from) h(\(4.24\))g(that)971 830 y FF(j)999 809 y Fz(e)994 830 y FH(\020)1030 842 y FA(")1061 850 y Fu(j)1097 830 y FG(\()p FF(\001)p FG(\))19 b FF(\000)1291 809 y Fz(e)1286 830 y FH(\020)7 b FG(\()p FF(\001)p FG(\))p FF(j)1439 845 y FA(L)1485 853 y Fr(1)1546 845 y FD(\(0)p FA(;T)i FD(;)p FA(H)1750 829 y Ft(1)1783 845 y FD(\))1859 783 y(in)18 b(Prob)p FA(:)1909 830 y FF(\000)-14 b(!)1902 880 y FA(")1933 888 y Fu(j)r Fr(!)p Ft(0)2139 830 y FG(0)14 b FH(:)759 b FI(\(4.26\))0 1053 y(By)27 b(Prop)r(osition)g(2.18)f(the)i (map)978 1236 y FG(\010)1038 1248 y FA(T)1113 1236 y FG(:)23 b FH(L)1216 1202 y FD(0)1216 1257 y Fw(1)1286 1236 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)1571 1202 y FD(1)1607 1236 y FG(\))23 b FF(!)g FH(L)1825 1248 y Fw(1)1895 1236 y FG(\(0)p FH(;)14 b(T)e FG(;)i FH(H)7 b FG(\))765 b FI(\(4.27\))0 1419 y(is)26 b(lo)r(cally)f(Lipsc)n(hitz.)37 b(\(W)-7 b(e)26 b(recall)f(that)i FH(L)1359 1389 y FD(0)1359 1439 y Fw(1)1455 1419 y FI(stands)e(for)h(the)h(linear)e(subspace)g(of) h FH(L)2704 1431 y Fw(1)2774 1419 y FI(,)h(formed)e(b)n(y)0 1518 y(functions,)j(con)n(tin)n(uous)e(at)i FH(t)23 b FG(=)f(0)p FI(\).)37 b(No)n(w)27 b(\(4.25\)\025\(4.27\))f(imply)i(the)f (\034rst)h(theorem's)f(assertion.)125 1618 y(Pro)r(of)g(of)g(the)h (second)f(is)h(left)g(as)f(an)g(exercise.)1517 b Fs(\003)125 1767 y FI(Both)26 b(the)g(theorem's)g(assertion)e(and)i(its)h(pro)r(of) e(b)r(ecome)h(more)g(natural)f(b)n(y)h(ev)n(oking)f(the)h(Sk)n(o-)0 1867 y(rokho)r(d)20 b(spaces)g(\(formed)i(b)n(y)f(discon)n(tin)n(uous)f (functions)h(with)h(jumps,)h(see)e([Bil99)o(,)g(LS89)o(]\).)36 b(Guided)0 1967 y(b)n(y)27 b FC(Oc)l(c)l(am's)j(R)l(azor,)681 1937 y Fo(3)747 1967 y FI(w)n(e)d(decided)h(not)f(to)h(in)n(tro)r(duce) f(this)h(notion.)p 0 4733 1276 4 v 92 4786 a Fn(3)151 4810 y Fm(\020En)n(tities)23 b(are)i(not)f(to)g(b)r(e)g(m)n(ultiplied)e (b)r(ey)n(ond)j(necessit)n(y)-6 b(.\021)1553 5059 y FI(43)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 44 43 bop 0 523 a FK(5)131 b(Preliminaries)45 b(from)f(the)g(measure)f(theory)0 705 y FI(Ev)n(erywhere)21 b(b)r(elo)n(w)h FH(M)31 b FI(is)23 b(a)f(complete)g(separable)f(metric) h(space)g(\(e.g.)35 b FH(M)d FG(=)22 b FH(H)7 b FI(,)24 b(or)d FH(M)31 b FI(is)23 b(a)f(closed)0 804 y(ball)27 b(in)h FE(R)318 774 y FA(n)363 804 y FI(\).)0 965 y Fy(5.1)112 b(W)-9 b(eak)38 b(con)m(v)m(ergence)f(of)h(measures)g(and)g(Lipsc)m (hitz-dual)e(distance)0 1119 y FI(A)23 b(sequence)f(of)g(measures)f FH(\026)914 1131 y FA(n)983 1119 y FF(2)i(P)29 b FC(we)l(akly)e(c)l (onver)l(ges)c FI(to)f FH(\026)h FF(2)h(P)7 b FI(,)23 b(denotes)45 b FH(\026)2495 1131 y FA(n)2563 1119 y FH(*)23 b(\026)p FI(,)h(if)f FG(\()p FH(\026)2919 1131 y FA(n)2964 1119 y FH(;)14 b(f)9 b FG(\))23 b FF(!)0 1218 y FG(\()p FH(\026;)14 b(f)9 b FG(\))28 b FI(for)f(eac)n(h)g FH(f)k FF(2)24 b FH(C)759 1188 y FA(b)792 1218 y FG(\()p FH(M)9 b FG(\))p FI(.)125 1318 y(F)-7 b(or)39 b(an)n(y)g FH(f)53 b FF(2)44 b FH(C)6 b FG(\()p FH(M)j FG(\))p FI(,)44 b(Lip)p FG(\()p FH(f)9 b FG(\))45 b FF(\024)e(1)d FI(is)h(the)f(Lipsc)n(hitz)g (constan)n(t)f(of)i FH(f)9 b FI(.)74 b(W)-7 b(e)40 b(de\034ne)h(the)0 1417 y FC(Lipschitz-norm)29 b FI(of)e FH(f)37 b FI(as)1087 1517 y FF(j)p FH(f)9 b FF(j)1183 1529 y FA(L)1255 1517 y FG(=)23 b(sup)1371 1587 y FA(M)1495 1517 y FF(j)p FH(f)9 b FG(\()p FH(m)p FG(\))p FF(j)19 b FG(+)f(Lip\()p FH(f)9 b FG(\))14 b FH(;)0 1710 y FI(and)27 b(set)1073 1810 y FH(O)e FG(=)e FF(f)p FH(f)31 b FF(2)23 b FH(C)1506 1776 y FA(b)1540 1810 y FG(\()p FH(M)9 b FG(\))p FF(j)23 b(j)p FH(f)9 b FF(j)1836 1822 y FA(L)1909 1810 y FF(\024)22 b FG(1)p FF(g)14 b FH(:)901 b FI(\(5.1\))0 1976 y FJ(Prop)s(osition)47 b(5.1.)i FC(A)44 b(se)l(quenc)l(e)f(of)h(me)l(asur)l(es)g FH(\026)1720 1988 y FA(n)1808 1976 y FC(we)l(akly)h(c)l(onver)l(ges)f (to)g FH(\026)f FC(if)i(and)f(only)g(if)0 2076 y FG(\()p FH(\026)82 2088 y FA(n)128 2076 y FH(;)14 b(f)9 b FG(\))22 b FF(!)h FG(\()p FH(\026;)14 b(f)9 b FG(\))30 b FC(for)h(e)l(ach)f FH(f)i FF(2)23 b FH(O)r FC(.)0 2242 y(Pr)l(o)l(of.)43 b FI(The)26 b(`only)g(if)6 b(')27 b(part)e(of)h(the)h(statemen)n(t)f (is)g(ob)n(vious.)35 b(If)26 b FH(M)35 b FI(is)26 b(compact,)g(then)g (the)g(`if)6 b(')28 b(part)0 2341 y(follo)n(ws)j(since)g(an)n(y)g(con)n (tin)n(uous)g(function)h(on)f FH(M)41 b FI(ma)n(y)31 b(b)r(e)h(appro)n(ximated)e(in)i(the)g(sup\025norm)e(b)n(y)0 2441 y(b)r(ounded)e(Lipsc)n(hitz)f(functions.)37 b(F)-7 b(or)27 b(the)h(non-compact)f(case)f(see)i([Dud89)o(,)g(Shi96)o(].)p 3128 2441 4 57 v 3132 2388 50 4 v 3132 2441 V 3182 2441 4 57 v 125 2607 a(F)-7 b(or)26 b FH(\026;)14 b(\027)29 b FF(2)23 b(P)7 b FG(\()p FH(M)i FG(\))p FI(,)28 b(de\034ne)1011 2790 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1292 2755 y Fw(\003)1292 2810 y FA(L)1364 2790 y FG(=)23 b(sup)1488 2859 y FA(O)1605 2790 y FF(j)p FG(\()p FH(f)t(;)14 b(\026)p FG(\))19 b FF(\000)f FG(\()p FH(f)t(;)c(\027)5 b FG(\))p FF(j)14 b FH(:)840 b FI(\(5.2\))0 3021 y(This)28 b(is)f(the)h FC(Lipschitz-dual)j(distanc)l(e)e FI(on)e FF(P)7 b FG(\()p FH(M)i FG(\))p FI(.)0 3187 y FJ(Exercise)37 b(5.2.)44 b FI(Let)33 b FH(M)40 b FG(=)32 b FH(H)40 b FI(\(with)33 b(the)h(usual)e(distance\))h(and)g FF(k)21 b(\001)h(k)2334 3157 y Fw(\003)2334 3210 y FA(L)2417 3187 y FI(b)r(e)33 b(the)g(corresp)r(onding)0 3287 y(Lipsc)n(hitz-dual)27 b(distance.)36 b(No)n(w,)27 b(for)g FG(0)c FH(<)g(d)g FF(\024)g FG(1)k FI(w)n(e)g(set)1092 3470 y FG(dist\()p FH(u)1306 3482 y FD(1)1343 3470 y FH(;)14 b(u)1428 3482 y FD(2)1465 3470 y FG(\))23 b(=)g FF(j)p FH(u)1679 3482 y FD(1)1734 3470 y FF(\000)18 b FH(u)1865 3482 y FD(2)1902 3470 y FF(j)g(^)h FH(d)14 b(:)0 3652 y FI(It)25 b(is)g(immediate)g (that)h(this)f(is)g(a)g(distance)f(whic)n(h)h(de\034nes)g(on)g FH(H)32 b FI(the)25 b(same)f(top)r(ology)-7 b(.)35 b(Let)25 b FF(k)13 b(\001)g(k)3085 3622 y Fw(\003)3085 3676 y FA(L;d)0 3752 y FI(b)r(e)28 b(the)g(corresp)r(onding)d(Lipsc)n (hitz-dual)i(distance.)37 b(Pro)n(v)n(e)26 b(that)872 3969 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1153 3935 y Fw(\003)1153 3990 y FA(L;d)1280 3969 y FF(\024)23 b(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1649 3935 y Fw(\003)1649 3990 y FA(L)1721 3969 y FF(\024)1820 3913 y FG(2)p 1819 3950 44 4 v 1819 4026 a FH(d)1895 3969 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)2176 3935 y Fw(\003)2176 3990 y FA(L;d)2294 3969 y FH(:)701 b FI(\(5.3\))0 4180 y FJ(Theorem)46 b(5.3.)131 b FI(a\))42 b FC(The)h(sp)l(ac)l(e)g FF(P)7 b FG(\()p FH(M)i FG(\))p FC(,)45 b(given)d(the)g(distanc)l(e)h(dist)p FG(\()p FH(\026;)14 b(\027)5 b FG(\))46 b(=)f FF(k)p FH(\026)27 b FF(\000)g FH(\027)5 b FF(k)3010 4149 y Fw(\003)3010 4202 y FA(L)3059 4180 y FC(,)46 b(is)0 4279 y(c)l(omplete.)0 4415 y FI(b\))30 b FC(The)g(we)l(ak)h(c)l(onver)l(genc)l(e)e(in)h FF(P)7 b FG(\()p FH(M)i FG(\))29 b FC(is)h(e)l(quivalent)g(to)f(the)h (c)l(onver)l(genc)l(e)g(in)g(the)f(Lipschitz-dual)0 4514 y(distanc)l(e.)1553 5059 y FI(44)p eop end PStoPSsaved restore %%Page: (44,45) 23 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 45 44 bop 0 523 a FC(Pr)l(o)l(of.)43 b FI(\(when)32 b FH(M)41 b FI(is)31 b(compact\).)49 b(a\))32 b(Let)f FH(L)h FI(b)r(e)g(the)g(space)f(of)g(Lipsc)n(hitz)h(functions)g (on)f FH(M)41 b FI(with)0 623 y(the)33 b(norm)g FF(j)22 b(\001)g(j)484 635 y FA(L)534 623 y FI(.)54 b(Let)33 b FH(L)822 593 y Fw(\003)893 623 y FI(b)r(e)g(the)h(dual)f(space.)52 b(The)33 b(norm)g(in)g FH(L)2178 593 y Fw(\003)2249 623 y FI(is)g(giv)n(en)f(b)n(y)h(\(5.2\).)53 b(Clearly)0 722 y FF(P)7 b FG(\()p FH(M)i FG(\))24 b FF(\032)h FH(L)390 692 y Fw(\003)427 722 y FI(.)40 b(Let)29 b FF(f)p FH(\026)732 734 y FA(n)777 722 y FF(g)f FI(b)r(e)h(a)f(sequence)g(in)h FF(P)7 b FG(\()p FH(M)i FG(\))28 b FI(whic)n(h)h(is)f(a)g(Cauc)n(h)n(y) g(sequence)g(in)h(the)g(norm)0 822 y FF(k)18 b(\001)g(k)143 792 y Fw(\003)143 845 y FA(L)193 822 y FI(.)37 b(Then)27 b(there)h(is)f FH(\026)c FF(2)h FH(L)974 792 y Fw(\003)1039 822 y FI(suc)n(h)j(that)1341 1005 y FH(\026)1391 1017 y FA(n)1459 1005 y FF(!)c FH(\026)g FF(2)h FH(L)1774 970 y Fw(\003)1825 1005 y FH(:)0 1187 y FI(W)-7 b(e)28 b(ha)n(v)n(e)e(to)h(sho)n(w)f(that)i FH(\026)23 b FF(2)g(P)7 b FG(\()p FH(M)i FG(\))p FI(.)37 b(Since)27 b FH(C)6 b FG(\()p FH(M)j FG(\))1685 1157 y Fw(\003)1747 1187 y FG(=)23 b FF(M)p FG(\()p FH(M)9 b FG(\))27 b FI(\(the)h(space)e(of)i (signed)e(measures)0 1287 y(on)h FH(M)9 b FI(\),)28 b(w)n(e)f(ha)n(v)n (e)f(to)i(c)n(hec)n(k)f(that)1157 1469 y FH(\026)c FF(\025)g FG(0)46 b FI(and)g FH(\026)23 b FF(2)g FH(C)6 b FG(\()p FH(M)j FG(\))1956 1435 y Fw(\003)2009 1469 y FH(:)986 b FI(\(5.4\))0 1652 y FJ(Exercise)31 b(5.4.)40 b FC(Pr)l(ove)31 b(that)f(\(5.4\))g(fol)t(lows)i(fr)l(om)e(the)g(fol)t(lowing)i(c)l (ondition:)947 1835 y FG(0)23 b FF(\024)f FG(\()p FH(\026;)14 b(f)9 b FG(\))23 b FF(\024)g FG(1)84 b FF(8)p FH(f)32 b FF(2)24 b FH(L;)36 b FG(0)23 b FF(\024)f FH(f)32 b FF(\024)23 b FG(1)14 b FH(:)775 b FI(\(5.5\))3124 2017 y Fs(\003)125 2167 y FI(Since)27 b(\(5.5\))h(holds)f(for)g FH(\026)g FI(replaced)g(b)n(y)g(an)n(y)g FH(\026)1611 2179 y FA(n)1656 2167 y FI(,)h(then)g(it)g(also)f(holds)g(for)g FH(\026)p FI(.)0 2267 y(b\))f(The)g(implication)g(`)p FF(\()-14 b FG(=)p FI(')26 b(follo)n(ws)f(from)g(Prop)r(osition)g(5.1,) h(and)f(`)p FG(=)-14 b FF(\))p FI(')26 b(holds)g(since)f(for)g(all)h FH(")d(>)f FG(0)0 2366 y FI(the)28 b(ball)f FF(f)p FH(f)22 b FF(j)14 b(j)p FH(f)9 b FF(j)542 2378 y FA(L)615 2366 y FF(\024)22 b FG(1)p FF(g)27 b FI(has)g(a)g(\034nite)h FH(")p FI(-net)g(in)f FH(C)1607 2378 y FA(b)1641 2366 y FG(\()p FH(M)9 b FG(\))p FI(.)125 2466 y(F)-7 b(or)26 b(the)i(case)f(when)h FH(M)36 b FI(is)28 b(a)f(separable)f(metric)h (space,)g(see)g([Dud89,)h(Shi96)o(].)p 3128 2466 4 57 v 3132 2413 50 4 v 3132 2466 V 3182 2466 4 57 v 125 2632 a(Note)34 b(that)h(the)g(relation)f(\(5.2\))g(also)g(de\034nes)h(a)f (distance)g(on)h(the)g(space)f FF(M)p FG(\()p FH(M)9 b FG(\))p FI(.)58 b(But)35 b(this)0 2731 y(distance)27 b FJ(is)k(not)h(complete)p FI(.)0 2964 y Fy(5.2)112 b(V)-9 b(ariational)35 b(distance)0 3117 y FI(F)-7 b(or)27 b FH(\026;)14 b(\027)28 b FF(2)c(P)7 b FG(\()p FH(M)i FG(\))27 b FI(w)n(e)g(de\034ne)493 3300 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)774 3312 y FD(v)l(ar)891 3300 y FG(=)94 b(sup)979 3373 y FD(\000)p Fw(\032)p FA(B)s FD(\()p FA(M)6 b FD(\))1260 3300 y FF(j)p FH(\026)p FG(\(\000\))19 b FF(\000)f FH(\027)5 b FG(\(\000\))p FF(j)24 b FG(=)f(sup)2000 3232 y Fz(\000)2038 3300 y FH(\026)p FG(\(\000\))c FF(\000)f FH(\027)5 b FG(\(\000\))2468 3232 y Fz(\001)2530 3300 y FF(\024)23 b FG(1)14 b FH(:)321 b FI(\(5.6\))0 3545 y(This)28 b(is)f(the)h FC(variational)k(distanc)l(e)c FI(b)r(et)n(w)n(een)g(the)g(t)n(w)n(o)e(measures.)0 3711 y FJ(Example)38 b(5.5.)43 b FI(Let)35 b FH(M)42 b FG(=)34 b FE(R)p FI(,)h FH(\026)f FG(=)g FH(\016)1311 3723 y FD(0)1382 3711 y FI(and)g FH(\027)1591 3723 y FA(n)1670 3711 y FG(=)f FH(\016)1805 3726 y FD(1)p FA(=n)1917 3711 y FI(,)j FH(n)e FF(\025)f FG(1)p FI(.)56 b(Then)34 b FF(k)p FH(\026)23 b FF(\000)f FH(\027)2745 3723 y FA(n)2790 3711 y FF(k)34 b FG(=)f(1)p FI(,)i(but)0 3819 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)234 3831 y FA(n)279 3819 y FF(k)321 3788 y Fw(\003)321 3841 y FA(L)393 3819 y FG(=)495 3786 y FD(1)p 491 3800 42 4 v 491 3847 a FA(n)542 3819 y FI(.)0 3985 y FJ(Exercise)44 b(5.6.)j FC(Pr)l(ove)41 b(that)g(if)g FH(\026)f FC(and)h FH(\027)k FC(ar)l(e)c(absolutely)g(c)l (ontinuous)f(with)g(r)l(esp)l(e)l(ct)g(to)h(some)0 4084 y(me)l(asur)l(e)29 b FH(\032)23 b FF(2)h(P)7 b FG(\()p FH(M)i FG(\))p FC(,)30 b(and)g FH(\026)23 b FG(=)g FH(p)1104 4096 y FA(\026)1148 4084 y FG(\()p FH(m)p FG(\))14 b FH(\032)p FG(\()p FH(dm)p FG(\))p FC(,)31 b FH(\027)d FG(=)23 b FH(p)1777 4096 y FA(\027)1818 4084 y FG(\()p FH(m)p FG(\))14 b FH(\032)p FG(\()p FH(dm)p FG(\))p FC(,)31 b(then)795 4312 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1076 4324 y FD(v)l(ar)1194 4312 y FG(=)1291 4256 y(1)p 1291 4293 V 1291 4369 a(2)1380 4199 y Fz(Z)1426 4388 y FA(M)1527 4312 y FF(j)p FH(p)1592 4324 y FA(\026)1637 4312 y FG(\()p FH(m)p FG(\))19 b FF(\000)f FH(p)1918 4324 y FA(\027)1959 4312 y FG(\()p FH(m)p FG(\))p FF(j)c FH(\032)p FG(\()p FH(dm)p FG(\))g FH(:)625 b FI(\(5.7\))0 4549 y FC(Note)37 b(that)g FH(\026)g FC(and)g FH(\027)42 b FC(ar)l(e)c(absolutely)f(c)l(ontinuous)g(with)g(r)l(esp)l(e)l(ct)g (to)g FH(\032)f FG(=)2444 4517 y FD(1)p 2444 4531 34 4 v 2444 4578 a(2)2501 4549 y FG(\()p FH(\026)24 b FG(+)f FH(\027)5 b FG(\))p FC(.)61 b(So)37 b(`\(5.7\))0 4649 y(always)31 b(applies'.)1553 5059 y FI(45)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 46 45 bop 125 523 a FI(Due)28 b(to)f(\(5.7\))g(for)g(an)n (y)g(con)n(tin)n(uous)g(b)r(ounded)h(function)g FH(f)36 b FI(w)n(e)27 b(ha)n(v)n(e)118 748 y FF(j)p FG(\()p FH(\026;)14 b(f)9 b FG(\))18 b FF(\000)g FG(\()p FH(\027)q(;)c(f)9 b FG(\))p FF(j)23 b FG(=)747 653 y Fz(\014)747 702 y(\014)747 752 y(\014)788 635 y(Z)834 824 y FA(M)922 681 y Fz(\000)960 748 y FH(p)1002 760 y FA(\026)1046 748 y FG(\()p FH(m)p FG(\))c FF(\000)f FH(p)1327 760 y FA(\027)1368 748 y FG(\()p FH(m)p FG(\))1505 681 y Fz(\001)1558 748 y FH(f)9 b FG(\()p FH(m)p FG(\))14 b FH(\032)p FG(\()p FH(dm)p FG(\))1982 653 y Fz(\014)1982 702 y(\014)1982 752 y(\014)770 970 y FF(\024)22 b(j)p FH(f)9 b FF(j)953 985 y FA(C)1001 994 y Fu(b)1031 985 y FD(\()p FA(M)d FD(\))1184 857 y Fz(Z)1295 970 y FF(j)p FH(p)1360 982 y FA(\026)1423 970 y FF(\000)18 b FH(p)1548 982 y FA(\027)1589 970 y FF(j)c FH(\032)p FG(\()p FH(dm)p FG(\)[1)p FH(ex)p FG(])23 b FF(\024)g FG(2)p FF(j)p FH(f)9 b FF(j)2272 985 y FA(C)2320 994 y Fu(b)2349 985 y FD(\()p FA(M)d FD(\))2489 970 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)2770 982 y FD(v)l(ar)2878 970 y FH(:)3018 859 y FI(\(5.8\))0 1195 y(Hence,)22 b(the)e(space)g FF(P)7 b FG(\()p FH(M)i FG(\))p FI(,)21 b(pro)n(vided)e(with)i(the)f(v)-5 b(ariational)19 b(distance,)i(is)f(con)n(tin)n(uously)f(em)n(b)r(edded)0 1295 y(in)28 b FH(C)156 1307 y FA(b)190 1295 y FG(\()p FH(M)9 b FG(\))344 1265 y Fw(\003)382 1295 y FI(.)37 b(Besides,)27 b(using)g(\(5.8\))g(in)h(\(5.2\))f(w)n(e)h(see)f(that) 1140 1477 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1421 1443 y Fw(\003)1421 1498 y FA(L)1494 1477 y FF(\024)22 b FG(2)14 b FF(k)p FH(\026)k FF(\000)g FH(\027)5 b FF(k)1918 1489 y FD(v)l(ar)2026 1477 y FH(:)0 1710 y Fy(5.3)112 b(Coupling)0 1863 y FJ(De\034nition)26 b(5.7.)36 b FC(A)25 b(p)l(air)i(of)f(r)l(andom)g(variables)i FH(\030)1649 1875 y FD(1)1686 1863 y FH(;)14 b(\030)1759 1875 y FD(2)1797 1863 y FC(,)27 b(de\034ne)l(d)f(on)f(the)h(same)g(pr)l(ob)l(ability)i (sp)l(ac)l(e,)0 1963 y(is)i(c)l(al)t(le)l(d)h(a)f(c)l(oupling)h(for)f (given)h(me)l(asur)l(es)e FH(\026)1469 1975 y FD(1)1506 1963 y FH(;)14 b(\026)1593 1975 y FD(2)1654 1963 y FF(2)23 b(P)7 b FG(\()p FH(M)i FG(\))29 b FC(if)i FF(D)r FG(\()p FH(\030)2195 1975 y FA(j)2231 1963 y FG(\))23 b(=)g FH(\026)2424 1975 y FA(j)2459 1963 y FC(,)30 b FH(j)e FG(=)23 b(1)p FH(;)14 b FG(2)p FC(.)125 2129 y FI(Giv)n(en)27 b(a)h(coupling)g FG(\()p FH(\030)836 2141 y FD(1)873 2129 y FH(;)14 b(\030)946 2141 y FD(2)984 2129 y FG(\))p FI(,)29 b(let)f(us)g(consider)f(the)h (random)g(v)-5 b(ariable)27 b FH(\030)h FG(=)23 b FH(\030)2568 2141 y FD(1)2624 2129 y FF(\002)c FH(\030)2744 2141 y FD(2)2805 2129 y FF(2)25 b FH(M)i FF(\002)18 b FH(M)9 b FI(.)0 2228 y(Denoting)28 b FH(m)22 b FG(=)h FF(D)r FG(\()p FH(\030)t FG(\))p FI(,)29 b(w)n(e)e(ha)n(v)n(e)1082 2411 y FH(\026)1132 2423 y FD(1)1193 2411 y FG(=)22 b FH(\031)1327 2423 y FD(1)1383 2411 y FF(\016)c FH(m;)97 b(\026)1686 2423 y FD(2)1746 2411 y FG(=)23 b FH(\031)1881 2423 y FD(2)1937 2411 y FF(\016)18 b FH(m;)0 2594 y FI(where)30 b FH(\031)290 2606 y FD(1)328 2594 y FG(\()p FH(m)433 2606 y FD(1)470 2594 y FH(;)14 b(m)580 2606 y FD(2)617 2594 y FG(\))29 b(=)f FH(m)844 2606 y FD(1)882 2594 y FI(,)j FH(\031)983 2606 y FD(2)1021 2594 y FG(\()p FH(m)1126 2606 y FD(1)1163 2594 y FH(;)14 b(m)1273 2606 y FD(2)1310 2594 y FG(\))29 b(=)g FH(m)1538 2606 y FD(2)1575 2594 y FI(.)47 b(An)31 b(equiv)-5 b(alen)n(t)31 b(de\034nition)g(of)g(the)g (coupling)f(is)0 2693 y(that)24 b(this)f(is)g(a)g(measure)f FH(m)i FI(on)1004 2626 y Fz(\000)1042 2693 y FH(M)18 b FF(\002)10 b FH(M)f FI(,)24 b FF(B)s FG(\()p FH(M)18 b FF(\002)10 b FH(M)f FG(\))1739 2626 y Fz(\001)1799 2693 y FI(whic)n(h)24 b(satis\034es)e(the)i(t)n(w)n(o)e(relations)g(ab) r(o)n(v)n(e.)0 2793 y(Indeed,)32 b(if)f(suc)n(h)g(a)g(measure)e (exists,)j(w)n(e)f(can)f(tak)n(e)g FG(\()p FH(M)g FF(\002)20 b FH(M)t(;)14 b FF(B)s FH(;)g(m)p FG(\))30 b FI(for)h(the)g(probabilit) n(y)f(space,)0 2892 y(and)d(c)n(ho)r(ose)g FH(\030)461 2904 y FD(1)521 2892 y FG(=)c FH(\031)656 2904 y FD(1)694 2892 y FI(,)k FH(\030)780 2904 y FD(2)841 2892 y FG(=)c FH(\031)976 2904 y FD(2)1013 2892 y FI(.)125 2992 y(If)28 b FG(\()p FH(\030)276 3004 y FD(1)313 2992 y FH(;)14 b(\030)386 3004 y FD(2)424 2992 y FG(\))28 b FI(is)f(a)g(coupling)g (for)g FG(\()p FH(\026)1177 3004 y FD(1)1215 2992 y FH(;)14 b(\026)1302 3004 y FD(2)1339 2992 y FG(\))p FI(,)28 b(then)g(for)f(an)n (y)g FG(\000)c FF(2)g(B)s FG(\()p FH(M)9 b FG(\))27 b FI(w)n(e)h(ha)n(v)n(e)485 3175 y FH(\026)535 3187 y FD(1)572 3175 y FG(\(\000\))19 b FF(\000)f FH(\026)840 3187 y FD(2)878 3175 y FG(\(\000\))23 b(=)g Fk(E)1168 3107 y Fz(\000)1205 3175 y FH(I)1241 3187 y FD(\000)1287 3175 y FG(\()p FH(\030)1355 3187 y FD(1)1393 3175 y FG(\))c FF(\000)f FH(I)1563 3187 y FD(\000)1608 3175 y FG(\()p FH(\030)1676 3187 y FD(2)1714 3175 y FG(\))1746 3107 y Fz(\001)1017 3333 y FG(=)23 b Fk(E)1168 3241 y Fz(\020)1217 3333 y FH(I)1253 3348 y Fw(f)p FA(\030)1317 3356 y Ft(1)1350 3348 y Fw(6)p FD(=)p FA(\030)1431 3356 y Ft(2)1464 3348 y Fw(g)1502 3266 y Fz(\000)1540 3333 y FH(I)1576 3345 y FD(\000)1622 3333 y FG(\()p FH(\030)1690 3345 y FD(1)1727 3333 y FG(\))c FF(\000)f FH(I)1897 3345 y FD(\000)1943 3333 y FG(\()p FH(\030)2011 3345 y FD(2)2049 3333 y FG(\))2081 3266 y Fz(\001)2119 3241 y(\021)2192 3333 y FF(\024)k Fk(P)p FG(\()p FH(\030)2412 3345 y FD(1)2474 3333 y FF(6)p FG(=)g FH(\030)2597 3345 y FD(2)2635 3333 y FG(\))14 b FH(:)0 3536 y FI(Therefore,)1100 3635 y Fk(P)p FG(\()p FH(\030)1233 3647 y FD(1)1294 3635 y FF(6)p FG(=)23 b FH(\030)1418 3647 y FD(2)1455 3635 y FG(\))h FF(\025)e(k)p FH(\026)1690 3647 y FD(1)1746 3635 y FF(\000)c FH(\026)1879 3647 y FD(2)1916 3635 y FF(k)1958 3647 y FD(v)l(ar)2066 3635 y FH(:)0 3785 y FI(A)28 b(coupling)f FG(\()p FH(\030)490 3797 y FD(1)528 3785 y FH(;)14 b(\030)601 3797 y FD(2)638 3785 y FG(\))28 b FI(is)g(called)f FC(maximal)h FI(if)1091 3967 y Fk(P)p FF(f)p FH(\030)1234 3979 y FD(1)1294 3967 y FF(6)p FG(=)23 b FH(\030)1418 3979 y FD(2)1455 3967 y FF(g)g FG(=)g FF(k)p FH(\026)1700 3979 y FD(1)1755 3967 y FF(\000)18 b FH(\026)1888 3979 y FD(2)1925 3967 y FF(k)1967 3979 y FD(v)l(ar)2075 3967 y FH(:)0 4150 y FI(The)28 b(follo)n(wing)e(result)h(is)h(often)g(kno)n(wn)e(as)h(the) h FC(Dobrushin)i(lemma)p FI(:)0 4316 y FJ(Lemma)f(5.8.)40 b FC(F)-6 b(or)30 b(any)g(two)g(me)l(asur)l(es)g FH(\026)1403 4328 y FD(1)1440 4316 y FH(;)14 b(\026)1527 4328 y FD(2)1587 4316 y FF(2)24 b(P)7 b FG(\()p FH(M)i FG(\))29 b FC(a)h(maximal)h(c)l (oupling)f FG(\()p FH(\030)2714 4328 y FD(1)2752 4316 y FH(;)14 b(\030)2825 4328 y FD(2)2862 4316 y FG(\))30 b FC(exists.)125 4482 y FI(F)-7 b(or)26 b(a)i(pro)r(of)f(see,)g(sa)n(y) -7 b(,)26 b([Lin92)o(])i(and)g(the)g(App)r(endix)g(in)g([KS01a)n(].)125 4681 y(This)36 b(lemma)h(sho)n(ws)e(that)i(a)f(coupling)g(can)g(b)r(e)h (used)g(as)f(a)g(to)r(ol)g(to)h(study)g(the)g(v)-5 b(ariational)0 4781 y(distance)19 b(b)r(et)n(w)n(een)h(measures.)33 b(A)20 b(coupling)f(is)g(also)f(suited)i(to)g(study)g(the)g(Lipsc)n (hitz-dual)e(distance:)1553 5059 y(46)p eop end PStoPSsaved restore %%Page: (46,47) 24 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 47 46 bop 0 523 a FJ(Exercise)21 b(5.9.)32 b FI(If)18 b(the)h(measures)e FH(\026)1154 535 y FD(1)1192 523 y FH(;)d(\026)1279 535 y FD(2)1339 523 y FF(2)23 b(P)7 b FG(\()p FH(M)i FG(\))18 b FI(admit)h(a)f(coupling)g FG(\()p FH(\030)2336 535 y FD(1)2374 523 y FH(;)c(\030)2447 535 y FD(2)2484 523 y FG(\))19 b FI(suc)n(h)f(that)37 b Fk(P)p FF(fk)p FH(\030)3087 535 y FD(1)3124 523 y FF(\000)0 623 y FH(\030)36 635 y FD(2)74 623 y FF(k)22 b FH(>)h(")p FF(g)f(\024)h FH(\022)16 b(;)28 b FI(for)f(some)g FH(";)14 b(\022)24 b(>)f FG(0)p FI(,)k(then)56 b FF(k)p FH(\026)1486 635 y FD(1)1541 623 y FF(\000)18 b FH(\026)1674 635 y FD(2)1712 623 y FF(k)1754 593 y Fw(\003)1754 646 y FA(L)1826 623 y FF(\024)23 b FG(2)p FH(\022)d FG(+)e FH(")c(:)0 780 y FC(Pr)l(o)l(of.)43 b FI(F)-7 b(or)27 b(an)n(y)g FH(f)k FF(2)24 b FH(O)30 b FI(\(see)e(\(5.1\)\))f(w)n(e)g(ha)n(v)n(e) 442 951 y FF(j)p FG(\()p FH(\026)547 963 y FD(1)603 951 y FF(\000)18 b FH(\026)736 963 y FD(2)774 951 y FH(;)c(f)9 b FG(\))p FF(j)22 b(\024)h(j)p Fk(E)14 b FH(I)1162 963 y FA(Q)1219 884 y Fz(\000)1257 951 y FH(f)9 b FG(\()p FH(\030)1375 963 y FD(1)1412 951 y FG(\))19 b FF(\000)f FH(f)9 b FG(\()p FH(\030)1664 963 y FD(2)1701 951 y FG(\))1733 884 y Fz(\001)1772 881 y(\014)1772 930 y(\014)1818 951 y FG(+)1901 881 y Fz(\014)1901 930 y(\014)1929 951 y Fk(E)14 b FH(I)2042 963 y FA(Q)2094 947 y Fu(c)2129 884 y Fz(\000)2167 951 y FH(f)9 b FG(\()p FH(\030)2285 963 y FD(1)2323 951 y FG(\))18 b FF(\000)g FH(f)9 b FG(\()p FH(\030)2574 963 y FD(2)2612 951 y FG(\))2644 884 y Fz(\001)2682 881 y(\014)2682 930 y(\014)2724 951 y FH(;)0 1123 y FI(where)29 b FH(Q)c FG(=)h FF(f)p FG(dist)o(\()p FH(\030)667 1135 y FD(1)705 1123 y FH(;)14 b(\030)778 1135 y FD(2)816 1123 y FG(\))26 b FH(>)f(")p FF(g)p FI(.)42 b(This)29 b(estimate)g(implies)h(that)f FF(j)p FG(\()p FH(\026)2205 1135 y FD(1)2262 1123 y FF(\000)19 b FH(\026)2396 1135 y FD(2)2434 1123 y FH(;)14 b(f)9 b FG(\))p FF(j)25 b(\024)h FG(2)p FH(\022)21 b FG(+)f FH(")p FI(,)29 b(so)g(the)0 1222 y(assertion)d(follo)n(ws.)p 3128 1222 4 57 v 3132 1169 50 4 v 3132 1222 V 3182 1222 4 57 v 0 1453 a Fy(5.4)112 b(Kan)m(toro)m(vic)m(h)36 b(functionals)0 1606 y FI(Let)28 b FH(f)190 1618 y FA(K)281 1606 y FI(b)r(e)g(an)n(y)f(measurable)f (function)i(on)f FH(M)g FF(\002)18 b FH(M)37 b FI(suc)n(h)27 b(that)597 1777 y FH(f)638 1789 y FA(K)702 1777 y FG(\()p FH(m)807 1789 y FD(1)844 1777 y FH(;)14 b(m)954 1789 y FD(2)991 1777 y FG(\))24 b(=)e FH(f)1175 1789 y FA(K)1239 1777 y FG(\()p FH(m)1344 1789 y FD(2)1381 1777 y FH(;)14 b(m)1491 1789 y FD(1)1528 1777 y FG(\))24 b FF(\025)e FG(dist\()p FH(m)1910 1789 y FD(1)1948 1777 y FH(;)14 b(m)2058 1789 y FD(2)2095 1777 y FG(\))97 b FF(8)14 b FH(m)2357 1789 y FD(1)2408 1777 y FH(;)g(m)2518 1789 y FD(2)2569 1777 y FH(:)0 1949 y FI(W)-7 b(e)28 b(de\034ne)h(the)f FC(Kantor)l(ovich)k(functional)p FI(,)d(corresp)r(onding)d(to)i FH(f)2087 1961 y FA(K)2151 1949 y FI(,)g(as)f(the)i(follo)n(wing)e (function)i FF(K)0 2048 y FI(on)e FF(P)7 b FG(\()p FH(M)i FG(\))18 b FF(\002)g(P)7 b FG(\()p FH(M)i FG(\))p FI(:)1044 2148 y FF(K)q FG(\()p FH(\026)1190 2160 y FD(1)1228 2148 y FH(;)14 b(\026)1315 2160 y FD(2)1352 2148 y FG(\))23 b(=)g(inf)7 b FF(f)p Fk(E)14 b FH(f)1756 2160 y FA(K)1818 2148 y FG(\()p FH(\030)1886 2160 y FD(1)1924 2148 y FH(;)g(\030)1997 2160 y FD(2)2035 2148 y FG(\))p FF(g)g FH(;)0 2291 y FI(where)27 b(the)h(in\034m)n(um)g(is)g(tak)n(en)e(o)n(v)n(er)g(all)i (couplings)e FG(\()p FH(\030)1743 2303 y FD(1)1781 2291 y FH(;)14 b(\030)1854 2303 y FD(2)1892 2291 y FG(\))28 b FI(for)f FG(\()p FH(\026)2161 2303 y FD(1)2198 2291 y FH(;)14 b(\026)2285 2303 y FD(2)2322 2291 y FG(\))p FI(.)0 2448 y FJ(Lemma)29 b(5.10.)40 b FC(F)-6 b(or)30 b(any)g FH(\026)940 2460 y FD(1)977 2448 y FH(;)14 b(\026)1064 2460 y FD(2)1125 2448 y FF(2)23 b(P)7 b FG(\()p FH(M)i FG(\))p FC(,)1146 2619 y FF(k)p FH(\026)1238 2631 y FD(1)1293 2619 y FF(\000)18 b FH(\026)1426 2631 y FD(2)1464 2619 y FF(k)1506 2585 y Fw(\003)1506 2640 y FA(L)1578 2619 y FF(\024)23 b(K)q FG(\()p FH(\026)1812 2631 y FD(1)1850 2619 y FH(;)14 b(\026)1937 2631 y FD(2)1974 2619 y FG(\))g FH(:)975 b FI(\(5.9\))0 2791 y FC(Pr)l(o)l(of.)43 b FI(Let)28 b FH(\030)445 2803 y FD(1)482 2791 y FH(;)14 b(\030)555 2803 y FD(2)620 2791 y FI(b)r(e)28 b(an)n(y)f(coupling)g(for)g FH(\026)1399 2803 y FD(1)1437 2791 y FH(;)14 b(\026)1524 2803 y FD(2)1561 2791 y FI(,)27 b(and)h FH(g)d FF(2)f FH(O)r FI(.)37 b(Then)454 2962 y FG(\()p FH(\026)536 2974 y FD(1)592 2962 y FF(\000)19 b FH(\026)726 2974 y FD(2)763 2962 y FH(;)14 b(g)s FG(\))23 b(=)f Fk(E)1048 2895 y Fz(\000)1086 2962 y FH(g)s FG(\()p FH(\030)1197 2974 y FD(1)1234 2962 y FG(\))d FF(\000)f FH(g)s FG(\()p FH(\030)1479 2974 y FD(2)1517 2962 y FG(\))1549 2895 y Fz(\001)1610 2962 y FF(\024)23 b Fk(E)14 b FG(dist)o(\()p FH(\030)1976 2974 y FD(1)2014 2962 y FH(;)g(\030)2087 2974 y FD(2)2125 2962 y FG(\))23 b FF(\024)g Fk(E)14 b FH(f)2386 2974 y FA(K)2449 2962 y FG(\()p FH(\030)2517 2974 y FD(1)2555 2962 y FH(;)g(\030)2628 2974 y FD(2)2665 2962 y FG(\))g FH(:)0 3133 y FI(T)-7 b(aking)21 b(\034rst)h(the)g (suprem)n(um)g(in)g FH(g)k FF(2)d FH(O)i FI(and)d(next)g(the)h(in\034m) n(um)f(in)h FG(\()p FH(\030)2250 3145 y FD(1)2288 3133 y FH(;)14 b(\030)2361 3145 y FD(2)2398 3133 y FG(\))23 b FI(w)n(e)e(reco)n(v)n(er)f(\(5.9\).)p 3128 3133 V 3132 3081 50 4 v 3132 3133 V 3182 3133 4 57 v 125 3297 a(Let)26 b(us)g(mo)r(dify)g(\(5.2\))g(and)g(in)n(tro)r(duce)f(the)h(new)h (distance)e(in)h FF(P)7 b FG(\()p FH(M)i FG(\))p FI(,)27 b(called)e(the)i FC(Kantor)l(ovich)0 3397 y(distanc)l(e)p FI(:)937 3496 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1218 3508 y FA(K)1305 3496 y FG(=)96 b(sup)1392 3570 y FD(Lip\()p FA(f)7 b FD(\))p Fw(\024)p FD(1)1678 3496 y FF(j)p FG(\()p FH(f)t(;)14 b(\026)p FG(\))19 b FF(\000)f FG(\()p FH(f)t(;)c(\027)5 b FG(\))p FF(j)14 b FH(:)0 3707 y FI(It)33 b(also)f(p)r(ossesses)g(the)i(prop)r(erties,)f (sp)r(eci\034ed)g(in)h(Theorem)e(5.3.)53 b(In)33 b(addition,)h(no)n(w)f (\(5.9\))g(with)0 3806 y FH(f)41 3818 y FA(K)105 3806 y FG(\()p FH(m)210 3818 y FD(1)247 3806 y FH(;)14 b(m)357 3818 y FD(2)394 3806 y FG(\))28 b FI(=dist)14 b FG(\()p FH(m)772 3818 y FD(1)809 3806 y FH(;)g(m)919 3818 y FD(2)956 3806 y FG(\))28 b FI(strengthens)f(as)g(follo)n(ws:)1027 3978 y FF(k)p FH(\026)18 b FF(\000)g FH(\027)5 b FF(k)1308 3990 y FA(K)1394 3978 y FG(=)23 b(inf)7 b FF(f)p Fk(E)14 b FG(dist)o(\()p FH(\030)1903 3990 y FD(1)1941 3978 y FH(;)g(\030)2014 3990 y FD(2)2052 3978 y FG(\))p FF(g)g FH(;)0 4149 y FI(where)29 b(the)g(in\034m)n(um)h(is)f(tak)n(en)f(o)n(v) n(er)g(all)h(couplings)f FG(\()p FH(\030)1756 4161 y FD(1)1794 4149 y FH(;)14 b(\030)1867 4161 y FD(2)1904 4149 y FG(\))30 b FI(for)f FG(\()p FH(\026;)14 b(\027)5 b FG(\))p FI(.)42 b(Moreo)n(v)n(er,)27 b(the)j(in\034m)n(um)0 4249 y(is)d(attained)f(at)h(some)f(coupling)g FG(\()p FH(\030)1116 4261 y FD(1)1154 4249 y FH(;)14 b(\030)1227 4261 y FD(2)1265 4249 y FG(\))p FI(.)37 b(This)27 b(is)f(the)h (assertion)f(of)g(the)i(celebrated)e FC(Kantor)l(ovich)0 4348 y(The)l(or)l(em)p FI(,)j(see)e([KA77)o(])g(and)h([Dud89].)1290 4318 y Fo(4)p 0 4418 1276 4 v 92 4471 a Fn(4)127 4495 y Fm(In)d(his)g(researc)n(h)i(on)e(the)h(mass-transfer)e(problem)h(L.)f (Kan)n(toro)n(vic)n(h)k(in)n(terpreted)f Fd(E)12 b Fg(dist)q(\()p Fi(\030)2626 4504 y Fc(1)2661 4495 y Fi(;)f(\030)2723 4504 y Fc(2)2758 4495 y Fg(\))25 b Fm(as)g(the)g(w)n(ork,)0 4573 y(needed)g(to)f(transp)r(ort)i(mass)c(p)r(oin)n(ts)h Fi(\030)1048 4582 y Fc(1)1083 4573 y Fg(\()p Fi(!)r Fg(\))h Fm(to)g(the)h(p)r(oin)n(ts)e Fi(\030)1657 4582 y Fc(2)1692 4573 y Fg(\()p Fi(!)r Fg(\))p Fm(,)h(and)g(used)g(the)g(equalit)n(y)g (ab)r(o)n(v)n(e)h(to)f(estimate)f(the)0 4652 y(w)n(ork)g(via)e(the)i (distance)f(b)r(et)n(w)n(een)i(the)e(measures)g Fi(\026)g Fm(and)g Fi(\027)k Fm(\(w)n(e)c(use)g(the)g(equalit)n(y)h(other)g(w)n (a)n(y)f(round,)h(i.e.)30 b(as)22 b(a)f(to)r(ol)0 4731 y(to)h(estimate)f(this)h(distance\).)32 b(This)21 b(researc)n(h)i(is)e (an)h(imp)r(ortan)n(t)g(part)h(of)e(the)h(study)g(of)f(the)i(optimal)d (pro)r(duction)k(for)0 4810 y(whic)n(h)g(in)f(1975)i(L.)e(Kan)n(toro)n (vic)n(h)j(w)n(as)d(giv)n(en)i(the)f(Nob)r(el)g(prise)g(in)f (economics.)1553 5059 y FI(47)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 48 47 bop 0 523 a FK(6)131 b(Uniqueness)44 b(of)g(a)g(stationary)g(measure:)57 b(kic)l(k-forces)0 721 y Fy(6.1)112 b(The)38 b(main)e(lemma)0 875 y FI(Consider)27 b(the)g(randomly)g(kic)n(k)n(ed)g(NSE)h(\(3.3\))f(and)h(the)g(corresp)r (onding)d(discrete)i(time)h(RDS:)714 1053 y FH(u)p FG(\()p FH(k)s FG(\))23 b(=)g FH(S)1039 985 y Fz(\000)1077 1053 y FH(u)p FG(\()p FH(k)e FF(\000)d FG(1\))1378 985 y Fz(\001)1434 1053 y FG(+)g FH(\021)1558 1065 y FA(k)1613 1053 y FH(;)180 b(\021)1857 1065 y FA(k)1921 1053 y FG(=)2009 974 y Fz(X)2053 1148 y FA(s)2156 1053 y FH(b)2192 1065 y FA(s)2241 1053 y FH(\030)2277 1065 y FA(sk)2364 1053 y FH(e)2403 1065 y FA(s)2452 1053 y FH(:)543 b FI(\(6.1\))0 1289 y(In)31 b(this)g(section)f(w)n(e)g(pro)n(v)n(e)f(that)i(the)g(system)f(\(6.1\)) g(is)h FC(exp)l(onential)t(ly)i(mixing)e FI(in)g(the)g(sense)f(that)0 1388 y(distributions)21 b(of)f(all)h(its)g(solutions)f(con)n(v)n(erge)f (exp)r(onen)n(tially)h(fast)h(to)f(a)h(stationary)e(measure.)34 b(Next)0 1488 y(w)n(e)c(deriv)n(e)g(from)g(this)h(result)g(some)f (immediate)g(consequences.)45 b(The)31 b(pro)r(of)f(of)h(the)g(exp)r (onen)n(tial)0 1587 y(mixing)f(is)f(the)h(most)g(tec)n(hnical)f(part)g (of)h(the)g(b)r(o)r(ok.)43 b(T)-7 b(o)29 b(simplify)i(its)f(reading)e (w)n(e)h(conclude)h(this)0 1687 y(section)d(with)h(an)f(app)r(endix,)h (con)n(taining)f(a)g(summary)g(of)g(the)h(pro)r(of.)125 1787 y(Let)f(us)h(denote)f(the)h(underlying)f(probabilit)n(y)g(space)f FG(\(\012)1941 1799 y FD(1)1979 1787 y FH(;)14 b FF(F)2076 1799 y FD(1)2113 1787 y FH(;)g Fk(P)2215 1799 y FD(1)2252 1787 y FG(\))p FI(.)125 1886 y(Recall)33 b(that)g(the)h(time-one)g (transition)e(function)i(for)g(the)f(corresp)r(onding)f(Mark)n(o)n(v)f (c)n(hain)i(is)0 1986 y FH(P)53 1998 y FD(1)91 1986 y FG(\()p FH(u;)14 b FF(\001)p FG(\))23 b(=)f FF(D)r FG(\()p FH(S)5 b FG(\()p FH(u)p FG(\))19 b(+)f FH(\021)782 1998 y FD(1)820 1986 y FG(\))28 b FI(\(see)f(\(3.10\))o(\).)0 2135 y FJ(Lemma)35 b(6.1.)42 b FC(Ther)l(e)35 b(exists)e(a)i(pr)l(ob)l (ability)h(sp)l(ac)l(e)e FG(\(\012)p FH(;)14 b FF(F)8 b FH(;)14 b Fk(P)p FG(\))34 b FC(such)g(that)g FF(8)p FH(R)e FF(\025)e FG(1)p FC(,)k(ther)l(e)g(exists)0 2234 y FH(N)e FG(=)22 b FH(N)9 b FG(\()p FH(R)q FG(\))24 b FF(\025)e FG(1)29 b FC(with)i(the)f(fol)t(lowing)i(pr)l(op)l(erty:)39 b(If)1192 2396 y FH(b)1228 2408 y FA(s)1286 2396 y FF(6)p FG(=)23 b(0)84 b FF(8)14 b(j)p FH(s)p FF(j)23 b(\024)g FH(N)9 b FG(\()p FH(R)q FG(\))14 b FH(;)1021 b FI(\(6.2\))0 2557 y FC(then)29 b(for)h(any)f FH(u)522 2569 y FD(1)559 2557 y FH(;)14 b(u)644 2569 y FD(2)704 2557 y FF(2)23 b FB(B)855 2569 y FA(R)910 2557 y FG(\()p FH(H)7 b FG(\))23 b(=:)g FB(B)1257 2569 y FA(R)1341 2557 y FC(the)29 b(me)l(asur)l(es)g FH(\026)1883 2569 y FD(1)1943 2557 y FG(=)23 b FH(P)2084 2569 y FD(1)2121 2557 y FG(\()p FH(u)2201 2569 y FD(1)2238 2557 y FH(;)14 b FF(\001)p FG(\))p FC(,)30 b FH(\026)2435 2569 y FD(2)2495 2557 y FG(=)23 b FH(P)2636 2569 y FD(1)2673 2557 y FG(\()p FH(u)2753 2569 y FD(2)2791 2557 y FH(;)14 b FF(\001)p FG(\))29 b FC(admit)h(a)0 2657 y(c)l(oupling)h FG(\()p FH(V)405 2669 y FD(1)442 2657 y FH(;)14 b(V)527 2669 y FD(2)565 2657 y FG(\))p FC(,)31 b(wher)l(e)f FH(V)935 2669 y FA(j)993 2657 y FG(=)23 b FH(V)1129 2669 y FA(j)1164 2657 y FG(\()p FH(u)1244 2669 y FD(1)1281 2657 y FH(;)14 b(u)1366 2669 y FD(2)1403 2657 y FG(;)g FH(!)s FG(\))30 b(\()p FH(j)e FG(=)23 b(1)p FH(;)14 b FG(2\))p FC(,)29 b(and)0 2756 y FI(\(a\))h FC(the)g(maps)g FH(V)538 2768 y FD(1)576 2756 y FH(;)14 b(V)661 2768 y FD(2)721 2756 y FG(:)23 b FB(B)840 2768 y FA(R)914 2756 y FF(\002)18 b FB(B)1070 2768 y FA(R)1143 2756 y FF(\002)g FG(\012)23 b FF(!)g FH(H)37 b FC(ar)l(e)30 b(me)l(asur)l(able,)0 2856 y FI(\(b\))g FC(denoting)h FH(d)23 b FG(=)g FF(j)p FH(u)703 2868 y FD(1)758 2856 y FF(\000)18 b FH(u)889 2868 y FD(2)926 2856 y FF(j)29 b FC(we)h(have)1077 3056 y Fk(P)1142 2964 y Fz(n)1197 3056 y FF(j)p FH(V)1268 3068 y FD(1)1325 3056 y FF(\000)18 b FH(V)1456 3068 y FD(2)1493 3056 y FF(j)23 b(\025)1637 3000 y FG(1)p 1637 3037 42 4 v 1637 3113 a(2)1712 3056 y FH(d)1755 2964 y Fz(o)1833 3056 y FF(\024)g FH(C)1980 3068 y Fw(\003)2032 3056 y FH(d)14 b(;)0 3241 y FC(wher)l(e)30 b FH(C)293 3253 y Fw(\003)362 3241 y FC(dep)l(ends)g(on)g FH(R)q FC(,)g FH(B)972 3253 y FD(0)1039 3241 y FC(and)g FH(b)1236 3253 y FD(1)1273 3241 y FH(;)14 b(:)g(:)g(:)f(;)h(b)1493 3253 y FA(N)1556 3241 y FC(.)0 3390 y(Pr)l(o)l(of.)43 b FI(Let)28 b(us)f(denote)h(b)n(y)f FH(P)951 3402 y FA(N)1042 3390 y FI(and)g FH(Q)1269 3402 y FA(N)1360 3390 y FI(the)h(orthogonal)d (pro)5 b(jectors)903 3551 y FH(P)956 3563 y FA(N)1042 3551 y FG(:)p FH(H)30 b FF(\000)-14 b(!)23 b FG(span)o FF(f)p FH(e)1568 3563 y FA(s)1603 3551 y FH(;)28 b FF(j)p FH(s)p FF(j)23 b(\024)g FH(N)9 b FF(g)22 b FG(=)h FH(H)2147 3566 y FD(\()p FA(N)6 b FD(\))2275 3551 y FH(;)891 3729 y(Q)957 3741 y FA(N)1042 3729 y FG(:)p FH(H)30 b FF(\000)-14 b(!)p 1321 3656 646 4 v 23 w FG(span)o FF(f)p FH(e)1568 3741 y FA(s)1603 3729 y FH(;)28 b FF(j)p FH(s)p FF(j)23 b FH(>)g(N)9 b FF(g)k FH(:)0 3890 y FI(W)-7 b(e)28 b(are)f(lo)r(oking)f (for)h(random)g(v)-5 b(ariables)26 b FH(V)1396 3902 y FD(1)1434 3890 y FH(;)14 b(V)1519 3902 y FD(2)1584 3890 y FI(in)28 b(the)g(follo)n(wing)e(form)958 4051 y FH(V)1006 4063 y FD(1)1067 4051 y FG(=)c FH(S)5 b FG(\()p FH(u)1290 4063 y FD(1)1327 4051 y FG(\))19 b(+)f FH(\030)1497 4063 y FD(1)1535 4051 y FH(;)60 b(V)1666 4063 y FD(2)1726 4051 y FG(=)23 b FH(S)5 b FG(\()p FH(u)1950 4063 y FD(2)1987 4051 y FG(\))18 b(+)g FH(\030)2156 4063 y FD(2)2208 4051 y FH(;)787 b FI(\(6.3\))0 4212 y(where)36 b FH(\030)285 4224 y FD(1)323 4212 y FH(;)14 b(\030)396 4224 y FD(2)471 4212 y FF(2)39 b FH(H)k FI(are)36 b(random)f(v)-5 b(ariables)35 b(on)i FG(\012)p FI(,)i(distributed)e(as)f FH(\021)2324 4224 y FD(1)2361 4212 y FI(.)64 b(Clearly)35 b FG(\()p FH(V)2825 4224 y FD(1)2863 4212 y FH(;)14 b(V)2948 4224 y FD(2)2986 4212 y FG(\))37 b FI(is)f(a)0 4312 y(coupling)29 b(for)h FG(\()p FH(\026)546 4324 y FD(1)583 4312 y FH(;)14 b(\026)670 4324 y FD(2)708 4312 y FG(\))p FI(.)44 b(T)-7 b(o)30 b(de\034ne)g(the)g(random)f(v)-5 b(ariables)29 b FH(\030)2009 4324 y FD(1)2046 4312 y FH(;)14 b(\030)2119 4324 y FD(2)2187 4312 y FI(w)n(e)29 b(sp)r(ecify)i(their)e(pro)5 b(jections)0 4412 y FH(P)53 4424 y FA(N)130 4412 y FH(\030)166 4424 y FA(j)229 4412 y FI(and)27 b FH(Q)456 4424 y FA(N)533 4412 y FH(\030)569 4424 y FA(j)632 4412 y FG(\()p FH(j)h FG(=)23 b(1)p FH(;)14 b FG(2\))p FI(,)27 b(where)g FH(N)36 b FI(is)28 b(to)f(b)r(e)h(c)n(hosen)f(b)r(elo)n(w.)125 4511 y(W)-7 b(e)19 b(construct)g(the)h(probabilit)n(y)f(space)g FG(\(\012)p FH(;)14 b FF(F)8 b FH(;)14 b Fk(P)p FG(\))20 b FI(for)f FH(\030)1888 4523 y FD(1)1925 4511 y FH(;)14 b(\030)1998 4523 y FD(2)2055 4511 y FI(as)19 b FG(\012)k(=)g(\012)2380 4523 y FD(1)2419 4511 y FF(\002)r FG(\012)2546 4523 y FD(2)2584 4511 y FI(,)e(where)e(the)h(space)0 4611 y FG(\(\012)92 4623 y FD(2)129 4611 y FH(;)14 b FF(F)226 4623 y FD(2)263 4611 y FH(;)g Fk(P)365 4623 y FD(2)403 4611 y FG(\))27 b FI(is)g(de\034ned)h(b)r(elo)n(w,)f(and)g(extend)g FH(\021)1560 4623 y FD(1)1625 4611 y FI(in)g(the)h(natural)e(w)n(a)n(y) g(to)h(a)g(random)f(v)-5 b(ariable)p 3108 4565 45 4 v 27 w FH(\021)3152 4631 y FD(1)0 4710 y FI(on)27 b FG(\(\012)p FH(;)14 b FF(F)8 b FH(;)14 b Fk(P)p FG(\))p FI(.)38 b(W)-7 b(e)28 b(set)1137 4810 y FH(Q)1203 4822 y FA(N)1280 4810 y FH(\030)1316 4822 y FD(1)1376 4810 y FG(=)23 b FH(Q)1530 4822 y FA(N)1607 4810 y FH(\030)1643 4822 y FD(2)1703 4810 y FG(=)g FH(Q)1857 4822 y FA(N)p 1933 4764 V 1933 4810 a FH(\021)1977 4830 y FD(1)2029 4810 y FH(:)1553 5059 y FI(48)p eop end PStoPSsaved restore %%Page: (48,49) 25 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 49 48 bop 0 523 a FI(T)-7 b(o)27 b(de\034ne)h FH(P)415 535 y FA(N)492 523 y FH(\030)528 535 y FD(1)566 523 y FI(,)f FH(P)669 535 y FA(N)746 523 y FH(\030)782 535 y FD(2)848 523 y FI(let)h(us)f(denote)h FH(\027)1383 535 y FA(j)1441 523 y FG(=)22 b FH(P)1581 535 y FA(N)1663 523 y FF(\016)c FH(\026)1773 535 y FA(j)1808 523 y FI(,)28 b FH(j)g FG(=)22 b(1)p FH(;)14 b FG(2)p FI(.)36 b(W)-7 b(e)28 b(claim)f(that)1226 706 y FF(k)p FH(\027)1309 718 y FD(1)1364 706 y FF(\000)18 b FH(\027)1488 718 y FD(2)1525 706 y FF(k)1567 718 y FD(v)l(ar)1684 706 y FF(\024)23 b FH(C)1831 718 y Fw(\003)1883 706 y FH(d)14 b(:)1055 b FI(\(6.4\))0 888 y(Indeed,)28 b(let)g(us)f(denote)h FH(v)827 900 y FA(j)885 888 y FG(=)23 b FH(P)1026 900 y FA(N)1103 888 y FH(S)5 b FG(\()p FH(u)1239 900 y FA(j)1273 888 y FG(\))p FI(,)28 b FH(j)g FG(=)23 b(1)p FH(;)14 b FG(2)p FI(.)36 b(Then,)27 b(due)h(to)g(\(S2\),)1244 1071 y FF(j)p FH(v)1307 1083 y FD(1)1363 1071 y FF(\000)18 b FH(v)1486 1083 y FD(2)1524 1071 y FF(j)23 b(\024)f FH(C)6 b FG(\()p FH(R)q FG(\))14 b FH(d)g(:)1074 b FI(\(6.5\))0 1254 y(Since)34 b FH(b)259 1266 y FA(s)328 1254 y FF(6)p FG(=)f(0)g FI(for)h FF(j)p FH(s)p FF(j)f(\024)h FH(N)42 b FI(and)34 b FF(D)r FG(\()p FH(\030)1263 1266 y FA(sk)1336 1254 y FG(\))g(=)f FH(p)1542 1266 y FA(s)1577 1254 y FG(\()p FH(r)r FG(\))p FH(dr)r FI(,)38 b(where)c(the)g(functions)g FH(p)2625 1266 y FA(s)2694 1254 y FI(are)f(Lipsc)n(hitz,)0 1353 y(then)1038 1453 y FF(D)r FG(\()p FH(P)1189 1465 y FA(N)1267 1453 y FH(\021)1308 1465 y FD(1)1345 1453 y FG(\))23 b(=)g FH(q)s FG(\()p FH(x)p FG(\))p FH(dx;)62 b(x)23 b FF(2)h FE(R)2023 1419 y FA(N)2076 1427 y Fr(\003)2128 1453 y FH(:)0 1602 y FI(Here)j FH(N)263 1614 y Fw(\003)324 1602 y FG(=)c(dim)14 b FH(H)633 1617 y FD(\()p FA(N)6 b FD(\))776 1602 y FI(and)27 b FH(q)k FI(is)c(a)g(Lipsc)n(hitz)h (function.)37 b(Hence,)919 1785 y FF(D)17 b FH(\027)1041 1797 y FA(j)1099 1785 y FG(=)22 b FF(D)r FG(\()p FH(v)1324 1797 y FA(j)1379 1785 y FG(+)c FH(P)1515 1797 y FA(N)1592 1785 y FH(\021)1633 1797 y FD(1)1670 1785 y FG(\))24 b(=)e FH(q)s FG(\()p FH(x)d FF(\000)f FH(v)2074 1797 y FA(j)2110 1785 y FG(\))p FH(dx)c(;)0 1968 y FI(and)27 b(\(6.4\))h(follo)n(ws)e(from)h(\(6.5\))h(and)f(\(5.3\).)125 2067 y(Due)21 b(to)g(\(6.4\))g(and)g(the)h(Dobrushin)f(lemma,)h(the)f (measures)f FH(\027)2082 2079 y FD(1)2120 2067 y FH(;)14 b(\027)2198 2079 y FD(2)2256 2067 y FI(admit)21 b(a)g(maximal)g (coupling)0 2167 y FG(\(\004)87 2179 y FD(1)125 2167 y FH(;)14 b FG(\004)217 2179 y FD(2)254 2167 y FG(\))p FI(,)35 b(de\034ned)e(on)g(a)f(probabilit)n(y)g(space)g FG(\(\012)1579 2179 y FD(2)1617 2167 y FH(;)14 b FF(F)1714 2179 y FD(2)1751 2167 y FH(;)g Fk(P)1853 2179 y FD(2)1890 2167 y FG(\))p FI(,)35 b(and)d(dep)r(ending)i(on)e(the)i(parameter)0 2267 y FG(\()p FH(u)80 2279 y FD(1)117 2267 y FH(;)14 b(u)202 2279 y FD(2)239 2267 y FG(\))p FI(.)37 b(I.e.,)28 b FG(\004)550 2279 y FA(j)608 2267 y FG(=)23 b(\004)751 2279 y FA(j)786 2267 y FG(\()p FH(!)870 2279 y FD(2)907 2267 y FG(;)14 b FH(u)992 2279 y FD(1)1029 2267 y FH(;)g(u)1114 2279 y FD(2)1151 2267 y FG(\))28 b FI(and)933 2449 y Fk(P)998 2461 y FD(2)1036 2382 y Fz(\000)1074 2449 y FG(\004)1129 2461 y FD(1)1189 2449 y FF(6)p FG(=)23 b(\004)1332 2461 y FD(2)1370 2382 y Fz(\001)1431 2449 y FG(=)f FF(k)p FH(\027)1601 2461 y FD(1)1656 2449 y FF(\000)c FH(\027)1780 2461 y FD(2)1818 2449 y FF(k)1860 2461 y FD(v)l(ar)1977 2449 y FF(\024)k FH(C)2123 2461 y Fw(\003)2176 2449 y FH(d)14 b(:)762 b FI(\(6.6\))0 2632 y(Moreo)n(v)n(er,)25 b(the)j(maps)f FG(\004)803 2644 y FA(j)861 2632 y FG(:)c(\012)967 2644 y FD(2)1023 2632 y FF(\002)18 b FB(B)1179 2644 y FA(R)1252 2632 y FF(\002)g FB(B)1408 2644 y FA(R)1486 2632 y FF(!)23 b FH(H)1661 2647 y FD(\()p FA(N)6 b FD(\))1804 2632 y FI(are)27 b(measurable)f(\(see)h([KS01a)n(]\).)125 2743 y(Denoting)g(b)n(y)p 598 2676 56 4 v 27 w FG(\004)654 2755 y FA(j)689 2743 y FI(,)g FH(j)h FG(=)23 b(1)p FH(;)14 b FG(2)p FI(,)27 b(the)h(natural)e(extension)h(of)h FG(\004)2010 2755 y FA(j)2073 2743 y FI(to)f FG(\012)p FI(,)h(w)n(e)f(set)1158 2926 y FH(P)1211 2938 y FA(N)1288 2926 y FH(\030)1324 2938 y FA(j)1382 2926 y FG(=)p 1470 2859 V 23 w(\004)1525 2938 y FA(j)1579 2926 y FF(\000)18 b FH(P)1715 2938 y FA(N)1792 2926 y FH(S)5 b FG(\()p FH(u)1928 2938 y FA(j)1962 2926 y FG(\))14 b FH(:)0 3108 y FI(Since)33 b FH(P)275 3120 y FA(N)352 3108 y FH(\030)388 3120 y FA(j)456 3108 y FI(and)f FH(Q)688 3120 y FA(N)764 3108 y FH(\030)800 3120 y FA(j)868 3108 y FI(are)f(sp)r(eci\034ed,)j(w)n(e)e(no)n(w)g(can) g(construct)g(the)h(random)e(v)-5 b(ariable)32 b FH(V)3029 3120 y FD(1)3066 3108 y FH(;)14 b(V)3151 3120 y FD(2)0 3208 y FI(\(see)27 b(\(6.3\)\).)37 b(Clearly)-7 b(,)p 734 3141 V 27 w FG(\004)789 3220 y FD(1)850 3208 y FG(=)p 937 3141 V 22 w(\004)993 3220 y FD(2)1057 3208 y FI(implies)28 b(that)g FH(P)1572 3220 y FA(N)1649 3208 y FH(V)1697 3220 y FD(1)1758 3208 y FG(=)22 b FH(P)1898 3220 y FA(N)1976 3208 y FH(V)2024 3220 y FD(2)2061 3208 y FI(,)28 b(so)f(that)h(in)g (this)f(case)652 3391 y FH(V)700 3403 y FD(1)756 3391 y FF(\000)18 b FH(V)887 3403 y FD(2)948 3391 y FG(=)k FH(Q)1101 3403 y FA(N)1178 3391 y FH(V)1226 3403 y FD(1)1282 3391 y FF(\000)c FH(Q)1431 3403 y FA(N)1507 3391 y FH(V)1555 3403 y FD(2)1616 3391 y FG(=)23 b FH(Q)1770 3403 y FA(N)1846 3391 y FH(S)5 b FG(\()p FH(u)1982 3403 y FD(1)2019 3391 y FG(\))19 b FF(\000)f FH(Q)2219 3403 y FA(N)2295 3391 y FH(S)5 b FG(\()p FH(u)2431 3403 y FD(2)2468 3391 y FG(\))14 b FH(:)0 3573 y FI(Using)27 b(\(S3\),)h(w)n(e)f(ha)n(v)n(e)g (for)p 877 3506 V 27 w FG(\004)932 3585 y FD(1)992 3573 y FG(=)p 1080 3506 V 23 w(\004)1135 3585 y FD(2)1173 3573 y FI(,)266 3756 y FF(j)p FH(Q)355 3768 y FA(N)432 3756 y FH(S)5 b FG(\()p FH(u)568 3768 y FD(1)604 3756 y FG(\))19 b FF(\000)f FH(Q)804 3768 y FA(N)881 3756 y FH(S)5 b FG(\()p FH(u)1017 3768 y FD(2)1053 3756 y FG(\))p FF(j)24 b(\024)e FH(N)1295 3722 y Fw(\000)p FD(1)1398 3756 y FF(k)p FH(Q)1506 3768 y FA(N)1582 3756 y FH(S)5 b FG(\()p FH(u)1718 3768 y FD(1)1755 3756 y FG(\))18 b FF(\000)g FH(Q)1954 3768 y FA(N)2031 3756 y FH(S)5 b FG(\()p FH(u)2167 3768 y FD(2)2204 3756 y FG(\))p FF(k)2278 3768 y FD(1)2338 3756 y FF(\024)22 b FH(N)2501 3722 y Fw(\000)p FD(1)2604 3756 y FH(C)2663 3768 y FD(1)2701 3756 y FG(\()p FH(R)q FG(\))14 b FH(d)g(:)0 3950 y FI(Cho)r(osing)30 b FH(N)37 b FF(\025)29 b FG(2)14 b FH(C)679 3962 y FD(1)716 3950 y FG(\()p FH(R)q FG(\))31 b FI(w)n(e)g(see)f(that)h FF(j)p FH(V)1392 3962 y FD(1)1451 3950 y FF(\000)20 b FH(V)1584 3962 y FD(2)1622 3950 y FF(j)29 b(\024)1777 3918 y FD(1)p 1777 3932 34 4 v 1777 3979 a(2)1834 3950 y FH(d)p FI(,)j(if)p 2012 3884 56 4 v 32 w FG(\004)2067 3962 y FD(1)2133 3950 y FG(=)p 2226 3884 V 28 w(\004)2282 3962 y FD(2)2319 3950 y FI(.)47 b(No)n(w)31 b(the)g(assertion)e(\(b\))0 4061 y(follo)n(ws)d(from)h(\(6.6\).)36 b(Since)28 b(the)f(maps)p 1274 3994 V 27 w FG(\004)1329 4073 y FD(1)1367 4061 y FH(;)p 1404 3994 V 14 w FG(\004)1459 4073 y FD(2)1523 4061 y FI(are)g(measurable,)f(then)h FH(V)2355 4073 y FD(1)2393 4061 y FH(;)14 b(V)2478 4073 y FD(2)2543 4061 y FI(are)26 b(measurable)g(as)0 4160 y(w)n(ell,)h(and)h(the)g(lemma)f (is)h(pro)n(v)n(ed.)p 3128 4160 4 57 v 3132 4108 50 4 v 3132 4160 V 3182 4160 4 57 v 125 4326 a(W)-7 b(e)27 b(call)h(the)g(measurable)e(maps)h FH(V)1262 4338 y FD(1)1300 4326 y FH(;)14 b(V)1385 4338 y FD(2)1450 4326 y FI(the)28 b FC(c)l(oupling)i(maps)p FI(.)0 4492 y FJ(Exercise)21 b(6.2.)32 b FC(Pr)l(ove)22 b(that)g(\(6.4\))g(stil)t(l)h(holds)f(if)h FH(p)1612 4504 y FA(s)1647 4492 y FG(\()p FH(r)r FG(\))g FC(ar)l(e)f(functions)f(of)h(b)l(ounde)l(d)g(total)g(variation.)1553 5059 y FI(49)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 50 49 bop 0 523 a FC(Pr)l(o)l(of.)43 b FI(Assume)28 b(\034rst)f(that)h FH(q)j FI(is)c(smo)r(oth.)37 b(Then)501 616 y Fz(Z)548 804 y FA(H)602 812 y Fu(N)673 729 y FF(j)p FH(q)s FG(\()p FH(x)20 b FF(\000)e FH(v)958 741 y FD(1)995 729 y FG(\))h FF(\000)f FH(q)s FG(\()p FH(x)h FF(\000)f FH(v)1390 741 y FD(2)1428 729 y FG(\))p FF(j)c FH(dx)501 1006 y FF(\024)23 b(j)p FH(v)652 1018 y FD(1)708 1006 y FF(\000)18 b FH(v)831 1018 y FD(2)869 1006 y FF(j)919 893 y Fz(Z)965 1082 y FA(H)1019 1090 y Fu(N)1114 893 y Fz(Z)1197 914 y FD(1)1160 1082 y(0)1248 1006 y FF(jr)p FH(q)s FG(\()p FH(x)i FF(\000)e FH(\022)r(v)1643 1018 y FD(1)1699 1006 y FG(+)g(\(1)g FF(\000)g FH(\022)r FG(\))p FH(v)2070 1018 y FD(2)2108 1006 y FG(\))c FF(j)g FH(d\022)j(dx)501 1307 y FG(=)23 b FF(j)p FH(v)652 1319 y FD(1)708 1307 y FF(\000)18 b FH(v)831 1319 y FD(2)869 1307 y FF(j)906 1194 y Fz(Z)952 1383 y FA(H)1006 1391 y Fu(N)1077 1307 y FF(jr)p FH(q)s FG(\()p FH(x)p FG(\))p FF(j)c FH(dx)25 b FF(\024)e(j)p FH(v)1623 1319 y FD(1)1679 1307 y FF(\000)18 b FH(v)1802 1319 y FD(2)1839 1307 y FF(j)1893 1202 y FA(N)1946 1210 y Fr(\003)1876 1228 y Fz(X)1879 1405 y FA(j)s FD(=1)2033 1194 y Fz(Z)2079 1383 y FA(R)2147 1307 y FF(j)p FH(@)2214 1319 y FA(x)2252 1327 y Fu(j)2301 1307 y FH(p)2343 1319 y FA(j)2378 1307 y FG(\()p FH(x)2457 1319 y FA(j)2493 1307 y FG(\))c FH(dx)2629 1319 y FA(j)2665 1307 y FF(j)501 1576 y FG(=)23 b FF(j)p FH(v)652 1588 y FD(1)708 1576 y FF(\000)18 b FH(v)831 1588 y FD(2)869 1576 y FF(j)919 1497 y Fz(X)1067 1576 y FG(v)-5 b(ar)13 b FH(q)1231 1588 y FA(j)1289 1576 y FG(=)23 b FF(j)p FH(v)1440 1588 y FD(1)1496 1576 y FF(\000)18 b FH(v)1619 1588 y FD(2)1656 1576 y FF(j)1707 1497 y Fz(X)1855 1576 y FH(b)1891 1540 y Fw(\000)p FD(1)1891 1599 y FA(j)1993 1576 y FG(v)-5 b(ar)13 b FH(p)2162 1588 y FA(j)2211 1576 y FH(;)0 1779 y FI(where)40 b FH(q)s FG(\()p FH(x)p FG(\))46 b(=)559 1717 y Fz(Q)637 1737 y FA(N)690 1745 y Fr(\003)637 1804 y FA(j)s FD(=1)770 1779 y FH(q)807 1791 y FA(j)842 1779 y FG(\()p FH(x)921 1791 y FA(j)957 1779 y FG(\))41 b FI(with)g FH(q)1269 1791 y FA(j)1304 1779 y FG(\()p FH(x)1383 1791 y FA(j)1418 1779 y FG(\))k(=)g FH(b)1641 1743 y Fw(\000)p FD(1)1641 1802 y FA(j)1743 1779 y FH(p)1785 1791 y FA(j)1820 1779 y FG(\()p FH(x)1899 1791 y FA(j)1935 1779 y FH(=b)2013 1791 y FA(j)2047 1779 y FG(\))p FI(.)76 b(If)41 b(the)g(functions)f FH(p)2842 1791 y FA(j)2918 1779 y FI(are)f(not)0 1879 y(smo)r(oth,)30 b(w)n(e)g(appro)n(ximate)e(them)i(b)n(y)g(smo)r(oth)f (functions)i(and)e(go)g(to)h(the)g(limit)h(in)f(the)g(estimate)0 1978 y(ab)r(o)n(v)n(e.)p 3128 1978 4 57 v 3132 1926 50 4 v 3132 1978 V 3182 1978 4 57 v 0 2208 a Fy(6.2)112 b(W)-9 b(eak)38 b(solution)e(of)h(\(6.1\))0 2361 y FJ(De\034nition)h (6.3.)43 b FC(A)35 b(pr)l(o)l(c)l(ess)h FF(f)p FH(u)p FG(\()p FH(k)s FG(\))c FF(2)h FH(H)7 b FC(,)37 b FH(k)f FF(\025)c FH(k)1707 2373 y FD(0)1745 2361 y FF(g)p FC(,)k(de\034ne)l(d) g(on)f(some)h(pr)l(ob)l(ability)h(sp)l(ac)l(e,)h(is)0 2461 y(c)l(al)t(le)l(d)30 b(a)f(we)l(ak)g(solution)f(of)i(\(6.1\),)g (if)f(it)g(satis\034es)f(\(6.1\))i(with)f(the)g(pr)l(o)l(c)l(ess)f(of)i (kicks)f FF(f)p FH(\021)2793 2473 y FA(k)2834 2461 y FF(g)f FC(r)l(eplac)l(e)l(d)0 2561 y(by)i(some)g(other)h(pr)l(o)l(c)l (ess)f FF(f)p FH(\021)903 2530 y Fw(0)900 2584 y FA(k)940 2561 y FF(g)p FC(,)g(distribute)l(d)g(as)g FF(f)p FH(\021)1632 2573 y FA(k)1673 2561 y FF(g)p FC(.)125 2715 y FI(Let)21 b FF(f)p FH(u)357 2727 y FD(1)394 2715 y FG(\()p FH(k)s FG(\))p FF(g)h FI(and)f FF(f)p FH(u)813 2727 y FD(2)849 2715 y FG(\()p FH(k)s FG(\))p FF(g)h FI(b)r(e)g(t)n(w)n(o)f(w)n(eak)g (solutions)g(de\034ned)h(on)g(the)g(same)f(probabilit)n(y)g(space.)0 2814 y(P)n(ossibly)-7 b(,)21 b(the)g(corresp)r(onding)e(t)n(w)n(o)h (pro)r(cesses)f FF(f)p FH(\021)1586 2784 y Fw(0)1583 2838 y FA(k)1623 2814 y FF(g)h FI(are)g(di\033eren)n(t)h(\(but)g(they)g (ma)n(y)f(b)r(e)h(correlated\).)0 2914 y(W)-7 b(e)28 b(set)687 3013 y FH(d)p FG(\()p FH(k)s FG(\))c(=)f FF(j)p FH(u)1023 3025 y FD(1)1060 3013 y FG(\()p FH(k)s FG(\))18 b FF(\000)g FH(u)1319 3025 y FD(2)1356 3013 y FG(\()p FH(k)s FG(\))p FF(j)p FH(;)84 b(R)q FG(\()p FH(k)s FG(\))23 b(=)g FF(j)p FH(u)1952 3025 y FD(1)1988 3013 y FG(\()p FH(k)s FG(\))p FF(j)c FG(+)f FF(j)p FH(u)2294 3025 y FD(2)2331 3013 y FG(\()p FH(k)s FG(\))p FF(j)c FH(:)0 3154 y FI(Let)28 b FH(u)p FG(\()p FH(k)s FG(\))f FI(b)r(e)h FH(u)495 3166 y FD(1)532 3154 y FG(\()p FH(k)s FG(\))g FI(or)f FH(u)820 3166 y FD(2)857 3154 y FG(\()p FH(k)s FG(\))p FI(.)37 b(Then)28 b(a.s.)36 b(w)n(e)27 b(ha)n(v)n(e)682 3361 y FF(j)p FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))p FF(j)23 b(\024)g FH(e)1179 3326 y Fw(\000)p FD(1)1281 3361 y FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)c FG(+)f FF(j)p FH(\021)1651 3373 y FA(k)q FD(+1)1776 3361 y FF(j)23 b(\024)g FH(e)1949 3326 y Fw(\000)p FD(1)2052 3361 y FF(j)p FH(u)p FG(\()p FH(k)s FG(\))p FF(j)18 b FG(+)2367 3305 y(1)p 2367 3342 42 4 v 2367 3418 a(2)2458 3340 y Fz(e)2442 3361 y FH(C)517 b FI(\(6.7\))0 3575 y(where)259 3554 y Fz(e)242 3575 y FH(C)33 b FG(=)26 b(2)467 3511 y FF(p)p 536 3511 101 4 v 64 x FH(B)599 3587 y FD(0)636 3575 y FI(,)31 b(see)e(Section)h(3.2)f(\(this)h(constan)n(t)f(also)f (dep)r(ends)j(on)e(the)h(viscosit)n(y)e FH(\027)5 b FI(,)31 b(whic)n(h)0 3685 y(no)n(w)g(is)g(assumed)g(to)g(b)r(e)h(one\).)49 b(A)n(ccordingly)-7 b(,)31 b Fk(E)14 b FH(R)q FG(\()p FH(k)23 b FG(+)e(1\))29 b FF(\024)g FH(e)2093 3655 y Fw(\000)p FD(1)2196 3685 y Fk(E)14 b FH(R)q FG(\()p FH(k)s FG(\))21 b(+)2570 3664 y Fz(e)2553 3685 y FH(C)7 b FI(.)48 b(Let)32 b(us)f(c)n(ho)r(ose)0 3785 y(an)n(y)c(n)n(um)n(b)r(er)1270 3919 y FH(R)1333 3931 y FD(0)1394 3919 y FF(\025)22 b FG(1)c FF(_)1676 3863 y FG(2)p FH(e)p 1625 3900 182 4 v 1625 3976 a(e)g FF(\000)g FG(1)1847 3898 y Fz(e)1830 3919 y FH(C)7 b(:)0 4089 y FI(Then)667 4207 y Fk(E)14 b FH(R)q FG(\()p FH(k)s FG(\))23 b FF(\025)g FH(R)1092 4219 y FD(0)1152 4207 y FG(=)-14 b FF(\))23 b Fk(E)14 b FH(R)q FG(\()p FH(`)p FG(\))22 b FF(\024)1669 4151 y FG(1)p 1669 4188 42 4 v 1669 4264 a(2)1734 4207 y Fk(E)14 b FH(R)q FG(\()p FH(k)s FG(\))p FH(;)97 b FF(8)p FH(`)22 b FF(\025)h FH(k)e FG(+)d(1)c FH(:)496 b FI(\(6.8\))0 4375 y(Similarly)-7 b(,)27 b(\(6.7\))g(implies)h(that)873 4543 y FH(R)q FG(\()p FH(k)s FG(\))23 b FF(\024)f FH(R)1220 4555 y FD(0)1281 4543 y FG(=)-14 b FF(\))22 b FH(R)q FG(\()p FH(`)p FG(\))h FF(\024)g FH(R)1774 4555 y FD(0)1811 4543 y FH(;)51 b FI(a.s.)36 b FF(8)p FH(`)23 b FF(\025)g FH(k)16 b(:)702 b FI(\(6.9\))125 4710 y(Let)38 b(us)g(\034x)g(some)f FG(0)j FH(<)g(d)977 4722 y FD(0)1055 4710 y FF(\024)g FG(1)p FI(.)68 b(Let)38 b FF(f)p FH(u)1542 4722 y FD(1)1578 4710 y FG(\()p FH(k)s FG(\))p FF(g)g FI(and)g FF(f)p FH(u)2030 4722 y FD(2)2066 4710 y FG(\()p FH(k)s FG(\))p FF(g)g FI(b)r(e)h(t)n(w)n(o)e(w)n(eak)g(solutions)g(of)0 4810 y(\(6.1\),)d(corresp)r(onding)c(to)j(the)g(same)f(kic)n(ks)g FF(f)p FH(\021)1531 4780 y Fw(0)1554 4830 y FA(k)1595 4810 y FF(g)p FI(.)52 b(Assume)32 b(that)h FH(R)q FG(\(0\))f FF(\024)f FH(R)2570 4822 y FD(0)2607 4810 y FI(.)53 b(Then,)34 b FF(j)p FH(u)3000 4822 y FD(1)3037 4810 y FG(\(0\))p FF(j)p FI(,)1553 5059 y(50)p eop end PStoPSsaved restore %%Page: (50,51) 26 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 51 50 bop 0 523 a FF(j)p FH(u)71 535 y FD(2)108 523 y FG(\(0\))p FF(j)28 b(\024)f FH(R)420 535 y FD(0)457 523 y FI(.)45 b(So,)31 b(if)g FG(0)c(=)g FH(\021)951 493 y Fw(0)948 544 y FD(1)1013 523 y FG(=)g FF(\001)14 b(\001)g(\001)28 b FG(=)f FH(\021)1366 493 y Fw(0)1363 546 y FA(T)1415 523 y FI(,)k(then)g FF(j)p FH(u)1732 535 y FA(j)1767 523 y FG(\()p FH(T)12 b FG(\))p FF(j)27 b(\024)g FG(exp\()p FF(\000)p FH(T)12 b FG(\))p FH(R)2414 535 y FD(0)2450 523 y FI(,)32 b(and)e FF(j)p FH(u)2740 535 y FA(j)2774 523 y FG(\()p FH(T)12 b FG(\))p FF(j)28 b(\024)3052 490 y FD(1)p 3052 504 34 4 v 3052 552 a(4)3109 523 y FH(d)3152 535 y FD(0)0 623 y FI(for)f FH(j)h FG(=)23 b(1)p FH(;)14 b FG(2)p FI(,)26 b(if)1171 722 y FH(T)35 b FG(=)22 b([ln)14 b(\(4)g FH(R)1599 734 y FD(0)1650 722 y FH(d)1693 687 y Fw(\000)p FD(1)1693 744 y(0)1782 722 y FG(\)])19 b(+)f(1)c FH(:)0 871 y FI(Since)28 b Fk(P)p FG(\()p FF(j)p FH(\021)381 841 y Fw(0)378 895 y FA(k)419 871 y FF(j)c(\024)e FH(")p FG(\))h FH(>)g FG(0)k FI(for)g(eac)n(h)g FH(")c(>)f FG(0)27 b FI(\(see)h(Section)f (3.1\),)g(w)n(e)h(ha)n(v)n(e)e(that)1180 1054 y Fk(P)p FG(\()p FH(d)p FG(\()p FH(T)12 b FG(\))23 b FF(\024)g FH(d)1599 1066 y FD(0)1636 1054 y FG(\))g FF(\025)g FH(\022)i(>)e FG(0)14 b FH(;)967 b FI(\(6.10\))0 1236 y(where)27 b FH(\022)j FI(dep)r(ends)e(on)f FH(R)808 1248 y FD(0)873 1236 y FI(and)h FH(d)1078 1248 y FD(0)1115 1236 y FI(.)0 1468 y Fy(6.3)112 b(The)38 b(theorem)0 1622 y FI(Let)28 b(us)f(denote)h(b)n(y)f FF(P)696 1634 y FD(1)733 1622 y FG(\()p FH(H)7 b FG(\))28 b FI(the)g(class)e(of)i(measures)e FH(\026)i FI(with)g(the)g(\034nite)g(\034rst)f(momen)n(t)1103 1846 y FH(M)1184 1858 y FD(1)1221 1846 y FG(\()p FH(\026)p FG(\))c(=)1446 1733 y Fz(Z)1557 1846 y FF(j)p FH(u)p FF(j)14 b FH(\026)p FG(\()p FH(du)p FG(\))22 b FH(<)h FF(1)p FH(:)0 2071 y FJ(Theorem)38 b(6.4.)44 b FC(L)l(et)35 b(the)h(kicks)h FF(f)p FH(\021)1193 2083 y FA(k)1234 2071 y FF(g)e FC(satisfy)i(the)f(assumptions)g(of)g(Se)l(ction)g(3.1.) 58 b(Then)37 b(ther)l(e)0 2171 y(exists)29 b FH(N)j FF(\025)23 b FG(1)p FC(,)29 b(dep)l(ending)i(on)f FH(\027)35 b FC(and)30 b FH(B)1315 2183 y FD(0)1353 2171 y FC(,)g(such)g(that)f(if)1256 2353 y FH(b)1292 2365 y FA(s)1350 2353 y FF(6)p FG(=)23 b(0)84 b FF(8)14 b(j)p FH(s)p FF(j)23 b(\024)g FH(N)g(;)0 2535 y FC(then)30 b(for)g(any)g FH(\026)526 2547 y FD(1)563 2535 y FH(;)14 b(\026)650 2547 y FD(2)711 2535 y FF(2)23 b(P)847 2547 y FD(1)884 2535 y FG(\()p FH(H)7 b FG(\))30 b FC(and)g(e)l(ach)h FH(k)26 b FF(2)d FE(N)30 b FC(we)g(have)693 2717 y FF(kB)793 2683 y Fw(\003)790 2738 y FA(k)844 2717 y FH(\026)894 2729 y FD(1)950 2717 y FF(\000)18 b(B)1091 2683 y Fw(\003)1088 2738 y FA(k)1142 2717 y FH(\026)1192 2729 y FD(2)1229 2717 y FF(k)1271 2683 y Fw(\003)1271 2738 y FA(L)1344 2717 y FF(\024)k FH(C)1496 2650 y Fz(\000)1534 2717 y FG(1)c(+)g FH(M)1758 2729 y FD(1)1795 2717 y FG(\()p FH(\026)1877 2729 y FD(1)1915 2717 y FG(\))h(+)f FH(M)2130 2729 y FD(1)2166 2717 y FG(\()p FH(\026)2248 2729 y FD(2)2286 2717 y FG(\))2318 2650 y Fz(\001)2370 2717 y FH(\024)2418 2683 y FA(k)2473 2717 y FH(;)481 b FI(\(6.11\))0 2900 y FC(wher)l(e)30 b FH(\024)23 b(<)g FG(1)29 b FC(and)h FH(C)g FF(\025)22 b FG(1)30 b FC(dep)l(end)g(on)g FH(\027)5 b FC(,)30 b FH(B)1431 2912 y FD(0)1498 2900 y FC(and)h FF(f)p FH(b)1738 2912 y FA(s)1772 2900 y FH(;)14 b FF(j)p FH(s)p FF(j)23 b(\024)g FH(N)9 b FF(g)p FC(.)0 3065 y(Pr)l(o)l(of.)43 b FJ(Step)32 b(1)27 b FI(\(coupling\).)37 b(Let)28 b(us)f(denote)989 3248 y FH(\026)1039 3260 y FA(j)1074 3248 y FG(\()p FH(n)p FG(\))d(=)e FF(B)1357 3213 y Fw(\003)1354 3268 y FA(n)1399 3248 y FG(\()p FH(\026)1481 3260 y FA(j)1516 3248 y FG(\))p FH(;)60 b(j)28 b FG(=)23 b(1)p FH(;)14 b FG(2)p FH(;)36 b(n)23 b FF(\025)g FG(0)14 b FH(:)0 3430 y FI(T)-7 b(o)27 b(estimate)g(the)g(distances)g FF(k)p FH(\026)1044 3442 y FD(1)1080 3430 y FG(\()p FH(n)p FG(\))18 b FF(\000)f FH(\026)1344 3442 y FD(2)1381 3430 y FG(\()p FH(n)p FG(\))p FF(k)1537 3400 y Fw(\003)1537 3453 y FA(L)1614 3430 y FI(w)n(e)27 b(shall)f(construct)h(sp)r(ecial)g(couplings)2928 3363 y Fz(\000)2966 3430 y FH(U)3023 3442 y FD(1)3060 3430 y FG(\(1\))p FI(,)0 3538 y FH(U)57 3550 y FD(2)94 3538 y FG(\(1\))200 3470 y Fz(\001)238 3538 y FI(,)292 3470 y Fz(\000)330 3538 y FH(U)387 3550 y FD(1)424 3538 y FG(\(2\))p FH(;)14 b(U)624 3550 y FD(2)661 3538 y FG(\(2\))767 3470 y Fz(\001)805 3538 y FH(;)g(:)g(:)g(:)44 b FI(for)30 b(the)h(measures)1618 3470 y Fz(\000)1656 3538 y FH(\026)1706 3550 y FD(1)1743 3538 y FG(\(1\))p FH(;)14 b(\026)1936 3550 y FD(2)1973 3538 y FG(\(1\))2079 3470 y Fz(\001)2117 3538 y FH(;)g(:)g(:)g(:)g FI(.)45 b(The)31 b(construction)e(starts)0 3637 y(with)h(an)n(y)e(coupling)g FG(\()p FH(U)771 3649 y FD(1)809 3637 y FH(;)14 b(U)903 3649 y FD(2)939 3637 y FG(\))30 b FI(for)e FG(\()p FH(\026)1211 3649 y FD(1)1249 3637 y FH(;)14 b(\026)1336 3649 y FD(2)1373 3637 y FG(\))p FI(,)30 b(de\034ned)f(on)g(a)g(space)f FG(\(\012)2248 3607 y FD(0)2285 3637 y FH(;)14 b FF(F)2390 3607 y FD(0)2428 3637 y FH(;)g Fk(P)2530 3607 y FD(0)2567 3637 y FG(\))29 b FI(whic)n(h)g(is)g(a)g(cop)n(y)0 3737 y(of)h(the)h(space)f FG(\(\012)p FH(;)14 b FF(F)8 b FH(;)14 b Fk(P)p FG(\))31 b FI(from)f(Lemma)g(6.1.)45 b(Belo)n(w)29 b(w)n(e)h(use)h(other)f (copies)f FG(\(\012)2582 3707 y FA(j)2618 3737 y FH(;)14 b FF(F)2723 3707 y FA(j)2757 3737 y FH(;)g Fk(P)2859 3707 y FA(j)2895 3737 y FG(\))30 b FI(of)h(this)0 3837 y(space)c(as)g(w)n(ell.)125 3936 y(Let)g(us)h(apply)f(Lemma)g(6.1)g (with)h FH(R)c FG(=)e FH(R)1464 3948 y FD(0)1529 3936 y FI(and)28 b(denote)1306 4153 y FH(d)1349 4165 y FD(0)1410 4153 y FG(=)23 b(1)18 b FF(^)1718 4097 y FG(1)p 1641 4134 195 4 v 1641 4210 a(16)c FH(C)1798 4222 y Fw(\003)1859 4153 y FH(:)0 4381 y FI(Let)20 b FH(V)189 4393 y FD(1)227 4381 y FG(\()p FH(u)307 4393 y FD(1)344 4381 y FH(;)14 b(u)429 4393 y FD(2)465 4381 y FG(;)g FH(!)557 4351 y FD(1)594 4381 y FG(\))p FI(,)22 b FH(V)719 4393 y FD(2)757 4381 y FG(\()p FH(u)837 4393 y FD(1)874 4381 y FH(;)14 b(u)959 4393 y FD(2)995 4381 y FG(;)g FH(!)1087 4351 y FD(1)1124 4381 y FG(\))20 b FI(b)r(e)h(the)f(coupling-maps)e(as)h(in) h(Lemma)g(6.1.)33 b(They)20 b(are)f(de\034ned)0 4491 y(for)27 b FF(j)p FH(u)198 4503 y FD(1)235 4491 y FF(j)p FH(;)14 b FF(j)p FH(u)366 4503 y FD(2)403 4491 y FF(j)23 b(\024)f FH(R)599 4503 y FD(0)637 4491 y FI(.)37 b(W)-7 b(e)28 b(de\034ne)f(maps)1307 4470 y Fz(b)1297 4491 y FH(V)1345 4503 y FD(1)1382 4491 y FH(;)1430 4470 y Fz(b)1419 4491 y FH(V)1467 4503 y FD(2)1532 4491 y FI(as)g(follo)n(ws:)575 4727 y Fz(b)565 4748 y FH(V)613 4760 y FA(j)648 4748 y FG(\()p FH(u)728 4760 y FD(1)765 4748 y FH(;)14 b(u)850 4760 y FD(2)887 4748 y FG(;)g FH(!)979 4714 y FD(1)1016 4748 y FG(\))23 b(=)1159 4606 y Fz(\()1244 4679 y FH(V)1292 4691 y FA(j)1327 4679 y FI(,)28 b(if)g FF(j)p FH(u)1525 4691 y FD(1)1580 4679 y FF(\000)18 b FH(u)1711 4691 y FD(2)1748 4679 y FF(j)23 b(\024)g FH(d)1925 4691 y FD(0)1962 4679 y FH(;)37 b FF(j)p FH(u)2093 4691 y FD(1)2130 4679 y FF(j)18 b FG(+)g FF(j)p FH(u)2325 4691 y FD(2)2362 4679 y FF(j)23 b(\024)g FH(R)2559 4691 y FD(0)1244 4815 y FH(S)5 b FG(\()p FH(u)1380 4827 y FA(j)1414 4815 y FG(\))19 b(+)f FH(\021)s FG(\()p FH(!)1679 4785 y FD(1)1716 4815 y FG(\))37 b FI(otherwise)13 b FH(:)1553 5059 y FI(51)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 52 51 bop 0 523 a FI(Here)27 b FH(\021)s FG(\()p FH(!)327 493 y FD(1)364 523 y FG(\))h FI(is)f(a)f(random)g(v)-5 b(ariable,)27 b(distributed)g(as)f(the)i(kic)n(k)e FH(\021)2095 535 y FD(1)2133 523 y FI(.)37 b(No)n(w)26 b(w)n(e)h(de\034ne)g(the)h (coupling)0 555 y Fz(\000)38 623 y FH(U)95 635 y FD(1)132 623 y FG(\(1\))p FH(;)14 b(U)332 635 y FD(2)369 623 y FG(\(1\))475 555 y Fz(\001)545 623 y FI(on)31 b(the)i(probabilit)n(y)d (space)h FG(\012)1526 593 y FD(0)1585 623 y FF(\002)21 b FG(\012)1731 593 y FD(1)1800 623 y FI(\(equipp)r(ed)33 b(with)f(the)g(pro)r(duct)g FH(\033)s FI(-algebra)0 722 y(and)27 b(the)h(pro)r(duct)g(measure\),)f(b)n(y)g(the)h(follo)n(wing)e (relation:)928 901 y FH(U)994 861 y FA(!)1038 836 y Ft(0)1070 861 y FA(;!)1134 836 y Ft(1)985 924 y FA(j)1170 901 y FG(\(1\))d(=)1397 880 y Fz(b)1387 901 y FH(V)1435 913 y FA(j)1470 901 y FG(\()p FH(U)1559 913 y FD(1)1596 901 y FG(\()p FH(!)1683 867 y FD(0)1721 901 y FG(\))p FH(;)37 b(U)1870 913 y FD(2)1907 901 y FG(\()p FH(!)1994 867 y FD(0)2031 901 y FG(\);)14 b FH(!)2155 867 y FD(1)2192 901 y FG(\))g FH(:)0 1059 y FI(Then,)28 b(b)n(y)f(the)h(de\034nition)g (of)f(the)h(coupling)f(maps)h FG(\()p FH(V)1734 1071 y FD(1)1772 1059 y FH(;)14 b(V)1857 1071 y FD(2)1894 1059 y FG(\))p FI(,)896 1238 y FH(U)962 1198 y FA(!)1006 1173 y Ft(0)1038 1198 y FA(;!)1102 1173 y Ft(1)953 1261 y FA(j)1139 1238 y FG(\(1\))23 b(=)f FH(S)5 b FG(\()p FH(U)1509 1203 y FA(!)1553 1178 y Ft(0)1500 1258 y FA(j)1590 1238 y FG(\))18 b(+)g FH(\021)1767 1203 y FA(!)1811 1178 y Ft(1)1764 1258 y FD(1)p FA(;j)1866 1238 y FH(;)97 b(j)28 b FG(=)23 b(1)p FH(;)14 b FG(2)g FH(;)0 1395 y FI(where)27 b FF(D)r FG(\()p FH(\021)379 1407 y FD(1)p FA(;)p FD(1)470 1395 y FG(\))c(=)g FF(D)r FG(\()p FH(\021)752 1407 y FD(1)p FA(;)p FD(2)843 1395 y FG(\))g(=)g FF(D)r FG(\()p FH(\021)1125 1407 y FD(1)1163 1395 y FG(\))p FI(.)37 b(In)28 b(particular,)1180 1577 y FF(D)1246 1510 y Fz(\000)1284 1577 y FH(U)1350 1537 y FA(!)1394 1512 y Ft(0)1426 1537 y FA(;!)1490 1512 y Ft(1)1341 1600 y FA(j)1526 1577 y FG(\(1\))1632 1510 y Fz(\001)1694 1577 y FG(=)22 b FH(\026)1831 1589 y FA(j)1866 1577 y FG(\(1\))14 b FH(:)0 1759 y FI(W)-7 b(e)31 b(iterate)f(this)h(pro)r(cedure)f FH(T)42 b FI(times,)31 b(using)g(successiv)n(ely)e(coupling)h(maps)2545 1738 y Fz(b)2535 1759 y FH(V)2583 1771 y FA(j)2619 1759 y FG(\()p FH(u)2699 1771 y FD(1)2736 1759 y FH(;)14 b(u)2821 1771 y FD(2)2857 1759 y FG(;)g FH(!)2949 1729 y FD(2)2986 1759 y FG(\))p FH(;)g(:)g(:)g(:)g FI(,)10 1851 y Fz(b)0 1872 y FH(V)48 1884 y FA(j)83 1872 y FG(\()p FH(u)163 1884 y FD(1)200 1872 y FH(;)g(u)285 1884 y FD(2)322 1872 y FI(;)27 b FH(!)427 1842 y FA(T)479 1872 y FG(\))g(\()p FH(j)h FG(=)23 b(1)p FH(;)14 b FG(2\))p FI(,)26 b(where)g FH(!)1216 1842 y FA(`)1271 1872 y FF(2)d FG(\012)1409 1842 y FA(`)1441 1872 y FI(.)36 b(In)27 b(this)g(w)n(a)n(y)e(for)h FH(`)d FG(=)f(1)p FH(;)14 b FG(2)p FH(;)g(:)g(:)g(:)f(;)h(T)38 b FI(w)n(e)26 b(get)g(couplings)0 1904 y Fz(\000)38 1972 y FH(U)95 1984 y FD(1)132 1972 y FG(\()p FH(`)p FG(\))p FH(;)14 b(U)325 1984 y FD(2)362 1972 y FG(\()p FH(`)p FG(\))461 1904 y Fz(\001\000)537 1972 y FH(u)585 1984 y FD(1)622 1972 y FH(;)g(u)707 1984 y FD(2)744 1972 y FG(;)g(\()p FH(!)868 1942 y FD(0)905 1972 y FH(;)g(!)997 1942 y FD(1)1034 1972 y FH(;)g(:)g(:)g(:)f(;)h(!)1273 1942 y FA(`)1305 1972 y FG(\))1337 1904 y Fz(\001)1403 1972 y FI(suc)n(h)27 b(that)791 2138 y FH(U)848 2150 y FA(j)883 2138 y FG(\()p FH(`)p FG(\))c(=)g FH(S)1149 2070 y Fz(\000)1186 2138 y FH(U)1243 2150 y FA(j)1278 2138 y FG(\()p FH(`)18 b FF(\000)g FG(1\))1520 2070 y Fz(\001)1577 2138 y FG(+)g FH(\021)1701 2150 y FA(`;j)1797 2138 y FH(;)97 b FF(D)r FG(\()p FH(\021)2056 2150 y FA(`;j)2140 2138 y FG(\))23 b(=)f FH(\021)2323 2150 y FD(1)2375 2138 y FH(:)0 2295 y FI(W)-7 b(e)28 b(view)f(the)h(couplings)f(as)g(random)f (v)-5 b(ariables,)27 b(de\034ned)h(on)f(the)h(same)f(probabilit)n(y)f (space)1210 2453 y FG(\012)1270 2465 y FA(T)1345 2453 y FG(=)d(\012)1493 2419 y FD(0)1549 2453 y FF(\002)18 b(\001)c(\001)g(\001)k(\002)g FG(\012)1890 2419 y FA(T)1956 2453 y FH(:)0 2611 y FI(Then)32 b(the)g(kic)n(ks)f FF(f)p FH(\021)661 2623 y FA(`;j)744 2611 y FF(g)g FI(also)g(are)g(de\034ned)h (on)g FG(\012)1601 2623 y FA(T)1653 2611 y FI(.)50 b(The)32 b(pro)r(cesses)e FF(f)p FH(U)2363 2623 y FD(1)2400 2611 y FG(\()p FH(`)p FG(\))p FH(;)14 b(`)30 b FG(=)g(0)p FH(;)14 b(:)g(:)g(:)f(;)h(T)e FF(g)30 b FI(and)0 2711 y FF(f)p FH(U)99 2723 y FD(2)135 2711 y FG(\()p FH(`)p FG(\))p FH(;)14 b(`)27 b FG(=)f(0)p FH(;)14 b(:)g(:)g(:)f(;)h(T)e FF(g)28 b FI(are)g(w)n(eak)h(solutions)g(of)g(\(6.1\).)43 b(As)30 b(b)r(efore,)f(w)n(e)h(denote)f FH(d)p FG(\()p FH(k)s FG(\))e(=)f FF(j)p FH(U)2957 2723 y FD(1)2994 2711 y FG(\()p FH(k)s FG(\))20 b FF(\000)0 2810 y FH(U)57 2822 y FD(2)94 2810 y FG(\()p FH(k)s FG(\))p FF(j)28 b FI(and)f FH(R)q FG(\()p FH(k)s FG(\))d(=)e FF(j)p FH(U)781 2822 y FD(1)818 2810 y FG(\()p FH(k)s FG(\))p FF(j)d FG(+)f FF(j)p FH(U)1133 2822 y FD(2)1170 2810 y FG(\()p FH(k)s FG(\))p FF(j)p FI(,)28 b FG(0)23 b FF(\024)g FH(k)i FF(\024)e FH(T)12 b FI(.)125 2910 y(Belo)n(w)26 b(w)n(e)h(also)g(write) g(the)h(probabilit)n(y)e(space)h FG(\012)1720 2922 y FA(T)1800 2910 y FI(as)930 3068 y FG(\012)990 3080 y FA(T)1066 3068 y FG(=)22 b(\012)1213 3033 y FD(0)1269 3068 y FF(\002)c FG(\012)1412 3033 y Fw(0)1435 3068 y FH(;)60 b FG(\012)1578 3033 y Fw(0)1625 3068 y FG(=)22 b(\012)1772 3033 y FD(1)1828 3068 y FF(\002)c(\001)c(\001)g(\001)k (\002)g FG(\012)2169 3033 y FA(T)2235 3068 y FH(:)0 3225 y FI(Let)28 b(us)f(mak)n(e)g(some)g(observ)-5 b(ations)26 b(ab)r(out)h(the)h(w)n(eak)f(solutions)f FH(U)2145 3237 y FD(1)2210 3225 y FI(and)h FH(U)2428 3237 y FD(2)2465 3225 y FI(:)125 3325 y(i\))k(If)h FH(d)p FG(\(0\))e(=)f FF(j)p FH(U)651 3337 y FD(1)709 3325 y FF(\000)21 b FH(U)852 3337 y FD(2)889 3325 y FF(j)30 b(\024)f FH(d)1079 3337 y FD(0)1148 3325 y FI(and)i FH(R)q FG(\(0\))e FF(\024)g FH(R)1669 3337 y FD(0)1707 3325 y FI(,)j(then)g(b)n(y)g(Lemma)f(6.1)f (the)i(probabilit)n(y)f(\(in)0 3424 y FG(\012)60 3394 y Fw(0)83 3424 y FG(\))d FI(that)g FH(d)p FG(\()p FH(T)12 b FG(\))23 b FF(\024)g FG(2)644 3394 y Fw(\000)p FA(T)747 3424 y FH(d)790 3436 y FD(0)855 3424 y FI(is)718 3582 y FF(\025)f FG(1)c FF(\000)g FH(C)1007 3594 y Fw(\003)1060 3582 y FH(d)p FG(\(1)g(+)g(2)1320 3548 y Fw(\000)p FD(1)1427 3582 y FG(+)g FF(\001)c(\001)g(\001)19 b FG(+)f(2)1751 3548 y Fw(\000)p FA(T)9 b FD(+1)1938 3582 y FG(\))24 b FF(\025)e FG(1)c FF(\000)g FG(2)c FH(C)2339 3594 y Fw(\003)2391 3582 y FH(d)g(:)125 3859 y FI(ii\))28 b(If)g FH(d)p FG(\(0\))23 b FH(>)g(d)617 3871 y FD(0)682 3859 y FI(and)k FH(R)q FG(\(0\))c FF(\024)g FH(R)1187 3871 y FD(0)1224 3859 y FI(,)28 b(then)g(b)n(y)g(\(6.10\))55 b Fk(P)1913 3829 y FD(\012)1960 3804 y Fr(0)1987 3859 y FG(\()p FH(d)p FG(\()p FH(T)12 b FG(\))23 b FF(\024)g FH(d)2341 3871 y FD(0)2378 3859 y FG(\))g FF(\025)g FH(\022)16 b(:)125 3979 y FJ(Step)31 b(2)d FI(\(Kan)n(toro)n(vic)n(h)d (functional\).)37 b(Let)27 b(us)h(in)n(tro)r(duce)f(in)h FH(H)34 b FI(the)28 b(distance)1152 4137 y FG(dist\()p FH(u;)14 b(v)s FG(\))23 b(=)g FF(j)p FH(u)18 b FF(\000)g FH(v)s FF(j)h(^)g FH(d)1963 4149 y FD(0)2014 4137 y FH(;)0 4294 y FI(and)36 b(denote)g(b)n(y)f FF(k)24 b(\001)g(k)724 4264 y Fw(\003)724 4318 y FA(L;d)825 4326 y Ft(0)896 4294 y FI(the)36 b(corresp)r(onding)e(Lipsc)n(hitz-dual)h(distance)h (\(see)f(Exercise)h(5.2)f(in)0 4394 y(Section)28 b(5.1\).)36 b(Consider)26 b(the)i(follo)n(wing)f(function)h FH(f)1695 4406 y FA(K)1758 4394 y FG(\()p FH(u)1838 4406 y FD(1)1876 4394 y FH(;)14 b(u)1961 4406 y FD(2)1997 4394 y FG(\))p FI(:)872 4687 y FH(f)913 4699 y FA(K)976 4687 y FG(\()p FH(u)1056 4699 y FD(1)1093 4687 y FH(;)g(u)1178 4699 y FD(2)1215 4687 y FG(\))23 b(=)1358 4492 y Fz(8)1358 4567 y(>)1358 4592 y(<)1358 4741 y(>)1358 4766 y(:)1450 4551 y FH(d;)115 b FI(if)28 b FH(d)23 b FF(\024)g FH(d)1904 4563 y FD(0)1941 4551 y FI(,)28 b FH(R)23 b FF(\024)g FH(R)2229 4563 y FD(0)2266 4551 y FI(,)1450 4686 y FG(2)p FH(d)1535 4698 y FD(0)1572 4686 y FH(;)36 b FI(if)28 b FH(d)23 b(>)g(d)1904 4698 y FD(0)1941 4686 y FI(,)28 b FH(R)23 b FF(\024)g FH(R)2229 4698 y FD(0)2266 4686 y FI(,)1450 4822 y FH(R)q(;)94 b FI(if)28 b FH(R)23 b(>)g(R)1944 4834 y FD(0)1981 4822 y FI(,)1553 5059 y(52)p eop end PStoPSsaved restore %%Page: (52,53) 27 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 53 52 bop 0 523 a FI(where)25 b FH(d)e FG(=)f FF(j)p FH(u)462 535 y FD(1)513 523 y FF(\000)14 b FH(u)640 535 y FD(2)676 523 y FF(j)p FI(,)26 b FH(R)d FG(=)g FF(j)p FH(u)993 535 y FD(1)1030 523 y FF(j)14 b FG(+)g FF(j)p FH(u)1217 535 y FD(2)1252 523 y FF(j)p FI(.)37 b(Since)25 b FH(R)1612 535 y FD(0)1672 523 y FH(>)e FG(2)p FH(d)1845 535 y FD(0)1882 523 y FI(,)j(then)f FH(f)2158 535 y FA(K)2222 523 y FG(\()p FH(u)2302 535 y FD(1)2339 523 y FH(;)14 b(u)2424 535 y FD(2)2461 523 y FG(\))23 b FF(\025)g FG(dist\()p FH(u)2818 535 y FD(1)2855 523 y FH(;)14 b(u)2940 535 y FD(2)2977 523 y FG(\))p FI(.)36 b(Let)0 623 y FF(K)29 b FI(b)r(e)f(the)g(corresp)r(onding)e (Kan)n(toro)n(vic)n(h)e(functional.)125 722 y(Let)g(us)h(tak)n(e)f(an)n (y)f(coupling)h FG(\()p FH(U)1122 734 y FD(1)1160 722 y FH(;)14 b(U)1254 734 y FD(2)1290 722 y FG(\))25 b FI(for)f FG(\()p FH(\026)1553 734 y FD(1)1591 722 y FH(;)14 b(\026)1678 734 y FD(2)1715 722 y FG(\))25 b FI(and)f(apply)h(the)g(construction)e (from)h(Step)h(1)0 822 y(to)f(get)f(the)h(couplings)732 755 y Fz(\000)770 822 y FH(U)827 834 y FD(1)864 822 y FG(\()p FH(`)p FG(\))p FH(;)14 b(U)1057 834 y FD(2)1094 822 y FG(\()p FH(`)p FG(\))1193 755 y Fz(\001)1232 822 y FI(,)24 b FG(1)f FF(\024)g FH(`)f FF(\024)h FH(T)12 b FI(.)35 b(W)-7 b(e)24 b(wish)f(to)h(estimate)g Fk(E)14 b FH(f)2565 834 y FA(K)2628 755 y Fz(\000)2666 822 y FH(U)2723 834 y FD(1)2760 822 y FG(\()p FH(T)e FG(\))p FI(,)24 b FH(U)2989 834 y FD(2)3026 822 y FG(\()p FH(T)12 b FG(\))3151 755 y Fz(\001)0 922 y FI(in)28 b(terms)f(of)h FH(A)23 b FG(=)g Fk(E)14 b FH(f)714 934 y FA(K)777 922 y FG(\()p FH(U)866 934 y FD(1)903 922 y FH(;)g(U)997 934 y FD(2)1034 922 y FG(\))p FI(.)37 b(W)-7 b(e)28 b(abbreviate)729 1090 y FH(f)770 1102 y FA(K)834 1090 y FG(\(0\))23 b(=)g FH(f)1092 1102 y FA(K)1155 1090 y FG(\()p FH(U)1244 1102 y FD(1)1281 1090 y FH(;)14 b(U)1375 1102 y FD(2)1412 1090 y FG(\))p FH(;)38 b(f)1546 1102 y FA(K)1609 1090 y FG(\()p FH(`)p FG(\))23 b(=)g FH(f)1860 1102 y FA(K)1924 1023 y Fz(\000)1962 1090 y FH(U)2019 1102 y FD(1)2056 1090 y FG(\()p FH(`)p FG(\))p FH(;)14 b(U)2249 1102 y FD(2)2286 1090 y FG(\()p FH(`)p FG(\))2385 1023 y Fz(\001)2437 1090 y FH(;)0 1258 y FI(and)27 b(consider)g(the)h(three)f(cases,)g (accordingly)e(to)j(the)g(partition)f(of)g FG(\012)2257 1228 y FD(0)2322 1258 y FI(to)h(three)f(ev)n(en)n(ts:)125 1358 y(a\))34 b FH(!)288 1328 y FD(0)359 1358 y FF(2)h FH(Q)515 1370 y FD(1)587 1358 y FG(=)f FF(f)p FH(R)q FG(\(0\))g FH(>)g(R)1094 1370 y FD(0)1132 1358 y FF(g)p FI(.)57 b(Then)34 b FH(f)1518 1370 y FA(K)1582 1358 y FG(\(0\))h(=)f FH(R)q FG(\(0\))p FI(.)58 b(Applying)34 b(\(6.8\))g(and)h(using)f(that)0 1457 y FG(2)p FH(d)85 1469 y FD(0)145 1457 y FF(\024)243 1425 y FD(1)p 243 1439 34 4 v 243 1486 a(2)309 1457 y FH(R)372 1469 y FD(0)432 1457 y FH(<)530 1425 y FD(1)p 530 1439 V 530 1486 a(2)596 1457 y FH(R)q FG(\(0\))27 b FI(w)n(e)g(see)h(that)f(in)h(this)g(case) 1190 1673 y Fk(E)1253 1638 y FD(\012)1300 1613 y Fr(0)1327 1673 y FH(f)1368 1685 y FA(K)1431 1673 y FG(\()p FH(T)12 b FG(\))23 b FF(\024)1677 1617 y FG(1)p 1677 1654 42 4 v 1677 1730 a(2)1751 1673 y FH(f)1792 1685 y FA(K)1856 1673 y FG(\(0\))14 b FH(:)125 1874 y FI(b\))28 b FH(!)286 1844 y FD(0)346 1874 y FF(2)23 b FH(Q)490 1886 y FD(2)550 1874 y FG(=)g FF(f)p FH(d)p FG(\(0\))f FH(>)h(d)982 1886 y FD(0)1019 1874 y FH(;)37 b(R)q FG(\(0\))23 b FF(\024)g FH(R)1423 1886 y FD(0)1460 1874 y FF(g)p FI(.)125 1974 y(No)n(w)g FH(f)351 1986 y FA(K)415 1974 y FG(\(0\))g(=)g(2)p FH(d)717 1986 y FD(0)754 1974 y FI(.)35 b(By)24 b(\(6.9\),)h FH(R)q FG(\()p FH(T)12 b FG(\))22 b FF(\024)h FH(R)1520 1986 y FD(0)1582 1974 y FI(a.s.,)h(and)g(b)n(y)g(ii\))h(with)f (probabilit)n(y)f FF(\025)g FH(\022)r FI(,)i(w)n(e)f(ha)n(v)n(e)0 2073 y FH(f)41 2085 y FA(K)105 2073 y FG(\()p FH(T)12 b FG(\))22 b FF(\024)h FH(d)383 2085 y FD(0)420 2073 y FI(.)37 b(Hence,)28 b(no)n(w)382 2281 y Fk(E)445 2247 y FD(\012)492 2222 y Fr(0)519 2281 y FH(f)560 2293 y FA(K)623 2281 y FG(\()p FH(T)12 b FG(\))23 b FF(\024)g FG(\(1)18 b FF(\000)g FH(\022)r FG(\))c(2)p FH(d)1206 2293 y FD(0)1262 2281 y FG(+)k FH(\022)r(d)1429 2293 y FD(0)1489 2281 y FG(=)23 b(2)p FH(d)1662 2293 y FD(0)1699 2189 y Fz(\020)1749 2281 y FG(1)18 b FF(\000)1902 2225 y FG(1)p 1902 2262 V 1902 2338 a(2)1976 2281 y FH(\022)2017 2189 y Fz(\021)2090 2281 y FG(=)2178 2189 y Fz(\020)2227 2281 y FG(1)g FF(\000)2380 2225 y FG(1)p 2380 2262 V 2380 2338 a(2)2455 2281 y FH(\022)2496 2189 y Fz(\021)2559 2281 y FH(f)2600 2293 y FA(K)2664 2281 y FG(\(0\))c FH(:)125 2482 y FI(c\))27 b FH(!)276 2452 y FD(0)336 2482 y FF(2)d FH(Q)481 2494 y FD(3)541 2482 y FG(=)e FF(f)p FH(d)p FG(\(0\))h FF(\024)g FH(d)973 2494 y FD(0)1010 2482 y FH(;)37 b(R)24 b FF(\024)f FH(R)1308 2494 y FD(0)1345 2482 y FF(g)p FI(.)125 2582 y(Then)34 b FH(f)389 2594 y FA(K)453 2582 y FG(\(0\))h(=)f FH(d)p FG(\(0\))p FI(.)58 b(By)34 b(\(6.9\),)i FH(R)q FG(\()p FH(T)12 b FG(\))34 b FF(\024)g FH(R)1675 2594 y FD(0)1747 2582 y FI(a.s.,)i(and)e(b)n(y)h (i\))f(with)h(probabilit)n(y)f(greater)0 2682 y(than)28 b FG(\(1)18 b FF(\000)g FG(2)c FH(C)484 2694 y Fw(\003)536 2682 y FH(d)p FG(\(0\)\))p FI(,)28 b(w)n(e)f(ha)n(v)n(e)f FH(f)1122 2694 y FA(K)1186 2682 y FG(\()p FH(T)12 b FG(\))23 b FF(\024)f FG(2)1463 2652 y Fw(\000)p FA(T)1581 2682 y FH(d)1624 2694 y FD(0)1661 2682 y FI(.)37 b(So,)489 2862 y Fk(E)552 2828 y FD(\012)599 2802 y Fr(0)639 2862 y FH(f)680 2874 y FA(K)744 2862 y FG(\()p FH(T)12 b FG(\))23 b FF(\024)f FG(2)1021 2828 y Fw(\000)p FA(T)1139 2862 y FH(d)p FG(\(0\))1288 2795 y Fz(\000)1326 2862 y FG(1)c FF(\000)g FG(2)c FH(C)1584 2874 y Fw(\003)1636 2862 y FH(d)p FG(\(0\))1785 2795 y Fz(\001)1842 2862 y FG(+)k(2)p FH(C)2026 2874 y Fw(\003)2078 2862 y FH(d)p FG(\(0\)2)p FH(d)2312 2874 y FD(0)892 3035 y FF(\024)k FH(d)p FG(\(0\)\(2)1202 3000 y Fw(\000)p FA(T)1325 3035 y FG(+)c(4)p FH(C)1509 3047 y Fw(\003)1561 3035 y FH(d)1604 3047 y FD(0)1641 3035 y FG(\))24 b FF(\024)e FH(d)p FG(\(0\))1933 2942 y Fz(\020)1993 2978 y FG(1)p 1993 3015 V 1993 3092 a(2)2063 3035 y(+)2156 2978 y(1)p 2156 3015 V 2156 3092 a(4)2208 2942 y Fz(\021)2280 3035 y FG(=)2378 2978 y(3)p 2378 3015 V 2378 3092 a(4)2452 3035 y FH(f)2493 3047 y FA(K)2557 3035 y FG(\(0\))14 b FH(:)0 3231 y FI(Using)27 b(the)h(estimates)f(in)h (a\)\025c\))f(w)n(e)h(get)369 3424 y Fk(E)14 b FH(f)487 3436 y FA(K)550 3424 y FG(\()p FH(T)e FG(\))22 b(=)h Fk(E)848 3390 y FD(\012)895 3365 y Ft(0)932 3332 y Fz(\020)981 3424 y FH(I)1017 3436 y FA(Q)1069 3444 y Ft(1)1120 3424 y Fk(E)1183 3390 y FD(\012)1230 3365 y Fr(0)1271 3424 y FH(f)1312 3436 y FA(K)1376 3424 y FG(\()p FH(T)12 b FG(\))18 b(+)g FH(I)1638 3436 y FA(Q)1690 3444 y Ft(2)1741 3424 y Fk(E)1804 3390 y FD(\012)1851 3365 y Fr(0)1892 3424 y FH(f)1933 3436 y FA(K)1996 3424 y FG(\()p FH(T)12 b FG(\))18 b(+)g FH(I)2258 3436 y FA(Q)2310 3444 y Ft(3)2361 3424 y Fk(E)2424 3390 y FD(\012)2471 3365 y Fr(0)2512 3424 y FH(f)2553 3436 y FA(K)2617 3424 y FG(\()p FH(T)12 b FG(\))2742 3332 y Fz(\021)697 3621 y FF(\024)23 b Fk(E)848 3587 y FD(\012)895 3561 y Ft(0)932 3529 y Fz(\020)981 3621 y FH(I)1017 3633 y FA(Q)1069 3641 y Ft(1)1130 3565 y FG(1)p 1130 3602 V 1130 3678 a(2)1205 3621 y FH(f)1246 3633 y FA(K)1310 3621 y FG(\(0\))18 b(+)g FH(I)1553 3633 y FA(Q)1605 3641 y Ft(2)1643 3529 y Fz(\020)1692 3621 y FG(1)g FF(\000)1845 3565 y FG(1)p 1845 3602 V 1845 3678 a(2)1920 3621 y FH(\022)1961 3529 y Fz(\021)2024 3621 y FH(f)2065 3633 y FA(K)2129 3621 y FG(\(0\))g(+)g FH(I)2372 3633 y FA(Q)2424 3641 y Ft(3)2486 3565 y FG(3)p 2486 3602 V 2486 3678 a(4)2560 3621 y FH(f)2601 3633 y FA(K)2665 3621 y FG(\(0\))2771 3529 y Fz(\021)697 3816 y FF(\024)24 b Fz(e)-47 b FH(\024)14 b Fk(E)g FH(f)965 3828 y FA(K)1028 3816 y FG(\(0\))g FH(;)0 3994 y FI(where)28 b Fz(e)-47 b FH(\024)23 b FG(=)399 3927 y Fz(\000)437 3994 y FG(1)18 b FF(\000)590 3961 y FD(1)p 590 3975 34 4 v 590 4023 a(2)656 3994 y FH(\022)697 3927 y Fz(\001)753 3994 y FF(_)837 3961 y FD(3)p 837 3975 V 837 4023 a(4)903 3994 y FH(<)23 b FG(1)p FI(.)36 b(If)28 b FH(j)g FG(=)23 b FH(k)s(T)12 b FI(,)27 b(then)h(iterating)f(these)g(argumen)n(ts)f(w)n (e)i(ha)n(v)n(e)1183 4175 y Fk(E)14 b FH(f)1301 4187 y FA(K)1365 4175 y FG(\()p FH(j)5 b FG(\))23 b FF(\024)h Fz(e)-47 b FH(\024)1627 4140 y FA(k)1681 4175 y Fk(E)14 b FH(f)1799 4187 y FA(K)1863 4175 y FG(\(0\))g FH(:)0 4343 y FI(If)24 b FH(j)k FF(2)23 b FG([1)p FH(;)14 b(T)21 b FF(\000)10 b FG(1])p FI(,)24 b(then)g(the)g(argumen)n(ts)e(in)i(a\))f (and)g(c\))h(with)g FH(T)35 b FI(replaced)22 b(b)n(y)i FH(j)k FI(remain)23 b(unc)n(hanged,)0 4443 y(while)31 b(in)g(b\))g(w)n(e)g(cannot)f(an)n(ymore)f(claim)h(that)h Fk(P)p FF(f)p FH(d)p FG(\()p FH(j)5 b FG(\))29 b FF(\024)f FH(d)1995 4455 y FD(0)2033 4443 y FF(g)g FH(>)g FG(0)p FI(.)46 b(So)30 b(\(6.10\))g(do)r(es)g(not)h(hold,)0 4542 y(and)c(in)h(this)g(case)f Fk(E)14 b FH(f)714 4554 y FA(K)777 4542 y FG(\()p FH(j)5 b FG(\))24 b FF(\024)e Fk(E)14 b FH(f)1109 4554 y FA(K)1172 4542 y FG(\(0\))p FI(.)125 4642 y(Finally)-7 b(,)27 b(for)g(an)n(y)g(in)n(teger)g FH(t)c FG(=)f FH(k)s(T)30 b FG(+)18 b FH(j)5 b FI(,)27 b FG(0)c FF(\024)f FH(j)28 b(<)23 b(T)12 b FI(,)27 b(w)n(e)g(ha)n(v)n (e)911 4810 y Fk(E)14 b FH(f)1029 4822 y FA(K)1092 4810 y FG(\()p FH(t)p FG(\))24 b FF(\024)f Fz(e)-47 b FH(\024)1345 4776 y FA(k)1386 4810 y Fk(E)14 b FH(f)1504 4822 y FA(K)1567 4810 y FG(\(0\))23 b FF(\024)g FH(C)d(\024)1911 4776 y FA(t)1954 4810 y Fk(E)14 b FH(f)2072 4822 y FA(K)2135 4810 y FG(\(0\))g FH(;)1553 5059 y FI(53)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 54 53 bop 0 523 a FI(with)28 b FH(\024)23 b FG(=)g Fz(e)-46 b FH(\024)396 493 y FD(1)p FA(=T)538 523 y FH(<)23 b FG(1)k FI(and)g FH(C)i FG(=)24 b Fz(e)-47 b FH(\024)1080 493 y Fw(\000)p FA(j)s(=T)1272 523 y FH(>)23 b FG(1)p FI(.)0 623 y FJ(Step)30 b(3)25 b FI(\(end)h(of)g(the)g(pro)r (of)6 b(\).)36 b(Let)26 b(us)g(c)n(ho)r(ose)e(for)h(the)h(original)e (coupling)h FG(\()p FH(U)2494 635 y FD(1)2532 623 y FH(;)14 b(U)2626 635 y FD(2)2662 623 y FG(\))27 b FI(the)f(coupling,)0 722 y(formed)h(b)n(y)g(indep)r(enden)n(t)i(random)d(v)-5 b(ariables.)36 b(Then)484 874 y Fk(E)14 b FH(f)602 886 y FA(K)665 874 y FG(\()p FH(U)754 886 y FD(1)791 874 y FH(;)g(U)885 886 y FD(2)922 874 y FG(\))23 b FF(\024)g Fk(E)p FG(\(2)p FH(d)1245 886 y FD(0)1300 874 y FG(+)18 b FF(j)p FH(U)1463 886 y FD(1)1500 874 y FF(j)h FG(+)f FF(j)p FH(U)1705 886 y FD(2)1742 874 y FF(j)p FG(\))23 b FF(\024)g FG(1)18 b(+)g FH(M)2132 886 y FD(1)2169 874 y FG(\()p FH(\026)2251 886 y FD(1)2288 874 y FG(\))h(+)f FH(M)2503 886 y FD(1)2540 874 y FG(\()p FH(\026)2622 886 y FD(2)2659 874 y FG(\))0 1025 y FI(since)27 b FH(d)246 1037 y FD(0)307 1025 y FH(<)404 993 y FD(1)p 404 1007 34 4 v 404 1054 a(2)448 1025 y FI(.)36 b(Therefore,)549 1177 y FF(K)613 1110 y Fz(\000)651 1177 y FH(\026)701 1189 y FD(1)739 1177 y FG(\()p FH(t)p FG(\))p FH(;)14 b(\026)920 1189 y FD(2)957 1177 y FG(\()p FH(t)p FG(\))1051 1110 y Fz(\001)1113 1177 y FF(\024)23 b Fk(E)14 b FH(f)1319 1189 y FA(K)1382 1177 y FG(\()p FH(t)p FG(\))24 b FF(\024)e FH(C)1652 1110 y Fz(\000)1691 1177 y FG(1)17 b(+)h FH(M)1914 1189 y FD(1)1951 1177 y FG(\()p FH(\026)2033 1189 y FD(1)2071 1177 y FG(\))h(+)f FH(M)2286 1189 y FD(1)2322 1177 y FG(\()p FH(\026)2404 1189 y FD(2)2442 1177 y FG(\))2474 1110 y Fz(\001)2526 1177 y FH(\024)2574 1143 y FA(t)2617 1177 y FH(:)337 b FI(\(6.12\))0 1328 y(Com)n(bining)27 b(this)h(estimate)f(with)h(\(5.3\))g(and)f(Lemma)g(5.10)f(w)n(e)i(get)f (that)678 1522 y FF(k)p FH(\026)770 1534 y FD(1)807 1522 y FG(\()p FH(t)p FG(\))19 b FF(\000)f FH(\026)1053 1534 y FD(2)1091 1522 y FG(\()p FH(t)p FG(\))p FF(k)1227 1488 y Fw(\003)1227 1543 y FA(L)1299 1522 y FF(\024)1397 1466 y FG(2)p FH(C)p 1397 1503 107 4 v 1410 1579 a(d)1453 1591 y FD(0)1537 1455 y Fz(\000)1575 1522 y FG(1)g(+)g FH(M)1799 1534 y FD(1)1836 1522 y FG(\()p FH(\026)1918 1534 y FD(1)1955 1522 y FG(\))h(+)f FH(M)2170 1534 y FD(1)2207 1522 y FG(\()p FH(\026)2289 1534 y FD(2)2326 1522 y FG(\))2358 1455 y Fz(\001)2397 1522 y FH(\024)2445 1488 y FA(t)2487 1522 y FH(:)0 1714 y FI(Re-denoting)27 b FG(2)p FH(C)t(=d)657 1726 y FD(0)721 1714 y FI(b)n(y)g FH(C)34 b FI(w)n(e)27 b(get)h(the)g(theorem's)f(assertion.)p 3128 1714 4 57 v 3132 1662 50 4 v 3132 1714 V 3182 1714 4 57 v 125 1949 a(The)39 b(theorem)h(whic)n(h)f(w)n(e)h(ha)n(v)n(e)e (pro)n(v)n(ed)h(has)g(man)n(y)g(imp)r(ortan)n(t)g(corollaries,)i (discussed)e(in)0 2048 y(the)34 b(next)h(section.)56 b(Before)33 b(that)h(w)n(e)g(deriv)n(e)f(some)h(consequences)f(whic)n (h)h(follo)n(w)f(not)h(from)g(the)0 2148 y(theorem,)27 b(but)h(from)f(its)h(pro)r(of.)125 2248 y(Due)g(to)f(\(6.12\))g(w)n(e)g (can)g(construct)g(a)g(coupling)1680 2180 y Fz(\000)1718 2248 y FH(U)1775 2260 y FD(1)1812 2248 y FG(\()p FH(t)p FG(\))p FI(,)i FH(U)2015 2260 y FD(2)2052 2248 y FG(\()p FH(t)p FG(\))2146 2180 y Fz(\001)2212 2248 y FI(for)2339 2180 y Fz(\000)2377 2248 y FH(\026)2427 2260 y FD(1)2464 2248 y FG(\()p FH(t)p FG(\))p FH(;)14 b(\026)2645 2260 y FD(2)2683 2248 y FG(\()p FH(t)p FG(\))2777 2180 y Fz(\001)2843 2248 y FI(suc)n(h)27 b(that)471 2408 y Fk(E)14 b FH(f)589 2420 y FA(K)653 2340 y Fz(\000)691 2408 y FH(U)748 2420 y FD(1)785 2408 y FG(\()p FH(t)p FG(\))p FH(;)g(U)973 2420 y FD(2)1010 2408 y FG(\()p FH(t)p FG(\))1104 2340 y Fz(\001)1166 2408 y FF(\024)22 b FG(2)p FH(C)e(\024)1422 2373 y FA(t)1465 2408 y FH(K)1536 2420 y FD(0)1572 2408 y FH(;)84 b(K)1750 2420 y FD(0)1809 2408 y FG(=)23 b(1)18 b(+)g FH(M)2121 2420 y FD(1)2158 2408 y FG(\()p FH(\026)2240 2420 y FD(1)2277 2408 y FG(\))h(+)f FH(M)2492 2420 y FD(1)2529 2408 y FG(\()p FH(\026)2611 2420 y FD(2)2648 2408 y FG(\))c FH(:)0 2559 y FI(Since)20 b(for)g FF(j)p FH(U)409 2571 y FD(1)450 2559 y FF(\000)t FH(U)576 2571 y FD(2)612 2559 y FF(j)j FH(>)g(d)789 2571 y FD(0)847 2559 y FI(w)n(e)c(ha)n(v)n(e)g FH(f)1186 2571 y FA(K)1273 2559 y FF(\025)k FG(2)p FH(d)1446 2571 y FD(0)1503 2559 y FI(and)d(for)g FF(j)p FH(U)1857 2571 y FD(1)1894 2559 y FF(j)t FG(+)t FF(j)p FH(U)2070 2571 y FD(2)2106 2559 y FF(j)j(\025)g FH(R)2303 2571 y FD(0)2360 2559 y FI(w)n(e)d(ha)n(v)n (e)f FH(f)2700 2571 y FA(K)2787 2559 y FF(\025)j FH(R)2937 2571 y FD(0)2998 2559 y FH(>)g(d)3128 2571 y FD(0)3166 2559 y FI(,)0 2659 y(then)290 2810 y FH(P)343 2822 y FA(d)405 2810 y FG(:=)h Fk(P)581 2743 y Fz(\010)630 2810 y FF(j)p FH(U)710 2822 y FD(1)747 2810 y FG(\()p FH(t)p FG(\))c FF(\000)f FH(U)1000 2822 y FD(2)1036 2810 y FG(\()p FH(t)p FG(\))p FF(j)24 b FH(>)f(d)37 b FI(or)f FF(j)p FH(U)1536 2822 y FD(1)1573 2810 y FG(\()p FH(t)p FG(\))p FF(j)19 b FG(+)f FF(j)p FH(U)1872 2822 y FD(2)1909 2810 y FG(\()p FH(t)p FG(\))p FF(j)24 b(\025)e FH(R)2200 2822 y FD(0)2237 2743 y Fz(\011)2309 2810 y FF(\024)h FH(d)2440 2776 y Fw(\000)p FD(1)2543 2810 y FG(2)p FH(C)c(\024)2711 2776 y FA(t)2754 2810 y FH(K)2825 2822 y FD(0)2876 2810 y FH(;)0 2962 y FI(for)27 b(an)n(y)g FH(d)c FF(\024)g FH(d)481 2974 y FD(0)518 2962 y FI(.)37 b(Since)28 b FH(P)848 2974 y FA(d)910 2962 y FF(\024)23 b FH(P)1051 2974 y FA(d)1086 2982 y Ft(0)1150 2962 y FI(for)k FH(d)c(>)g(d)1474 2974 y FD(0)1511 2962 y FI(,)28 b(then)879 3113 y FH(P)932 3125 y FA(d)994 3113 y FF(\024)23 b FH(d)1125 3079 y Fw(\000)p FD(1)1214 3113 y FG(\(2)p FH(C)d(d)1410 3078 y Fw(\000)p FD(1)1410 3135 y(0)1499 3113 y FG(\))14 b FH(\024)1593 3079 y FA(t)1636 3113 y FH(K)1707 3125 y FD(0)1767 3113 y FG(=)23 b FH(d)1898 3079 y Fw(\000)p FD(1)2001 3113 y FH(C)2060 3125 y FD(1)2111 3113 y FH(\024)2159 3079 y FA(t)2202 3113 y FH(K)2273 3125 y FD(0)0 3265 y FI(for)k FH(d)c FF(\024)g FG(1)p FI(.)36 b(Cho)r(osing)1144 3364 y FH(d)24 b FG(=)e FH(\016)s FG(\()p FH(t)p FG(\))i(=)f(\()p FH(C)1635 3376 y FD(1)1686 3364 y FH(\024)1734 3330 y FA(t)1777 3364 y FH(K)1848 3376 y FD(0)1885 3364 y FG(\))1917 3330 y FD(1)p FA(=)p FD(2)2022 3364 y FH(;)0 3495 y FI(w)n(e)k(get)h (the)f(follo)n(wing)0 3636 y FJ(Corollary)33 b(6.5.)40 b FC(The)31 b(me)l(asur)l(es)e FH(\026)1191 3648 y FD(1)1228 3636 y FG(\()p FH(t)p FG(\))p FH(;)14 b(\026)1409 3648 y FD(2)1447 3636 y FG(\()p FH(t)p FG(\))30 b FC(admit)h(a)f(c)l (oupling)2203 3569 y Fz(\000)2241 3636 y FH(U)2298 3648 y FD(1)2335 3636 y FG(\()p FH(t)p FG(\))p FH(;)14 b(U)2523 3648 y FD(2)2561 3636 y FG(\()p FH(t)p FG(\))2655 3569 y Fz(\001)2723 3636 y FC(such)30 b(that)544 3796 y Fk(P)609 3729 y Fz(\010)658 3796 y FF(j)p FH(U)738 3808 y FD(1)775 3796 y FG(\()p FH(t)p FG(\))19 b FF(\000)f FH(U)1028 3808 y FD(2)1065 3796 y FG(\()p FH(t)p FG(\))p FF(j)23 b(\025)g FH(\016)s FG(\()p FH(t)p FG(\))37 b FC(or)29 b FF(j)p FH(U)1650 3808 y FD(1)1687 3796 y FG(\()p FH(t)p FG(\))p FF(j)19 b FG(+)f FF(j)p FH(U)1986 3808 y FD(2)2023 3796 y FG(\()p FH(t)p FG(\))p FF(j)23 b(\025)g FH(R)2314 3808 y FD(0)2351 3729 y Fz(\011)2423 3796 y FF(\024)f FH(\016)s FG(\()p FH(t)p FG(\))333 b FI(\(6.13\))0 3948 y FC(for)31 b(al)t(l)f(inte)l(ger)g FH(t)23 b FF(\025)g FG(0)p FC(.)125 4089 y FI(Denoting)k(the)h(ev)n(en)n(t)f(in)h(\(6.13\)) f(b)n(y)g FG(\012)1357 4101 y FA(t)1386 4089 y FI(,)h(it)g(follo)n(ws)e (from)i(Borel-Can)n(telli's)d(Lemma)i(that)1238 4265 y Fk(P)1303 4173 y Fz(\020)1382 4187 y(\\)1367 4365 y FA(k)q Fw(\025)p FD(0)1525 4187 y Fz([)1516 4365 y FA(l)p Fw(\025)p FA(k)1640 4265 y FG(\012)1700 4277 y FA(l)1726 4173 y Fz(\021)1798 4265 y FG(=)c(0)p FH(:)0 4507 y FI(Hence,)28 b(there)f(exists)g(a)g(random)g(v)-5 b(ariable)27 b FH(T)1458 4477 y Fw(0)1503 4507 y FF(\025)c FG(0)p FI(,)k(a.s.)36 b(\034nite,)28 b(and)g(suc)n(h)f(that)488 4659 y FF(j)p FH(U)568 4671 y FD(1)605 4659 y FG(\()p FH(`)p FG(\))18 b FF(\000)g FH(U)862 4671 y FD(2)899 4659 y FG(\()p FH(`)p FG(\))p FF(j)24 b FH(<)e(\016)s FG(\()p FH(`)p FG(\))37 b FI(and)g FF(j)p FH(U)1559 4671 y FD(1)1596 4659 y FG(\()p FH(`)p FG(\))p FF(j)19 b FG(+)f FF(j)p FH(U)1900 4671 y FD(2)1937 4659 y FG(\()p FH(`)p FG(\))p FF(j)23 b FH(<)f(R)2232 4671 y FD(0)2270 4659 y FH(;)97 b FF(8)p FH(`)22 b FF(\025)h FH(T)2642 4624 y Fw(0)2678 4659 y FH(:)276 b FI(\(6.14\))0 4810 y(W)-7 b(e)28 b(shall)f(need)h(this)g(relation)e(later.)1553 5059 y(54)p eop end PStoPSsaved restore %%Page: (54,55) 28 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 55 54 bop 0 523 a FJ(De\034nition)31 b(6.6.)40 b FC(The)30 b(Pr)l(okhor)l(ov)i(distanc)l(e)e(b)l(etwe)l(en)g FH(\026)1873 535 y FD(1)1910 523 y FH(;)14 b(\026)1997 535 y FD(2)2057 523 y FF(2)24 b(P)7 b FG(\()p FH(H)g FG(\))29 b FC(is)357 687 y FG(dist)491 699 y FD(Prok)634 687 y FG(\()p FH(\026)716 699 y FD(1)754 687 y FH(;)14 b(\026)841 699 y FD(2)878 687 y FG(\))23 b(=)g(inf)7 b FF(f)p FH(")22 b(>)h FG(0)14 b FF(j)g FH(\026)1456 699 y FD(1)1492 687 y FG(\()p FH(A)p FG(\))24 b FF(\024)f FH(\026)1780 699 y FD(2)1817 687 y FG(\()p FH(A)c FG(+)f FH(")p FG(\))h(+)f FH(";)98 b FF(8)p FH(A)24 b FF(2)f(B)s FG(\()p FH(H)7 b FG(\))p FF(g)14 b FH(;)0 850 y FC(wher)l(e)30 b FH(A)19 b FG(+)f FH(")30 b FC(stands)f(for)i(the)f FH(")p FC(\025neighb)l(ourho)l(o)l(d)h(of)g FH(A)f FC(in)g FH(H)7 b FC(.)125 1001 y FI(This)26 b(is)h(a)g(distance)g(on)f FF(P)7 b FG(\()p FH(H)g FG(\))27 b FI(whic)n(h)g(satis\034es)f(all)h (assertions)e(of)i(Theorem)f(5.3,)h(see)f([Dud89,)0 1101 y(Shi96)o(].)37 b(Moreo)n(v)n(er,)25 b FF(k)p FH(\026)761 1113 y FD(1)816 1101 y FF(\000)18 b FH(\026)949 1113 y FD(2)987 1101 y FF(k)1029 1071 y Fw(\003)1029 1124 y FA(L)1101 1101 y FF(\024)23 b FG(2)14 b(dist)1378 1113 y FD(Prok)1521 1101 y FG(\()p FH(\026)1603 1113 y FD(1)1640 1101 y FH(;)g(\026)1727 1113 y FD(2)1764 1101 y FG(\))p FH(:)0 1252 y FJ(Exercise)31 b(6.7.)40 b FI(Sho)n(w)27 b(that)h(\(6.13\))f(implies)h(that)1077 1415 y FG(dist)1211 1427 y FD(Prok)1354 1348 y Fz(\000)1392 1415 y FH(\026)1442 1427 y FD(1)1479 1415 y FG(\()p FH(t)p FG(\))p FH(;)14 b(\026)1660 1427 y FD(2)1698 1415 y FG(\()p FH(t)p FG(\))1792 1348 y Fz(\001)1853 1415 y FF(\024)23 b FH(\016)s FG(\()p FH(t)p FG(\))14 b FH(:)125 1579 y FI(W)-7 b(e)27 b(note)h(that)g(the)g (last)f(estimate)g(can)h(b)r(e)g(obtained)f(as)g(a)g(corollary)e(of)i (Theorem)g(6.4)g(since)743 1772 y FG(dist)877 1784 y FD(Prok)1020 1772 y FG(\()p FH(\026)1102 1784 y FD(1)1140 1772 y FH(;)14 b(\026)1227 1784 y FD(2)1264 1772 y FG(\))23 b FF(\024)g(k)p FH(\026)1499 1784 y FD(1)1554 1772 y FF(\000)18 b FH(\026)1687 1784 y FD(2)1724 1772 y FF(k)1766 1738 y Fw(\003)1766 1793 y FA(L)1834 1772 y FG(+)1917 1674 y Fz(q)p 2000 1674 409 4 v 98 x FF(k)p FH(\026)2092 1784 y FD(1)2147 1772 y FF(\000)g FH(\026)2280 1784 y FD(2)2318 1772 y FF(k)2360 1743 y Fw(\003)2360 1796 y FA(L)2423 1772 y FH(;)0 1955 y FI(see)27 b([Shi96],)g(Section)h(I)r(I)r (I.7.)0 2184 y Fy(6.4)112 b(Corollaries)35 b(from)i(the)g(theorem)0 2337 y FI(By)31 b(Theorem)f(3.5,)h(equation)g(\(6.1\))g(has)g(a)f (stationary)g(measure)g FH(\026)f FF(2)g(P)2365 2349 y FD(1)2402 2337 y FG(\()p FH(H)7 b FG(\))p FI(.)49 b(Applying)31 b(Theo-)0 2437 y(rem)c(6.4)g(w)n(e)g(obtain)0 2587 y FJ(Corollary)40 b(6.8.)k FC(Ther)l(e)37 b(exists)e(a)h(stationary)g(me) l(asur)l(e)f FH(\026)f FF(2)g(P)2140 2599 y FD(1)2177 2587 y FG(\()p FH(H)7 b FG(\))36 b FC(such)f(that)h(for)g(e)l(ach)h FH(\027)i FF(2)0 2687 y(P)58 2699 y FD(1)95 2687 y FG(\()p FH(H)7 b FG(\))30 b FC(we)g(have)993 2787 y FF(kB)1093 2752 y Fw(\003)1090 2807 y FA(k)1144 2787 y FH(\027)24 b FF(\000)18 b FH(\026)p FF(k)1384 2752 y Fw(\003)1384 2807 y FA(L)1456 2787 y FF(\024)k FH(C)1608 2719 y Fz(\000)1647 2787 y FG(1)c(+)g FH(M)1871 2799 y FD(1)1908 2787 y FG(\()p FH(\027)5 b FG(\))2018 2719 y Fz(\001)2070 2787 y FH(\024)2118 2752 y FA(k)2173 2787 y FH(:)125 2937 y FI(In)27 b(particular,)1133 3037 y FH(P)1186 3049 y FA(k)1228 3037 y FG(\()p FH(u;)14 b FF(\001)p FG(\))66 b FH(*)1423 3091 y FA(k)q Fw(!1)1615 3037 y FH(\026)83 b FF(8)p FH(u)23 b FF(2)g FH(H)e(:)0 3203 y FI(Therefore,)826 3303 y FF(B)881 3315 y FA(k)936 3303 y FH(f)9 b FG(\()p FH(u)p FG(\))22 b(=)1208 3236 y Fz(\000)1246 3303 y FH(f)t(;)14 b(P)1381 3315 y FA(k)1422 3303 y FG(\()p FH(u;)g FF(\001)p FG(\))1594 3236 y Fz(\001)1655 3303 y FF(!)23 b FG(\()p FH(f)t(;)14 b(\026)p FG(\))84 b FF(8)14 b FH(u)22 b FF(2)i FH(H)c(;)615 b FI(\(6.15\))0 3441 y(for)27 b(eac)n(h)g FH(f)k FF(2)24 b FH(C)524 3453 y FA(b)558 3441 y FG(\()p FH(H)7 b FG(\))p FI(.)0 3592 y FJ(Corollary)33 b(6.9.)40 b FC(The)31 b(system)e(\(6.1\))i(has)g(a)f (unique)f(stationary)h(me)l(asur)l(e)g FH(\026)23 b FF(2)g(P)7 b FG(\()p FH(H)g FG(\))p FC(.)0 3743 y(Pr)l(o)l(of.)43 b FI(If)33 b FH(\027)j FF(2)31 b(P)7 b FG(\()p FH(H)g FG(\))33 b FI(is)f(a)g(stationary)e(measure,)j(then)g(due)f(to)g (\(6.15\))g(for)g(an)n(y)f FH(f)40 b FF(2)31 b FH(C)2888 3755 y FA(b)2922 3743 y FG(\()p FH(H)7 b FG(\))32 b FI(w)n(e)0 3842 y(ha)n(v)n(e)603 3942 y FG(\()p FH(\027)q(;)14 b(f)9 b FG(\))23 b(=)f(\()p FF(B)996 3908 y Fw(\003)993 3963 y FA(k)1048 3942 y FH(\027)q(;)14 b(f)9 b FG(\))22 b(=)1319 3875 y Fz(\000)1357 3942 y FH(\027)q(;)14 b FF(B)1491 3954 y FA(k)1531 3942 y FH(f)1581 3875 y Fz(\001)1659 3942 y FF(\000)-14 b(!)1642 3996 y FA(k)q Fw(!1)1834 3875 y Fz(\000)1872 3942 y FH(\027)q(;)14 b FG(\()p FH(\026;)g(f)9 b FG(\))p Fk(1)2200 3875 y Fz(\001)2260 3942 y FG(=)23 b(\()p FH(\026;)14 b(f)9 b FG(\))14 b FH(:)0 4108 y FI(Hence,)28 b FH(\027)g FG(=)23 b FH(\026)p FI(.)p 3128 4108 4 57 v 3132 4056 50 4 v 3132 4108 V 3182 4108 4 57 v 125 4271 a(Let)32 b FH(\027)k FF(2)c(P)500 4283 y FD(1)537 4271 y FG(\()p FH(H)7 b FG(\))33 b FI(and)f FH(u)p FG(\()p FH(k)s FG(;)14 b FH(\027)5 b FG(\))32 b FI(b)r(e)h(a)f(solution)g(suc)n (h)g(that)h FF(D)16 b FH(u)p FG(\(0\))31 b(=)f FH(\027)5 b FI(.)52 b(Then)65 b FF(D)16 b FH(u)p FG(\()p FH(k)s FG(;)e FH(\027)5 b FG(\))31 b(=)0 4370 y FF(B)58 4340 y Fw(\003)55 4394 y FA(k)118 4370 y FH(\027)72 b(*)188 4424 y FA(k)q Fw(!1)403 4370 y FH(\026:)55 b FI(No)n(w)27 b(w)n(e)h(shall)f(get)g(a)g(stronger)f(v)n(ersion)g(of)i(this)f (result.)125 4511 y(Denote)c FE(Z)460 4523 y FD(+)538 4511 y FG(=)g FE(N)g FF([)9 b(f)p FG(0)p FF(g)p FI(,)23 b FF(A)1011 4523 y FD(+)1089 4511 y FG(=)g FH(H)1253 4481 y Fv(Z)1292 4489 y Ft(+)1365 4511 y FG(=)g FF(f)p FH(u)1543 4523 y FA(j)1600 4511 y FF(2)g FH(H)r(;)14 b(j)28 b FF(\025)23 b FG(0)p FF(g)p FI(.)34 b(Pro)n(vide)22 b FF(A)2444 4523 y FD(+)2523 4511 y FI(with)h(the)g(Tikhono)n(v)0 4611 y(top)r(ology)k(and)i(a)g(corresp)r(onding)d(norm.)40 b(F)-7 b(or)29 b FH(r)e FF(\025)e FG(0)k FI(consider)e(the)i(shift-map) g FH(T)2629 4623 y FA(r)2691 4611 y FG(:)c FF(A)2805 4623 y FD(+)2886 4611 y FF(7\000)-15 b(!)26 b(A)3111 4623 y FD(+)3166 4611 y FI(,)0 4710 y FG(\()p FH(u)80 4722 y FD(0)117 4710 y FH(;)14 b(u)202 4722 y FD(1)239 4710 y FH(;)g(:)g(:)g(:)f FG(\))36 b FF(7\000)-14 b(!)36 b FG(\()p FH(u)704 4722 y FA(r)740 4710 y FH(;)14 b(u)825 4722 y FA(r)r FD(+1)946 4710 y FH(;)g(:)g(:)g(:)f FG(\))p FI(.)60 b(Finally)-7 b(,)37 b(let)f(us)f(denote)g FH(u)p FG(\()p FH(k)s FG(;)14 b FH(\026)p FG(\))35 b(=)g FH(U)9 b FG(\()p FH(k)s FG(\))p FI(,)38 b(where)c FH(\026)i FI(is)f(the)0 4810 y(stationary)26 b(measure.)1553 5059 y(55)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 56 55 bop 0 523 a FJ(Exercise)41 b(6.10.)k FI(Pro)n(v)n(e)35 b(that)h(for)g(an)n(y)f FH(\027)43 b FF(2)37 b(P)7 b FG(\()p FH(H)g FG(\))p FI(,)38 b FF(D)r FG(\()p FH(T)1952 535 y FA(r)2003 523 y FH(u)2051 456 y Fz(\000)2089 523 y FF(\001)p FH(;)14 b(\027)5 b FG(\))2227 456 y Fz(\001)2344 523 y FH(*)2303 573 y FA(r)r Fw(!1)2505 523 y FF(D)2571 456 y Fz(\000)2609 523 y FH(U)k FG(\()p FF(\001)p FG(\))2762 456 y Fz(\001)2837 523 y FI(w)n(eakly)35 b(in)0 652 y FF(P)7 b FG(\()p FF(A)163 664 y FD(+)218 652 y FG(\))p FI(.)0 751 y FJ(Hin)m(t:)44 b FI(1\))31 b(By)g(Prop)r(osition)g(5.1)f(it)i(is)f(su\036cien)n(t)h(to)f(c)n(hec)n (k)g(that)g Fk(E)14 b FH(f)9 b FG(\()p FH(T)2338 763 y FA(r)2374 684 y Fz(\000)2412 751 y FH(u)p FG(\()p FF(\001)p FH(;)14 b(\027)5 b FG(\))2630 684 y Fz(\001)2698 751 y FH(*)29 b Fk(E)14 b FH(f)2937 684 y Fz(\000)2974 751 y FH(U)9 b FG(\()p FF(\001)p FG(\))3127 684 y Fz(\001)3166 751 y FI(,)0 851 y(where)27 b FH(f)36 b FI(is)28 b(a)f(b)r(ounded)h (Lipsc)n(hitz)f(functional)h(on)f FF(A)1734 863 y FD(+)1790 851 y FI(.)0 951 y(2\))g(It)h(su\036ces)g(to)f(c)n(hec)n(k)g(this)h (for)f FH(f)36 b FI(whic)n(h)27 b(dep)r(end)h(only)g(on)f(\034nitely)h (man)n(y)f(co)r(ordinates)f FH(u)2982 963 y FA(j)3016 951 y FI(.)0 1050 y(3\))h(T)-7 b(o)28 b(pro)n(v)n(e)e(this)h(use)h (Exercise)f(3.3.)1914 b Fs(\003)0 1215 y FJ(Exercise)45 b(6.11.)h FC(By)41 b(Exer)l(cise)g(3.7,)k FG(supp)14 b FH(\026)42 b FF(\032)g FB(B)1751 1227 y FA(r)1782 1235 y Ft(0)1819 1215 y FG(\()p FH(H)7 b FG(\))p FC(,)44 b FH(r)2065 1227 y FD(0)2145 1215 y FG(=)2304 1182 y FA(e)p 2262 1196 117 4 v 2262 1244 a(e)p Fw(\000)p FD(1)2431 1145 y FF(p)p 2500 1145 68 4 v 70 x FH(B)2567 1229 y FD(0)2604 1215 y FC(.)71 b(Use)40 b(The)l(or)l(em)0 1329 y(2.15)f(to)f(pr)l(ove)h(that)e(if)i FH(B)857 1341 y FA(k)935 1329 y FG(=)1038 1267 y Fz(P)1139 1329 y FF(j)p FH(s)p FF(j)1224 1299 y FD(2)p FA(k)1312 1329 y FH(b)1348 1299 y FD(2)1348 1350 y FA(s)1422 1329 y FH(<)e FF(1)p FC(,)k FH(k)f FF(\025)d FG(1)p FC(,)j(then)75 b FG(supp)38 b FH(\026)f FF(\032)g FB(B)2666 1341 y FA(r)2697 1350 y Fu(k)2738 1329 y FG(\()p FH(H)2846 1299 y FA(k)2887 1329 y FG(\))p FC(,)j(wher)l(e)0 1446 y FH(r)37 1458 y FA(k)101 1446 y FG(=)23 b FH(C)248 1458 y FA(k)289 1446 y FG(\()p FH(r)358 1458 y FD(0)415 1446 y FG(+)18 b FH(r)537 1402 y FD(2\()p FA(k)q FD(+1\))535 1468 y(0)766 1446 y FG(+)849 1381 y FF(p)p 918 1381 104 4 v 65 x FH(B)981 1458 y FA(k)1022 1446 y FG(\))c FH(:)0 1545 y FJ(Hin)m(t:)37 b FC(Use)30 b(that)g(for)g(e)l(ach)h FH(j)5 b FC(,)30 b FF(k)p FH(\021)1083 1557 y FA(j)1118 1545 y FF(k)1160 1515 y FD(2)1160 1569 y FA(k)1223 1545 y FF(\024)23 b FH(B)1374 1557 y FA(k)1415 1545 y FC(,)30 b(a.s.)0 1710 y FJ(Corollary)j(6.12.)40 b FC(If)30 b FH(B)814 1722 y FA(k)878 1710 y FH(<)22 b FF(1)30 b FC(for)h(al)t(l)f FH(k)c FF(\025)d FG(0)p FC(,)30 b(then)59 b FH(\026)p FG(\()p FH(H)25 b FF(\\)19 b FH(C)2112 1680 y Fw(1)2183 1710 y FG(\))k(=)g(1)14 b FH(:)125 1874 y FI(Let)26 b FH(B)335 1886 y FA(k)399 1874 y FH(<)d FF(1)k FI(and)g FH(f)k FF(2)24 b FH(C)6 b FG(\()p FH(H)1082 1844 y FA(k)1123 1874 y FG(\))27 b FI(b)r(e)g(a)f(function,)i(whic)n(h)f(is)f(Lipsc)n (hitz)h(on)f(b)r(ounded)i(subsets)e(of)0 1974 y FH(H)76 1944 y FA(k)117 1974 y FI(.)37 b(Then,)27 b(due)h(to)g(Theorem)e(2.15,) h FF(B)1279 1986 y FA(r)1315 1974 y FH(f)9 b FG(\()p FH(u)p FG(\))23 b(=)f(\()p FH(P)1672 1986 y FA(r)1710 1974 y FG(\()p FH(u;)14 b FF(\001)p FG(\))p FH(;)g(f)9 b FG(\))27 b FI(is)h(w)n(ell)f(de\034ned.)0 2139 y FJ(Prop)s(osition)j (6.13.)40 b FC(If)30 b FH(B)904 2151 y FA(k)q FD(+1)1052 2139 y FH(<)23 b FF(1)p FC(,)30 b(then)860 2328 y FF(jB)938 2340 y FA(r)974 2328 y FH(f)9 b FG(\()p FH(u)p FG(\))19 b FF(\000)1238 2261 y Fz(\000)1276 2328 y FH(f)t(;)14 b(\026)p FG(\))p FF(j)23 b(\024)g FH(C)d(C)1712 2340 y FA(f)1769 2328 y FH(\024)1900 2266 y Fu(r)p 1827 2275 177 3 v 1827 2309 a Ft(2\()p Fu(k)q Ft(+1\))2017 2328 y FH(;)60 b(r)26 b FF(\025)d FG(1)14 b FH(:)0 2509 y FC(Her)l(e)36 b FH(C)k FG(=)34 b FH(C)6 b FG(\()p FH(k)s(;)14 b FF(j)p FH(u)p FF(j)p FG(\))36 b FC(and)h FH(C)972 2521 y FA(f)1051 2509 y FC(is)f(the)g(Lipschitz-norm)h(of)g FH(f)44 b FC(r)l(estricte)l(d)36 b(to)g(a)h(b)l(al)t(l)f FB(B)2843 2521 y FA(R)2898 2509 y FG(\()p FH(H)3006 2479 y FA(k)q FD(+1)3131 2509 y FG(\))p FC(,)0 2609 y(wher)l(e)30 b FH(R)24 b FG(=)f FH(R)q FG(\()p FH(k)s(;)14 b FF(j)p FH(u)p FF(j)p FG(\))p FC(.)125 2773 y FI(F)-7 b(or)26 b(a)i(pro)r(of)f(see)g(Section)g(3)g(in)h([KS03)o(].)0 2938 y FJ(Example)d(6.14.)35 b FI(F)-7 b(or)23 b(an)n(y)f(t)n(w)n(o)g (p)r(oin)n(ts)h FH(x)1363 2950 y FD(1)1401 2938 y FH(;)14 b(x)1485 2950 y FD(2)1545 2938 y FF(2)24 b FE(T)1679 2908 y FD(2)1716 2938 y FI(,)g(w)n(e)f(set)g FH(f)2047 2950 y FA(ij)2105 2938 y FG(\()p FH(u)p FG(\))g(=)g FH(u)2376 2950 y FA(i)2403 2938 y FG(\()p FH(x)2482 2950 y FD(1)2520 2938 y FG(\))p FH(u)2600 2950 y FA(j)2635 2938 y FG(\()p FH(x)2714 2950 y FD(2)2752 2938 y FG(\))p FI(,)h FH(i;)14 b(j)27 b FG(=)c(1)p FH(;)14 b FG(2)p FI(.)0 3038 y(Since)32 b FH(H)297 3008 y FD(2)365 3038 y FF(\032)f FH(C)6 b FG(\()p FE(T)613 3008 y FD(2)651 3038 y FG(\))p FI(,)34 b(then)f FH(f)41 b FI(is)32 b(a)g(lo)r(cally)g(Lipsc)n(hitz)g(function) g(on)h FH(H)2331 3008 y FD(2)2368 3038 y FI(.)51 b(If)32 b FH(B)2592 3050 y FD(3)2661 3038 y FH(<)e FF(1)p FI(,)k(then)f(for)0 3137 y(an)n(y)27 b(solution)g(of)g(\(6.1\))h(with)g(\(sa)n(y\))f(a)g (deterministic)h(initial)f(data)g FH(u)2205 3149 y FD(0)2270 3137 y FI(w)n(e)g(ha)n(v)n(e)761 3361 y Fk(E)14 b FH(u)886 3373 y FA(i)913 3361 y FG(\()p FH(k)s(;)g(x)1075 3373 y FD(1)1112 3361 y FG(\))g FH(u)1206 3373 y FA(j)1241 3361 y FG(\()p FH(k)s(;)g(x)1403 3373 y FD(2)1441 3361 y FG(\))23 b FF(!)1602 3248 y Fz(Z)1648 3436 y FA(H)1739 3361 y FH(u)1787 3373 y FA(i)1814 3361 y FG(\()p FH(x)1893 3373 y FD(1)1931 3361 y FG(\))14 b FH(u)2025 3373 y FA(j)2060 3361 y FG(\()p FH(x)2139 3373 y FD(2)2177 3361 y FG(\))g FH(\026)p FG(\()p FH(du)p FG(\))0 3583 y FI(exp)r(onen)n(tially)34 b(fast.)57 b(The)34 b(l.h.s)h(is)f(called)g(the)g FC(c)l(orr)l(elation) j(tensor)f(of)h(the)f(solution)p FI(,)g(ev)-5 b(aluated)0 3683 y(at)40 b(time)h FH(k)s FI(,)j(and)d(the)g(r.h.s.)75 b(is)40 b(called)g(the)h FC(c)l(orr)l(elation)i(tensor)f(of)g(the)g (stationary)g(me)l(asur)l(e)p FI(.)0 3782 y(The)29 b(con)n(v)n(ergence) e(ab)r(o)n(v)n(e)h(is)h(one)g(of)g(p)r(ostulates)g(of)g(statistical)g (h)n(ydro)r(dynamic,)f(tak)n(en)h(there)g(for)0 3882 y(gran)n(ted.)36 b(See)27 b([Bat82)o(].)0 4114 y Fy(6.5)112 b(3D)37 b(NSE)g(with)f(small)g(random)i(kic)m(ks)0 4267 y FI(Consider)27 b(the)g(3D)h(NSE,)h(written)e(as)1164 4448 y FG(_)-38 b FH(u)18 b FG(+)g FH(Lu)g FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))p FH(:)0 4629 y FI(All)28 b(ob)5 b(jects)27 b(in)h(this)g(equation)f(are)f(natural)h(3D)h(analogues)d(of)j(those)f (in)h(Section)f(3.)37 b(As)27 b(b)r(efore,)1104 4810 y FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))23 b(=)g FH(S)5 b FG(\()p FH(u)p FG(\()p FH(k)s FG(\)\))19 b(+)f FH(\021)1937 4822 y FA(k)q FD(+1)2062 4810 y FH(:)892 b FI(\(6.16\))1553 5059 y(56)p eop end PStoPSsaved restore %%Page: (56,57) 29 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 57 56 bop 0 523 a FI(It)32 b(is)f(kno)n(wn)f(that)i FF(k)o FH(S)5 b FG(\()p FH(u)808 535 y FD(1)845 523 y FG(\))p FF(k)919 548 y FD(2)985 523 y FF(\024)29 b FH(C)20 b FF(k)p FH(u)1248 535 y FD(1)1284 523 y FF(k)1326 548 y FD(2)1394 523 y FI(and)31 b FF(k)p FH(S)5 b FG(\()p FH(u)1737 535 y FD(1)1774 523 y FG(\))18 b FF(\000)h FH(S)5 b FG(\()p FH(u)2044 535 y FD(2)2080 523 y FG(\))p FF(k)2154 548 y FD(2)2220 523 y FF(\024)29 b FH(C)20 b FF(k)p FH(u)2483 535 y FD(1)2538 523 y FF(\000)e FH(u)2669 535 y FD(2)2706 523 y FF(k)2747 548 y FD(2)2816 523 y FI(with)32 b(some)0 623 y FH(C)k(<)28 b FG(1)p FI(,)k(if)g FF(k)p FH(u)455 635 y FD(1)491 623 y FF(k)533 648 y FD(2)584 623 y FH(;)14 b FF(k)o FH(u)710 635 y FD(2)747 623 y FF(k)789 648 y FD(2)857 623 y FI(are)31 b(small)f(enough.)48 b(Therefore)30 b(if)i FH(B)2073 635 y FD(2)2141 623 y FI(\(de\034ned)g(as)f(in)g(Section)g(3.2\))g(is)0 722 y(su\036cien)n(tly)c(small,)g(then)h(\(6.16\))e(de\034nes)h(a)g(random) f(dynamical)h(system)f(\(and)i(a)e(Mark)n(o)n(v)f(c)n(hain\))0 822 y(in)35 b(a)e(small)h(ball)h FB(B)645 834 y FA(\017)677 822 y FG(\()p FH(H)785 792 y FD(2)822 822 y FG(\))p FI(.)57 b(A)35 b(natural)f(analogy)e(of)i(Theorem)g(6.4)f(holds)h(true)g(for)g (\(6.16\))g(with)0 922 y FH(u)23 b FF(2)g FB(B)222 934 y FA(\017)254 922 y FG(\()p FH(H)362 891 y FD(2)399 922 y FG(\))p FI(.)38 b(In)27 b(fact,)h(the)g(pro)r(of)f(m)n(uc)n(h)g (simpli\034es)h(since)f(no)n(w)g(the)h(map)g FH(S)k FI(is)c(a)f(con)n (traction.)0 1151 y Fy(6.6)112 b(Stationary)37 b(measures)h(and)g (random)g(attractors)0 1304 y FI(In)32 b(this)g(section)f(w)n(e)g (discuss)g(random)f(attractors)g(for)h(the)h(randomly)e(kic)n(k)n(ed)h (NSE,)i(in)n(terpreted)0 1404 y(as)27 b(the)h(system)f(\(6.1\).)125 1503 y(First)h(w)n(e)f(put)i(the)f(system)g(\(6.1\))g(in)h(the)f (classical)f(framew)n(ork)f(of)i(RDS)h([Arn98)o(].)38 b(T)-7 b(o)28 b(do)g(this)0 1603 y(w)n(e)33 b(tak)n(e)f(in\034nitely)i (man)n(y)e(copies)h FG(\(\012)1241 1615 y FA(j)1276 1603 y FH(;)28 b(F)1380 1615 y FA(j)1415 1603 y FH(;)g Fk(P)1531 1615 y FA(j)1566 1603 y FG(\))p FI(,)35 b FH(j)i FF(2)c FE(Z)p FI(,)i(of)e(the)h(probabilit)n(y)e(space,)h(on)g(whic)n(h)0 1702 y(are)24 b(de\034ned)h(the)g(random)f(v)-5 b(ariables)23 b FH(\021)1243 1714 y FA(k)1284 1702 y FI(,)j(and)e(de\034ne)h FG(\(\012)p FH(;)j FF(F)8 b FH(;)28 b Fk(P)p FG(\))e FI(as)e(the)h(pro)r(duct)g(of)f(these)h(spaces)0 1802 y(\(so)31 b FG(\012)e(=)g(\005)383 1814 y FA(j)s Fw(2)p Fv(Z)502 1802 y FG(\012)562 1814 y FA(j)597 1802 y FI(,)j(etc\).)48 b(In)n(tro)r(ducing)31 b(the)g(shift)h(op)r(erators)e FH(\022)2063 1814 y FA(k)2133 1802 y FG(:)f(\012)g FF(!)g FG(\012)p FI(,)47 b FH(k)32 b FF(2)d FE(Z)p FI(,)k(whic)n(h)e(send)p 0 1856 55 4 v 0 1902 a FH(!)26 b FG(=)c(\()p FH(!)249 1914 y FA(l)275 1902 y FH(;)28 b(l)c FF(2)f FE(Z)p FG(\))29 b FI(to)p 670 1856 V 27 w FH(!)725 1870 y Fw(0)771 1902 y FG(=)23 b(\()p 891 1856 V FH(!)946 1870 y Fw(0)946 1924 y FA(l)971 1902 y FH(;)28 b(l)c FF(2)g FE(Z)p FG(\))p FH(;)k(!)1343 1872 y Fw(0)1340 1925 y FA(l)1389 1902 y FG(=)23 b FH(!)1529 1914 y FA(l)p FD(+)p FA(k)1642 1902 y FI(,)k(w)n(e)h(write)f(the)h(system)f(\(6.1\))g(as)1143 2066 y FH(u)p FG(\()p FH(k)21 b FG(+)d(1\))23 b(=)f FH(\036)1603 1999 y Fz(\000)1642 2066 y FH(u)p FG(\()p FH(k)s FG(\))p FH(;)28 b(\022)1890 2078 y FA(k)p 1930 2020 V 1930 2066 a FH(!)1985 1999 y Fz(\001)2023 2066 y FH(;)931 b FI(\(6.17\))0 2230 y(where)30 b FH(\036)p FG(\()p FH(u;)p 423 2184 V 28 w(!)r FG(\))e(=)f FH(S)5 b FG(\()p FH(u)p FG(\))20 b(+)g FH(\021)943 2242 y FD(0)980 2230 y FG(\()p FH(!)1064 2242 y FD(0)1101 2230 y FG(\))p FI(.)45 b(Clearly)-7 b(,)30 b(\(6.17\))g(de\034nes)g(in)g FH(H)37 b FI(the)31 b(same)e(Mark)n(o)n(v)f(c)n(hain)i(as)0 2329 y(\(6.1\).)51 b(F)-7 b(or)31 b FH(k)j FF(\025)d FG(0)h FI(w)n(e)g(denote)g FH(\036)1093 2341 y FA(k)1134 2329 y FG(\()p 1166 2284 V FH(!)s FG(\))p FH(u)f FG(=)f FH(u)p FG(\()p FH(k)s FG(\))p FI(,)k(where)e FG(\()p FH(u)p FG(\()p FH(l)r FG(\))p FH(;)27 b(l)33 b FF(\025)d FG(0\))i FI(is)g(a)g(solution)g(of) 39 b(\(6.17\))o(,)0 2429 y(suc)n(h)27 b(that)h FH(u)p FG(\(0\))23 b(=)f FH(u)p FI(.)125 2529 y(A)28 b(closed)g(\(compact\))g (set)g FH(A)p 1050 2508 49 3 v 12 x FA(!)1123 2529 y FF(\032)c FH(H)7 b FI(,)28 b(dep)r(ending)g(on)g(a)g(random)g (parameter)p 2627 2483 55 4 v 26 w FH(!)f FF(2)e FG(\012)p FI(,)j(is)g(called)0 2628 y(a)f FC(close)l(d)k(\(c)l(omp)l(act\))f(r)l (andom)g(set)36 b FI(if)28 b(for)f(an)n(y)g(op)r(en)g(set)h FH(U)k FF(\032)22 b FH(H)35 b FI(w)n(e)27 b(ha)n(v)n(e)1185 2792 y FF(f)p 1227 2747 V FH(!)e FF(j)e FH(A)p 1412 2771 49 3 v 13 x FA(!)1479 2792 y FF(\\)18 b FH(U)32 b FF(6)p FG(=)23 b FF(;g)f(2)h(F)8 b FH(:)125 2956 y FI(A)25 b(random)g(compact) g(set)h FH(A)p 1035 2935 V 13 x FA(!)1108 2956 y FI(is)g(called)f(a)g FC(r)l(andom)k(p)l(oint)f(attr)l(actor)g(\(in)f(the)h(sense)g(of)h(c)l (onver-)0 3056 y(genc)l(e)22 b(in)g(pr)l(ob)l(ability\))i FI(if)c(for)e(eac)n(h)h FH(u)j FF(2)i FH(H)i FI(the)19 b(sequence)g(of)g(random)f(v)-5 b(ariables)17 b FH(d)p FG(\()p FH(\036)2657 3068 y FA(k)2699 3056 y FG(\()p 2731 3010 55 4 v FH(!)s FG(\))d FH(u;)27 b(A)2992 3068 y FA(\022)3024 3077 y Fu(k)p 3065 3010 V 3065 3056 a FH(!)r FG(\))3151 3026 y Fo(5)0 3156 y FI(con)n(v)n(erges)e(to)i(zero)g (in)h(probabilit)n(y)-7 b(.)35 b(I.e.,)1000 3320 y FG(lim)974 3374 y FA(k)q Fw(!1)1156 3320 y Fk(P)p FF(f)p FH(d)p FG(\()p FH(\036)1387 3332 y FA(k)1429 3320 y FG(\()p 1461 3274 V FH(!)s FG(\))14 b FH(u;)27 b(A)1722 3332 y FA(\022)1754 3341 y Fu(k)p 1791 3299 44 3 v 1791 3332 a FA(!)1839 3320 y FG(\))c FH(>)g(\016)s FF(g)f FG(=)h(0)0 3513 y FI(for)30 b(eac)n(h)f FH(\016)i(>)c FG(0)p FI(.)45 b(A)31 b(random)e(p)r(oin)n(t)i(attractors)d FH(A)p 1657 3492 49 3 v 12 x FA(!)1736 3513 y FI(is)i(said)g(to)h(b)r(e)f FC(minimal)i FI(if)f(for)e(an)n(y)h(random)0 3612 y(p)r(oin)n(t)c (attractor)e FH(A)626 3582 y Fw(0)p 626 3601 V 626 3635 a FA(!)701 3612 y FI(w)n(e)h(ha)n(v)n(e)g FH(A)p 1073 3591 V 13 x FA(!)1144 3612 y FF(\032)e FH(A)1294 3582 y Fw(0)p 1318 3591 V 1318 3625 a FA(!)1392 3612 y FI(for)i(a.a.)p 1682 3567 55 4 v 35 w FH(!)s FI(.)36 b(Clearly)-7 b(,)26 b(if)g(a)g(minimal)g(random)f(attractor)0 3712 y(exists,)i(then)h(it)g (is)g(unique.)125 3812 y(There)g(are)f(other)h(t)n(yp)r(es)g(of)h (random)e(attractors,)g(considered)h(in)h(mo)r(dern)f(mathematical)g (lit-)0 3911 y(erature,)c(see)f([Arn98)o(,)i(CDF97)o(,)f(Cra01)n(].)36 b(Among)24 b(them)g(the)h(random)e(attractors,)g(de\034ned)h(ab)r(o)n (v)n(e,)0 4011 y(are)j(the)h(smallest.)125 4110 y(Let)i FH(\026)h FI(b)r(e)g(a)g(stationary)e(measure)g(for)i(\(6.1\))o(,)h (and)e FF(F)1850 4122 y Fw(\000)1937 4110 y FI(b)r(e)h(a)f FH(\033)s FI(-algebra)f(in)i FF(F)8 b FI(,)32 b(generated)d(b)n(y)0 4210 y(random)k(v)-5 b(ariables)32 b FH(\030)t FG(\()p 733 4164 V FH(!)s FG(\))p FI(,)j(whic)n(h)f(dep)r(end)g(only)f(on)g FH(!)1777 4222 y FA(k)1818 4210 y FH(;)27 b(k)36 b FF(\024)d FG(0)p FI(.)54 b(F)-7 b(or)33 b(the)h(follo)n(wing)e(imp)r(ortan)n(t)0 4310 y(result)27 b(see)g([Led86)o(,)h(Le)g(87)o(,)f(Cra91)n(].)0 4461 y FJ(Prop)s(osition)j(6.15.)40 b FC(The)30 b(limit)1164 4625 y FH(\026)p 1214 4604 49 3 v 12 x FA(!)1285 4625 y FG(=)49 b(lim)1373 4679 y FA(k)q Fw(!1)1555 4625 y FH(\036)1604 4637 y FA(k)1646 4625 y FG(\()p FH(\022)1717 4637 y Fw(\000)p FA(k)p 1810 4579 55 4 v 1810 4625 a FH(!)r FG(\))19 b FF(\016)f FH(\026)p 0 4733 1276 4 v 92 4787 a Fn(5)127 4810 y Fm(for)24 b(a)g(p)r(oin)n(t)g Fi(t)c Fh(2)g Fi(H)28 b Fm(and)c(a)g(set)g Fi(B)f Fh(\032)d Fi(H)28 b Fm(w)n(e)c(denote)i Fi(d)p Fg(\()p Fi(z)s(;)d(B)s Fg(\))e(=)90 b Fm(inf)2046 4822 y Ff(\030)q Fb(2)p Ff(B)2170 4810 y Fh(j)p Fi(x)15 b Fh(\000)g Fi(\030)s Fh(j)1553 5059 y FI(57)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 58 57 bop 0 523 a FC(exists)29 b(almost)h(sur)l(ely)f(in) g(the)h(sense)f(of)h(we)l(ak)g(c)l(onver)l(genc)l(e)g(of)g(me)l(asur)l (es.)38 b(The)30 b(r)l(andom)g(me)l(asur)l(e)0 623 y FH(\026)50 635 y FA(!)134 623 y FC(is)37 b FF(F)290 635 y Fw(\000)346 623 y FC(\025me)l(asur)l(able)g(in)f(the)h(sense)f(that)g (for)h(any)g(Bor)l(el)g(set)f FH(Q)f FF(\032)g FH(H)43 b FC(the)36 b(r)l(andom)h(variable)0 722 y FH(\026)p 50 701 49 3 v 13 x FA(!)98 722 y FG(\()p FH(Q)p FG(\))30 b FC(is)g FF(F)407 734 y Fw(\000)492 722 y FC(me)l(asur)l(able,)h(and) 1102 692 y Fo(6)1422 822 y FH(\026)23 b FG(=)g Fk(E)p FH(\026)p 1696 801 V 12 x FA(!)1744 822 y FH(:)1210 b FI(\(6.18\))125 982 y(The)25 b(represen)n(tation)e(\(6.18\))h(is)h (called)f(the)i FC(Markov)j(disinte)l(gr)l(ation)j FI(of)25 b(a)g(stationary)e(measure)0 1082 y FH(\026)p FI(.)36 b(It)25 b(is)f(pro)n(v)n(ed)f(in)i([Cra01)n(])f(that)h(for)f(a)g(large) f(class)g(of)i(RDS,)g(whic)n(h)f(includes)h(the)g(system)f(\(6.17\))o (,)0 1182 y(the)i(disin)n(tegration)f FH(\026)p 714 1161 V 12 x FA(!)788 1182 y FI(is)h(supp)r(orted)g(b)n(y)f(eac)n(h)h(random) f(p)r(oin)n(t)h(attractor)e FH(A)2487 1152 y Fw(0)p 2511 1161 V 2511 1194 a FA(!)2559 1182 y FI(,)i(i.e.,)h FH(\026)p 2814 1161 V 12 x FA(!)2862 1182 y FG(\()p FH(A)2956 1152 y Fw(0)p 2956 1170 V 2956 1204 a FA(!)3004 1182 y FG(\))d(=)e(1)0 1281 y FI(a.s.)37 b(That)28 b(is)g(to)g(sa)n(y)-7 b(,)27 b(supp)13 b FH(\026)p 946 1260 V 13 x FA(!)1018 1281 y FF(\032)23 b FH(A)1168 1251 y Fw(0)p 1168 1270 V 1168 1304 a FA(!)1245 1281 y FI(a.s.)37 b(F)-7 b(or)27 b(the)h(sp)r (eci\034c)g(systems)g(\(6.18\))f(whic)n(h)h(w)n(e)f(consider)0 1381 y(a)g(m)n(uc)n(h)h(stronger)d(result)i(is)h(true:)0 1541 y FJ(Theorem)f(6.16.)38 b FC(Under)27 b(the)g(assumptions)h(of)g (The)l(or)l(em)g(6.4)g(supp)15 b FH(\026)p 2295 1520 V 13 x FA(!)2370 1541 y FC(is)28 b(the)f(minimal)h(r)l(andom)0 1641 y(p)l(oint)i(attr)l(actor)g(of)g(the)g(system)36 b FI(\(6.1\))p FC(.)125 1802 y FI(In)29 b(particular,)f(due)i(to)f(the) g(kno)n(wn)g(results)f(on)h(random)g(attractors)e(\(see)i([Deb98)o (]\),)i(a.s.)41 b(the)0 1901 y(Hausdor\033)27 b(dimension)g(of)h(supp) 14 b FH(\026)p 1101 1880 V 13 x FA(!)1176 1901 y FI(is)28 b(b)r(ounded)g(b)n(y)f(a)g(certain)g(deterministic)h(constan)n(t.)125 2001 y(Theorem)f(6.16)g(establishes)h(connection)g(b)r(et)n(w)n(een)g (t)n(w)n(o)g(imp)r(ortan)n(t)g(ob)5 b(jects,)28 b(related)g(to)g(ran-)0 2100 y(domly)g(p)r(erturb)r(ed)h(2D)g(NSE,)h(and)e(pro)n(vides)f(its)i (minimal)g(random)f(p)r(oin)n(t)h(attractor,)e(written)i(as)0 2200 y(supp)14 b FH(\026)p 235 2179 V 13 x FA(!)283 2200 y FI(,)37 b(with)e(the)g(natural)f(measure)f FH(\026)p 1367 2179 V 13 x FA(!)1415 2200 y FI(.)58 b(Since)35 b(dimension)g(of)f(an)h(attractor)e(go)r(es)g(to)i(in\034n-)0 2300 y(it)n(y)c(with)g(the)h(Reynolds)e(n)n(um)n(b)r(er,)h(then)h(the)f (measures)f FH(\026)p 1907 2279 V 12 x FA(!)1986 2300 y FI(ma)n(y)g(b)r(e)h(useful)h(for)e(applications)g(of)0 2399 y(the)e(random)f(p)r(oin)n(t)h(attractors)f(to)g(2D)h(\035o)n(ws)f (with)i(high)f(Reynolds)f(n)n(um)n(b)r(ers)g(as)h(it)g(pro)n(vides)f (the)0 2499 y(attractors)f(with)i(a)f(structure)g(whic)n(h)g(is)h(not)f (sensitiv)n(e)g(to)h(their)f(dimensions.)125 2599 y(The)20 b(theorem)f(is)h(pro)n(v)n(ed)f(in)h([KS04a)o(],)i(where)d(it)i(is)f (deriv)n(ed)f(from)h(a)g(general)e(abstract)i(theorem.)125 2745 y(F)-7 b(or)32 b(the)i(white\025forced)f(NSE)h(\(4.3\))f(a)g (natural)g(v)n(ersion)f(of)h(Theorem)g(6.16)f(holds)h(true.)54 b(See)0 2845 y([KS04a)n(].)0 3076 y Fy(6.7)112 b(App)s(endix:)50 b(Summary)37 b(of)h(the)f(pro)s(of)h(of)g(Theorem)f(6.4)0 3229 y FI(Denote)27 b FH(\026)334 3241 y FA(j)369 3229 y FG(\()p FH(k)s FG(\))c(=)g FF(B)648 3199 y Fw(\003)645 3253 y FA(k)685 3229 y FG(\()p FH(\026)767 3241 y FA(j)803 3229 y FG(\))p FI(,)k FH(j)h FG(=)22 b(1)p FH(;)14 b FG(2)p FI(,)26 b FH(k)g FG(=)d(1)p FH(;)14 b FG(2)p FH(;)g(:)g(:)g(:)e FI(.)37 b(T)-7 b(o)26 b(pro)n(v)n(e)f(the)i(Theorem)f(w)n(e)g (construct)g(w)n(eak)0 3329 y(solution)h(of)h(\(6.1\))f(suc)n(h)g(that) 1163 3429 y FF(D)16 b FH(U)1300 3441 y FA(j)1335 3429 y FG(\(0\))23 b(=)g FH(\026)1602 3441 y FA(j)1637 3429 y FH(;)60 b(j)28 b FG(=)22 b(1)p FH(;)14 b FG(2)g FH(:)950 b FI(\(6.19\))0 3574 y(Then)28 b FH(\026)267 3586 y FA(j)302 3574 y FG(\()p FH(k)s FG(\))23 b(=)g FF(D)16 b FH(U)660 3586 y FA(j)695 3574 y FG(\()p FH(k)s FG(\))p FI(,)28 b FH(k)e FF(\025)c FG(0)p FI(,)28 b FH(j)g FG(=)22 b(1)p FH(;)14 b FG(2)p FI(.)36 b(W)-7 b(e)28 b(wish)g(to)f(construct)g(them)h (in)g(suc)n(h)f(a)g(w)n(a)n(y)g(that)504 3750 y FH(d)p FG(\()p FH(k)s FG(\))d(:=)e FF(j)p FH(U)871 3762 y FD(1)908 3750 y FG(\()p FH(k)s FG(\))d FF(\000)f FH(U)1177 3762 y FD(2)1214 3750 y FG(\()p FH(k)s FG(\))p FF(j)24 b(\034)f FG(1)45 b FI(with)28 b(high)g(probabilit)n(y)-7 b(,)27 b(if)41 b FH(k)26 b FF(\035)d FG(1)292 b FI(\(6.20\))0 3925 y(This)28 b(w)n(ould)f(imply)g(that)h(the)g(measures)e FH(\026)1393 3937 y FD(1)1431 3925 y FG(\()p FH(k)s FG(\))p FI(,)i FH(\026)1642 3937 y FD(2)1679 3925 y FG(\()p FH(k)s FG(\))g FI(are)f(close.)125 4025 y(Denote)k FH(B)h FG(=)d FB(B)722 4029 y Fu(e)p 685 4038 102 3 v 685 4071 a(e)p Fr(\000)p Ft(1)808 4002 y Fw(p)p 862 4002 53 3 v 862 4051 a FA(B)915 4061 y Ft(0)952 4025 y FG(\()p FH(H)7 b FG(\))p FI(.)47 b(W)-7 b(e)31 b(kno)n(w)f(\(see)h(\(6.7\)\))g(that)g (if)h FH(U)2252 4037 y FA(j)2286 4025 y FG(\(0\))d FF(2)g FH(B)35 b FI(for)c FH(j)i FG(=)c(1)p FH(;)14 b FG(2)30 b FI(a.s.,)0 4144 y(then)918 4243 y FH(U)975 4255 y FA(j)1010 4243 y FG(\()p FH(k)s FG(\))23 b FF(2)g FH(B)32 b FI(for)27 b FH(j)h FG(=)23 b(1)p FH(;)14 b FG(2)p FI(,)27 b(and)g FH(k)f FF(\025)d FG(0)k FI(a.s.)705 b(\(6.21\))0 4388 y(T)-7 b(o)26 b(simplify)h(the)g(presen)n(tation)e(w)n(e)h(assume)f (that)i(supp)g FH(\026)1881 4400 y FA(j)1939 4388 y FF(\032)22 b FH(B)31 b FG(\()p FH(j)d FG(=)23 b(1)p FH(;)14 b FG(2\))p FI(.)36 b(Then)26 b(an)n(y)g(coupling)0 4488 y FG(\()p FH(U)89 4500 y FD(1)126 4488 y FG(\()p FH(k)s FG(\))p FH(;)14 b(U)330 4500 y FD(2)367 4488 y FG(\()p FH(k)s FG(\)\))29 b FI(for)e(the)h(measures)e FG(\()p FH(\026)1246 4500 y FD(1)1283 4488 y FG(\()p FH(k)s FG(\))p FH(;)14 b(\026)1480 4500 y FD(2)1518 4488 y FG(\()p FH(k)s FG(\)\))28 b FI(satis\034es)f(\(6.21\).)36 b(Argumen)n(ts)27 b(in)h(the)g(general) 0 4588 y(case)f(are)f(v)n(ery)h(similar.)p 0 4654 1276 4 v 92 4708 a Fn(6)127 4731 y Fm(The)h(in)n(tegral)g(in)f(\(6.18\))h (is)e(understo)r(o)r(d)k(in)d(the)h(w)n(eak)g(sense.)42 b(That)28 b(is,)f(for)g(an)n(y)h Fi(Q)d Fh(2)g(B)r Fg(\()p Fi(H)5 b Fg(\))p Fm(,)56 b Fg(\()p Fd(E)p Fi(\026)p 2925 4713 45 3 v 11 x Ff(!)2971 4731 y Fg(\)\()p Fi(Q)p Fg(\))26 b(=)0 4810 y Fd(E)p Fg(\()p Fi(\026)p 122 4792 V 11 x Ff(!)168 4810 y Fg(\()p Fi(Q)p Fg(\)\))p Fm(.)1553 5059 y FI(58)p eop end PStoPSsaved restore %%Page: (58,59) 30 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 59 58 bop 125 523 a FI(F)-7 b(rom)27 b(Lemma)g(6.1)g(w)n (e)g(kno)n(w)g(the)h(follo)n(wing:)0 623 y(Let)33 b FH(\026)204 635 y FD(1)272 623 y FG(=)d FH(\016)404 635 y FA(u)443 643 y Ft(1)480 623 y FI(,)k FH(\026)d FG(=)g FH(\016)751 635 y FA(u)790 643 y Ft(2)826 623 y FI(.)52 b(Denote)32 b FH(d)f FG(=)g FF(j)p FH(u)1431 635 y FD(1)1489 623 y FF(\000)22 b FH(u)1624 635 y FD(2)1660 623 y FF(j)p FI(.)52 b(Then)32 b(there)h(exist)f(w)n(eak)f(solutions)h FH(U)3019 635 y FD(1)3056 623 y FG(\()p FH(k)s FG(\))p FI(,)0 722 y FH(U)57 734 y FD(2)94 722 y FG(\()p FH(k)s FG(\))p FI(,)c FH(k)e FG(=)d(0)p FH(;)14 b FG(1)p FI(,)26 b(suc)n(h)h(that)h(with)g(probabilit)n(y)f FF(\025)22 b FG(1)c FF(\000)g FH(C)1852 734 y Fw(\003)1905 722 y FH(d)28 b FI(w)n(e)f(ha)n(v)n(e)1382 932 y FH(d)p FG(\(1\))d FF(\024)1652 876 y FG(1)p 1652 913 42 4 v 1652 989 a(2)1727 932 y FH(d)14 b(:)125 1130 y FF(\017)41 b FI(Let)c FH(U)423 1142 y FD(1)460 1130 y FG(\(0\))p FI(,)i FH(U)685 1142 y FD(2)722 1130 y FG(\(0\))e FI(b)r(e)g(an)n(y)f(coupling)h(for)f FG(\()p FH(\026)1713 1142 y FD(1)1751 1130 y FH(;)14 b(\026)1838 1142 y FD(2)1875 1130 y FG(\))p FI(.)65 b(These)36 b(are)g(our)g(w)n(eak)g(solution)h(for)208 1230 y FH(k)25 b FG(=)e(0)p FI(.)125 1391 y FF(\017)41 b FI(Assume)32 b(that)h(w)n(e)f(ha)n(v)n(e)f(constructed)h(w)n(eak)g(solutions)f FH(U)2102 1403 y FD(1)2139 1391 y FG(\()p FH(t)p FG(\))p FI(,)k FH(U)2348 1403 y FD(2)2384 1391 y FG(\()p FH(t)p FG(\))f FI(for)e FH(t)f FF(\024)g FH(k)s FI(.)51 b(Ho)n(w)32 b(to)208 1490 y(construct)f FH(U)634 1502 y FD(1)671 1490 y FG(\()p FH(k)24 b FG(+)d(1\))p FI(,)33 b FH(U)1043 1502 y FD(2)1080 1490 y FG(\()p FH(k)24 b FG(+)d(1\))p FI(?)50 b(The)32 b(strategy)e(is)i(Mark)n(o)n(v,)f(i.e.)49 b(it)33 b(dep)r(ends)f(only)f(on)208 1590 y FH(U)265 1602 y FD(1)301 1590 y FG(\()p FH(k)s FG(\))p FI(,)e FH(U)520 1602 y FD(2)557 1590 y FG(\()p FH(k)s FG(\))p FI(.)0 1746 y(Fix)f(a)f(threshold)g FH(d)626 1758 y FD(0)686 1746 y FF(2)d FG(\(0)p FH(;)14 b FG(1\))p FI(.)0 1846 y FJ(Recip)s(e)26 b(1)p FI(.)36 b(If)24 b FH(d)p FG(\()p FH(k)s FG(\))g FF(\024)f FH(d)805 1858 y FD(0)842 1846 y FI(,)i(then)g(`w)n(e)e(are)g(in)i(coupling',)f(and)g(w)n(e)f (construct)h FH(U)2516 1858 y FD(1)p FA(;)p FD(2)2606 1846 y FG(\()p FH(k)14 b FG(+)d(1\))24 b FI(`in)g(terms)0 1946 y(of)k FH(U)152 1958 y FD(1)188 1946 y FG(\()p FH(k)s FG(\))p FI(,)h FH(U)407 1958 y FD(2)444 1946 y FG(\()p FH(k)s FG(\))p FI(',)f(using)f(Lemma)g(6.1.)36 b(Then,)28 b(with)g(high)f(probabilit)n(y)-7 b(,)1290 2155 y FH(d)p FG(\()p FH(k)22 b FG(+)c(1\))23 b FF(\024)1707 2099 y FG(1)p 1707 2136 V 1707 2212 a(2)1782 2155 y FH(d)1825 2167 y FD(0)1876 2155 y FH(:)0 2348 y FI(If)f(THIS)g(happ)r(ens,)h (then)f(`w)n(e)g(are)e(in)i(coupling',)h(and)e(w)n(e)g(con)n(tin)n(ue)h (to)f(iterate.)35 b(W)-7 b(e)22 b(iterate)f(forev)n(er)0 2448 y(with)28 b(the)g(probabilit)n(y)182 2653 y FF(\025)23 b FG(1)18 b FF(\000)g FH(C)472 2665 y Fw(\003)510 2653 y FH(d)h FF(\000)f FH(C)714 2665 y Fw(\003)786 2596 y FG(1)p 786 2634 V 786 2710 a(2)860 2653 y FH(d)h FF(\000)f FH(C)1064 2665 y Fw(\003)1135 2596 y FG(1)p 1135 2634 V 1135 2710 a(4)1210 2653 y FH(d)h FF(\000)f(\001)c(\001)g(\001)23 b FG(=)f(1)c FF(\000)g FH(C)1764 2665 y Fw(\003)1803 2653 y FH(d)1846 2561 y Fz(\020)1896 2653 y FG(1)k(=)2058 2596 y(1)p 2058 2634 V 2058 2710 a(2)2128 2653 y(+)2221 2596 y(1)p 2221 2634 V 2221 2710 a(4)2291 2653 y(+)c FH(:)c(:)g(:)2484 2561 y Fz(\021)2557 2653 y FG(=)22 b(1)c FF(\000)g FG(2)p FH(C)2888 2665 y Fw(\003)2926 2653 y FH(d)c(:)0 2846 y FI(W)-7 b(e)28 b(`lose)f(coupling')g(with)h (probabilit)n(y)e FF(\024)d FG(2)p FH(C)1485 2858 y Fw(\003)1523 2846 y FH(d)p FI(.)0 2946 y FJ(Recip)s(e)29 b(2.)36 b FI(If)27 b FH(d)p FG(\()p FH(k)s FG(\))d FH(>)f(d)815 2958 y FD(0)852 2946 y FI(,)k(then)g(w)n(e)g(are)e(not)i(in)g (coupling.)36 b(This)26 b(usually)g(happ)r(ens)h(when)g FH(k)f FG(=)c(0)p FI(.)0 3046 y(Also,)27 b(it)h(happ)r(ens)g(when)g(w)n (e)f(lose)g(coupling.)36 b(Then:)125 3180 y FF(9)14 b FH(T)39 b FG(=)28 b FH(T)12 b FG(\()p FH(d)502 3192 y FD(0)539 3180 y FG(\))28 b FF(\025)g FG(1)p FI(,)j(suc)n(h)g(that)g (with)g(probabilit)n(y)f FH(\022)r FG(\()p FH(d)1898 3192 y FD(0)1936 3180 y FG(\))e FH(>)g FG(0)i FI(w)n(e)h(ha)n(v)n(e)e FF(j)p FH(U)2561 3192 y FD(1)2598 3180 y FG(\()p FH(k)24 b FG(+)c FH(T)12 b FG(\))p FF(j)28 b(\024)3029 3124 y FG(1)p 3029 3161 V 3029 3237 a(2)3109 3180 y FH(d)3152 3192 y FD(0)0 3355 y FI(and)f FF(j)p FH(U)241 3367 y FD(2)278 3355 y FG(\()p FH(k)22 b FG(+)c FH(T)12 b FG(\))p FF(j)22 b(\024)694 3298 y FG(1)p 694 3336 V 694 3412 a(2)769 3355 y FH(d)812 3367 y FD(0)863 3355 y FH(:)28 b FI(Then)1323 3482 y FH(d)p FG(\()p FH(k)22 b FG(+)c FH(T)12 b FG(\))22 b FF(\024)h FH(d)1792 3494 y FD(0)1843 3482 y FH(:)1111 b FI(\(6.22\))0 3624 y(In)28 b(\(6.10\))f(I)g(use)h(a) f(brutal)g(w)n(a)n(y)f(to)i(ac)n(hiev)n(e)e(\(6.22\),)h(but)h(I)g(do)f (not)g(kno)n(w)g(ho)n(w)g(to)h(do)f(this)h(b)r(etter!)125 3724 y(So,)e(after)h FH(T)38 b FI(steps)27 b(w)n(e)g(switc)n(h)f(to)h (Recip)r(e)h(1)e(with)i(probabilit)n(y)d FH(\022)r FI(.)37 b(W)-7 b(e)28 b(con)n(tin)n(ue)e(with)i(Recip)r(e)0 3823 y(2)f(with)h(probabilit)n(y)f FG(1)18 b FF(\000)g FH(\022)r FI(.)125 3923 y(This)27 b(strategy)f(results)h(in)h(\(6.20\).)125 4023 y(T)-7 b(o)27 b(put)h(this)g(argumen)n(ts)e(in)i(n)n(um)n(b)r (ers,)f(the)h(metho)r(d)g(of)f(Kan)n(toro)n(vic)n(h)e(functionals)i(is) h(used.)0 4212 y FJ(The)47 b(ph)m(ysical)g(relev)-5 b(ance)46 b(of)h(the)f(results.)74 b FI(The)41 b(con)n(v)n(ergences)c FF(B)2444 4224 y FA(k)2485 4212 y FH(f)9 b FG(\()p FH(u)p FG(\))44 b FF(!)g FG(\()p FH(f)t(;)14 b(\026)p FG(\))41 b FI(and)0 4312 y FF(B)58 4282 y Fw(\003)55 4335 y FA(k)95 4312 y FH(\027)29 b FF(!)23 b FH(\026)p FI(,)g(where)e FH(\026)h FI(is)g(the)g(unique)g(stationary)e(measure,)i(are)f(v)n(ery) g(imp)r(ortan)n(t)g(for)g(the)i(statistical)0 4411 y(h)n(ydro)r (dynamics,)j(where)g(they)h(are)f(p)r(ostulated)h(in)g(the)g(form)g (that)g(\020statistical)f(prop)r(erties)g(of)h(the)0 4511 y(turbulen)n(t)21 b(motion)f(of)h(\035uid)g(fast)g(b)r(ecome)f (indep)r(enden)n(t)i(of)e(the)h(initial)g(data\021,)2501 4481 y Fo(7)2561 4511 y FI(cf.)35 b(Example)20 b(6.14;)p 0 4575 1276 4 v 92 4629 a Fn(7)159 4652 y Fm(\020)6 b(.)11 b(.)g(.)g(w)n(e)33 b(put)g(our)g(faith)f(in)f(the)i(tendency)i(for)d (dynamical)f(systems)h(with)g(a)g(large)h(n)n(um)n(b)r(er)f(of)g (degrees)i(of)0 4731 y(freedom,)28 b(and)g(with)f(coupling)h(b)r(et)n (w)n(een)h(these)g(degrees)g(of)e(freedom,)h(to)g(approac)n(h)i(a)d (statistical)h(state)h(whic)n(h)f(is)0 4810 y Fl(indep)l(endent)c Fm(\(partially)-6 b(,)25 b(if)d(not)j(wholly\))f(from)e(the)j(initial)e (condition.\021)37 b(G.)23 b(K.)g(Batc)n(helor,)i([Bat82)q(],)e(pp.)g (6-7.)1553 5059 y FI(59)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 60 59 bop 0 523 a FI(see)21 b([Gal01)o(].)35 b(The)22 b(theorems)e(from)i(this)g(c)n(hapter)e(and)i(Section)f(7)g (rigorously)e(pro)n(v)n(e)h(this)i(p)r(ostulate)0 623 y(for)27 b(the)h(p)r(erio)r(dic)f(2D)h(turbulence,)g(driv)n(en)e(b)n(y) i(a)f(random)g(force.)125 722 y(The)e(v)-5 b(alidit)n(y)26 b(of)f(the)h(p)r(ostulate)f(ab)r(o)n(v)n(e)g(for)g(randomly)f(stirred)h (2D)g(NSE)i(is)e(ob)n(vious)f(to)i(ph)n(ysi-)0 822 y(cists.)53 b(In)33 b(con)n(trast,)g(its)g(rigorous)d(mathematical)j(pro)r(of,)g (giv)n(en)f(in)i(Theorem)e(6.4)g(\(kic)n(k\025forces\))0 922 y(and)j(Theorem)f(7.1)g(\(white\025forces\),)j(is)e(surprisingly)e (di\036cult.)61 b(Moreo)n(v)n(er,)34 b(it)h(uses)g(essen)n(tially)0 1021 y(the)d(additional)g(assumption)f(\(6.2\))o(=\(7.2\).)49 b(If)33 b(this)f(assumption)f(is)h(violated,)g(the)h(pro)r(of)e(has)g (to)0 1121 y(b)r(e)36 b(mo)r(di\034ed)f(signi\034can)n(tly)-7 b(,)36 b(and)f(the)h(correct)d(v)n(ersion)h(of)h(Theorem)f(6.4=Theorem) f(7.1)h(is)h(not)0 1220 y(kno)n(wn)30 b(y)n(et)g(\(see)h([HM04)o(])g (and)f(discussion)g(in)h(Section)f(12\).)45 b(In)31 b(particular,)f(w)n (e)g(do)h(not)f(exclude)0 1320 y(that)h(in)f(this)h(case)f(the)h(con)n (v)n(ergence)d(to)i(the)h(unique)g(stationary)d(measure)i(\(if)h(it)g (still)g(holds!\))45 b(is)0 1420 y(not)28 b(exp)r(onen)n(tial.)125 1519 y(The)37 b(random)g(attractors,)i(discussed)e(in)h(Section)g(6.6,) h(and)f(their)g(deterministic)f(coun)n(ter\025)0 1619 y(parts)32 b(\(see)i([BV92)o(,)f(CF88)o(]\))h(are)e(w)n(ell)h (established)g(to)r(ols)g(for)f(analytical)h(and)g(n)n(umerical)f (study)0 1719 y(of)f(the)f(2D)h(NSE)g(\(see)g([Gal01)o(]\).)46 b(Their)30 b(imp)r(ortance)g(is)g(due)h(to)g(the)f(fact)h(that)g(they)g (reduce)f(the)0 1818 y FC(in\034nite\025dimensional)24 b FI(dynamical)c(system,)j(de\034ned)f(b)n(y)f(the)h(NSE)h(in)f(a)f (function)h(space,)g(to)g FC(\034nite\025)0 1918 y(dimensional)k (systems)c FI(on)g(attractors.)33 b(Since)23 b(dimensions)e(of)h (attractors)e(gro)n(wth)h(to)h(in\034nit)n(y)h(with)0 2017 y(the)e(Reynolds)g(n)n(um)n(b)r(er,)h(then)g(it)f(is)g(hard)f(to)h (apply)g(them)h(to)f(study)g(2D)g(\035o)n(ws)f(with)i(high)f(Reynolds)0 2117 y(n)n(um)n(b)r(ers.)46 b(Theorem)30 b(6.16)f(impro)n(v)n(es)g(the) j(situation)e(a)h(bit)g(since)f(it)i(pro)n(vides)d(minimal)i(random)0 2217 y(p)r(oin)n(t)d(attractors)d(with)j(an)g(additional)f(structure)g (\026)g(the)h(natural)f(measure)f FH(\026)p 2570 2196 49 3 v 12 x FA(!)2619 2217 y FI(.)1553 5059 y(60)p eop end PStoPSsaved restore %%Page: (60,61) 31 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 61 60 bop 0 523 a FK(7)131 b(Uniqueness)44 b(of)g(a)g(stationary)g(measure:)57 b(white-forces)0 721 y Fy(7.1)112 b(The)38 b(main)e(theorem)0 875 y FI(Let)28 b(us)f(consider)g(the)h(2D)f(NSE,)i(p)r(erturb)r(ed)f(b)n(y)f(a)g (white)h(force:)484 1056 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)1185 1000 y FH(d)p 1170 1037 74 4 v 1170 1113 a(dt)1276 1056 y(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)181 b(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)2060 977 y Fz(X)2046 1163 y FA(s)p Fw(2)p Fv(Z)2161 1143 y Ft(2)2161 1181 y(0)2221 1056 y FH(b)2257 1068 y FA(s)2306 1056 y FH(\014)2353 1068 y FA(s)2388 1056 y FG(\()p FH(t)p FG(\))14 b FH(e)2535 1068 y FA(s)2571 1056 y FG(\()p FH(x)p FG(\))g FH(:)299 b FI(\(7.1\))0 1298 y(The)28 b(random)e(\034eld)i FH(\020)34 b FI(is)28 b(as)f(in)g(Section)h(4.)36 b(In)28 b(particular,)1195 1453 y FH(B)1258 1465 y FD(1)1319 1453 y FG(=)1406 1374 y Fz(X)1554 1453 y FF(j)p FH(s)p FF(j)1639 1419 y FD(2)1690 1453 y FH(b)1726 1419 y FD(2)1726 1474 y FA(s)1786 1453 y FH(<)23 b FF(1)14 b FH(:)125 1613 y FI(By)27 b FH(u)p FG(\()p FH(t)p FG(;)14 b FH(u)450 1625 y FD(0)486 1613 y FG(\))p FH(;)28 b(t)23 b FF(\025)g FG(0)p FI(,)k(w)n(e)g(denote)h(a)f(solution,)g(equal)g FH(u)1865 1625 y FD(0)1930 1613 y FI(at)g FH(t)c FG(=)g(0)p FI(.)0 1747 y FJ(Theorem)30 b(7.1.)40 b FC(Ther)l(e)31 b(exists)e FH(N)j FF(\025)23 b FG(1)p FC(,)29 b(dep)l(ending)i(on)f FH(\027)35 b FC(and)30 b FH(B)2149 1759 y FD(1)2187 1747 y FC(,)g(such)g(that)f(if)1256 1890 y FH(b)1292 1902 y FA(s)1350 1890 y FF(6)p FG(=)23 b(0)84 b FF(8)14 b(j)p FH(s)p FF(j)23 b(\024)g FH(N)g(;)1085 b FI(\(7.2\))0 2033 y FC(then)40 b(ther)l(e)h(exists)f(a)h(unique)f(stationary)h(me)l (asur)l(e)f FH(\026)i FF(2)h(P)7 b FG(\()p FH(H)g FG(\))p FC(,)44 b(and)d(for)g(any)g(solution)f FH(u)p FG(\()p FH(t)p FG(\))p FC(,)0 2133 y(satisfying)31 b Fk(E)14 b FF(j)p FH(u)p FG(\(0\))p FF(j)646 2103 y FD(2)706 2133 y FG(=:)22 b FH(M)897 2145 y FD(2)957 2133 y FH(<)h FF(1)p FC(,)30 b(we)g(have)950 2205 y Fz(\015)950 2255 y(\015)996 2276 y FH(\026)19 b FF(\000)f(D)1214 2209 y Fz(\000)1252 2276 y FH(u)p FG(\()p FH(t)p FG(\))1394 2209 y Fz(\001)1432 2205 y(\015)1432 2255 y(\015)1478 2226 y Fw(\003)1478 2309 y FA(L)1551 2276 y FF(\024)23 b FH(C)6 b FG(\()p FH(M)1817 2288 y FD(2)1873 2276 y FG(+)18 b(1\))c FH(e)2083 2242 y Fw(\000)p FA(\024t)2216 2276 y FH(;)779 b FI(\(7.3\))0 2419 y FC(for)31 b(al)t(l)f FH(t)23 b FF(\025)g FG(0)p FC(.)38 b(Her)l(e)30 b FH(\024)23 b(>)f FG(0)29 b FC(and)i FH(C)e FF(\025)23 b FG(1)29 b FC(dep)l(end)i(on)e FH(\027)q(;)14 b(B)1870 2431 y FD(1)1937 2419 y FC(and)30 b FF(f)p FH(b)2176 2431 y FA(s)2211 2419 y FH(;)14 b FF(j)p FH(s)p FF(j)23 b(\024)f FH(N)9 b FF(g)p FC(.)0 2553 y FJ(Remark)31 b(7.2.)40 b FC(The)30 b(c)l(onstant)f FH(N)39 b FC(dep)l(ends)31 b(on)e FH(\027)q(;)f(B)1743 2565 y FD(1)1810 2553 y FC(and)i FF(f)p FH(b)2049 2565 y FA(s)2083 2553 y FF(g)p FC(,)g(and)g(go)l(es)h (to)e FF(1)h FC(as)g FH(\027)e FF(!)23 b FG(0)p FC(.)0 2687 y FJ(Ampli\034cation)34 b(1.)47 b FI(If)31 b FH(B)871 2699 y FA(k)941 2687 y FH(<)d FF(1)p FI(,)33 b FH(k)e FF(\025)e FG(1)i FI(then)g(supp)h FH(\026)c FF(\032)h FH(H)2057 2657 y FA(k)2098 2687 y FI(.)47 b(Moreo)n(v)n(er,)29 b(let)j FH(f)40 b FI(b)r(e)31 b(a)g(lo)r(cally)0 2787 y(H\366lder)c(function)h(on)f FH(H)786 2757 y FA(k)q Fw(\000)p FD(1)940 2787 y FI(suc)n(h)g(that)900 2915 y FF(j)p FH(f)9 b FG(\()p FH(u)p FG(\))p FF(j)23 b(\024)g FH(C)1278 2927 y FD(1)1315 2848 y Fz(\000)1353 2915 y FG(1)18 b(+)g FF(k)p FH(u)p FF(k)1628 2878 y FA(C)1676 2886 y Ft(2)1628 2940 y FA(k)q Fw(\000)p FD(1)1753 2915 y FG(\))c FH(;)641 3093 y FF(j)p FH(f)9 b FG(\()p FH(u)p FG(\))19 b FF(\000)f FH(f)9 b FG(\()p FH(v)s FG(\))p FF(j)23 b(\024)g(k)p FH(u)17 b FF(\000)h FH(v)s FF(k)1494 3053 y FA(\015)1494 3118 y(k)q Fw(\000)p FD(1)1634 3093 y FH(C)1693 3105 y FD(1)1730 3093 y FG(\(1)g(+)g FF(k)p FH(u)p FF(k)2037 3056 y FA(C)2085 3064 y Ft(2)2037 3118 y FA(k)q Fw(\000)p FD(1)2180 3093 y FG(+)g FF(k)p FH(v)s FF(k)2390 3056 y FA(C)2438 3064 y Ft(2)2390 3118 y FA(k)q Fw(\000)p FD(1)2515 3093 y FG(\))3018 3003 y FI(\(7.4\))0 3247 y(for)29 b(all)h FH(u;)14 b(v)29 b FF(2)e FH(H)559 3217 y FA(k)q Fw(\000)p FD(1)685 3247 y FI(,)j(where)g FH(\015)h FF(2)c FG(\(0)p FH(;)14 b FG(1])29 b FI(and)h FH(C)1565 3259 y FD(1)1602 3247 y FH(;)14 b(C)1698 3259 y FD(2)1763 3247 y FH(>)26 b FG(0)p FI(.)43 b(Then)30 b(for)f(eac)n(h)g(non-random)f(v)n(ector)0 3347 y FH(u)48 3359 y FD(0)108 3347 y FF(2)23 b FH(H)35 b FI(w)n(e)27 b(ha)n(v)n(e)733 3490 y FF(j)p Fk(E)14 b FH(f)883 3422 y Fz(\000)920 3490 y FH(u)p FG(\()p FH(t)p FG(;)g FH(u)1115 3502 y FD(0)1152 3490 y FG(\))1184 3422 y Fz(\001)1241 3490 y FF(\000)k FG(\()p FH(f)t(;)c(\026)p FG(\))p FF(j)23 b(\024)g FH(C)1713 3502 y FD(3)1750 3490 y FG(\()p FH(\027)q(;)14 b(B)1924 3502 y FA(k)1965 3490 y FH(;)g(f)t(;)g FF(j)p FH(u)2155 3502 y FD(0)2192 3490 y FF(j)p FG(\))g FH(e)2300 3455 y Fw(\000)p FA(\024)2391 3464 y Fu(k)2427 3455 y FA(t)0 3633 y FI(for)35 b(an)n(y)f FH(t)i FF(\025)f FG(0)p FI(,)h(where)f FH(\024)862 3645 y FA(k)938 3633 y FH(>)h FG(0)p FI(.)59 b(If)35 b FH(k)k FG(=)c(1)p FI(,)i(then)e(the)h(same)e (is)h(true)g(if)h(in)f(\(7.4\))g FG(1)23 b(+)g FF(j)p FH(u)p FF(j)3014 3602 y FA(C)3062 3610 y Ft(2)3133 3633 y FI(is)0 3732 y(replaced)k(b)n(y)g FG(exp)14 b FH(")p FF(j)p FH(u)p FF(j)717 3702 y FD(2)753 3732 y FI(,)28 b(where)f FH(")c(>)f FG(0)28 b FI(is)f(a)g(su\036cien)n(tly)h(small)f (constan)n(t.)0 3866 y FJ(Ampli\034cation)41 b(2.)67 b FI(Consider)36 b(the)i(2D)g(NSE,)g(p)r(erturb)r(ed)g(b)n(y)g(a)f (force)g(with)h(a)f(deterministic)0 3965 y(comp)r(onen)n(t:)950 4104 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(f)9 b FG(\()p FH(x)p FG(\))19 b(+)1914 4048 y FH(d)p 1899 4085 V 1899 4161 a(dt)1996 4104 y(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)0 4257 y FI(where)28 b FH(\020)36 b FI(is)29 b(as)f(ab)r(o)n(v)n(e)f(and)i FH(f)9 b FG(\()p FH(x)p FG(\))26 b(=)1177 4195 y Fz(P)1278 4257 y FH(f)1319 4269 y FA(s)1354 4257 y FH(e)1393 4269 y FA(s)1428 4257 y FG(\()p FH(x)p FG(\))h FF(2)e FH(H)36 b FI(is)29 b(a)f(non-random)g (smo)r(oth)g(v)n(ector-\034eld.)40 b(If)0 4357 y FH(b)36 4369 y FA(s)104 4357 y FF(6)p FG(=)32 b(0)h FI(for)g(all)g FH(s)p FI(,)i(then)e(the)h(assertions)d(of)j(Theorem)e(7.1)h(and)g (Ampli\034cation)g(2)g(hold)g(true)h(for)0 4457 y(solutions)27 b(of)g(this)h(equation.)36 b(Pro)r(of)28 b(remains)e(essen)n(tially)h (the)h(same,)f(see)g([Shi05)o(].)125 4590 y(Belo)n(w)22 b(w)n(e)h(sk)n(etc)n(h)g(a)g(pro)r(of)g(of)h(Theorem)e(7.1)h(in)h (comparison)e(with)i(that)g(of)f(Theorem)g(6.4.)35 b(F)-7 b(or)0 4690 y(the)28 b(corresp)r(onding)e(complete)h(pro)r(of)g(see)g ([KS02a)n(,)h(Shi05],)f(and)h(for)f(a)g(pro)r(of)g(of)h (Ampli\034cation)f(1)0 4789 y(see)g([KS03)o(].)1553 5059 y(61)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 62 61 bop 546 510 a FJ(Kic)m(k-force)p 1561 575 4 136 v 974 w(White-force)p 0 579 2962 4 v 1561 714 4 136 v 50 784 a FI(1.)99 b FH(\026)264 796 y FD(1)301 784 y FH(;)14 b(\026)388 796 y FD(2)461 784 y FI(\025)35 b(initial)h(measures.)60 b(W)-7 b(e)36 b(ha)n(v)n(e)214 883 y(to)27 b(compare)387 1066 y FF(B)445 1036 y Fw(\003)442 1090 y FA(k)496 1066 y FH(\026)546 1078 y FD(1)611 1066 y FI(with)h FF(B)858 1036 y FA(k)855 1090 y(k)912 1066 y FH(\026)962 1078 y FD(2)999 1066 y FH(;)37 b(k)26 b FF(!)d(1)p FH(:)p 1561 1279 4 565 v 2111 784 a FI(same)50 1348 y(2.)99 b(T)-7 b(ak)n(e)29 b FH(k)h FF(2)e FE(N)20 b FF([)h(f)p FG(0)p FF(g)p FI(.)43 b(Assume)30 b(that)h(for)214 1448 y FG(0)23 b FF(\024)f FH(t)h FF(\024)g FH(k)k FI(w)n(e)d(ha)n(v)n (e)f(constructed)g(`suit-)214 1548 y(able')k(w)n(eak)g(solutions)214 1730 y FH(U)271 1742 y FD(1)308 1730 y FG(\()p FH(t)p FG(\))p FH(;)14 b(U)496 1742 y FD(2)533 1730 y FG(\()p FH(t)p FG(\))24 b FI(suc)n(h)j(that)c FF(D)17 b FH(U)1151 1742 y FA(j)1185 1730 y FG(\(0\))23 b(=)g FH(\026)1452 1742 y FA(j)1501 1730 y FH(:)p 1561 1943 4 665 v 2111 1348 a FI(same)214 1941 y(Set)83 b FH(d)p FG(\()p FH(t)p FG(\))24 b(=)f FF(j)p FH(U)741 1953 y FD(1)778 1941 y FG(\()p FH(t)p FG(\))c FF(\000)f FH(U)1031 1953 y FD(2)1068 1941 y FG(\()p FH(t)p FG(\))p FF(j)p FI(,)p 1561 1971 4 100 v 394 2041 a FH(R)q FG(\()p FH(t)p FG(\))23 b(=)g FF(j)p FH(U)743 2053 y FD(1)780 2041 y FG(\()p FH(t)p FG(\))p FF(j)c FG(+)f FF(j)p FH(U)1079 2053 y FD(2)1116 2041 y FG(\()p FH(t)p FG(\))p FF(j)p FI(.)p 1561 2178 4 207 v 50 2248 a(3.)99 b(If)383 2430 y FH(d)p FG(\()p FH(k)s FG(\))24 b FF(\024)f FH(d)691 2442 y FD(0)728 2430 y FH(;)37 b(R)q FG(\()p FH(k)s FG(\))23 b FF(\024)g FH(R)1136 2442 y FD(0)1343 2430 y FI(\(7.5\))214 2613 y(then)48 b(use)f(Lemma)g(6.1)f(to)h(construct)214 2712 y FH(U)271 2724 y FD(1)308 2712 y FG(\()p FH(k)15 b FG(+)d(1\))p FI(,)24 b FH(U)653 2724 y FD(2)690 2712 y FG(\()p FH(k)15 b FG(+)d(1\))24 b FI(suc)n(h)g(that)g FH(d)p FG(\()p FH(k)16 b FG(+)214 2812 y(1\))23 b FF(\024)408 2779 y FD(1)p 408 2793 34 4 v 408 2841 a(2)475 2812 y FH(d)p FG(\()p FH(k)s FG(\))28 b FI(with)g(high)f(probabilit)n(y)-7 b(.)p 1561 3400 4 1223 v 1613 2248 a(If)1661 2430 y FH(d)p FG(\()p FH(t)p FG(\))47 b FF(\024)23 b FH(d)1976 2442 y FD(0)2036 2430 y FI(and)g FH(R)q FG(\()p FH(t)p FG(\))g FF(\024)g FH(R)2525 2442 y FD(0)2562 2354 y Fz(p)p 2645 2354 219 4 v 76 x FH(t)c FF(\000)f FH(T)2826 2442 y Fw(\003)2741 2530 y FI(\(7.6\))1613 2629 y(for)41 b FH(T)1803 2641 y Fw(\003)1868 2629 y FG(+)27 b(1)46 b FF(\024)f FH(t)h FF(\024)g FH(k)e FI(\(with)e(a)f(suit-)1613 2729 y(able)24 b FH(T)1834 2741 y Fw(\003)1895 2729 y FF(\024)e FH(k)2025 2741 y FD(1)2063 2729 y FI(\),)j(then)g(use)g(an)f(`adjusted)1613 2829 y(Girsano)n(v)c(lemma')j(to)f(construct)g FH(U)2758 2841 y FD(1)2794 2829 y FG(\()p FH(t)p FG(\))p FI(,)1613 2928 y FH(U)1670 2940 y FD(2)1707 2928 y FG(\()p FH(t)p FG(\))p FI(,)27 b FH(k)e FF(\024)e FH(t)g FF(\024)g FH(k)17 b FG(+)d(1)p FI(,)26 b(suc)n(h)f(that)h(\(7.6\))1613 3028 y(holds)f(for)g FH(k)h FG(:=)d FH(k)17 b FG(+)e(1)p FI(,)25 b FH(d)2406 3040 y FD(0)2467 3028 y FG(:=)2588 2995 y FD(1)p 2588 3009 34 4 v 2588 3057 a(2)2644 3028 y FH(d)2687 3040 y FD(0)2751 3028 y FI(with)1613 3128 y(high)k(probabilit)n(y)-7 b(.)40 b(After)29 b(this,)h(replace)1613 3227 y FH(d)1656 3239 y FD(0)1721 3227 y FI(b)n(y)1846 3194 y FD(1)p 1846 3208 V 1846 3256 a(2)1903 3227 y FH(d)1946 3239 y FD(0)2011 3227 y FI(and)e FH(k)i FI(b)n(y)d FH(k)22 b FG(+)c(1)p FI(.)50 3470 y(4.)99 b(If)64 b(\(7.5\))f(violated,)72 b(then)64 b(construct)214 3569 y FH(U)271 3581 y FD(1)308 3569 y FG(\()p FH(k)17 b FG(+)d(1\))p FI(,)25 b FH(U)658 3581 y FD(2)695 3569 y FG(\()p FH(k)17 b FG(+)d(1\))24 b FI(in)i(`trivial)e(w)n(a)n(y',)214 3669 y(i.e.)47 b(using)32 b(\(6.1\))e(with)i(the)f(same)f(force)214 3769 y FH(\021)255 3781 y FA(k)q FD(+1)399 3769 y FI(for)18 b(b)r(oth)i(solutions.)33 b(Note)19 b(that)g(in)214 3868 y(this)30 b(case)g(with)g(small)g(p)r (ositiv)n(e)g(proba-)214 3968 y(bilit)n(y)-7 b(,)28 b(for)f FH(k)f FG(:=)c FH(k)g FG(+)c(1)27 b FI(w)n(e)g(ha)n(v)n(e)g(\(7.5\).)p 1561 3998 4 598 v 2111 3470 a(same)p 1561 4133 4 136 v 50 4203 a(5.)99 b(Iterate)86 b(the)g(pro)r(cedure.)212 b(Con-)214 4302 y(struct)34 b(a)g(suitable)h(Kan)n(toro)n(vic)n(h)c (func-)214 4402 y(tional)79 b(suc)n(h)g(that)g FF(K)1035 4335 y Fz(\000)1074 4402 y FH(U)1131 4414 y FD(1)1168 4402 y FG(\()p FH(k)56 b FG(+)c(1\))p FI(,)214 4510 y FH(U)271 4522 y FD(2)308 4510 y FG(\()p FH(k)35 b FG(+)c(1\))588 4443 y Fz(\001)682 4510 y FF(\024)56 b FH(\024)14 b FF(K)929 4443 y Fz(\000)968 4510 y FH(U)1025 4522 y FD(1)1062 4510 y FG(\()p FH(k)s FG(\))p FI(,)53 b FH(U)1305 4522 y FD(2)1342 4510 y FG(\()p FH(k)s FG(\))1452 4443 y Fz(\001)1490 4510 y FI(,)214 4610 y FH(\024)23 b(<)g FG(1)p FI(.)p 1561 4640 4 507 v 2111 4203 a(same)1553 5059 y(62)p eop end PStoPSsaved restore %%Page: (62,63) 32 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 63 62 bop 125 523 a FI(The)33 b(random)f(forces)g FH(\020)6 b FG(\()p FH(t;)14 b(x)p FG(\))35 b FI(whic)n(h)e(are)g (homogeneous)e(in)j FH(x)g FI(are)e(the)i(most)f(imp)r(ortan)n(t)g(for) 0 623 y(the)28 b(h)n(ydro)r(dynamics.)35 b(Due)28 b(to)g(Exercise)f (4.2,)g(this)g(is)h(the)g(case)f(if)h FH(b)2188 635 y FA(s)2246 623 y FG(=)22 b FH(b)2369 635 y Fw(\000)p FA(s)2484 623 y FI(for)27 b(all)g FH(s)p FI(.)0 767 y FJ(Theorem)33 b(7.3.)42 b FC(If)33 b(under)f(the)g(assumptions)g(of)h(The)l(or)l(em)g (7.1)h FH(b)2168 779 y FA(s)2231 767 y FG(=)27 b FH(b)2359 779 y Fw(\000)p FA(s)2478 767 y FC(for)33 b(al)t(l)g FH(s)27 b FF(2)h FE(Z)2938 737 y FD(2)2938 788 y(0)2976 767 y FC(,)33 b(then)0 867 y(the)d(me)l(asur)l(e)f FH(\026)h FC(is)g(homo)l(gene)l(ous,)h(i.e.)1199 1022 y FH(T)1248 1034 y FA(h)1309 1022 y FF(\016)18 b FH(\026)23 b FG(=)g FH(\026;)99 b FF(8)p FH(h)23 b FF(2)g FE(T)1952 988 y FD(2)0 1177 y FC(wher)l(e)29 b FH(T)282 1189 y FA(h)338 1177 y FH(u)p FG(\()p FH(x)p FG(\))24 b(=)f FH(u)p FG(\()p FH(x)16 b FG(+)f FH(h)p FG(\))p FC(.)38 b(Mor)l(e)l(over,)30 b FH(S)21 b FF(\016)15 b FH(\026)23 b FG(=)g FH(\026)p FC(,)29 b(wher)l(e)g FH(S)18 b(u)p FG(\()p FH(x)p FG(\))24 b(=)e FF(\000)p FH(u)p FG(\()p FF(\000)p FH(x)p FG(\))p FC(.)38 b(If)29 b(in)g(addition,)0 1277 y FH(b)36 1289 y FA(s)94 1277 y FG(=)23 b FH(b)218 1294 y FA(s)249 1277 y Fr(?)38 b FC(for)30 b(al)t(l)h FH(s)23 b FF(2)g FE(Z)778 1247 y FD(2)778 1298 y(0)816 1277 y FC(,)30 b(then)g FH(R)19 b FF(\016)f FH(\026)23 b FG(=)g FH(\026)p FC(,)30 b(wher)l(e)g FH(R)15 b(u)p FG(\()p FH(x)p FG(\))23 b(=)g FH(u)2094 1247 y Fw(?)2149 1277 y FG(\()p FF(\000)p FH(x)2293 1247 y Fw(?)2350 1277 y FG(\))p FC(.)0 1421 y(Pr)l(o)l(of.)43 b FI(Let)37 b FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))p FI(,)41 b FH(t)e FF(\025)g FG(0)p FI(,)g(b)r(e)f(a)f(stationary)f (solution)g(of)h(\(7.1\).)66 b(Then)38 b FH(T)2573 1433 y FA(h)2629 1421 y FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))38 b FI(satis\034es)0 1521 y(\(7.1\))g(with)g FH(\020)45 b FI(replaced)37 b(b)n(y)h FH(T)1002 1533 y FA(h)1045 1521 y FH(\020)6 b FI(.)69 b(Since)38 b FF(D)1472 1454 y Fz(\000)1511 1521 y FH(T)1560 1533 y FA(h)1602 1521 y FH(\020)6 b FG(\()p FF(\001)p FG(\))1731 1454 y Fz(\001)1811 1521 y FG(=)40 b FF(D)1982 1454 y Fz(\000)2021 1521 y FH(\020)6 b FG(\()p FF(\001)p FG(\))2150 1454 y Fz(\001)2189 1521 y FI(,)41 b(then)d FH(T)2501 1533 y FA(h)2544 1521 y FH(U)47 b FI(is)38 b(a)g(stationary)0 1620 y(pro)r(cess)27 b(whic)n(h)i(is)f(a)h(w)n(eak)e(solution)h(of)h(\(7.1\).)40 b(Hence,)29 b FF(D)16 b FH(T)1937 1632 y FA(h)1979 1620 y FH(U)9 b FG(\(0\))29 b FI(is)f(a)h(stationary)e(measure)g(and)0 1720 y FH(T)49 1732 y FA(h)110 1720 y FF(\016)18 b FH(\026)23 b FG(=)g FF(D)16 b FH(T)460 1732 y FA(h)502 1720 y FH(U)9 b FG(\(0\))23 b(=)g FH(\026)28 b FI(b)n(y)f(the)h(uniqueness.)125 1820 y(Since)h FF(D)16 b FH(S)5 b FG(\()p FH(\020)h FG(\))27 b(=)f FF(D)r FH(\020)6 b FI(,)31 b(then)f FH(S)24 b FF(\016)19 b FH(\026)27 b FG(=)e FH(\026)30 b FI(b)n(y)f(the)h(same)f(reasons.)40 b(Pro)r(of)29 b(of)h(the)g(last)f(assertion)0 1919 y(is)e(the)h(same.)p 3128 1919 4 57 v 3132 1867 50 4 v 3132 1919 V 3182 1919 4 57 v 125 2080 a(Let)33 b(us)g(in)n(terpret)f(the)i(torus)e FE(T)1162 2050 y FD(2)1233 2080 y FI(as)g(the)i(square)d FG([)p FF(\000)p FH(\031)s(;)14 b(\031)s FG(])2003 2050 y FD(2)2074 2080 y FI(with)34 b(iden)n(ti\034ed)f(opp)r(osite)g(edges.) 0 2180 y(Then)28 b(the)g(rotation)e(b)n(y)h(the)h(angle)f FF(\000)p FH(\031)s(=)p FG(2)p FI(,)1140 2271 y Fz(\022)1201 2332 y FH(x)1248 2302 y FD(1)1201 2445 y FH(x)1248 2421 y FD(2)1286 2271 y Fz(\023)1370 2388 y FF(7\000)-14 b(!)1527 2271 y Fz(\022)1588 2332 y FF(\000)p FH(x)1700 2302 y FD(2)1650 2445 y FH(x)1697 2421 y FD(1)1737 2271 y Fz(\023)1821 2388 y FG(=)23 b FH(x)1956 2354 y Fw(?)2026 2388 y FH(;)0 2599 y FI(maps)29 b FE(T)274 2569 y FD(2)341 2599 y FI(to)h(itself.)43 b(This)29 b(rotation)g(induces)g(the)h(transformation)e FH(R)i FI(of)g(the)g(v)n(ector)e(\034elds)h FH(u)p FG(\()p FH(x)p FG(\))p FI(.)0 2698 y(So)35 b(the)g(last)g(assertion)e(of)i(the) g(theorem)g(means)f(that)h(the)h(rotation)d(b)n(y)i(the)g(angle)f FF(\000)p FH(\031)s(=)p FG(2)g FI(do)r(es)0 2798 y(not)d(c)n(hange)f (the)i(stationary)d(measure)i(\(if)h FH(b)1450 2810 y FA(s)1514 2798 y FF(\021)d FH(b)1644 2815 y FA(s)1675 2798 y Fr(?)8 b FI(\).)48 b(The)31 b(theory)g(of)g(homogeneous)e (isotropic)0 2898 y(turbulence)40 b(deals)g(with)g(measures)f(on)h(the) g(space)f(of)h(v)n(ector)f(\034elds)h FH(u)p FG(\()p FH(x)p FG(\))h FI(that)f(are)f(in)n(v)-5 b(arian)n(t)0 2997 y(under)40 b(the)h(translations)d FH(T)918 3009 y FA(h)1001 2997 y FI(and)i(the)h(rotations)e FH(R)1758 3009 y FA(\022)1835 2997 y FI(b)n(y)h(an)n(y)g(angle)f FH(\022)k FI(\(see)d([Bat82)n(,)h(F)-7 b(ri95)o(]\).)0 3097 y(Unfortunately)g(,)29 b(the)h(p)r(erio)r(dic)f(v)n(ector)f (\034elds)h(\(whic)n(h)g(w)n(e)g(in)n(terpret)f(as)h(v)n(ector)e (\034elds)j(on)e FE(T)2965 3067 y FD(2)3003 3097 y FI(\))h(can)0 3196 y(b)r(e)21 b(in)n(v)-5 b(arian)n(t)19 b(only)h(under)h(rotations)e (b)n(y)h(angles)f(prop)r(ortional)g(to)h FH(\031)s(=)p FG(2)p FI(,)i(since)e(only)g(these)h(rotations)0 3296 y(preserv)n(e)26 b(the)i(torus.)0 3524 y Fy(7.2)112 b(Stationary)56 b(measures)g(for)g(equation,)61 b(p)s(erturb)s(ed)c(b)m(y)f(high)f (fre-)255 3640 y(quency)38 b(kic)m(ks)0 3793 y FI(In)31 b(Section)f(4.5)f(w)n(e)h(compared)g(solutions)f(of)i(equation)e (\(7.1\))h(with)h FH(\027)i FG(=)28 b(1)i FI(and)g(solutions)f(of)i (the)0 3893 y(2D)d(NSE,)g(p)r(erturb)r(ed)g(b)n(y)f(the)h (high\025frequency)f(kic)n(k-force)e FH(\021)1983 3905 y FA(")2019 3893 y FI(,)744 4060 y FH(\021)785 4072 y FA(")821 4060 y FG(\()p FH(t)p FG(\))e(=)1026 3996 y FF(p)p 1095 3996 39 4 v 64 x FH(")1161 3981 y Fz(X)1203 4160 y FA(k)1295 4060 y FH(\021)1339 4026 y FA(!)1336 4081 y(k)1387 4060 y FH(\016)s FG(\()p FH(t)c FF(\000)f FH("k)s FG(\))p FH(;)97 b(\021)1869 4072 y FA(k)1933 4060 y FG(=)2021 3981 y Fz(X)2154 4060 y FH(b)2190 4072 y FA(s)2226 4060 y FH(\030)2262 4072 y FA(sk)2334 4060 y FH(e)2373 4072 y FA(s)2422 4060 y FH(;)0 4300 y FI(where)27 b Fk(E)14 b FH(\030)353 4312 y FA(sk)448 4300 y FF(\021)23 b FG(0)k FI(and)g FH(B)829 4312 y FD(1)890 4300 y FG(=)977 4238 y Fz(P)1079 4300 y FF(j)p FH(s)p FF(j)1164 4270 y FD(2)1201 4300 y FH(b)1237 4270 y FD(2)1237 4321 y FA(s)1297 4300 y FH(<)c FF(1)p FI(.)37 b(W)-7 b(e)28 b(pro)n(v)n(ed)d(that)782 4455 y FF(D)r FG(\()p FH(u)928 4467 y FA(")964 4455 y FG(\()p FH(T)12 b FG(;)i FH(u)1142 4467 y FD(0)1178 4455 y FG(\)\))23 b FH(*)g FF(D)r FG(\()p FH(u)p FG(\()p FH(T)12 b FG(;)i FH(u)1695 4467 y FD(0)1732 4455 y FG(\)\))83 b FF(8)14 b FH(u)1987 4467 y FD(0)2038 4455 y FH(;)37 b FF(8)14 b FH(T)34 b(>)22 b FG(0)14 b FH(;)611 b FI(\(7.7\))0 4611 y(where)34 b FH(u)295 4623 y FA(")365 4611 y FI(is)g(a)g(solution)g(of)h(the)g(equation,)g(p)r (erturb)r(ed)g(b)n(y)g(the)f(force)g FH(\021)2390 4623 y FA(")2426 4611 y FI(,)i(and)f FH(u)f FI(is)g(a)h(solution)0 4710 y(of)i(\(7.1\).)45 b(Let)31 b FH(\026)545 4722 y FA(")611 4710 y FI(b)r(e)g(a)f(stationary)f(measure)g(of)i(the)g(kic)n (k)n(ed)e(equation,)i(and)f FH(\026)h FI(b)r(e)g(a)f(stationary)0 4810 y(measure)c(of)34 b(\(7.1\).)j(It)28 b(turns)f(out)h(that)g(the)g (measures)e FH(\026)1839 4822 y FA(")1902 4810 y FI(con)n(v)n(erge)f (to)j FH(\026)p FI(:)1553 5059 y(63)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 64 63 bop 0 523 a FJ(Theorem)31 b(7.4.)41 b FC(L)l(et)30 b FH(\027)g FG(=)25 b(1)30 b FC(and)h(assume)f(that)h FH(b)1634 535 y FA(s)1694 523 y FF(6)p FG(=)24 b(0)30 b FC(for)i(al)t(l)f FH(s)g FC(and)g FH(B)2403 535 y FD(1)2465 523 y FH(<)24 b FF(1)p FC(.)42 b(Then)31 b FH(\026)2971 535 y FA(")3031 523 y FH(*)25 b(\026)0 623 y FC(in)30 b FF(P)7 b FG(\()p FH(H)g FG(\))29 b FC(as)h FH(")23 b FF(!)g FG(0)p FC(.)125 789 y(Sketch)37 b(of)i(the)f(pr)l(o)l(of.)64 b FI(Let)36 b FH(U)1126 801 y FA(")1162 789 y FG(\()p FH(t)p FG(\))g FI(b)r(e)h(a)f(stationary)e(solution)i(of)g(the)h(kic)n (k)n(ed)e(equation)g(\(so)0 888 y FF(D)r FH(U)123 900 y FA(")159 888 y FG(\()p FH(k)s(")p FG(\))23 b(=)f FH(\026)468 900 y FA(")532 888 y FI(for)27 b(eac)n(h)f FH(k)s FI(\).)37 b(Then)326 1071 y Fk(E)p FF(k)p FH(U)488 1083 y FA(")522 1071 y FG(\(\()p FH(k)22 b FG(+)c(1\))p FH(")p FG(\))p FF(k)921 1037 y FD(2)921 1092 y(1)980 1071 y FG(=)23 b Fk(E)p FF(k)p FH(S)1224 1083 y FA(")1259 1071 y FG(\()p FH(U)1348 1083 y FA(")1383 1071 y FG(\()p FH(k)s(")p FG(\)\))c(+)f FH(\021)1707 1083 y FA(k)q FD(+1)1832 1071 y FF(k)1874 1037 y FD(2)1874 1092 y(1)1934 1071 y FF(\024)23 b FH(e)2061 1037 y Fw(\000)p FD(2)p FA(")2181 1071 y Fk(E)p FF(k)p FH(U)2343 1083 y FA(")2377 1071 y FG(\()p FH(k)s FG(\))p FF(k)2529 1037 y FD(2)2529 1092 y(1)2585 1071 y FG(+)18 b FH(D)2737 1083 y FD(1)2788 1071 y FH(")c(;)0 1254 y FI(where)27 b FH(D)309 1266 y FD(1)369 1254 y FG(=)c Fk(E)p FF(k)p FH(\021)603 1266 y FA(k)q FD(+1)727 1254 y FF(k)769 1224 y FD(2)769 1274 y(1)829 1254 y FF(\024)g FH(B)980 1266 y FD(1)1017 1254 y FI(.)37 b(Therefore,)970 1366 y Fz(Z)1067 1479 y FF(k)p FH(u)p FF(k)1199 1444 y FD(2)1199 1499 y(1)1248 1479 y FH(\026)1298 1491 y FA(")1334 1479 y FG(\()p FH(du)p FG(\))23 b(=)g Fk(E)p FF(k)p FH(U)1762 1491 y FA(")1796 1479 y FG(\()p FH(k)s FG(\))p FF(k)1948 1444 y FD(2)1948 1499 y(1)2008 1479 y FF(\024)g FH(B)2159 1491 y FD(1)2196 1479 y FH(;)799 b FI(\(7.8\))0 1704 y(uniformly)36 b(in)g FG(0)h FH(<)g(")g FF(\024)g FG(1)f FI(\(cf.)64 b(\(3.17\))o(\).)f(Hence,)38 b(ev)n(ery)d(sequence)h FH(\026)2355 1716 y FD(~)-37 b FA(")2382 1724 y Fu(j)2417 1704 y FI(,)44 b FG(~)-47 b FH(")2518 1716 y FA(j)2590 1704 y FF(!)37 b FG(0)p FI(,)h(con)n(tains)d(a)0 1803 y(con)n(v)n(erging)28 b(in)i FF(P)7 b FG(\()p FH(H)g FG(\))30 b FI(subsequence)g FH(\026)1272 1815 y FA(")1303 1823 y Fu(j)1366 1803 y FH(*)d(\026)1526 1815 y FD(0)1563 1803 y FI(.)45 b(It)31 b(remains)e(to)h(c)n(hec)n(k)f (that)i FH(\026)2599 1815 y FD(0)2663 1803 y FG(=)c FH(\026)j FI(\(for)g(ev)n(ery)0 1903 y(sequence)d FF(f)p FH(")425 1915 y FA(j)459 1903 y FF(g)g FI(as)g(ab)r(o)n(v)n(e\).)125 2003 y(Let)e(us)g(denote)h(b)n(y)f FF(B)812 1973 y Fw(\003)809 2023 y FA(t)849 2003 y FH(;)39 b(t)23 b FF(\025)g FG(0)p FI(,)i(the)h(semigroup)e(in)i(measures)e(for)g(\(7.1\),)i(and)f(denote) g(b)n(y)g FF(B)3066 1958 y FA(")3097 1966 y Fu(j)3128 1958 y Fw(\003)3063 2028 y FA(k)3166 2003 y FI(,)0 2102 y FG(0)e FF(\024)f FH(k)k FF(2)e FE(Z)p FI(,)k(the)g(semigroup)e(for)h (the)h(kic)n(k)n(ed)e(equation.)37 b(Then)1332 2285 y FF(B)1390 2240 y FA(")1421 2248 y Fu(j)1452 2240 y Fw(\003)1387 2310 y FA(k)1490 2285 y FH(\026)1540 2297 y FA(")1571 2305 y Fu(j)1629 2285 y FG(=)23 b FH(\026)1767 2297 y FA(")1798 2305 y Fu(j)1834 2285 y FH(:)1161 b FI(\(7.9\))0 2468 y(Denote)33 b FH(\022)329 2480 y FA(")396 2468 y FG(=)e([)p FH(")554 2437 y Fw(\000)p FD(1)643 2468 y FG(])p FI(.)53 b(It)33 b(is)f(not)h(di\036cult)h(to)e(deriv)n(e)g(from) g(\(7.9\))h(with)g FH(k)h FG(=)e FH(\022)2556 2480 y FA(")2591 2468 y FI(,)i(\(7.8\))f(and)f(\(7.7\))0 2567 y(that)27 b FF(B)237 2537 y Fw(\003)234 2588 y FD(1)274 2567 y FH(\026)324 2579 y FD(0)385 2567 y FG(=)22 b FH(\026)522 2579 y FD(0)587 2567 y FI(\(see)27 b([KS03)n(])g(for)g(details\).)36 b(No)n(w)27 b(b)n(y)f(the)i(uniqueness)e(of)h(a)g(stationary)e(measure) 0 2667 y(w)n(e)i(ha)n(v)n(e)g FH(\026)364 2679 y FD(0)424 2667 y FG(=)c FH(\026)p FI(.)p 3128 2667 4 57 v 3132 2614 50 4 v 3132 2667 V 3182 2667 4 57 v 125 2816 a(The)k(results,)g (obtained)g(in)h(Sections)g(6,)f(7,)g(mak)n(e)g(v)n(ery)f(plausible)h (the)h(follo)n(wing)125 2966 y FJ(Conjecture.)36 b FI(Consider)27 b(the)h(2D)f(NSE,)i(forced)e(b)n(y)g(the)h(random)f(force)958 3097 y FF(p)p 1027 3097 39 4 v 63 x FH(")1093 3081 y Fz(X)1138 3256 y FA(s)1227 3160 y FH(b)1263 3172 y FA(s)1298 3160 y FH(\030)1334 3172 y FA(s)1370 3160 y FG(\()p FH(t=")p FG(\))p FH(e)1584 3172 y FA(s)1619 3160 y FG(\()p FH(x)p FG(\))14 b FH(;)98 b FG(0)22 b FH(<)h(")g FF(\024)f FG(1)p FH(;)0 3418 y FI(where)38 b FH(b)287 3430 y FA(s)364 3418 y FF(6)p FG(=)j(0)e FI(for)f(all)h FH(s)f FI(and)h FH(\030)1102 3430 y FA(s)1177 3418 y FI(are)e(i.i.d.)71 b(stationary)37 b(pro)r(cesses)h(that)h(are)e(mixing)i(\(in)g(an)0 3517 y(appropriate)26 b(sense\))h(and)g(satisfy)h Fk(E)p FH(\030)1213 3529 y FA(s)1248 3517 y FG(\()p FH(t)p FG(\))c FF(\021)e FG(0)p FI(,)28 b Fk(E)p FH(\030)1649 3487 y FD(2)1645 3538 y FA(s)1686 3517 y FG(\()p FH(t)p FG(\))23 b FF(\021)g FG(1)p FI(.)36 b(Then)125 3617 y(i\))28 b(the)h(equation)e (has)h(a)g(unique)g(stationary)e(measure)h FH(\026)1949 3629 y FA(")1985 3617 y FI(,)h(and)g(all)g(solutions)g(con)n(v)n(erge)d (to)j FH(\026)3153 3629 y FA(")0 3717 y FI(in)g(distribution)f(as)g FH(t)c FF(!)g(1)p FI(;)125 3816 y(ii\))j FH(\026)279 3828 y FA(e)337 3816 y FH(*)d(\026)j FI(as)f FH(")e FF(!)g FG(0)p FI(,)i(where)g FH(\026)h FI(is)f(the)h(stationary)e(measure)g (for)h(the)h(white\025forced)f(equation)0 3916 y(\(7.1\).)1553 5059 y(64)p eop end PStoPSsaved restore %%Page: (64,65) 33 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 65 64 bop 0 523 a FK(8)131 b(Ergo)t(dicit)l(y)46 b(and)e(the)g(strong)f(la)l(w)i(of)e(large)h(n)l(um)l(b)t(ers)0 705 y FI(The)22 b(results)g(and)g(their)h(pro)r(ofs)e(whic)n(h)h(w)n(e) g(discuss)g(in)h(this)f(section)g(are)f(v)n(ery)h(similar)f(for)h(kic)n (k-)f(and)0 804 y(white-forced)29 b(equations,)h(and)g(w)n(e)g (restrict)f(ourselv)n(es)f(to)i(the)g(kic)n(k)n(ed-forced)e(NSE)j (\(3.3\),)g(where)0 904 y(the)d(kic)n(ks)f FH(\021)390 916 y FA(j)452 904 y FI(are)g(as)g(in)h(the)g(Section)f(3.1.)125 1004 y(The)g(corresp)r(onding)f(random)g(dynamical)h(system)1120 1186 y FH(u)p FG(\()p FH(j)c FG(+)18 b(1\))23 b(=)g FH(S)1581 1119 y Fz(\000)1618 1186 y FH(u)p FG(\()p FH(j)5 b FG(\))1769 1119 y Fz(\001)1826 1186 y FG(+)18 b FH(\021)1950 1198 y FA(j)s FD(+1)3018 1186 y FI(\(8.1\))0 1369 y(de\034nes)40 b(a)g(Mark)n(o)n(v)e(pro)r(cess)i(with)g(the)h(transition)f(function)h FH(P)2125 1381 y FA(k)2166 1369 y FG(\()p FH(x;)14 b FF(\001)p FG(\))p FI(,)45 b(and)40 b(has)g(a)g(stationary)0 1469 y(measure)35 b FH(\026)p FI(.)61 b(The)36 b(solution)f FF(f)p FH(u)p FG(\()p FH(j)5 b FG(\))p FH(;)14 b(j)41 b FF(\025)36 b FG(0)p FF(g)f FI(suc)n(h)g(that)h FF(D)16 b FH(u)p FG(\(0\))36 b(=)h FH(\026)e FI(is)h(a)f(stationary)f(pro)r (cess.)0 1568 y(Moreo)n(v)n(er,)26 b(one)j(of)g(the)g(basic)f (constructions)g(from)g(the)i(theory)e(of)g(Mark)n(o)n(v)f(pro)r (cesses,)h(pro)n(vides)0 1668 y(us)h(with)g(a)f(stationary)g(pro)r (cess)f FF(f)p FH(U)9 b FG(\()p FH(j)c FG(\))p FI(,)29 b FH(j)h FF(2)c FE(Z)p FF(g)p FI(,)j(suc)n(h)f(that)i FF(D)2048 1601 y Fz(\000)2086 1668 y FH(U)9 b FG(\()p FH(j)c FG(\))2255 1601 y Fz(\001)2318 1668 y FG(=)25 b FH(\026)k FI(for)f(eac)n(h)g FH(j)5 b FI(,)29 b(whic)n(h)g(is)0 1768 y(a)e(w)n(eak)g(solution)g(of)g(\(8.1\))h(\(e.g.,)f(see)g([DZ96)o (,)h(KS00)o(]\).)125 1867 y(Let)k(us)g(denote)f FF(A)g FG(=)f FH(H)928 1837 y Fv(Z)971 1867 y FI(,)j(equip)f FF(A)g FI(with)h(the)f(Tikhono)n(v)f(top)r(ology)f(and)i(the)h(Borel)d (sigma-)0 1967 y(algebra)c FF(B)g FG(=)e FF(B)s FG(\()p FF(A)p FG(\))p FI(.)39 b(The)28 b(distribution)g(of)g(the)h(pro)r(cess) e FH(U)9 b FI(,)28 b Fk(P)c FG(=)g FF(D)2225 1900 y Fz(\000)2263 1967 y FH(U)9 b FG(\()p FF(\001)p FG(\))2416 1900 y Fz(\001)2454 1967 y FI(,)29 b(is)f(a)g(measure)f(on)h FF(A)p FI(,)0 2066 y(in)n(v)-5 b(arian)n(t)26 b(under)i(the)g(shifts)413 2249 y FH(T)462 2261 y FA(`)516 2249 y FG(:)23 b FF(A)h(!)f(A)p FH(;)180 b FG(\()p FH(:)14 b(:)g(:)g(;)g(u)1255 2261 y FD(0)1292 2249 y FH(;)g(u)1377 2261 y FD(1)1413 2249 y FH(;)g(:)g(:)g(:)g FG(\))23 b FF(7!)g FG(\()p FH(:)14 b(:)g(:)g(;)g(u)1950 2215 y Fw(0)1950 2270 y FD(0)2010 2249 y FG(=)23 b FH(u)2146 2261 y FA(`)2177 2249 y FH(;)14 b(u)2262 2215 y Fw(0)2262 2270 y FD(1)2322 2249 y FG(=)23 b FH(u)2458 2261 y FA(`)p FD(+1)2573 2249 y FH(;)14 b(:)g(:)g(:)g FG(\))p FH(:)0 2432 y FI(W)-7 b(e)28 b(ha)n(v)n(e)e(got)h(the)g (probabilit)n(y)g(space)f FG(\()p FF(A)p FH(;)14 b FF(B)s FH(;)g Fk(P)p FG(\))28 b FI(and)f(the)h(group)e(of)h (measure-preserving)d(trans-)0 2531 y(formations)i FF(f)p FH(T)504 2543 y FA(`)535 2531 y FF(g)p FI(.)37 b(See,)27 b(e.g.)37 b([Ros71)n(],)28 b(p.)37 b(95-96,)25 b(and)j([DZ96)o(],)g (Chapter)f(2.)0 2697 y FJ(De\034nition)k(8.1.)40 b FC(A)29 b(set)h FH(A)23 b FF(2)g(B)s FG(\()p FH(H)7 b FG(\))29 b FC(is)h(c)l(al)t(le)l(d)h(invariant)g(for)g(\(8.1\))f(if)1077 2880 y FH(P)1130 2892 y FD(1)1168 2880 y FG(\()p FH(x;)14 b(A)p FG(\))24 b(=)f(1)29 b FC(for)i(a.e.)39 b FH(x)24 b FF(2)f FH(A)14 b(:)0 3063 y FC(A)30 b(set)243 3042 y FG(~)221 3063 y FH(A)24 b FF(2)g(A)30 b FC(is)h(c)l(al)t(le)l(d)g (invariant)g(for)f(the)h(gr)l(oup)f FF(f)p FH(T)1751 3075 y FA(`)1782 3063 y FF(g)p FC(,)g(if)h Fk(P)p FG(\()2079 3042 y(~)2057 3063 y FH(A)14 b FG(\001)g FH(T)2265 3075 y FA(`)2319 3042 y FG(~)2297 3063 y FH(A)p FG(\))24 b(=)f(0)29 b FC(for)i(e)l(ach)g FH(`)p FC(,)f(wher)l(e)0 3162 y FG(\001)g FC(stands)g(for)g(the)g(symmetric)g(di\033er)l(enc)l(e)g(of)h (sets.)125 3328 y FI(It)f(is)f(ob)n(vious)g(that)h(if)g FH(A)h FI(is)e(an)h(in)n(v)-5 b(arian)n(t)29 b(set)g(for)h(\(8.1\),)g (then)g FH(A)2250 3298 y Fv(Z)2320 3328 y FF(\032)d(A)j FI(is)g(in)n(v)-5 b(arian)n(t)28 b(for)i(the)0 3428 y(group)23 b FF(f)p FH(T)323 3440 y FA(`)354 3428 y FF(g)p FI(.)36 b(Con)n(v)n(ersely)-7 b(,)23 b(if)983 3407 y FG(~)962 3428 y FH(A)g FF(2)g(B)k FI(is)d(an)g(in)n(v)-5 b(arian)n(t)24 b(set)g(for)g FF(f)p FH(T)2086 3440 y FA(`)2117 3428 y FF(g)p FI(,)h(then)g(there)f(exists)g FH(A)f FF(2)h(B)s FG(\()p FH(H)7 b FG(\))0 3528 y FI(suc)n(h)26 b(that)386 3507 y FG(~)364 3528 y FH(A)h FI(equals)f FH(A)766 3497 y Fv(Z)835 3528 y FI(up)h(to)f(a)g(n)n(ull-set,)g(see)g([Ros71)n(,)h (DZ96)o(].)36 b(Clearly)26 b(the)g(set)g FH(A)h FI(is)f(in)n(v)-5 b(arian)n(t)0 3627 y(for)32 b(\(8.1\).)53 b(This)33 b(relation)f(b)r (et)n(w)n(een)h(in)n(v)-5 b(arian)n(t)31 b(sets)i(of)g(\(8.1\))f(and)h (the)g(group)f FF(f)p FH(T)2681 3639 y FA(`)2712 3627 y FF(g)h FI(implies)g(the)0 3727 y(follo)n(wing)26 b(classical)h (result:)0 3893 y FJ(Theorem)36 b(8.2.)43 b FC(If)34 b FH(\026)h FC(is)f(a)h(unique)e(stationary)i(me)l(asur)l(e)f(for)h (\(8.1\),)i(then)d(the)g(tr)l(ansformation)0 3993 y FH(T)49 4005 y FD(1)115 3993 y FC(is)d(er)l(go)l(dic.)0 4159 y(Pr)l(o)l(of.)43 b FI(Let)425 4138 y FG(~)403 4159 y FH(A)23 b FI(b)r(e)f(an)g(in)n(v)-5 b(arian)n(t)21 b(set)h(of)g FH(T)1310 4171 y FD(1)1347 4159 y FI(,)h(and)f FH(A)h FI(b)r(e)f(the)h(corresp)r(onding)d(in)n(v)-5 b(arian)n(t)21 b(set)h(for)f(\(8.1\).)0 4258 y(Assume)28 b(that)f FH(\026)p FG(\()p FH(A)p FG(\))d FF(6)p FG(=)f(0)p FI(.)36 b(Then)28 b(the)g(conditional)f(measure)f FH(\026)p FG(\()p FF(\001)14 b(j)p FH(A)p FG(\))p FI(,)1082 4441 y FH(\026)p FG(\()p FH(B)k FF(j)c FH(A)p FG(\))24 b(=)f FH(\026)p FG(\()p FH(B)g FF(\\)18 b FH(A)p FG(\))c FH(=)g(\026)p FG(\()p FH(A)p FG(\))g FH(;)0 4623 y FI(is)28 b(also)f(stationary)f(for)h (\(8.1\).)38 b(So)28 b FH(\026)p FG(\()p FH(A)p FG(\))h FI(is)e FG(0)h FI(or)f FG(1)p FI(.)38 b(A)n(ccordingly)-7 b(,)26 b FH(P)12 b FG(\()2278 4603 y(~)2256 4623 y FH(A)q FG(\))28 b FI(also)f(is)h FG(0)f FI(or)g FG(1)p FI(.)38 b(So)27 b FH(T)3068 4635 y FD(1)3133 4623 y FI(is)0 4723 y(an)g(ergo)r(dic)g(transformation.)p 3128 4723 4 57 v 3132 4670 50 4 v 3132 4723 V 3182 4723 4 57 v 1553 5059 a(65)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 66 65 bop 125 523 a FI(Due)21 b(to)f(this)h(theorem,)h (the)f(transformation)e FH(T)1626 535 y FD(1)1683 523 y FI(satis\034es)h(the)h(Strong)f(La)n(w)g(of)g(Large)f(Num)n(b)r(ers)0 623 y(\(SLLN\),)29 b(i.e.)37 b(the)28 b(Birkho\033\025Hinc)n(hin)f (ergo)r(dic)g(theorem)g(\(see)h([Ros71)n(,)g(DZ96)o(]\).)38 b(In)28 b(terms)g(of)g(the)0 722 y(pro)r(cess)e FF(f)p FH(U)9 b FG(\()p FH(j)c FG(\))p FF(g)27 b FI(w)n(e)g(ha)n(v)n(e)836 918 y Fz(\012)875 985 y FH(f)9 b FG(\()p FH(U)g FG(\))1055 918 y Fz(\013)1094 935 y FA(T)1094 1019 y FD(0)1169 985 y FG(:=)1299 929 y(1)p 1290 966 61 4 v 1290 1042 a FH(T)1397 882 y FA(T)g Fw(\000)p FD(1)1403 906 y Fz(X)1406 1083 y FA(j)s FD(=0)1558 985 y FH(f)1608 918 y Fz(\000)1645 985 y FH(U)g FG(\()p FH(j)c FG(\))1814 918 y Fz(\001)1876 985 y FF(!)23 b(h)p FH(\026;)14 b(f)9 b FF(i)46 b FI(a.e.)664 b(\(8.2\))0 1256 y(if)30 b FH(f)k FF(2)25 b FH(L)290 1268 y FD(1)327 1256 y FG(\()p FH(H)r(;)14 b(d\026)p FG(\))p FI(,)30 b(and)f(in)g(particular,)f(if)i FH(f)k FF(2)26 b FH(C)1607 1268 y FA(b)1640 1256 y FG(\()p FH(H)7 b FG(\))p FI(.)42 b(It)29 b(turns)g(out)g(that)g(the)h(SLLN)f(holds)g (for)0 1355 y(all)e(solutions)g(of)h(\(8.1\))f(\(not)h(only)f(for)g (stationary\):)0 1521 y FJ(Theorem)35 b(8.3.)42 b FC(L)l(et)33 b FH(b)785 1533 y FA(s)850 1521 y FF(6)p FG(=)c(0)k FC(for)h FF(j)p FH(s)p FF(j)c(\024)f FH(N)9 b FC(,)35 b(wher)l(e)f FH(N)42 b FC(is)34 b(as)f(in)h(The)l(or)l(em)g(6.4,)i(and)e(let)f FH(f)42 b FC(b)l(e)34 b(a)0 1620 y(lo)l(c)l(al)t(ly)i(Lipschitz)f (function)f(on)g FH(H)7 b FC(.)51 b(Then)35 b(any)f(solution)g FF(f)p FH(u)p FG(\()p FH(j)5 b FG(\))p FF(g)33 b FC(of)i(\(8.1\))g (such)f(that)g FF(D)16 b FH(u)p FG(\(0\))31 b FF(2)0 1720 y(P)58 1732 y FD(1)95 1720 y FG(\()p FH(H)7 b FG(\))30 b FC(satis\034es)1194 1771 y Fz(\012)1234 1839 y FH(f)9 b FG(\()p FH(u)p FG(\))1396 1771 y Fz(\013)1434 1789 y FA(T)1434 1872 y FD(0)1510 1839 y FF(!)23 b FG(\()p FH(\026;)14 b(f)9 b FG(\))46 b FC(a.e.)1025 b FI(\(8.3\))0 2004 y FC(Pr)l(o)l(of.)43 b FI(W)-7 b(e)36 b(ha)n(v)n(e)f(to)g(sho)n(w) g(that)h(the)g(con)n(v)n(ergence)d(\(8.3\))j(holds)f(for)g(a.e.)60 b FH(~)-41 b(u)36 b FG(=)g(\()p FH(u)2758 2016 y FD(0)2795 2004 y FH(;)14 b(u)2880 2016 y FD(1)2917 2004 y FH(;)g(:)g(:)g(:)g FG(\))37 b FF(2)0 2104 y(A)66 2116 y FD(+)150 2104 y FG(=)28 b FH(H)319 2074 y Fw(f)p FD(0)p Fw(g[)p Fv(N)511 2104 y FI(,)k(with)f(resp)r(ect)f(to)h(the)g(measure)f FF(D)1689 2036 y Fz(\000)1728 2104 y FH(u)p FG(\()p FF(\001)p FG(\))1863 2036 y Fz(\001)1901 2104 y FI(.)47 b(T)-7 b(o)30 b(c)n(hec)n(k)g(this)h(w)n(e)g(can)f(replace)g FH(u)p FG(\()p FF(\001)p FG(\))0 2203 y FI(b)n(y)f(an)n(y)f(w)n(eak)h (solution)f FH(u)848 2215 y FD(1)885 2203 y FG(\()p FF(\001)p FG(\))i FI(of)f(\(8.1\),)h(equal)e FH(u)p FG(\(0\))h FI(at)g FH(t)d FG(=)g(0)p FI(.)41 b(Similarly)-7 b(,)29 b(\(8.2\))g(remains)f(true)i(if)0 2303 y(w)n(e)d(replace)g FH(U)9 b FG(\()p FH(j)c FG(\))p FI(,)28 b FH(j)g FF(\025)22 b FG(0)p FI(,)28 b(b)n(y)f(a)g(pro)r(cess)f FH(u)1387 2315 y FD(2)1424 2303 y FG(\()p FH(j)5 b FG(\))p FI(,)28 b FH(j)g FF(\025)23 b FG(0)p FI(,)k(distributed)h(as)f FH(U)9 b FI(.)125 2403 y(F)-7 b(or)19 b(an)n(y)g(real-v)-5 b(alued)19 b(random)g(sequence)g FF(f)p FH(x)p FG(\()p FH(j)5 b FG(\))p FH(;)14 b(j)29 b FF(\025)22 b FG(0)p FF(g)e FI(and)f(an)n(y)h(a.s.)33 b(\034nite)21 b(random)e(v)-5 b(ariable)0 2502 y FH(T)61 2472 y Fw(0)106 2502 y FF(\025)23 b FG(0)p FI(,)k(w)n(e)g(set)949 2665 y Fz(\012)988 2732 y FH(x)1035 2665 y Fz(\013)1075 2682 y FA(T)1075 2766 y(T)1123 2749 y Fr(0)1173 2732 y FG(=)1260 2537 y Fz(8)1260 2612 y(>)1260 2637 y(<)1260 2786 y(>)1260 2811 y(:)1371 2622 y FG(1)p 1362 2659 V 1362 2735 a FH(T)1521 2575 y FA(T)1485 2600 y Fz(X)1469 2778 y FA(j)s FD(=)p FA(T)1598 2761 y Fr(0)1635 2678 y FH(x)p FG(\()p FH(j)5 b FG(\))p FH(;)37 b FI(if)28 b FH(T)34 b(>)23 b(T)2153 2648 y Fw(0)1352 2867 y FG(0)p FH(;)428 b FI(if)28 b FH(T)34 b FF(\024)23 b FH(T)2153 2837 y Fw(0)2189 2867 y FH(:)0 3012 y FI(Then)28 b(a.e.)36 b(w)n(e)27 b(ha)n(v)n(e)1165 3111 y FG(lim)1132 3165 y FA(T)9 b Fw(!1)1350 3044 y Fz(\012)1389 3111 y FH(x)1436 3044 y Fz(\013)1476 3061 y FA(T)1476 3144 y(T)1524 3128 y Fr(0)1573 3111 y FG(=)56 b(lim)1661 3165 y FA(T)9 b Fw(!1)1878 3044 y Fz(\012)1918 3111 y FH(x)1965 3044 y Fz(\013)2004 3061 y FA(T)2004 3144 y FD(0)0 3293 y FI(\(i.e.)49 b(if)32 b(one)f(limit)h(exists,)g(then)g(the)g(second)f (also)f(exists)h(and)g(they)h(are)e(equal\).)48 b(Let)32 b(us)f(c)n(ho)r(ose)0 3393 y(for)d FH(u)176 3405 y FD(1)213 3393 y FG(\()p FH(k)s FG(\))p FI(,)h FH(u)423 3405 y FD(2)460 3393 y FG(\()p FH(k)s FG(\))g FI(the)g(w)n(eak)f(solutions)g (as)g(in)h(Corollary)d(6.5)i(\(de\034ned)h(on)f(the)h(same)f (probabilit)n(y)0 3503 y(space\).)51 b(Let)471 3482 y Fz(e)455 3503 y FH(f)40 b FF(2)32 b FH(C)682 3515 y FA(b)716 3503 y FG(\()p FH(H)7 b FG(\))33 b FI(b)r(e)g(a)f(Lipsc)n(hitz)g (function,)j(equal)d FH(f)41 b FI(for)32 b FF(j)p FH(u)p FF(j)g(\024)f FH(r)2492 3515 y FD(0)2561 3503 y FG(=)2709 3471 y FA(e)p 2667 3485 117 4 v 2667 3532 a(e)p Fw(\000)p FD(1)2825 3433 y FF(p)p 2894 3433 68 4 v 70 x FH(B)2961 3517 y FD(0)2998 3503 y FI(,)j(and)0 3603 y(let)29 b FH(T)182 3573 y Fw(0)233 3603 y FI(b)r(e)g(the)g(random)f(v)-5 b(ariable)27 b(from)i(\(6.14\).)39 b(Then)29 b(w)n(e)f(ha)n(v)n(e)f (the)j(follo)n(wing)d(equalities)h(whic)n(h)0 3703 y(hold)f(a.e.:)757 3802 y FF(h)p FH(\026;)14 b(f)9 b FF(i)23 b FG(=)g FF(h)p FH(\026;)1204 3780 y Fz(e)1188 3802 y FH(f)9 b FF(i)23 b FG(=)g(lim)1510 3735 y Fz(\012)1565 3780 y(e)1549 3802 y FH(f)9 b FG(\()p FH(u)1679 3814 y FD(2)1716 3802 y FG(\))1748 3735 y Fz(\013)1787 3752 y FA(T)1787 3836 y FD(0)1863 3802 y FG(=)22 b(lim)2080 3735 y Fz(\012)2134 3780 y(e)2119 3802 y FH(f)8 b FG(\()p FH(u)2248 3814 y FD(2)2285 3802 y FG(\))2317 3735 y Fz(\013)2357 3752 y FA(T)2357 3836 y(T)2405 3819 y Fr(0)0 3951 y FI(\(here)27 b(the)h(\034rst)g(equalit)n(y)e(holds)i(b)n(y)f(Exercise)g(3.7,)g(and)g (the)h(second)f(b)n(y)g(\(8.2\)\).)125 4062 y(Next,)g(since)569 4040 y Fz(e)554 4062 y FH(f)36 b FI(is)28 b(a)f(Lipsc)n(hitz)g (function,)h(then)g(b)n(y)g(\(6.14\))e(w)n(e)i(ha)n(v)n(e)459 4263 y FG(lim)588 4196 y Fz(\012)643 4241 y(e)627 4263 y FH(f)9 b FG(\()p FH(u)757 4275 y FD(2)794 4263 y FG(\))826 4196 y Fz(\013)865 4213 y FA(T)865 4296 y(T)913 4280 y Fr(0)963 4263 y FG(=)23 b(lim)1180 4196 y Fz(\012)1235 4241 y(e)1219 4263 y FH(f)9 b FG(\()p FH(u)1349 4275 y FD(1)1386 4263 y FG(\))1418 4196 y Fz(\013)1457 4213 y FA(T)1457 4296 y(T)1505 4280 y Fr(0)1555 4263 y FG(=)23 b(lim)1772 4196 y Fz(\012)1811 4263 y FH(f)9 b FG(\()p FH(u)1941 4275 y FD(1)1978 4263 y FG(\))2010 4196 y Fz(\013)2049 4213 y FA(T)2049 4296 y(T)2097 4280 y Fr(0)2147 4263 y FG(=)23 b(lim)2364 4196 y Fz(\012)2403 4263 y FH(f)9 b FG(\()p FH(u)2533 4275 y FD(1)2570 4263 y FG(\))2602 4196 y Fz(\013)2641 4213 y FA(T)2641 4296 y FD(0)2707 4263 y FH(:)0 4445 y FI(So)27 b(\(8.3\))h(is)f(pro)n(v)n(ed.)p 3128 4445 4 57 v 3132 4392 50 4 v 3132 4445 V 3182 4445 4 57 v 125 4611 a(No)n(w)32 b(let)i(us)f(discuss)f(the)i(white-forced)e (NSE)i(\(7.1\).)53 b(If)34 b(the)f(assumptions)g(of)g(Theorem)f(7.1)0 4710 y(hold,)j(then)f(an)n(y)e(solution)h FH(u)p FG(\()p FH(t)p FG(\))h FI(with)f(a)g(non-random)f(initial)i(data)e FH(u)p FG(\(0\))i FI(satis\034es)e(\(8.3\),)j(where)0 4810 y(the)28 b(sum)g(should)f(b)r(e)h(replaced)f(b)n(y)g(an)g(in)n (tegral)f(in)i(the)g(de\034nition)g(of)g FF(h)p FH(f)9 b FG(\()p FH(u)p FG(\))p FF(i)2485 4780 y FA(T)2485 4831 y FD(0)2537 4810 y FI(:)1553 5059 y(66)p eop end PStoPSsaved restore %%Page: (66,67) 34 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 67 66 bop 0 523 a FJ(Theorem)24 b(8.4.)35 b FC(Under)25 b(the)g(assumptions)g(of)h(The)l(or)l(em)g(7.1,)h(for)f (any)f(solution)g FH(u)p FG(\()p FH(t)p FG(\))g FC(of)h(e)l(q.)37 b(\(7.1\))0 623 y(such)30 b(that)f Fk(E)14 b FF(j)p FH(u)p FG(\(0\))p FF(j)635 593 y FD(2)695 623 y FH(<)23 b FF(1)p FC(,)30 b(and)g(any)g(Lipschitz)h FH(f)h FF(2)23 b FH(C)1802 635 y FA(b)1836 623 y FG(\()p FH(H)7 b FG(\))30 b FC(we)g(have)1194 757 y Fz(\012)1234 825 y FH(f)9 b FG(\()p FH(u)p FG(\))1396 757 y Fz(\013)1434 774 y FA(T)1434 858 y FD(0)1510 825 y FF(!)23 b FG(\()p FH(\026;)14 b(f)9 b FG(\))46 b FC(a.e.)1025 b FI(\(8.4\))0 1019 y FC(Mor)l(e)l(over,)32 b(if)f FH(B)535 1031 y FA(k)599 1019 y FH(<)24 b FF(1)30 b FC(for)h(some)f FH(k)d FF(\025)c FG(1)p FC(,)30 b(then)g FF(h)p FH(f)9 b FG(\()p FH(u)p FG(\))p FF(i)1812 989 y FA(T)1812 1039 y FD(1)1889 1019 y FF(!)23 b FG(\()p FH(\026;)14 b(f)9 b FG(\))30 b FC(a.e.,)j(for)d(any)h(function)f FH(f)39 b FC(as)0 1119 y(in)30 b(the)g(A)n(mpli\034c)l(ation)g(to)g(The)l(or)l (em)g(7.1.)1369 1088 y Fo(8)125 1285 y FI(Pro)r(of)d(remains)g(the)g (same)g(\(see)h([Kuk02a)n(]\).)125 1384 y(In)23 b([Shi06])g(the)h(rate) f(of)g(con)n(v)n(ergence)e(in)j(\(8.3\))f(and)g(\(8.4\))g(w)n(as)g(sp)r (eci\034ed,)h(using)f(the)h(tec)n(hnique)0 1484 y(of)k(martingale)e (appro)n(ximation)f(\(see)j(the)g(next)g(section\).)0 1683 y FJ(The)f(ph)m(ysical)h(relev)-5 b(ance)27 b(of)g(the)g(results.) 34 b FI(The)23 b(SLLN,)h(pro)n(v)n(ed)e(in)i(this)g(section,)f(sho)n (ws)g(that)0 1783 y(for)30 b(the)g(2D)g(turbulence,)h(driv)n(en)e(b)n (y)h(a)g(random)f(force,)h(the)g(time)h(a)n(v)n(erage)c(equals)i(the)h (ensem)n(ble)0 1882 y(a)n(v)n(erage.)j(This)26 b(equalit)n(y)g(is)g(p)r (ostulated)g(b)n(y)g(the)g(theory)f(of)h(turbulence)g(\(e.g.,)h(see)e ([Bat82)o(],)i(p.)f(17,)0 1982 y(and)34 b([F)-7 b(ri95)o(],)37 b(p.)e(58\),)h(where)e(it)h(is)f(imp)r(ortan)n(t)g(since)h(it)g(allo)n (ws)e(to)i(calculate)f(v)-5 b(arious)33 b(a)n(v)n(eraged)0 2082 y(c)n(haracteristics)18 b(of)h(a)h(turbulen)n(t)g(\035o)n(w)f(b)n (y)h(running)g(one)f(exp)r(erimen)n(t)h(for)f(a)h(long)f(time)h(\(exp)r (erimen)n(ts)0 2181 y(with)26 b(turbulen)n(t)g(\035o)n(ws)f(are)g(exp)r (ensiv)n(e,)g(and)h(it)g(is)g(practically)e(imp)r(ossible)i(to)f (calculate)g(ensem)n(ble)0 2281 y(a)n(v)n(erages)f(directly\).)p 0 4731 1276 4 v 92 4786 a Fn(8)127 4810 y Fm(W)-6 b(e)24 b(replace)i Fh(h)p Fi(f)7 b Fg(\()p Fi(u)p Fg(\))p Fh(i)678 4786 y Ff(T)678 4830 y Fc(0)752 4810 y Fm(b)n(y)24 b Fh(h)p Fi(f)7 b Fg(\()p Fi(u)p Fg(\))p Fh(i)1040 4786 y Ff(T)1040 4830 y Fc(1)1114 4810 y Fm(since,)23 b(p)r(ossibly)-6 b(,)23 b Fi(f)7 b Fg(\()p Fi(u)p Fg(\()p Fi(t)p Fg(\)\))22 b Fh(!)d(1)k Fm(as)h Fi(t)c Fh(!)f Fg(0)p Fm(.)1553 5059 y FI(67)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 68 67 bop 0 523 a FK(9)131 b(The)44 b(martingale)g(appro) l(ximation)h(and)f(CL)-11 b(T)0 705 y FI(Let)28 b(us)f(consider)g(the)h (white-forced)f(NSE:)219 900 y FG(_)-37 b FH(u)18 b FG(+)g FH(Lu)f FG(+)h FH(B)t FG(\()p FH(u)p FG(\))24 b(=)874 844 y FH(d)p 859 881 74 4 v 859 957 a(dt)966 900 y(\020)6 b FG(\()p FH(t)p FG(\))p FH(;)61 b(\020)29 b FG(=)1339 821 y Fz(X)1486 900 y FH(b)1522 912 y FA(s)1571 900 y FH(\014)1618 912 y FA(s)1654 900 y FG(\()p FH(t)p FG(\))14 b FH(e)1801 912 y FA(s)1836 900 y FG(\()p FH(x)p FG(\))p FH(;)61 b(b)2067 912 y FA(s)2125 900 y FF(6)p FG(=)23 b(0)83 b FF(8)p FH(s;)37 b(B)2546 912 y FD(1)2606 900 y FH(<)23 b FF(1)14 b FH(;)204 b FI(\(9.1\))0 1101 y(\(the)33 b(results)f(remain)g(true)h(for)f(the)h(kic)n(k)n(ed)f(equation,)h(but) g(the)g(notations)f(b)r(ecome)g(more)g(cum-)0 1201 y(b)r(ersome\).)k(F) -7 b(or)27 b(simplicit)n(y)h(w)n(e)f(assume)g(that)1402 1374 y FH(u)p FG(\(0\))22 b(=)h FH(u)1714 1386 y FD(0)1764 1374 y FH(;)0 1547 y FI(where)35 b FH(u)296 1559 y FD(0)370 1547 y FF(2)j FH(H)43 b FI(is)36 b(a)f(non-random)g(v)n(ector.)61 b(Let)36 b FH(\026)g FI(b)r(e)g(the)h(unique)f(stationary)e(measure)h (and)0 1646 y FH(f)d FF(2)23 b FH(C)210 1658 y FA(b)244 1646 y FG(\()p FH(H)7 b FG(\))28 b FI(b)r(e)g(a)f(Lipsc)n(hitz)g (functional)h(suc)n(h)f(that)1402 1819 y FG(\()p FH(f)t(;)14 b(\026)p FG(\))23 b(=)g(0)14 b FH(:)0 1992 y FI(Denoting)1224 2145 y FH(S)1275 2157 y FA(t)1328 2145 y FG(=)1415 2032 y Fz(Z)1498 2053 y FA(t)1461 2221 y FD(0)1555 2145 y FH(f)1605 2078 y Fz(\000)1643 2145 y FH(u)p FG(\()p FH(s)p FG(\))1794 2078 y Fz(\001)1846 2145 y FH(ds)g(;)0 2341 y FI(w)n(e)27 b(ha)n(v)n(e)g FH(t)344 2311 y Fw(\000)p FD(1)447 2341 y FH(S)498 2353 y FA(t)550 2341 y FF(!)c FG(0)k FI(a.s.)36 b(as)27 b FH(t)c FF(!)g(1)28 b FI(b)n(y)f(the)h (SLLN.)125 2440 y(Our)35 b(goal)g(in)h(this)h(section)e(is)h(to)g(pro)n (v)n(e)f(that)h(the)h(random)e(v)-5 b(ariables)35 b FH(t)2520 2410 y Fw(\000)p FD(1)p FA(=)p FD(2)2690 2440 y FH(S)2741 2452 y FA(t)2806 2440 y FI(satisfy)h(the)0 2540 y(Cen)n(tral)27 b(Limit)h(Theorem)e(\(CL)-7 b(T\):)0 2698 y FJ(Theorem)30 b(9.1.)40 b FC(Under)30 b(the)g(assumptions)g(ab)l(ove)h(we)f(have)907 2871 y FF(D)r FG(\()p FH(T)1066 2837 y Fw(\000)p FD(1)p FA(=)p FD(2)1236 2871 y FH(S)1287 2883 y FA(T)1339 2871 y FG(\))24 b FH(*)h Fz(b)-48 b FH(\033)17 b FF(N)12 b FG(\(0)p FH(;)i FG(1\))37 b FC(as)30 b FH(T)k FF(!)23 b(1)14 b FH(;)736 b FI(\(9.2\))0 3044 y FC(for)31 b(some)h Fz(b)-48 b FH(\033)26 b FF(\025)d FG(0)p FC(,)29 b(indep)l(endent)i(of) f FH(u)1208 3056 y FD(0)1245 3044 y FC(.)125 3203 y FI(Let)f FH(U)9 b FG(\()p FH(t)p FG(\))p FI(,)30 b FH(t)c FF(\025)f FG(0)p FI(,)30 b(b)r(e)f(a)g(stationary)f(solution)h(of)g(\(9.1\).)42 b(Then)29 b(the)h(CL)-7 b(T)29 b(for)g(stationary)e(pro-)0 3302 y(cesses)j(as)h(in)h([Dur91)o(])f(applies)g(to)g(the)h(pro)r(cess) e FH(f)1625 3235 y Fz(\000)1663 3302 y FH(U)9 b FG(\()p FH(t)p FG(\))1823 3235 y Fz(\001)1861 3302 y FI(,)32 b(and)g(the)f(theorem)g(can)g(b)r(e)h(pro)n(v)n(ed)d(b)n(y)0 3410 y(comparing)i(the)i(pro)r(cesses)e FH(f)973 3343 y Fz(\000)1011 3410 y FH(u)p FG(\()p FH(t)p FG(\))1153 3343 y Fz(\001)1224 3410 y FI(and)h FH(f)1440 3343 y Fz(\000)1478 3410 y FH(U)9 b FG(\()p FH(t)p FG(\))1638 3343 y Fz(\001)1709 3410 y FI(similar)32 b(to)h(the)g(pro)r(of)f(of)g (Theorem)g(8.2,)h(see)0 3510 y([Kuk02a)n(].)51 b(Here)32 b(w)n(e)g(presen)n(t)g(another)f(pro)r(of)h(of)g(the)g(theorem,)h (based)f(on)g(the)h(martingale)e(ap-)0 3609 y(pro)n(ximation)25 b(for)g(the)i(pro)r(cess)e FH(f)1070 3542 y Fz(\000)1107 3609 y FH(u)p FG(\()p FH(t)p FG(\))1249 3542 y Fz(\001)1287 3609 y FI(,)i(follo)n(wing)e([Shi06)o(].)37 b(This)26 b(appro)n(ximation)e(is)i(a)g(p)r(o)n(w)n(erful)0 3709 y(tec)n(hnical)h(to)r(ol)f(to)h(study)g(Mark)n(o)n(v)e(pro)r(cesses)g (and)i(random)f(dynamical)g(systems.)36 b(In)27 b(particular,)0 3809 y(it)34 b(can)e(b)r(e)i(used)f(to)g(get)g(for)g(the)g(pro)r(cess)f FH(u)p FG(\()p FH(t)p FG(\))i FI(a)f(v)n(ersion)e(of)i(the)h(SLLN)f (with)h(a)f(con)n(trol)f(of)h(the)0 3908 y(rate)c(of)h(con)n(v)n (ergence)e([Shi06)o(])i(and)g(to)g(pro)n(v)n(e)f(the)h(La)n(w)f(of)h (Iterated)g(Logarithm)e(for)i(the)g(pro)r(cess)0 4008 y FH(f)50 3941 y Fz(\000)88 4008 y FH(u)p FG(\()p FH(t)p FG(\))230 3941 y Fz(\001)295 4008 y FI([Den04].)125 4107 y(F)-7 b(or)26 b FH(t)e FF(\025)e FG(0)27 b FI(let)h(us)g(set)472 4318 y FH(M)553 4330 y FA(t)605 4318 y FG(=)692 4205 y Fz(Z)775 4225 y Fw(1)739 4394 y FD(0)873 4318 y Fk(E)936 4251 y Fz(\000)974 4318 y FH(f)1024 4251 y Fz(\000)1062 4318 y FH(u)p FG(\()p FH(s)p FG(\))1213 4251 y Fz(\001)1265 4247 y(\014)1265 4297 y(\014)1292 4318 y FF(F)1352 4330 y FA(t)1381 4251 y Fz(\001)1433 4318 y FH(ds)33 b FF(\000)e FH(I)21 b(;)180 b(I)30 b FG(=)2058 4205 y Fz(Z)2141 4225 y Fw(1)2104 4394 y FD(0)2225 4318 y Fk(E)14 b FH(f)2352 4251 y Fz(\000)2389 4318 y FH(u)2437 4251 y Fz(\000)2475 4318 y FH(s)p FG(\))2546 4251 y Fz(\001)2598 4318 y FH(ds)g(;)301 b FI(\(9.3\))0 4537 y(where)35 b FF(fF)350 4549 y FA(s)385 4537 y FH(;)14 b(s)37 b FF(\025)f FG(0)p FF(g)f FI(is)h(the)g (\034ltration)f(of)h FH(\033)s FI(-algebras)e(corresp)r(onding)g(to)h (the)i(Wiener)e(pro)r(cess)0 4637 y FH(\020)6 b FG(\()p FH(t)p FG(\))p FI(.)38 b(By)27 b(the)h(Mark)n(o)n(v)d(prop)r(ert)n(y)-7 b(,)27 b(for)g FH(s)c FF(\025)g FH(t)k FI(w)n(e)g(ha)n(v)n(e)496 4740 y Fz(\014)496 4789 y(\014)524 4810 y Fk(E)587 4743 y Fz(\000)624 4810 y FH(f)674 4743 y Fz(\000)712 4810 y FH(u)p FG(\()p FH(s)p FG(\))863 4743 y Fz(\001)915 4740 y(\014)915 4789 y(\014)943 4810 y FF(F)1003 4822 y FA(t)1032 4743 y Fz(\001)1070 4740 y(\014)1070 4789 y(\014)1120 4810 y FG(=)1208 4740 y Fz(\014)1208 4789 y(\014)1236 4810 y FG(\()p FF(B)1323 4822 y FA(s)p Fw(\000)p FA(t)1449 4810 y FH(f)9 b FG(\))1531 4743 y Fz(\000)1569 4810 y FH(u)p FG(\()p FH(t)p FG(\))1711 4743 y Fz(\001)1749 4740 y(\014)1749 4789 y(\014)1800 4810 y FF(\024)22 b FH(C)e(e)2005 4776 y Fw(\000)p FA(\024)p FD(\()p FA(s)p Fw(\000)p FA(t)p FD(\))2260 4810 y FG(\(1)e(+)g FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)2623 4776 y FD(2)2661 4810 y FG(\))1553 5059 y FI(68)p eop end PStoPSsaved restore %%Page: (68,69) 35 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 69 68 bop 0 523 a FI(\(see)29 b(Theorem)f(7.1\).)41 b(In)29 b(particular,)f(the)h(in)n(tegral)f(in)h(\(9.3\))g(con)n(v)n (erges)e(and)h Fk(E)p FF(j)p FH(M)2685 535 y FA(t)2714 523 y FF(j)e FH(<)f FF(1)p FI(.)41 b(Since)0 623 y FF(F)60 635 y FD(0)125 623 y FI(is)27 b(the)h(trivial)f FH(\033)s FI(-algebra,)f(then)997 833 y FH(M)1078 845 y FD(0)1138 833 y FG(=)1226 720 y Fz(Z)1309 741 y Fw(1)1272 909 y FD(0)1393 833 y Fk(E)14 b FH(f)9 b FG(\()p FH(u)p FG(\()p FH(s)p FG(\)\))14 b FH(ds)32 b FF(\000)18 b FH(I)44 b FG(=)22 b(0)14 b FH(:)0 1043 y FI(Next,)28 b(for)f FH(t)383 1055 y FD(1)443 1043 y FH(>)c(t)561 1055 y FD(2)626 1043 y FI(w)n(e)k(ha)n(v)n(e)631 1254 y Fk(E)p FG(\()p FH(M)807 1266 y FA(t)832 1274 y Ft(1)882 1254 y FF(j)14 b(F)979 1266 y FA(t)1004 1274 y Ft(2)1040 1254 y FG(\))23 b(=)1183 1141 y Fz(Z)1266 1161 y Fw(1)1229 1329 y FD(0)1364 1161 y Fz(\020)1414 1254 y Fk(E)1477 1186 y Fz(\000)1515 1254 y FH(f)9 b FG(\()p FH(u)p FG(\()p FH(s)p FG(\))1748 1186 y Fz(\001)1799 1183 y(\014)1799 1233 y(\014)1841 1254 y FF(F)1901 1266 y FA(t)1926 1274 y Ft(2)1962 1186 y Fz(\001)2014 1254 y FH(ds)32 b FF(\000)18 b FH(I)44 b FG(=)23 b FH(M)2460 1266 y FA(t)2485 1274 y Ft(2)2535 1254 y FH(:)0 1477 y FI(So)37 b FF(f)p FH(M)248 1489 y FA(t)276 1477 y FF(g)g FI(is)h(a)f(martingale)f(with)i(resp)r(ect)f (to)g(the)h(\034ltration)f FF(fF)2168 1489 y FA(s)2203 1477 y FF(g)p FI(.)66 b(Since)38 b Fk(E)2624 1409 y Fz(\000)2661 1477 y FH(f)9 b FG(\()p FH(u)p FG(\()p FH(s)p FG(\))2894 1409 y Fz(\001)2932 1406 y(\014)2932 1456 y(\014)2960 1477 y FF(F)3020 1489 y FA(t)3049 1409 y Fz(\001)3124 1477 y FG(=)0 1585 y FH(f)50 1517 y Fz(\000)88 1585 y FH(u)p FG(\()p FH(s)p FG(\))239 1517 y Fz(\001)304 1585 y FI(for)27 b FH(s)c FF(\024)g FH(t)p FI(,)28 b(then)880 1808 y FH(M)961 1820 y FA(t)1013 1808 y FG(=)23 b FH(S)1152 1820 y FA(t)1199 1808 y FG(+)1282 1695 y Fz(Z)1365 1716 y Fw(1)1329 1884 y FA(t)1463 1808 y Fk(E)1526 1741 y Fz(\000)1564 1808 y FH(f)9 b FG(\()p FH(u)p FG(\()p FH(s)p FG(\))1797 1741 y Fz(\001)1849 1737 y(\014)1849 1787 y(\014)1891 1808 y FF(F)1951 1820 y FA(t)1979 1741 y Fz(\001)2031 1808 y FH(ds)19 b FF(\000)32 b FH(I)21 b(:)709 b FI(\(9.4\))0 2018 y(The)26 b(second)f(term)g(in)h(the)g(r.h.s.)36 b(is)25 b(a)g(random)g(v)-5 b(ariable)24 b(suc)n(h)h(that)h(the)g(exp)r (ectation)f(of)h(its)g(norm)0 2117 y(is)h(b)r(ounded)h(uniformly)g(in)g FH(t)p FI(.)36 b(Hence,)859 2286 y FF(kD)r FG(\()p FH(t)1029 2251 y Fw(\000)p FD(1)p FA(=)p FD(2)1185 2286 y FH(M)1266 2298 y FA(t)1295 2286 y FG(\))19 b FF(\000)f(D)r FG(\()p FH(t)1557 2251 y Fw(\000)p FD(1)p FA(=)p FD(2)1714 2286 y FH(S)1765 2298 y FA(t)1794 2286 y FG(\))p FF(k)1868 2251 y Fw(\003)1868 2306 y FA(L)1940 2286 y FF(\024)23 b FH(C)d(t)2137 2251 y Fw(\000)p FD(1)p FA(=)p FD(2)2307 2286 y FH(:)688 b FI(\(9.5\))0 2454 y(That)29 b(is,)h(asymptotical)f (prop)r(erties)f(of)i(distribution)f(of)h FH(S)1903 2466 y FA(t)1961 2454 y FI(are)f(the)g(same)g(as)g(of)h(the)f(martingale)0 2554 y FH(M)81 2566 y FA(t)110 2554 y FI(.)37 b(The)27 b(relation)g(\(9.4\))g(de\034nes)h(the)g FC(martingale)i(appr)l (oximation)i(of)e(the)g(pr)l(o)l(c)l(ess)e FH(S)2785 2566 y FA(t)2814 2554 y FI(.)0 2697 y FC(Pr)l(o)l(of)j(of)f(The)l(or)l (em)h(9.1.)39 b FI(Without)28 b(lost)f(of)h(generalit)n(y)e(w)n(e)h (assume)g(that)h FF(j)p FH(f)9 b FF(j)22 b(\024)h FG(1)p FI(.)125 2797 y(Since)28 b FF(j)p FH(f)9 b FF(j)24 b(\024)g FG(1)p FI(,)k(then)g(it)h(is)f(su\036cien)n(t)g(to)h(pro)n(v)n(e)d(the) j(con)n(v)n(ergence)c(\(9.2\))j(for)g(in)n(teger)f FH(t)p FI(.)39 b(More-)0 2896 y(o)n(v)n(er,)26 b(due)i(to)f(\(9.5\))g(w)n(e)h (can)f(replace)f FH(S)1264 2908 y FA(t)1321 2896 y FI(b)n(y)h(the)h (martingale)e FH(M)2075 2908 y FA(t)2104 2896 y FI(.)37 b(Let)28 b(us)f(consider)g(the)h(function)1097 3112 y FH(g)s FG(\()p FH(v)s FG(\))c(=)1358 2999 y Fz(Z)1441 3019 y Fw(1)1404 3187 y FD(0)1525 3112 y Fk(E)14 b FH(f)1652 3044 y Fz(\000)1690 3112 y FH(u)p FG(\()p FH(s)p FG(;)g FH(v)s FG(\))1921 3044 y Fz(\001)1973 3112 y FH(ds)g(;)0 3326 y FI(where)31 b FH(u)p FG(\()p FH(s)p FG(;)14 b FH(v)s FG(\))33 b FI(is)f(the)g(solution)f(of)h(\(9.1\))g(with)g (initial)h(condition)e FH(v)s FI(.)51 b(In)32 b(particular,)g FH(g)s FG(\()p FH(u)2929 3338 y FD(0)2965 3326 y FG(\))f(=)f FH(I)7 b FI(.)0 3426 y(W)-7 b(e)28 b(ha)n(v)n(e)420 3621 y FF(j)p FH(g)s FG(\()p FH(v)s FG(\))p FF(j)c(\024)727 3508 y Fz(Z)810 3528 y Fw(1)773 3696 y FD(0)894 3550 y Fz(\014)894 3600 y(\014)922 3621 y Fk(E)14 b FH(f)1049 3553 y Fz(\000)1086 3621 y FH(u)p FG(\()p FH(s)p FG(;)g FH(v)s FG(\))1317 3553 y Fz(\001)1369 3550 y(\014)1369 3600 y(\014)1411 3621 y FH(ds)640 3883 y FF(\024)727 3770 y Fz(Z)810 3790 y Fw(1)773 3958 y FD(0)894 3815 y Fz(\000)933 3883 y FH(C)20 b(e)1051 3848 y Fw(\000)p FA(\024s)1177 3883 y FG(\(1)e(+)g FF(j)p FH(v)s FF(j)1441 3848 y FD(2)1478 3883 y FG(\))h FF(^)g FG(1)1645 3815 y Fz(\001)1696 3883 y FH(ds)24 b FF(\024)e FH(\024)1937 3848 y Fw(\000)p FD(1)2026 3815 y Fz(\000)2064 3883 y FG(1)c(+)g(ln)c FH(C)6 b FG(\(1)19 b(+)f FF(j)p FH(u)p FF(j)2625 3848 y FD(2)2662 3883 y FG(\))2694 3815 y Fz(\001)2746 3883 y FH(:)0 4097 y FI(W)-7 b(riting)28 b(\(9.4\))f(as)624 4323 y FH(M)705 4335 y FA(t)756 4323 y FG(=)c FH(S)895 4335 y FA(t)943 4323 y FG(+)18 b FH(g)1069 4256 y Fz(\000)1106 4323 y FH(u)p FG(\()p FH(t)p FG(\))1248 4256 y Fz(\001)1305 4323 y FF(\000)g FH(I)44 b FG(=)1569 4210 y Fz(Z)1652 4231 y FA(t)1615 4399 y FD(0)1695 4323 y FH(f)9 b FG(\()p FH(u)p FG(\()p FH(s)p FG(\)\))14 b FH(ds)19 b FG(+)f FH(g)s FG(\()p FH(u)p FG(\()p FH(t)p FG(\)\))h FF(\000)f FH(I)0 4533 y FI(and)27 b(denoting)h FH(X)572 4545 y FA(j)629 4533 y FG(=)23 b FH(M)798 4545 y FA(j)851 4533 y FF(\000)18 b FH(M)1015 4545 y FA(j)s Fw(\000)p FD(1)1135 4533 y FI(,)27 b FH(j)h FF(2)c FE(N)p FI(,)j(w)n(e)h(ha)n(v)n(e)750 4765 y FH(X)819 4777 y FA(j)877 4765 y FG(=)964 4652 y Fz(Z)1047 4672 y FA(j)1010 4840 y(j)s Fw(\000)p FD(1)1144 4765 y FH(f)1194 4697 y Fz(\000)1232 4765 y FH(u)p FG(\()p FH(s)p FG(\))1383 4697 y Fz(\001)1435 4765 y FH(ds)18 b FG(+)g FH(g)1661 4697 y Fz(\000)1699 4765 y FH(u)p FG(\()p FH(j)5 b FG(\))1850 4697 y Fz(\001)1907 4765 y FF(\000)18 b FH(g)2033 4697 y Fz(\000)2070 4765 y FH(u)p FG(\()p FH(j)23 b FF(\000)18 b FG(1\))2364 4697 y Fz(\001)2416 4765 y FH(:)1553 5059 y FI(69)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 70 69 bop 0 523 a FI(Clearly)26 b FF(f)p FH(X)399 535 y FA(j)433 523 y FF(g)h FI(is)g(a)g(martingale)f (di\033erence)h(sequence,)g(and)g FH(M)2049 535 y FA(n)2117 523 y FG(=)22 b FH(X)2273 535 y FD(1)2328 523 y FG(+)17 b FF(\001)d(\001)g(\001)k FG(+)f FH(X)2676 535 y FA(n)2721 523 y FI(.)37 b(It)28 b(is)f(kno)n(wn)0 623 y(\(see)34 b([HH80)o(,)g(LS89)o(,)g(Dur91)o(]\))h(that)f(the)g(sequence)f FF(f)p FH(M)1821 635 y FA(n)1865 623 y FF(g)h FI(satis\034es)f(the)h (CL)-7 b(T)34 b(\(i.e.)56 b(\(9.2\))33 b(holds)0 722 y(with)27 b FH(S)239 734 y FA(t)295 722 y FI(replaced)f(b)n(y)h FH(M)818 734 y FA(t)846 722 y FI(,)g FH(t)d FF(2)f FE(N)p FI(\),)k(pro)n(vided)f(that)h(certain)f(conditions)g(are)g (ful\034lled.)38 b(As)26 b FF(j)p FH(X)3043 734 y FA(j)3078 722 y FF(j)d(\024)0 822 y FH(C)59 834 y FD(1)97 822 y FG(\(1)18 b(+)g(ln)355 755 y Fz(\000)393 822 y FG(1)g(+)g FF(j)p FH(u)p FG(\()p FH(j)23 b FF(\000)18 b FG(1\))p FF(j)876 792 y FD(2)913 822 y FG(\))h(+)f(ln\(1)g(+)g FF(j)p FH(u)p FG(\()p FH(j)5 b FG(\))p FF(j)1488 792 y FD(2)1526 822 y FG(\))1558 755 y Fz(\001)1624 822 y FI(and)27 b Fk(E)p FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)2036 792 y FD(2)2096 822 y FF(\024)2194 789 y FD(1)p 2194 803 34 4 v 2194 851 a(2)2237 822 y FH(B)2300 834 y FD(0)2356 822 y FG(+)18 b FH(e)2478 792 y Fw(\000)p FD(2)p FA(t)2592 822 y FF(j)p FH(u)2663 834 y FD(0)2699 822 y FF(j)2722 792 y FD(2)2787 822 y FI(\(see)28 b(\(4.16\))o(,)0 922 y(then)g(only)f(one)g(of)h(these)f(conditions)h(is)f(non-trivial:) 125 1046 y FJ(H\))55 b FH(n)342 1016 y Fw(\000)p FD(1)444 1046 y FH(V)492 1058 y FA(n)561 1046 y FF(!)25 b Fz(b)-48 b FH(\033)717 1016 y FD(2)801 1046 y FI(in)28 b(probabilit)n(y)-7 b(,)26 b(where)h FH(V)1626 1058 y FA(n)1695 1046 y FG(=)1783 984 y Fz(P)1870 1004 y FA(n)1870 1071 y(j)s FD(=1)2017 1046 y Fk(E)p FG(\()p FH(X)2188 1016 y FD(2)2181 1068 y FA(j)2239 1046 y FF(j)14 b(F)2336 1058 y FA(j)s Fw(\000)p FD(1)2455 1046 y FG(\))p FI(.)125 1161 y(By)27 b(the)h(Mark)n(o)n(v)d (prop)r(ert)n(y)-7 b(,)27 b Fk(E)p FG(\()p FH(X)1227 1131 y FD(2)1220 1183 y FA(j)1277 1161 y FF(j)14 b(F)1374 1173 y FA(j)s Fw(\000)p FD(1)1493 1161 y FG(\))24 b(=)e FH(h)1684 1094 y Fz(\000)1722 1161 y FH(u)p FG(\()p FH(j)i FF(\000)18 b FG(1\))2017 1094 y Fz(\001)2054 1161 y FI(,)28 b(where)f FH(h)h FI(is)f(the)h(function)723 1415 y FH(h)p FG(\()p FH(v)s FG(\))c(=)e Fk(E)1052 1323 y Fz(\020)1116 1302 y(Z)1199 1323 y FD(1)1162 1491 y(0)1250 1415 y FH(f)1300 1348 y Fz(\000)1337 1415 y FH(u)p FG(\()p FH(s)p FG(;)14 b FH(v)s FG(\))1568 1348 y Fz(\001)1620 1415 y FH(ds)19 b FG(+)f FH(f)1854 1348 y Fz(\000)1892 1415 y FH(u)p FG(\(1\))2046 1348 y Fz(\001)2102 1415 y FF(\000)g FH(f)9 b FG(\()p FH(v)s FG(\))2342 1323 y Fz(\021)2392 1340 y FD(2)2443 1415 y FH(:)0 1650 y FI(So)35 b FH(n)173 1620 y Fw(\000)p FD(1)262 1650 y FH(V)310 1662 y FA(n)391 1650 y FG(=)g FH(n)541 1620 y Fw(\000)p FD(1)644 1588 y Fz(P)731 1608 y FA(n)731 1675 y(j)s FD(=1)864 1650 y FH(h)p FG(\()p FH(u)p FG(\()p FH(j)28 b FF(\000)23 b FG(1\)\))p FI(.)60 b(It)35 b(can)g(b)r(e)g(c)n(hec)n(k)n(ed)f(\(see)h ([Shi06)o(]\))h(that)f(Theorem)f(8.4)0 1765 y(applies)27 b(to)h FH(h)427 1698 y Fz(\000)464 1765 y FH(u)p FG(\()p FH(s)p FG(\))615 1698 y Fz(\001)654 1765 y FI(.)36 b(Hence,)28 b(H\))g(follo)n(ws)f(from)g(the)h(SLLN)g(and)f(Theorem)g(9.1)g(is)g (pro)n(v)n(ed.)144 b Fs(\003)0 1964 y FJ(The)42 b(ph)m(ysical)h(relev) -5 b(ance)42 b(of)g(the)g(results.)63 b FI(The)37 b(CL)-7 b(T,)36 b(pro)n(v)n(ed)f(ab)r(o)n(v)n(e,)j(sho)n(ws)d(that)i(\(for)0 2064 y(randomly)22 b(forced)h(2D)g(\035o)n(ws\))f(`in)i(large)e (time\025scales)g(the)h(turbulen)n(t)h(\035o)n(w)e(is)h(a)g(Gaussian)f (pro)r(cess'.)0 2163 y(This)33 b(result)h(is)f(w)n(ell)g(kno)n(wn)g (from)g(exp)r(erimen)n(ts.)54 b(In)33 b(particular,)h(the)g(PDF)g(of)g (the)g(v)n(elo)r(cit)n(y)e(of)0 2263 y(turbulen)n(t)27 b(\035uid)f(at)g(an)g(arbitrary)e(p)r(oin)n(t,)j(calculated)f(b)n(y)g (A.)g(A.)h(T)-7 b(o)n(wnsend)26 b(in)g(the)h(middle)f(of)h(the)0 2363 y(last)g(cen)n(tury)g(using)h(certain)f(mec)n(hanical)f(device,)i (surprisingly)e(turned)i(out)f(to)h(b)r(e)g(v)n(ery)e(close)h(to)0 2462 y(the)e(Gaussian)f(probabilit)n(y)g(densit)n(y)g(\(see)h(in)g ([Bat82)o(],)g(p.)g(169\).)2077 2432 y Fo(9)2150 2462 y FI(W)-7 b(e)25 b(b)r(eliev)n(e)g(that)g(this)g(parado)n(x)0 2562 y(is)k(explained)g(b)n(y)h(the)f(CL)-7 b(T)30 b(since)f(an)n(y)g (mec)n(hanical)f(device)h(measures)f(not)i(the)g(instan)n(t)f(v)n(elo)r (cit)n(y)-7 b(,)0 2662 y(but)28 b(its)g(a)n(v)n(erage)d(o)n(v)n(er)g (some)i(time\025in)n(terv)-5 b(al.)p 0 4575 1276 4 v 92 4629 a Fn(9)127 4652 y Fm(T)f(o)n(wnsend's)23 b(exp)r(erimen)n(ts)h (deal)g(with)f(3D)g(\035o)n(ws,)g(describ)r(ed)h(b)n(y)g(the)g(3D)f (NSE.)f(But)i(n)n(umerical)f(in)n(v)n(estigation)0 4731 y(of)h(the)g(randomly)g(forced)h(2D)f(NSE,)g(made)f(in)h(1980's)g(and)g (1990's,)g(sho)n(w)g(for)h(the)f(PDF)g(of)g(their)h(solutions)f(a)g(v)n (ery)0 4810 y(similar)e(picture.)1553 5059 y FI(70)p eop end PStoPSsaved restore %%Page: (70,71) 36 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 71 70 bop 0 523 a FK(10)131 b(The)44 b(Eulerian)h(limit)0 721 y Fy(10.1)112 b(White-forces,)33 b(prop)s(ortional)f(to)g(the)h (square-ro)s(ot)g(of)g(the)g(viscos-)311 838 y(it)m(y)0 991 y FI(Let)28 b(us)f(consider)g(the)h(NSE)g(with)g(the)g(force)f (prop)r(ortional)f(to)2026 931 y FF(p)p 2095 931 47 4 v 60 x FH(\027)61 b FI(\()p FH(\027)29 b(>)22 b FG(0)27 b FI(is)h(the)g(viscosit)n(y:)1064 1166 y FG(_)-37 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)1741 1102 y FF(p)p 1810 1102 V 64 x FH(\027)28 b(\021)s FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)839 b FI(\(10.1\))0 1341 y(where)1124 1463 y FH(\021)26 b FG(=)1304 1407 y FH(d)p 1289 1444 74 4 v 1289 1520 a(dt)1409 1384 y Fz(X)1453 1559 y FA(s)1557 1463 y FH(b)1593 1475 y FA(s)1642 1463 y FH(\014)1689 1475 y FA(s)1724 1463 y FG(\()p FH(t)p FG(\))e FH(e)1881 1475 y FA(s)1916 1463 y FG(\()p FH(x)p FG(\))14 b FH(:)913 b FI(\(10.2\))125 1678 y(W)-7 b(e)37 b(assume)g(that)g FH(B)826 1690 y FD(1)903 1678 y FH(<)h FF(1)g FI(and)f FH(b)1334 1690 y FA(s)1408 1678 y FF(6)p FG(=)i(0)e FI(for)f(all)h FH(s)p FI(.)66 b(Let)37 b FH(\026)2188 1690 y FA(\027)2267 1678 y FI(b)r(e)h(the)f(unique)h(stationary)0 1778 y(measure)24 b(and)i FH(u)529 1790 y FA(\027)570 1778 y FG(\()p FH(t)p FG(\))g FI(b)r(e)g(the)g(corresp)r(onding)d(stationary)h(solution,)i (i.e.)36 b FF(D)2410 1710 y Fz(\000)2448 1778 y FH(u)2496 1790 y FA(\027)2537 1778 y FG(\()p FH(t)p FG(\))2631 1710 y Fz(\001)2693 1778 y FG(=)23 b FH(\026)2831 1790 y FA(\027)2898 1778 y FI(for)i(all)g FH(t)p FI(.)0 1886 y(Applying)30 b(Ito's)g(form)n(ula)f(to)h FH(')1017 1818 y Fz(\000)1056 1886 y FH(u)1104 1898 y FA(\027)1145 1886 y FG(\()p FH(t)p FG(\))1239 1818 y Fz(\001)1277 1886 y FI(,)h FH(')p FG(\()p FH(u)p FG(\))d(=)f FF(j)p FH(u)p FF(j)1711 1856 y FD(2)1778 1886 y FI(\(see)j(\(4.15\))o(\))h(and)f (taking)g(exp)r(ectation)f(w)n(e)0 1985 y(get)455 2213 y FH(E)18 b FF(j)p FH(u)605 2225 y FA(\027)646 2213 y FG(\()p FH(t)p FG(\))p FF(j)763 2179 y FD(2)820 2213 y FF(\000)g Fk(E)c FF(j)p FH(u)1051 2225 y FA(\027)1091 2213 y FG(\(0\))p FF(j)1220 2179 y FD(2)1281 2213 y FG(=)22 b FF(\000)p FG(2)p FH(\027)1534 2100 y Fz(Z)1618 2121 y FA(t)1581 2289 y FD(0)1674 2213 y Fk(E)p FF(h)p FH(Lu)1874 2225 y FA(\027)1915 2213 y FH(;)14 b(u)2000 2225 y FA(\027)2040 2213 y FF(i)g FH(d\034)473 2450 y FF(\000)k FG(2)625 2337 y Fz(Z)708 2358 y FA(t)671 2526 y FD(0)765 2450 y Fk(E)p FF(h)p FH(B)t FG(\()p FH(u)1007 2462 y FA(\027)1048 2450 y FG(\))p FH(;)c(u)1165 2462 y FA(\027)1206 2450 y FF(i)g FH(d\034)29 b FG(+)1453 2394 y(1)p 1453 2431 42 4 v 1453 2507 a(2)1518 2450 y FH(\027)1592 2337 y Fz(Z)1675 2358 y FA(t)1638 2526 y FD(0)1732 2450 y Fk(E)1823 2372 y Fz(X)1867 2546 y FA(s)1970 2450 y FH(b)2006 2416 y FD(2)2006 2471 y FA(s)2057 2450 y FH(d)2100 2416 y FD(2)2137 2450 y FH(')p FG(\()p FH(u)2271 2462 y FA(\027)2313 2450 y FG(\)\()p FH(e)2416 2462 y FA(s)2451 2450 y FH(;)14 b(e)2527 2462 y FA(s)2562 2450 y FG(\))g FH(d\034)24 b(:)0 2695 y FI(Noting)j(that)h(the)g(l.h.s.)37 b(v)-5 b(anishes)27 b(as)g(w)n(ell)g(as)g(the)h(second)f(term)h(in)f(the)h (r.h.s.,)g(and)f(that)870 2848 y FG(1)p 870 2886 V 870 2962 a(2)950 2826 y Fz(X)994 3000 y FA(s)1097 2905 y FH(b)1133 2870 y FD(2)1133 2925 y FA(s)1184 2905 y FH(d)1227 2870 y FD(2)1278 2905 y FH(')p FG(\()p FH(u)1412 2917 y FA(\027)1454 2905 y FG(\))14 b(\()p FH(e)1571 2917 y FA(s)1606 2905 y FH(;)g(e)1682 2917 y FA(s)1717 2905 y FG(\))24 b(=)1860 2826 y Fz(X)2008 2905 y FH(b)2044 2870 y FD(2)2044 2925 y FA(s)2104 2905 y FG(=)e FH(B)2254 2917 y FD(0)2305 2905 y FH(;)0 3174 y FI(w)n(e)27 b(ha)n(v)n(e)g FG(2)370 3107 y Fz(R)424 3128 y FA(t)408 3204 y FD(0)481 3174 y Fk(E)14 b FF(h)p FH(Lu)695 3186 y FA(\027)735 3174 y FH(;)g(u)820 3186 y FA(\027)861 3174 y FF(i)g FH(d\034)33 b FG(=)23 b FH(tB)1200 3186 y FD(0)1237 3174 y FI(.)37 b(That)27 b(is,)776 3397 y Fk(E)14 b FF(k)p FH(u)943 3409 y FA(\027)983 3397 y FG(\()p FH(t)p FG(\))p FF(k)1119 3363 y FD(2)1119 3418 y(1)1179 3397 y FG(=)1277 3341 y(1)p 1277 3378 V 1277 3454 a(2)1351 3397 y FH(B)1414 3409 y FD(0)1507 3397 y FI(for)27 b(eac)n(h)f FH(t)e FF(\025)e FG(0)27 b FI(and)h FH(\027)g(>)23 b FG(0)o FH(:)564 b FI(\(10.3\))0 3608 y(Similarly)-7 b(,)30 b(applying)f(Ito's) h(form)n(ula)f(to)g(the)i(functional)f FH(')1905 3620 y FD(1)1942 3541 y Fz(\000)1980 3608 y FH(u)2028 3620 y FA(\027)2069 3608 y FG(\()p FH(t)p FG(\))2163 3541 y Fz(\001)2229 3608 y FG(=)c FF(k)p FH(u)2410 3620 y FA(\027)2450 3608 y FG(\()p FH(t)p FG(\))p FF(k)2586 3578 y FD(2)2586 3629 y(1)2654 3608 y FI(and)j(using)h(that)0 3708 y FF(h)p FH(B)t FG(\()p FH(u)p FG(\))p FH(;)14 b(Lu)p FF(i)23 b(\021)f FG(0)p FI(,)28 b(w)n(e)f(ha)n(v)n(e)687 3941 y FG(2)743 3828 y Fz(Z)825 3848 y FA(t)789 4016 y FD(0)882 3941 y Fk(E)p FF(h)p FH(L)1034 3906 y FD(2)1071 3941 y FH(u)1119 3953 y FA(\027)1160 3941 y FH(;)14 b(u)1245 3953 y FA(\027)1286 3941 y FF(i)g FH(d\034)33 b FG(=)1531 3828 y Fz(Z)1614 3848 y FA(t)1578 4016 y FD(0)1681 3862 y Fz(X)1725 4036 y FA(s)1828 3941 y FH(b)1864 3953 y FA(s)1913 3941 y FF(k)p FH(e)1994 3953 y FA(s)2029 3941 y FF(k)2071 3906 y FD(2)2071 3961 y(1)2121 3941 y FH(d\034)g FG(=)23 b FH(t)14 b(B)2428 3953 y FD(1)2479 3941 y FH(:)0 4185 y FI(Therefore)1238 4303 y Fk(E)g FF(k)p FH(u)1405 4315 y FA(\027)1445 4303 y FG(\()p FH(t)p FG(\))p FF(k)1581 4269 y FD(2)1581 4324 y(2)1642 4303 y FG(=)1739 4247 y(1)p 1739 4284 V 1739 4360 a(2)1814 4303 y FH(B)1877 4315 y FD(1)1928 4303 y FH(:)1026 b FI(\(10.4\))0 4496 y(Simple)36 b(analysis)d(of)i(the)h(deriv)-5 b(ation)34 b(in)h(Exercise)g(4.2)f(the)h(estimate)g Fk(E)14 b FH(e)2435 4466 y FA(\033)r Fw(k)p FA(u)p FD(\()p FA(t)p FD(;0\))p Fw(k)2711 4441 y Ft(2)2711 4482 y(1)2783 4496 y FF(\024)35 b FH(C)42 b FI(\(v)-5 b(alid)0 4595 y(for)33 b(eac)n(h)g FH(t)g FF(\025)f FG(0)p FI(\),)j(sho)n(ws)d(that)i(it)g(holds)f(for)g (solutions)g(of)g(\(10.1\),)i(uniformly)e(in)h FH(\027)k(>)33 b FG(0)p FI(.)54 b(Since)0 4695 y FF(D)r FH(u)p FG(\()p FH(t)p FG(;)14 b(0\))23 b FH(*)g(\026)466 4707 y FA(\027)507 4695 y FI(,)28 b(then)1027 4810 y Fk(E)14 b FH(e)1143 4776 y FA(\033)r Fw(k)p FA(u)1256 4784 y Fu(\027)1292 4776 y FD(\()p FA(t)p FD(\))p Fw(k)1403 4751 y Ft(2)1403 4792 y(1)1463 4810 y FF(\024)22 b FH(C)90 b FF(8)p FH(t)23 b FF(\025)g FG(0)p FH(;)14 b(\027)27 b(>)c FG(0)814 b FI(\(10.5\))1553 5059 y(71)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 72 71 bop 0 523 a FI(\(to)26 b(get)g(\(10.5\))f(from)h (the)g(estimate)g(for)g FG(exp\()p FH(\033)s FF(k)p FH(u)p FG(\()p FH(t)p FG(;)14 b(0\))p FF(k)1814 493 y FD(2)1814 544 y(1)1850 523 y FG(\))27 b FI(one)e(should)h(use)g(F)-7 b(atou's)25 b(lemma)h(as)g(in)0 623 y(the)i(pro)r(of)f(of)g(Theorem)g (4.3.\))125 722 y(Since)f FF(k)p FH(u)430 734 y FA(\027)470 722 y FF(k)512 692 y FD(2)512 743 y(1)572 722 y FF(\024)c(j)p FH(u)730 734 y FA(\027)771 722 y FF(j)16 b(\001)f(k)p FH(u)938 734 y FA(\027)978 722 y FF(k)1020 734 y FD(2)1057 722 y FI(,)27 b(then)f FG(\()p Fk(E)14 b FF(k)p FH(u)1493 734 y FA(\027)1533 722 y FF(k)1575 692 y FD(2)1575 743 y(1)1612 722 y FG(\))1644 692 y FD(2)1705 722 y FF(\024)22 b Fk(E)14 b FF(j)p FH(u)1940 734 y FA(\027)1981 722 y FF(j)2004 692 y FD(2)2056 722 y FF(\001)i Fk(E)e FF(k)p FH(u)2262 734 y FA(\027)2301 722 y FF(k)2343 692 y FD(2)2343 743 y(2)2380 722 y FI(.)37 b(Using)25 b(this)i(inequalit)n(y)0 822 y(and)g(the)h(estimates)g(\(10.3\))o(,)f(\(10.4\))g(w)n(e)g(see)h (that)719 1001 y FG(1)p 719 1038 42 4 v 719 1114 a(2)827 1001 y FH(B)894 971 y FD(2)890 1022 y(0)p 827 1038 105 4 v 829 1114 a FH(B)892 1126 y FD(1)964 1057 y FF(\024)23 b Fk(E)14 b FF(j)p FH(u)1200 1069 y FA(\027)1240 1057 y FG(\()p FH(t)p FG(\))p FF(j)1357 1023 y FD(2)1418 1057 y FF(\024)1516 1001 y FG(1)p 1516 1038 42 4 v 1516 1114 a(2)1590 1057 y FH(B)1653 1069 y FD(0)1774 1057 y FI(for)27 b(all)g FH(t)14 b(;)60 b FI(all)27 b FH(\027)h(>)23 b FG(0)p FH(:)497 b FI(\(10.6\))0 1280 y(The)36 b(Reynolds)e(n)n(um)n(b)r (er)i FH(R)q FG(\()p FH(u)p FG(\))f FI(of)h(a)f(v)n(elo)r(cit)n (y-\034eld)f FH(u)p FG(\()p FH(x)p FG(\))i FI(w)n(as)f(de\034ned)h(in)f (Section)h(2.7.)60 b(F)-7 b(or)34 b(a)0 1380 y(random)c(\034eld)h FH(u)f FI(the)i(c)n(haracteristic)c(scale)i(for)h FH(u)f FI(is)h FG(\()p Fk(E)p FF(j)p FH(u)p FF(j)1916 1349 y FD(2)1953 1380 y FG(\))1985 1349 y FD(1)p FA(=)p FD(2)2090 1380 y FI(.)46 b(Therefore,)31 b(due)g(to)g(\(10.6\),)f(w)n(e)0 1479 y(ha)n(v)n(e)1344 1579 y FH(R)q FG(\()p FH(u)1488 1591 y FA(\027)1529 1579 y FG(\))24 b FF(\030)e FH(\027)1718 1545 y Fw(\000)p FD(1)1821 1579 y FH(:)0 1728 y FI(A)e(traditional)e (de\034nition)i(of)f(a)g(turbulen)n(t)g(v)n(elo)r(cit)n(y)g(\034eld)g (is)h(that)f(this)h(is)f(one)g(with)h(a)e(high)i(Reynolds)0 1828 y(n)n(um)n(b)r(er.)37 b(A)n(ccordingly)-7 b(,)26 b(the)i(v)n(elo)r(cit)n(y)e(\034eld)i FH(u)1487 1840 y FA(\027)1528 1828 y FG(\()p FH(t;)14 b(x)p FG(\))29 b FI(b)r(ecomes)e(turbulen)n(t)h(when)f FH(\027)33 b FI(is)28 b(small.)125 1977 y(Since)34 b(the)h(estimates)e (\(10.3\)\025\(10.5\))g(hold)h(uniformly)g(in)g FH(\027)40 b(>)34 b FG(0)p FI(,)h(they)g(allo)n(w)e(to)h(pass)f(to)h(a)0 2077 y(limit)28 b(as)f FH(\027)i FF(!)23 b FG(0)k FI(along)f (sequences:)0 2242 y FJ(Theorem)k(10.1.)40 b FI(1\))30 b FC(A)n(ny)f(se)l(quenc)l(e)i Fz(e)-48 b FH(\027)1301 2254 y FA(j)1359 2242 y FF(!)23 b FG(0)29 b FC(c)l(ontains)h(a)g(subse) l(quenc)l(e)f FH(\027)2432 2254 y FA(j)2490 2242 y FF(!)23 b FG(0)30 b FC(such)f(that)889 2425 y FF(D)16 b FH(u)1017 2437 y FA(\027)1050 2445 y Fu(j)1085 2425 y FG(\()p FF(\001)p FG(\))24 b FH(*)f FF(D)16 b FH(U)9 b FG(\()p FF(\001)p FG(\))37 b FC(in)g FF(P)1746 2357 y Fz(\000)1784 2425 y FH(C)6 b FG(\(0)p FH(;)14 b FF(1)p FG(;)g FH(H)2156 2390 y FD(1)2193 2425 y FG(\))2225 2357 y Fz(\001)2277 2425 y FH(:)677 b FI(\(10.7\))0 2607 y FC(The)31 b(limiting)f(pr)l(o)l (c)l(ess)g FH(U)9 b FG(\()p FH(t)p FG(\))24 b FF(2)f FH(H)1096 2577 y FD(1)1163 2607 y FC(is)30 b(stationary)g(and)h(p)l (ossesses)f(the)g(fol)t(lowing)i(pr)l(op)l(erties:)0 2756 y FI(2\))d FC(Every)i(tr)l(aje)l(ctory)f(of)h(the)f(pr)l(o)l(c)l (ess)g FH(U)38 b FC(is)30 b(such)g(that)811 2938 y FH(U)9 b FG(\()p FF(\001)p FG(\))24 b FF(2)f FH(L)1123 2950 y FD(2)p FA(;)11 b FD(lo)r(c)1274 2938 y FG(\(0)p FH(;)j FF(1)p FG(;)g FH(H)1581 2907 y FD(2)1618 2938 y FG(\))g FH(;)835 3041 y FG(_)811 3062 y FH(U)9 b FG(\()p FF(\001)p FG(\))24 b FF(2)f FH(L)1123 3074 y FD(1)p FA(;)11 b FD(lo)r(c)1274 3062 y FG(\(0)p FH(;)j FF(1)p FG(;)g FH(H)1581 3032 y FD(1)1618 3062 y FG(\))k FF(\\)h FH(L)1799 3074 y Fw(1)p FA(;)11 b FD(lo)r(c)1983 3062 y FG(\(0)p FH(;)j FF(1)p FG(;)g FH(L)2271 3074 y FA(p)2309 3062 y FG(\))g FH(;)0 3240 y FC(for)31 b(every)f FH(p)23 b(<)g FG(2)p FC(.)38 b(It)29 b(satis\034es)h(the)g(Euler)g(e)l(quation:)1341 3401 y FG(_)1318 3422 y FH(U)d FG(+)18 b FH(B)t FG(\()p FH(U)9 b FG(\))23 b(=)g(0)14 b FH(;)1105 b FI(\(10.8\))0 3604 y FC(and)40 b FF(j)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(j)40 b FC(and)g FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)832 3616 y FD(1)908 3604 y FC(ar)l(e)40 b(time-indep)l(endent)g(c) l(onstants.)68 b(Besides,)44 b(if)c FH(g)k FG(:)d FE(R)g FF(!)g FE(R)f FC(is)g(a)0 3704 y(b)l(ounde)l(d)30 b(c)l(ontinuous)f (function,)h(then)g(the)f(quantity)1222 3811 y Fz(Z)1269 4000 y Fv(T)1308 3983 y Ft(2)1358 3924 y FH(g)s FG(\(curl)13 b FH(U)c FG(\()p FH(t;)14 b(x)p FG(\)\))g FH(dx)1012 b FI(\(10.9\))0 4158 y FC(also)31 b(is)f(time-indep)l(endent.)942 4128 y Fo(10)0 4327 y FI(3\))24 b FC(F)-6 b(or)24 b(e)l(ach)i FH(t)d FF(\025)f FG(0)i FC(we)g(have)50 b Fk(E)14 b FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)1283 4296 y FD(2)1283 4347 y(2)1342 4327 y FF(\024)1440 4294 y FD(1)p 1440 4308 34 4 v 1440 4355 a(2)1497 4327 y FH(B)1560 4339 y FD(1)1597 4327 y FC(,)1682 4294 y FD(1)p 1682 4308 V 1682 4355 a(2)1749 4284 y FA(B)1802 4259 y Ft(2)1799 4301 y(0)p 1749 4307 86 4 v 1750 4355 a FA(B)1800 4363 y Ft(1)1867 4327 y FF(\024)23 b Fk(E)14 b FF(j)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(j)2238 4296 y FD(2)2298 4327 y FF(\024)2396 4294 y FD(1)p 2396 4308 34 4 v 2396 4355 a(2)2439 4327 y FH(B)2502 4339 y FD(0)2553 4327 y FH(;)48 b Fk(E)14 b FH(e)2740 4296 y FA(\033)r Fw(k)p FA(U)6 b FD(\()p FA(t)p FD(\))p Fw(k)2976 4271 y Ft(2)2976 4313 y(1)3036 4327 y FF(\024)23 b FH(C)0 4426 y FC(and)1254 4544 y Fk(E)14 b FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)1575 4510 y FD(2)1575 4565 y(1)1635 4544 y FG(=)1732 4488 y(1)p 1732 4525 42 4 v 1732 4601 a(2)1797 4544 y FH(B)1860 4556 y FD(0)1912 4544 y FG(;)1000 b FI(\(10.10\))p 0 4654 1276 4 v 62 4708 a Fn(10)127 4731 y Fm(In)35 b(other)h(w)n (ords,)i(the)d(push-forw)n(ard)h(of)e(the)i(Leb)r(esgue)g(measure)f(on) g(the)g(torus)h(b)n(y)f(the)g(map)f Fa(T)2940 4708 y Fc(2)3013 4731 y Fh(!)j Fa(R)p Fm(,)0 4810 y Fi(x)19 b Fh(7!)g Fg(curl)12 b Fi(U)7 b Fg(\()p Fi(t;)12 b(x)p Fg(\))p Fm(,)23 b(is)g(a)h(time-indep)r(enden)n(t)h(measure.)1553 5059 y FI(72)p eop end PStoPSsaved restore %%Page: (72,73) 37 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 73 72 bop 0 523 a FC(in)30 b(p)l(articular,)h FH(U)38 b FC(is)30 b(not)g(the)g(zer)l(o-pr)l(o)l(c)l(ess.)0 667 y FI(4\))j FC(If)g FH(b)233 679 y FA(s)297 667 y FF(\021)28 b FH(b)426 679 y Fw(\000)p FA(s)513 667 y FC(,)34 b(then)f(the)g(pr)l(o)l(c)l(ess)g FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))34 b FC(is)f(homo)l(gene)l(ous)h(in)f FH(x)p FC(.)49 b(If,)34 b(in)f(addition,)j FH(B)2921 679 y FD(6)2987 667 y FH(<)29 b FF(1)p FC(,)0 767 y(then)810 867 y Fk(E)14 b FH(e)926 832 y FA(\033)964 840 y Ft(1)996 832 y Fw(j)p FA(U)6 b FD(\()p FA(t;x)p FD(\))p Fw(j)1244 867 y FG(+)18 b Fk(E)c FH(e)1443 832 y FA(\033)1481 840 y Ft(1)1513 832 y Fw(jr)p FA(U)6 b FD(\()p FA(t;x)p FD(\))p Fw(j)1794 807 y Ft(1)p Fu(=)p Ft(2)1912 867 y FF(\024)23 b FH(C)2059 879 y FD(1)2181 867 y FF(8)p FH(t;)14 b(x)g(;)557 b FI(\(10.11\))0 1008 y FC(wher)l(e)30 b FH(\033)281 1020 y FD(1)349 1008 y FC(and)g FH(C)569 1020 y FD(1)636 1008 y FC(ar)l(e)g(suitable)h(p)l(ositive)g(c)l(onstants.)0 1164 y(Pr)l(o)l(of.)43 b FI(1\))28 b(Let)f(us)h(denote)924 1334 y FH(Z)981 1346 y FA(n)1049 1334 y FG(=)22 b FH(C)6 b FG(\(0)p FH(;)14 b(n)p FG(;)g FH(H)1475 1300 y FD(1)1512 1334 y FG(\))p FH(;)84 b(Z)28 b FG(=)23 b FH(C)6 b FG(\(0)p FH(;)14 b FF(1)p FG(;)g FH(H)2196 1300 y FD(1)2233 1334 y FG(\))0 1504 y FI(\(the)28 b(space)f FH(Z)33 b FI(is)28 b(endo)n(w)n(ed)e(with)i(the)g(Tikhono)n(v)f(top)r(ology\))f(and)h(set) 693 1684 y FH(m)766 1649 y FA(n)766 1704 y(\027)835 1684 y FG(=)22 b FF(D)r FG(\()p FH(u)1068 1696 y FA(\027)1110 1684 y FF(j)1133 1699 y FD([0)p FA(;n)p FD(])1268 1684 y FG(\))i FF(2)f(P)7 b FG(\()p FH(Z)1556 1696 y FA(n)1601 1684 y FG(\))p FH(;)60 b(\027)29 b FF(2)1871 1663 y Fz(e)1864 1684 y FG(\006)23 b(=)g FF(f)r Fz(e)-48 b FH(\027)2118 1696 y FD(1)2154 1684 y FH(;)17 b Fz(e)-49 b FH(\027)2232 1696 y FD(2)2270 1684 y FH(;)14 b(:)g(:)g(:)f FF(g)h FH(:)0 1873 y FI(W)-7 b(e)32 b(sho)n(w)f(\034rst)h(that)g(for)f(eac)n (h)g FH(n)h FI(the)g(sequence)f FF(f)p FH(m)1731 1842 y FA(n)1731 1893 y(\027)1776 1873 y FH(;)d(\027)35 b FF(2)1995 1852 y Fz(e)1988 1873 y FG(\006)p FF(g)d FI(con)n(tains)f(a)g (w)n(eakly)g(con)n(v)n(erging)0 1972 y(subsequence.)36 b(Let)28 b(us)f(denote)815 2142 y FH(v)855 2154 y FA(\027)897 2142 y FG(\()p FH(t)p FG(\))c(=)g FH(u)1150 2154 y FA(\027)1191 2142 y FG(\()p FH(t)p FG(\))c FF(\000)1387 2078 y(p)p 1456 2078 47 4 v 64 x FH(\027)g(\020)6 b FG(\()p FH(t)p FG(\))p FH(;)61 b FG(0)23 b FF(\024)f FH(t)h FF(\024)g FH(n;)k(\027)i FF(2)2284 2121 y Fz(e)2277 2142 y FG(\006)14 b FH(:)0 2312 y FI(Then)1157 2411 y FG(_)-35 b FH(v)1185 2423 y FA(\027)1245 2411 y FF(\000)18 b FH(\027)h(Lu)1493 2423 y FA(\027)1552 2411 y FG(+)f FH(B)t FG(\()p FH(u)1782 2423 y FA(\027)1823 2411 y FG(\))23 b(=)g(0)14 b FH(:)890 b FI(\(10.12\))0 2553 y(Since)28 b FH(B)280 2565 y FD(1)340 2553 y FH(<)23 b FF(1)p FI(,)k(then)h(w)n(e)g(get)f(from)g(\(10.4\))g (that)1194 2723 y Fk(E)14 b FF(j)p FH(u)1342 2735 y FA(\027)1382 2723 y FF(j)1405 2689 y FD(2)1405 2745 y FA(L)1451 2753 y Ft(2)1483 2745 y FD(\(0)p FA(;n)p FD(;)p FA(H)1680 2729 y Ft(2)1713 2745 y FD(\))1766 2723 y FF(\024)23 b FH(C)1913 2735 y FA(n)1972 2723 y FH(:)940 b FI(\(10.13\))0 2893 y(Due)28 b(to)g(\(10.5\),)1194 2992 y Fk(E)14 b FF(j)p FH(u)1342 3004 y FA(\027)1382 2992 y FF(j)1405 2958 y FD(4)1405 3015 y FA(L)1451 3023 y Ft(4)1483 3015 y FD(\(0)p FA(;n)p FD(;)p FA(H)1680 2998 y Ft(1)1713 3015 y FD(\))1766 2992 y FF(\024)23 b FH(C)1913 3004 y FA(n)1972 2992 y FH(:)940 b FI(\(10.14\))p 3128 3134 4 57 v 3132 3081 50 4 v 3132 3134 V 3182 3134 4 57 v 0 3297 a FJ(Lemma)38 b(10.2.)45 b FC(The)37 b(nonline)l(arity)h FH(B)j FC(de\034nes)36 b(c)l(ontinuous)g(quadr)l(atic)i(maps)f FH(H)2716 3267 y FD(2)2789 3297 y FF(!)f FH(H)2984 3267 y FD(1)3057 3297 y FC(and)0 3397 y FH(H)76 3367 y FD(1)136 3397 y FF(!)23 b FH(L)299 3409 y FA(p)337 3397 y FG(\()p FE(T)424 3367 y FD(2)462 3397 y FG(;)14 b FE(R)559 3367 y FD(2)596 3397 y FG(\))p FC(,)30 b(for)h(any)f FH(p)23 b(<)g FG(2)p FC(.)125 3553 y FI(The)29 b(pro)r(of)f(of)i(these)f (assertions)e(follo)n(ws)h(from)h(t)n(w)n(o)g(w)n(ell)f(kno)n(wn)h (facts.)42 b(First,)29 b(the)h(m)n(ultipli-)0 3652 y(cation)d(of)g (functions)h(de\034nes)g(con)n(tin)n(uous)e(maps)h FH(H)1682 3622 y FD(2)1719 3652 y FG(\()p FE(T)1806 3622 y FD(2)1844 3652 y FG(\))19 b FF(\002)f FH(H)2054 3622 y FD(1)2091 3652 y FG(\()p FE(T)2178 3622 y FD(2)2215 3652 y FG(\))24 b FF(!)f FH(H)2453 3622 y FD(1)2490 3652 y FG(\()p FE(T)2577 3622 y FD(2)2615 3652 y FG(\))28 b FI(and)f FH(H)2912 3622 y FD(1)2949 3652 y FG(\()p FE(T)3036 3622 y FD(2)3074 3652 y FG(\))18 b FF(\002)0 3752 y FH(L)57 3764 y FD(2)94 3752 y FG(\()p FE(T)181 3722 y FD(2)219 3752 y FG(\))41 b FF(!)g FH(L)473 3764 y FA(p)511 3752 y FG(\()p FE(T)598 3722 y FD(2)636 3752 y FG(\))e(\()p FH(p)i(<)g FG(2\))p FI(,)g(and)d(second,)j(the)d(Lera)n(y)f(pro)5 b(jection)38 b FG(\005)g FI(is)h(con)n(tin)n(uous)e(in)i(the)0 3852 y FH(L)57 3864 y FA(p)95 3852 y FI(-norm)27 b(for)g FG(1)22 b FH(<)h(p)g(<)f FF(1)p FI(.)125 3951 y(This)27 b(lemma)h(and)f (\(10.12\)\025\(10.14\))d(imply)k(that)1175 4121 y Fk(E)14 b FF(j)e FG(_)-35 b FH(v)1315 4133 y FA(\027)1356 4121 y FF(j)1379 4087 y FD(2)1379 4144 y FA(L)1425 4152 y Ft(2)1457 4144 y FD(\(0)p FA(;n)p FD(;)p FA(L)1642 4155 y Ft(3)p Fu(=)p Ft(2)1732 4144 y FD(\))1785 4121 y FF(\024)23 b FH(C)1932 4133 y FA(n)1991 4121 y FH(:)0 4301 y FI(Let)28 b(us)f(denote)h(b)n(y)f FF(C)32 b FI(the)c(space)824 4471 y FF(C)g FG(=)23 b FF(f)p FH(v)i FF(2)f FH(L)1227 4483 y FD(2)1264 4471 y FG(\(0)p FH(;)14 b(n)p FG(;)g FH(H)1538 4436 y FD(2)1574 4471 y FG(\))g FF(j)26 b FG(_)-35 b FH(v)26 b FF(2)e FH(L)1859 4483 y FD(2)1896 4471 y FG(\(0)p FH(;)14 b(n)p FG(;)g FH(L)2151 4486 y FD(3)p FA(=)p FD(2)2254 4471 y FG(\))p FF(g)g FH(:)0 4640 y FI(Then)28 b FF(C)f Fs(b)c FH(Z)433 4652 y FA(n)506 4640 y FI(\(see)k([Lio69)n(],)h(Section)g(1.5,)f(cf.)37 b(Lemma)27 b(2.5\),)g(and)1345 4810 y Fk(E)14 b FF(j)p FH(v)1485 4822 y FA(\027)1526 4810 y FF(j)1549 4776 y FD(2)1549 4831 y Fw(C)1615 4810 y FF(\024)23 b FH(C)1768 4776 y Fw(0)1762 4831 y FA(n)1821 4810 y FH(:)1553 5059 y FI(73)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 74 73 bop 0 523 a FI(This)28 b(estimate)f(and)g(Prokhoro) n(v's)e(theorem)i(imply)h(that)1210 706 y FF(D)16 b FH(v)1330 718 y FA(\027)1363 726 y Fu(j)1422 706 y FH(*)23 b(m)1601 671 y FA(n)1669 706 y FF(2)h(P)7 b FG(\()p FH(Z)1902 718 y FA(n)1946 706 y FG(\))957 b FI(\(10.15\))0 888 y(for)27 b(a)g(suitable)h(subsequence)f FF(f)p FH(\027)1058 900 y FA(j)1092 888 y FF(g)p FI(.)37 b(Let)27 b FH(f)37 b FI(b)r(e)27 b(a)h(b)r(ounded)g(Lipsc)n(hitz)f(functional)h(on)f FH(Z)2857 900 y FA(n)2902 888 y FI(.)37 b(Since)864 1071 y FG(\()p FH(f)t(;)14 b(m)1051 1037 y FA(n)1051 1092 y(\027)1084 1100 y Fu(j)1120 1071 y FG(\))23 b(=)g Fk(E)14 b FH(f)9 b FG(\()p FH(u)1470 1083 y FA(\027)1503 1091 y Fu(j)1537 1071 y FG(\))24 b(=)e Fk(E)14 b FH(f)9 b FG(\()p FH(v)1879 1083 y FA(\027)1912 1091 y Fu(j)1966 1071 y FG(+)2049 1007 y FF(p)p 2118 1007 47 4 v 64 x FH(\027)2164 1091 y FA(j)2213 1071 y FH(\020)d FG(\))14 b FH(;)0 1254 y FI(w)n(e)27 b(obtain)279 1416 y FF(j)p FG(\()p FH(f)t(;)14 b(m)489 1381 y FA(n)489 1436 y(\027)522 1444 y Fu(j)558 1416 y FG(\))19 b FF(\000)f FG(\()p FH(f)t(;)c(m)879 1381 y FA(n)924 1416 y FG(\))p FF(j)23 b FG(=)g FF(j)14 b Fk(E)1190 1348 y Fz(\000)1227 1416 y FH(f)9 b FG(\()p FH(v)1349 1428 y FA(\027)1382 1436 y Fu(j)1437 1416 y FG(+)1520 1352 y FF(p)p 1589 1352 V 64 x FH(\027)1635 1436 y FA(j)1684 1416 y FH(\020)d FG(\))19 b FF(\000)f FH(f)9 b FG(\()p FH(v)1982 1428 y FA(\027)2015 1436 y Fu(j)2051 1416 y FG(\))2083 1348 y Fz(\001)609 1556 y FG(+)692 1489 y Fz(\000)730 1556 y Fk(E)14 b FH(f)9 b FG(\()p FH(v)929 1568 y FA(\027)962 1576 y Fu(j)997 1556 y FG(\))18 b FF(\000)g FG(\()p FH(f)t(;)c(m)1317 1522 y FA(n)1363 1556 y FG(\))1395 1489 y Fz(\001)1433 1556 y FF(j)23 b(\024)g FH(C)1632 1492 y FF(p)p 1701 1492 V 64 x FH(\027)1748 1576 y FA(j)1796 1556 y Fk(E)p FF(j)p FH(\020)6 b FF(j)1947 1568 y FA(Z)1992 1576 y Fu(n)2056 1556 y FG(+)18 b FF(j)p FG(\()p FH(f)t(;)c FF(D)r FH(v)2382 1568 y FA(\027)2415 1576 y Fu(j)2452 1556 y FG(\))k FF(\000)g FG(\()p FH(f)t(;)c(m)2772 1522 y FA(n)2818 1556 y FG(\))p FF(j)g FH(:)0 1741 y FI(The)29 b(\034rst)f(term)h(in)g(the)g(r.h.s.)40 b(go)r(es)28 b(to)g(zero)g(since)g Fk(E)14 b FF(j)p FH(\020)6 b FF(j)1839 1753 y FA(Z)1884 1761 y Fu(n)1954 1741 y FH(<)25 b FF(1)p FI(,)k(and)g(the)g(second)f(go)r(es)g(to)g(zero)0 1841 y(b)n(y)e(\(10.15\).)35 b(So)25 b FH(m)613 1811 y FA(n)613 1861 y(\027)646 1869 y Fu(j)705 1841 y FH(*)e(m)884 1811 y FA(n)929 1841 y FI(.)36 b(No)n(w)26 b(w)n(e)f(can)h(use)g(the)g (diagonalisation)d(pro)r(cedure)i(to)h(construct)f(a)0 1940 y(new)j(subsequence)f FF(f)p FH(\027)723 1952 y FA(j)757 1940 y FF(g)g FI(suc)n(h)g(that)1028 2123 y FH(m)1101 2089 y FA(n)1101 2144 y(\027)1134 2152 y Fu(j)1192 2123 y FH(*)c(m)1371 2089 y FA(n)1440 2123 y FF(2)g(P)7 b FG(\()p FH(Z)1672 2135 y FA(n)1717 2123 y FG(\))83 b FI(for)27 b(all)g FH(n)14 b(:)0 2306 y FI(The)27 b(measures)g FF(f)p FH(m)642 2276 y FA(n)686 2306 y FF(g)g FI(form)g(a)g(compatible) g(family)h(and)f(de\034ne)h(a)f(measure)f FH(m)d FF(2)g(P)7 b FG(\()p FH(Z)f FG(\))28 b FI(with)g(the)0 2405 y(prop)r(ert)n(y)860 2505 y FG(\()p FF(D)r FH(u)1006 2517 y FA(\027)1039 2525 y Fu(j)1074 2505 y FH(;)14 b(f)9 b FG(\))23 b(=)g(\()p FH(m)1409 2471 y FA(n)1409 2525 y(\027)1442 2533 y Fu(j)1477 2505 y FH(;)14 b(f)9 b FG(\))23 b FF(!)g FG(\()p FH(m)1830 2471 y FA(n)1876 2505 y FH(;)14 b(f)9 b FG(\))23 b(=)f(\()p FH(m;)14 b(f)9 b FG(\))0 2654 y FI(for)29 b(ev)n(ery)f FH(n)h FI(and)h(ev)n(ery)e(functional)h FH(f)38 b FI(on)29 b FH(Z)36 b FI(of)29 b(the)h(form)f FH(f)35 b FG(=)25 b FH(f)2140 2666 y FA(n)2185 2654 y FG(\()p FH(u)p FF(j)2288 2669 y FD([0)p FA(;n)p FD(])2424 2654 y FG(\))p FI(,)30 b FH(f)2550 2666 y FA(n)2621 2654 y FF(2)c FH(C)2761 2666 y FA(b)2795 2654 y FG(\()p FH(Z)2884 2666 y FA(n)2929 2654 y FG(\))p FI(.)43 b(This)0 2754 y(relation)27 b(and)g(Prop)r (osition)g(5.1)g(imply)g(that)1133 2937 y FF(D)r FH(u)1247 2949 y FA(\027)1280 2957 y Fu(j)1316 2937 y FG(\()p FF(\001)p FG(\))d FH(*)f(m)82 b FI(in)i FF(P)7 b FG(\()p FH(Z)f FG(\))p FH(:)0 3119 y FI(F)-7 b(or)27 b(details)g(see)g([Kuk04)o(],)h (p.)g(485.)125 3219 y(By)39 b(Sk)n(orokho)r(d's)f(theorem)h(w)n(e)h (can)g(\034nd)g(random)f(pro)r(cesses)f FH(U)2304 3231 y FA(\027)2337 3239 y Fu(j)2372 3219 y FG(\()p FH(t)p FG(\))p FI(,)44 b FH(j)49 b FF(\025)43 b FG(1)p FI(,)g(and)d FH(U)9 b FG(\()p FH(t)p FG(\))p FI(,)0 3319 y(de\034ned)28 b(on)f(the)h(same)f(\(new\))h(probabilit)n(y)f(space,)g(suc)n(h)g(that) 521 3501 y FF(D)16 b FH(U)658 3513 y FA(\027)691 3521 y Fu(j)749 3501 y FG(=)23 b FF(D)16 b FH(u)965 3513 y FA(\027)998 3521 y Fu(j)1089 3501 y FI(for)27 b(all)g FH(j)5 b FI(,)27 b FF(D)r FH(U)33 b FG(=)22 b FH(m)42 b FI(and)f FH(U)2010 3513 y FA(\027)2043 3521 y Fu(j)2101 3501 y FF(!)23 b FH(U)37 b FI(in)27 b FH(Z)6 b FI(,)28 b(a.s.)13 b FH(:)267 b FI(\(10.16\))0 3684 y(Since)30 b(the)h(pro)r(cesses)e FH(U)784 3696 y FA(\027)817 3704 y Fu(j)882 3684 y FI(are)g(stationary)-7 b(,)30 b(then)g FH(U)40 b FI(is)30 b(stationary)e(as)i(w)n(ell.)44 b(The)31 b(\034rst)f(assertion)0 3783 y(of)e(the)f(theorem)g(is)h(pro)n(v)n(ed.) 0 3933 y(2\))f(The)h(estimate)g(\(10.13\),)e(and)h(F)-7 b(atou's)27 b(lemma)h(imply)g(that)1116 4116 y Fk(E)14 b FF(j)p FH(U)9 b FF(j)1305 4081 y FD(2)1305 4138 y FA(L)1351 4146 y Ft(2)1382 4138 y FD(\(0)p FA(;n)p FD(;)p FA(H)1579 4122 y Ft(2)1612 4138 y FD(\))1665 4116 y FF(\024)23 b FH(C)1812 4128 y FA(n)1940 4116 y FF(8)p FH(n)14 b(:)0 4298 y FI(So)31 b(the)g(\034rst)g(assertion)e(in)i(2\))g(follo)n(ws.)46 b(An)n(y)31 b(pro)r(cess)e FH(U)1840 4310 y FA(\027)1873 4318 y Fu(j)1908 4298 y FG(\()p FH(t)p FG(\))j FI(is)f(a)f(w)n(eak)g (solution)g(of)h(\(10.1\).)46 b(So)0 4398 y(for)27 b(an)n(y)g FH(n)g FI(and)h(an)n(y)e FG(0)d FF(\024)g FH(t)862 4410 y FD(1)922 4398 y FF(\024)g FH(t)1040 4410 y FD(2)1100 4398 y FF(\024)g FH(n)k FI(w)n(e)g(ha)n(v)n(e)379 4633 y FH(U)436 4645 y FA(\027)477 4633 y FG(\()p FH(t)539 4645 y FD(2)577 4633 y FG(\))19 b FF(\000)f FH(U)768 4645 y FA(\027)809 4633 y FG(\()p FH(t)871 4645 y FD(1)908 4633 y FG(\))h(+)1042 4520 y Fz(Z)1125 4541 y FA(t)1150 4549 y Ft(2)1088 4709 y FA(t)1113 4717 y Ft(1)1214 4633 y FH(B)1281 4566 y Fz(\000)1319 4633 y FH(U)1376 4645 y FA(\027)1417 4633 y FG(\()p FH(s)p FG(\))1520 4566 y Fz(\001)1573 4633 y FH(ds)k FG(=)g FF(\000)1831 4570 y(p)p 1899 4570 V 1899 4633 a FH(\027)1959 4520 y Fz(Z)2042 4541 y FA(t)2067 4549 y Ft(2)2006 4709 y FA(t)2031 4717 y Ft(1)2132 4633 y FH(AU)2251 4645 y FA(\027)2292 4633 y FG(\()p FH(s)p FG(\))14 b FH(ds)p FG(+)379 4746 y FF(p)p 448 4746 V 59 x FH(\027)495 4738 y Fz(\000)533 4805 y FH(\020)569 4817 y FA(\027)610 4805 y FG(\()p FH(t)672 4817 y FD(2)710 4805 y FG(\))19 b FF(\000)f FH(\020)880 4817 y FA(\027)921 4805 y FG(\()p FH(t)983 4817 y FD(1)1021 4805 y FG(\))1053 4738 y Fz(\001)1091 4805 y FH(;)203 b(\027)29 b FF(2)23 b FG(\006)g(=)g FF(f)p FH(\027)1719 4817 y FA(j)1753 4805 y FF(g)14 b FH(;)2935 4691 y FI(\(10.17\))1553 5059 y(74)p eop end PStoPSsaved restore %%Page: (74,75) 38 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 75 74 bop 0 523 a FI(where)29 b FH(\020)278 535 y FA(\027)320 523 y FG(\()p FH(T)12 b FG(\))30 b FI(is)g(a)f(v)n(ersion)g(of)h(the)g(pro)r(cess)f FH(\020)6 b FG(\()p FH(t)p FG(\))p FI(.)45 b(Since)30 b(the)h(map)f FH(B)h FG(:)c FH(H)34 b FF(!)27 b FH(H)2640 493 y Fw(\000)p FD(2)2759 523 y FI(is)j(quadratic)0 623 y(and)d(con)n(tin)n(uous)g (\(see)g(Section)h(2.2\),)f(then)h(the)g(l.h.s.)37 b(of)27 b(\(10.17\))g(con)n(v)n(erges)e(a.s.)36 b(in)28 b FH(H)2841 593 y Fw(\000)p FD(2)2957 623 y FI(to)987 863 y FH(U)9 b FG(\()p FH(t)1115 875 y FD(2)1152 863 y FG(\))19 b FF(\000)f FH(U)9 b FG(\()p FH(t)1414 875 y FD(1)1451 863 y FG(\))19 b(+)1585 750 y Fz(Z)1668 770 y FA(t)1693 778 y Ft(2)1631 939 y FA(t)1656 947 y Ft(1)1757 863 y FH(B)1824 796 y Fz(\000)1862 863 y FH(U)9 b FG(\()p FH(s)p FG(\))2031 796 y Fz(\001)2083 863 y FH(ds)14 b(:)733 b FI(\(10.18\))0 1100 y(Due)28 b(to)g(\(10.13\))e(and)h(\(4.1\),)815 1341 y Fk(E)892 1249 y Fz(\020)942 1245 y(\014)942 1295 y(\014)942 1345 y(\014)983 1228 y(Z)1066 1248 y FA(t)1091 1256 y Ft(2)1029 1416 y FA(t)1054 1424 y Ft(1)1142 1341 y FH(AU)1261 1353 y FA(\027)1316 1341 y FH(ds)1398 1245 y Fz(\014)1398 1295 y(\014)1398 1345 y(\014)1444 1341 y FG(+)18 b FF(j)p FH(\020)1586 1353 y FA(\027)1628 1341 y FG(\()p FH(t)1690 1353 y FD(2)1727 1341 y FG(\))h FF(\000)f FH(\020)1897 1353 y FA(\027)1939 1341 y FG(\()p FH(t)2001 1353 y FD(1)2038 1341 y FG(\))p FF(j)2093 1249 y Fz(\021)2166 1341 y FH(<)23 b FF(1)14 b FH(:)0 1578 y FI(Therefore)35 b(the)i(r.h.s.)64 b(of)37 b(\(10.17\))e(con)n(v)n(erges)f(to)j(zero)f (in)g FH(H)44 b FI(in)37 b(probabilit)n(y)-7 b(.)63 b(Hence,)39 b(\(10.18\))0 1678 y(v)-5 b(anishes)37 b(a.s.)66 b(for)37 b(an)n(y)g(\034xed)h FG(0)h FF(\024)g FH(t)1253 1690 y FD(1)1330 1678 y FF(\024)g FH(t)1464 1690 y FD(2)1502 1678 y FI(.)67 b(Since)37 b(the)h(tra)5 b(jectories)36 b(of)h(the)h(pro)r(cess)f FH(U)46 b FI(are)0 1778 y(con)n(tin)n(uous)23 b(in)h FH(H)581 1747 y FD(1)618 1778 y FI(,)h(then)g(\(10.18\))e(holds) g(a.s.)35 b(for)24 b(all)g FG(0)e FF(\024)h FH(t)1917 1790 y FD(1)1977 1778 y FF(\024)g FH(t)2095 1790 y FD(2)2132 1778 y FI(.)36 b(That)24 b(is,)h(\(10.8\))e(holds)h(almost)0 1877 y(surely)-7 b(.)64 b(Rede\034ning)36 b FH(U)9 b FG(\()p FH(t)p FG(\))37 b FI(to)g(b)r(e)g(zero)f(on)g(the)h (exceptional)f(n)n(ull-set,)j(w)n(e)d(ac)n(hiev)n(e)f(that)i(ev)n(ery)0 1977 y(tra)5 b(jectory)28 b FH(U)9 b FG(\()p FH(t)p FG(\))31 b FI(b)r(elongs)f(to)g FH(L)1043 1989 y FD(2)p FA(;)11 b FD(lo)r(c)1194 1977 y FG(\(0)p FH(;)j FF(1)p FG(;)g FH(H)1501 1947 y FD(2)1537 1977 y FG(\))31 b FI(and)f(satis\034es)g (\(10.8\).)44 b(Since)2596 1956 y FG(_)2572 1977 y FH(U)36 b FG(=)28 b FF(\000)p FH(B)t FG(\()p FH(U)9 b FG(\))p FI(,)31 b(the)0 2076 y(second)e(assertion)f(in)i(2\))g(follo)n(ws)f(no) n(w)g(from)g(Lemma)g(10.2.)43 b(The)29 b(established)h(smo)r(othness)f (of)g FH(U)0 2176 y FI(and)187 2155 y FG(_)163 2176 y FH(U)39 b FI(easily)29 b(imply)h(that)f(b)r(oth)h FF(j)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(j)31 b FI(and)e FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)1748 2188 y FD(1)1814 2176 y FI(are)29 b(in)n(tegrals)f(of)i(motion,)f(as)g(w)n(ell)h(as)f (the)0 2276 y(quan)n(tities)e(\(10.9\))g(\(see)g([Kuk04)o(]\).)37 b(So)28 b(2\))f(is)g(pro)n(v)n(ed.)0 2425 y(3\))e(The)g(\034rst)g(and)g (the)h(third)f(inequalities)g(follo)n(w)f(from)h(\(10.16\),)g(the)g (corresp)r(onding)e(inequalities)0 2525 y(for)f FH(U)179 2537 y FA(\027)212 2545 y Fu(j)247 2525 y FG(\()p FH(t)p FG(\))i FI(and)e(F)-7 b(atou's)23 b(lemma.)35 b(T)-7 b(o)22 b(pro)n(v)n(e)f(\(10.10\))g(let)i(us)g(consider)f FH(U)2321 2495 y FA(N)2312 2545 y(\027)2383 2525 y FG(\()p FH(t)p FG(\))i(=)f FH(P)2642 2537 y FA(N)2719 2525 y FH(U)2776 2537 y FA(\027)2817 2525 y FG(\()p FH(t)p FG(\))p FI(,)h FH(\027)29 b FF(2)23 b FG(\006)p FI(,)0 2624 y(where)k FH(P)293 2636 y FA(N)384 2624 y FI(is)g(the)h(Galerkin)f(pro)5 b(jection.)36 b(By)27 b(\(10.16\),)1078 2807 y FF(k)p FH(U)1186 2773 y FA(N)1177 2828 y(\027)1248 2807 y FG(\()p FH(t)p FG(\))p FF(k)1384 2773 y FD(2)1384 2828 y(1)1444 2807 y FF(!)c(k)p FH(U)1658 2773 y FA(N)1721 2807 y FG(\()p FH(t)p FG(\))p FF(k)1857 2773 y FD(2)1857 2828 y(1)1945 2807 y FI(a.s.)f FH(:)0 2990 y FI(Due)28 b(to)g(\(10.5\),)e(the)i (random)f(v)-5 b(ariables)26 b FF(k)p FH(U)1438 2960 y FA(N)1429 3010 y(\027)1500 2990 y FG(\()p FH(t)p FG(\))p FF(k)1636 2960 y FD(2)1636 3010 y(1)1701 2990 y FI(are)h(uniformly)g (in)n(tegrable,)g(and)g(hence)861 3172 y Fk(E)14 b FF(k)p FH(U)1046 3138 y FA(N)1037 3193 y(\027)1107 3172 y FG(\()p FH(t)p FG(\))p FF(k)1243 3138 y FD(2)1243 3193 y(1)1304 3172 y FF(!)23 b Fk(E)14 b FF(k)p FH(U)1595 3138 y FA(N)1657 3172 y FG(\()p FH(t)p FG(\))p FF(k)1793 3138 y FD(2)1793 3193 y(1)1830 3172 y FH(;)60 b FG(\006)23 b FF(3)g FH(\027)29 b FF(!)23 b FG(0)14 b FH(:)0 3355 y FI(Due)28 b(to)g(\(10.3\))e(and)i (\(10.4\),)932 3521 y FG(1)p 932 3558 42 4 v 932 3634 a(2)1007 3577 y FH(B)1070 3589 y FD(1)1130 3577 y FF(\025)23 b Fk(E)14 b FF(k)p FH(U)1403 3542 y FA(N)1394 3597 y(\027)1465 3577 y FG(\()p FH(t)p FG(\))p FF(k)1601 3542 y FD(2)1601 3597 y(1)1661 3577 y FF(\025)1759 3521 y FG(1)p 1759 3558 V 1759 3634 a(2)1833 3577 y FH(B)1896 3589 y FD(1)1952 3577 y FF(\000)k FH(')p FG(\()p FH(N)9 b FG(\))14 b FH(;)0 3787 y FI(where)27 b FH(')p FG(\()p FH(N)9 b FG(\))24 b FF(!)f FG(0)k FI(when)h FH(N)k FF(!)23 b(1)p FI(.)37 b(So)932 3953 y FG(1)p 932 3990 V 932 4066 a(2)1007 4009 y FH(B)1070 4021 y FD(1)1130 4009 y FF(\025)23 b Fk(E)14 b FF(k)p FH(U)1403 3975 y FA(N)1465 4009 y FG(\()p FH(t)p FG(\))p FF(k)1601 3975 y FD(2)1601 4030 y(1)1661 4009 y FF(\025)1759 3953 y FG(1)p 1759 3990 V 1759 4066 a(2)1833 4009 y FH(B)1896 4021 y FD(1)1952 4009 y FF(\000)k FH(')p FG(\()p FH(N)9 b FG(\))14 b FH(;)0 4220 y FI(and)20 b(\(10.10\))f (follo)n(ws)g(from)h(the)h(monotone)e(con)n(v)n(ergence)f(theorem.)34 b(No)n(w)20 b(the)g(second)g(\(b)r(oth\025side\))0 4319 y(inequalit)n(y)27 b(follo)n(ws)g(b)n(y)g(the)h(same)f(argumen)n(ts)f (whic)n(h)h(pro)n(v)n(e)f(\(10.6\))o(.)0 4469 y(4\))33 b(If)g FH(b)231 4481 y FA(s)298 4469 y FF(\021)e FH(b)430 4481 y Fw(\000)p FA(s)517 4469 y FI(,)k(then)e(the)g(pro)r(cesses)e FH(u)1329 4481 y FA(\027)1370 4469 y FG(\()p FH(t;)14 b(x)p FG(\))34 b FI(and)f FH(U)1806 4481 y FA(\027)1847 4469 y FG(\()p FH(t;)14 b(x)p FG(\))34 b FI(are)d(homogeneous)g(in)i FH(x)h FI(b)n(y)e(The-)0 4568 y(orem)f(7.3.)49 b(Hence,)33 b FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))32 b FI(is)g(homogeneous)e(due) i(to)g(\(10.16\).)48 b(Pro)r(of)31 b(of)h(\(10.11\))f(is)g(non)n (trivial)0 4668 y(and)k(uses)g(obtained)g(b)r(elo)n(w)g(in)g(Section)h (11.1)d(explicit)j(algebraical)d(relations,)j(satis\034ed)f(b)n(y)g (the)0 4767 y(stationary)26 b(pro)r(cess)g FH(u)732 4779 y FA(\027)773 4767 y FI(.)37 b(See)28 b([Kuk05)n(].)1837 b Fs(\003)1553 5059 y FI(75)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 76 75 bop 125 523 a FI(W)-7 b(e)27 b(call)h(the)g (limiting)g(pro)r(cess)e FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))28 b FI(the)g FC(Eulerian)j(limit)p FI(.)125 672 y(Com)n(bining)39 b(assertions)g(of)h(Theorem)g(7.1)f(and)h(Theorem)g (10.1,)i(w)n(e)e(obtain)g(that)h(for)f(an)n(y)0 772 y FH(u)48 784 y FD(0)108 772 y FF(2)23 b FH(H)35 b FI(there)27 b(exists)g(the)h(follo)n(wing)f(double)g(limit:)978 955 y FF(D)16 b FH(u)1106 967 y FA(\027)1147 955 y FG(\()p FH(t)p FG(;)e FH(u)1294 967 y FD(0)1331 955 y FG(\))60 b FH(*)1386 1005 y FA(t)p Fw(!1)1590 955 y FH(\026)1640 967 y FA(\027)1843 955 y FH(*)1727 1012 y Fw(f)p FA(\027)1794 1020 y Fu(j)1826 1012 y Fw(g3)p FA(\027)t Fw(!)p FD(0)2087 955 y FH(\026)2137 967 y FD(0)2188 955 y FH(:)0 1190 y FJ(Remark)31 b(10.3.)40 b FI(Since)28 b FH(U)9 b FG(\()p FH(t)p FG(\))28 b FI(is)g(a)f(stationary)g(pro)r(cess)f(that)i (satis\034es)f(a.s.)37 b(the)29 b(Euler)f(equation,)0 1290 y(then)37 b FH(\026)248 1302 y FD(0)323 1290 y FG(=)g FF(D)r FH(U)9 b FG(\()p FH(t)p FG(\))37 b FI(is)f(a)g(stationary)e (measure)i(for)f(this)i(equation.)62 b(T)-7 b(o)36 b(pro)n(v)n(e)f (this)h(statemen)n(t)0 1390 y(w)n(e)f(need)g(to)g(kno)n(w)f(that)h FH(\026)901 1402 y FD(0)974 1390 y FI(is)g(supp)r(orted)f(b)n(y)h(a)g (certain)f(top)r(ological)g(space)g FF(K)q FI(,)j(where)e(\(10.8\))0 1489 y(de\034nes)h(a)g(measurable)f(dynamical)g(system.)62 b(Due)37 b(to)f(the)h(\034rst)e(relation)h(in)g(3\),)i FH(\026)2770 1501 y FD(0)2808 1489 y FG(\()p FH(H)2916 1459 y FD(2)2953 1489 y FG(\))f(=)g(1)p FI(.)0 1589 y(Unfortunately)-7 b(,)32 b(w)n(e)e(cannot)h(tak)n(e)f FF(K)g FG(=)e FH(H)1404 1559 y FD(2)1472 1589 y FI(since)i(it)i(is)e(unkno)n(wn)h(if)g(for)f FH(U)9 b FG(\(0\))29 b FF(2)g FH(H)2780 1559 y FD(2)2848 1589 y FI(the)i(Euler)0 1689 y(equation)25 b(has)f(a)h(unique)h (solution.)35 b(Instead)25 b(w)n(e)g(c)n(ho)r(ose)f(for)h FF(K)h FI(the)g(space,)f(formed)g(b)n(y)g(all)g(v)n(ectors)0 1788 y FH(u)p FG(\(0\))p FI(,)31 b(where)f FH(u)p FG(\()p FH(t)p FG(\))p FH(;)14 b(t)28 b FF(\025)f FG(0)p FI(,)k(is)f(a)g(con)n (tin)n(uous)f(in)i FH(H)1628 1758 y FD(1)1696 1788 y FI(solution)f(of)g(the)h(Euler)f(equation,)h(satisfying)0 1888 y(assertions)k(of)i(item)h(2\))f(of)g(Theorem)f(10.1.)65 b(This)37 b(is)g(a)g(w)n(ell-de\034ned)f(phase\025space,)i(suc)n(h)f (that)0 1987 y FH(H)76 1957 y FD(3)142 1987 y FF(\032)30 b(K)g(\032)g FH(H)501 1957 y FD(1)569 1987 y FI(\(see)h([Kuk04)o(],)i (Section)e(3.4\).)48 b(W)-7 b(e)32 b(do)f(not)g(kno)n(w)g(if)h(the)f (space)g FF(K)i FI(is)e(linear,)h(or)0 2087 y(not.)2981 b Fs(\003)0 2344 y FJ(Remark)41 b(10.4.)k FI(The)36 b(space)f FF(C)5 b FI(,)38 b(de\034ned)f(in)f(the)h(theorem's)e(pro)r(of,)j(is)e (compactly)g(em)n(b)r(edded)0 2444 y(in)h FH(C)171 2414 y FD(1)209 2444 y FG(\(0)p FH(;)14 b(n)p FG(;)g FH(H)483 2414 y FA(\024)525 2444 y FG(\))39 b(=:)g FH(Z)786 2414 y FA(\024)780 2465 y(n)866 2444 y FI(for)d(an)n(y)h FH(\024)h(<)h FG(2)p FI(.)65 b(If)38 b FH(B)1645 2456 y FD(2)1721 2444 y FH(<)g FF(1)p FI(,)i(then)e Fk(E)14 b FF(j)p FH(\020)45 b FF(j)2373 2459 y FD([0)p FA(;n)p FD(])2547 2444 y FF(j)2570 2456 y FA(Z)2619 2440 y Fu(\024)2615 2473 y(n)2701 2444 y FH(<)39 b FF(1)p FI(.)65 b(These)0 2544 y(observ)-5 b(ations)37 b(and)j(simple)f(analysis)f(of)h(the)h(theorem's)e(pro)r (of)h(sho)n(w)f(that)i(if)f FH(B)2692 2556 y FD(2)2772 2544 y FH(<)j FF(1)p FI(,)h(then)0 2643 y(for)34 b(an)n(y)g FH(\024)g(<)g FG(2)g FI(the)h(pro)r(cess)e FH(U)43 b FI(can)34 b(b)r(e)h(c)n(hosen)f(suc)n(h)g(that)g(the)h(con)n(v)n (ergence)d(\(10.7\))i(holds)g(in)0 2743 y FF(P)65 2676 y Fz(\000)103 2743 y FH(C)6 b FG(\(0)p FH(;)14 b FF(1)p FG(;)g FH(H)475 2713 y FA(\024)517 2743 y FG(\))549 2676 y Fz(\001)588 2743 y FI(.)36 b(Cho)r(osing)24 b FH(\024)e(>)h FG(1)i FI(and)g(using)g(\(1.2\))g(w)n(e)g(ac)n(hiev)n(e)e(that)j(a.e.) 35 b(realisation)23 b FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))0 2843 y FI(is)27 b(con)n(tin)n(uous)g(in)h FH(t)f FI(and)h FH(x)p FI(.)0 3125 y FJ(Remark)f(10.5.)36 b FI(By)23 b(the)i(item)f(3\))g(of)g(the)g(theorem,)g FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)1959 3137 y FD(1)2019 3125 y FF(\030)22 b(j)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(j)p FI(.)36 b(Therefore)23 b(the)h(Eulerian)0 3224 y(limit)38 b FH(U)46 b FI(is)37 b(mostly)g(supp)r(orted)f(b)n(y)h(lo)n (w)g(F)-7 b(ourier)36 b(mo)r(des.)65 b(F)-7 b(rom)37 b(other)f(hand,)k(w)n(e)d(kno)n(w)f(that)0 3324 y FF(j)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(j)23 b(2)h FH(H)384 3294 y FD(2)421 3324 y FI(,)h(and)e(our)h(attempts)g(to)g(impro)n(v)n (e)e(the)i(smo)r(othness)g(of)g FH(U)32 b FI(failed.)k(This)24 b(ma)n(y)f(indicate)0 3424 y(that)k(the)g(F)-7 b(ourier)25 b(co)r(e\036cien)n(ts)h(in)h FH(x)g FI(of)f FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))p FI(,)28 b FH(U)1641 3436 y FA(s)1676 3424 y FG(\()p FH(t)p FG(\))p FI(,)f(deca)n(y)f(with)h FF(j)p FH(s)p FF(j)f FI(algebraically)e(\(e.g.,)j(that)0 3523 y FF(j)p FH(U)80 3535 y FA(s)115 3523 y FG(\()p FH(t)p FG(\))p FF(j)d Fs(.)f FF(j)p FH(s)p FF(j)429 3493 y Fw(\000)p FD(2)518 3523 y FI(,)k(but)i FG(lim)14 b(sup)975 3544 y Fw(j)p FA(s)p Fw(j!1)1196 3523 y FF(j)p FH(U)1276 3535 y FA(s)1311 3523 y FF(jj)p FH(s)p FF(j)1419 3493 y FD(3+)p FA(\017)1558 3523 y FG(=)23 b FF(1)k FI(for)h FH(\017)22 b(>)h FG(0)p FI(\).)125 3750 y(Let)k(us)h(denote)f(b)n(y)g FH(K)34 b FI(the)28 b(set)f(of)h(all)f(Borel)f(measures)h FH(\026)g FI(in)h FH(H)2176 3720 y FD(2)2241 3750 y FI(suc)n(h)f(that) 125 3870 y(i\))208 3803 y Fz(R)247 3900 y FA(H)305 3883 y Ft(2)356 3778 y Fz(\020)406 3870 y FF(k)p FH(u)p FF(k)538 3840 y FD(2)538 3891 y(2)592 3870 y FG(+)18 b FH(e)714 3840 y FA(\033)r Fw(k)p FA(u)p Fw(k)861 3815 y Ft(2)861 3857 y(1)898 3778 y Fz(\021)961 3870 y FH(\026)p FG(\()p FH(du)p FG(\))24 b FH(<)e FF(1)p FI(,)125 4006 y(ii\))231 3939 y Fz(R)270 4036 y FA(H)328 4019 y Ft(2)379 4006 y FF(k)p FH(u)p FF(k)511 3976 y FD(2)511 4027 y(1)561 4006 y FH(\026)p FG(\()p FH(du)p FG(\))h(=)887 3973 y FD(1)p 887 3987 34 4 v 887 4035 a(2)930 4006 y FH(B)993 4018 y FD(0)1044 4006 y FH(;)1160 3973 y FD(1)p 1160 3987 V 1160 4035 a(2)1217 4006 y FH(B)1284 3976 y FD(2)1280 4027 y(0)1321 4006 y FH(B)1384 4018 y FD(1)1421 3970 y Fw(\000)p FD(1)1533 4006 y FF(\024)1621 3939 y Fz(R)1660 4036 y FA(H)1718 4019 y Ft(2)1770 4006 y FF(j)p FH(u)p FF(j)1864 3976 y FD(2)1914 4006 y FH(\026)p FG(\()p FH(du)p FG(\))h FF(\024)2240 3973 y FD(1)p 2240 3987 V 2240 4035 a(2)2283 4006 y FH(B)2346 4018 y FD(0)2384 4006 y FI(,)125 4106 y(iii\))55 b FH(\026)p FG(\()p FF(K)21 b(\\)d FH(H)596 4076 y FD(2)633 4106 y FG(\))24 b(=)f(1)k FI(\()p FF(K)i FI(is)f(de\034ned)g(in)f(Remark)g(10.3\),)g(and)g FH(\026)h FI(is)f(an)h(in)n(v)-5 b(arian)n(t)26 b(measure)h(for)0 4205 y(the)h(Euler)g(equation.)0 4371 y FJ(Corollary)36 b(10.6.)41 b FC(The)33 b(stationary)g(me)l(asur)l(es)f FH(\026)1644 4383 y FA(\027)1717 4371 y FC(c)l(onver)l(ge)h(to)f FH(K)37 b FC(in)c(the)f(Lipschitz\025dual)i(dis-)0 4471 y(tanc)l(e:)1182 4571 y FG(lim)1171 4622 y FA(\027)t Fw(!)p FD(0)1340 4571 y FG(inf)1321 4624 y FA(\032)p Fw(2)p FA(K)1474 4571 y FF(k)p FH(\026)1566 4583 y FA(\027)1626 4571 y FF(\000)18 b FH(\032)p FF(k)1794 4536 y Fw(\003)1794 4591 y FA(L)1866 4571 y FG(=)k(0)p FH(:)1553 5059 y FI(76)p eop end PStoPSsaved restore %%Page: (76,77) 39 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 77 76 bop 125 523 a FI(W)-7 b(e)31 b(stress)f(that)h(the) g(set)g FH(K)36 b FI(is)31 b(a)g(pre-compact)e(subset)i(of)g FF(P)7 b FG(\()p FH(H)2238 493 y FD(1)2275 523 y FG(\))31 b FI(of)g(in\034nite)g(co)r(dimension,)0 623 y(dep)r(ending)c(only)f (on)h(t)n(w)n(o)f(c)n(haracteristics)e(of)j(the)g(force)f FH(\021)s FI(,)h(whic)n(h)g(under)g(the)g(double)f(limit)i(\020time)0 722 y(to)k(in\034nit)n(y)-7 b(,)33 b(viscosit)n(y)e(to)g(zero")g (attracts)g(distribution)h(of)f(ev)n(ery)g(solution)g(of)h(the)g(NS)h (equation)0 822 y(\(10.1\))o(.)272 792 y Fo(11)342 822 y FI(.)0 1054 y Fy(10.2)112 b(One)38 b(negativ)m(e)f(result)0 1208 y FI(T)-7 b(o)39 b(study)h(further)g(prop)r(erties)f(of)g(the)h (Eulerian)g(limit)g FH(U)9 b FG(\()p FH(t;)14 b(x)p FG(\))41 b FI(and)e(the)i(limiting)f(stationary)0 1307 y(measure)c FH(\026)383 1319 y FD(0)458 1307 y FG(=)i FF(D)r FH(U)9 b FG(\()p FH(t)p FG(\))38 b FI(is)e(a)h(di\036cult)g(task.)64 b(A)37 b(natural)f(idea)g(to)h(resolv)n(e)e(it)i(is)g(\034rst)f(to)h (guess)0 1407 y(the)31 b(righ)n(t)f(answ)n(er)g(and)g(next)i(to)e(pro)n (v)n(e)g(that)h(the)g(guess)f(is)h(correct.)45 b(In)31 b(particular,)g(O.)f(P)n(enrose)0 1506 y(and)c(A.)h(Shnirelman)f (explained)g(us)g(that)h(some)e(ph)n(ysical)h(insigh)n(ts)f(on)h(the)h (2D\025turbulence)f(mak)n(e)0 1606 y(it)32 b(plausible)g(that)h(the)f (random)f(\034eld)h FH(U)41 b FI(is)32 b(a.s.)50 b(time\025indep)r (enden)n(t)33 b(\(that)f(is,)h(the)g(measure)e FH(\026)3152 1618 y FD(0)0 1706 y FI(is)36 b(supp)r(orted)f(b)n(y)g(time\025indep)r (enden)n(t)i(solutions)d(of)i(the)g(Euler)g(equation\).)61 b(W)-7 b(e)36 b(cannot)f(pro)n(v)n(e)0 1805 y(or)j(dispro)n(v)n(e)f (this)i(conjecture.)70 b(Belo)n(w)38 b(w)n(e)g(discuss)h(another)f (guess)f(concerning)h(the)h(measure)0 1905 y FH(\026)50 1917 y FD(0)87 1905 y FI(,)29 b(coming)e(from)h(the)g(statistical)g(ph) n(ysics,)f(whic)n(h)h(iden)n(ti\034es)g FH(\026)2092 1917 y FD(0)2158 1905 y FI(with)g(a)g(certain)g(w)n(eigh)n(ted)f(sum)0 2005 y(of)32 b(micro)r(canonical)f(measures)g(\(see)i(b)r(elo)n(w)f (for)g(the)h(relev)-5 b(an)n(t)31 b(de\034nitions\).)52 b(Imp)r(ortance)32 b(of)h(the)0 2104 y(canonical)27 b(and)h(micro)r (canonical)e(measures)g(for)i(the)g(statistical)g(h)n(ydro)r(dynamics)e (\(including)j(the)0 2204 y(theory)h(of)h(turbulence\))h(is)f(a)f(p)r (opular)g(ph)n(ysical)h(idea,)g(e.g.)47 b(see)31 b([KM80)n(],)h([F)-7 b(ri95)o(],)32 b(Section)f(9.7.2)0 2303 y(and)c([Gal01)o(].)125 2403 y(T)-7 b(o)24 b(explain)g(ho)n(w)g(the)h(idea)f(of)h(the)g(micro)r (canonical)e(ensem)n(ble)h(en)n(ters)g(the)h(problem)f(whic)n(h)g(w)n (e)0 2503 y(discuss,)k(let)h(us)g(consider)f(the)h(Galerkin)f(appro)n (ximation)e(to)j(the)g(equation)f(\(10.1\))o(.)40 b(That)29 b(is,)g(the)0 2602 y(equation)f(\(4.13\))g(with)i(the)f(op)r(erator)e FH(L)i FI(replaced)f(b)n(y)g FH(\027)5 b(L)p FI(,)30 b(and)e(with)i(the)f(factor)2675 2542 y FF(p)p 2744 2542 47 4 v 60 x FH(\027)34 b FI(in)c(fron)n(t)e(of)0 2702 y(the)k(force)e(in)i(the)g(r.h.s.)14 b(.)48 b(Theorems)30 b(7.1)h(and)g(10.1)f(apply)h(to)h(the)f(equation)g(`uniformly)g(in)h FH(N)9 b FI('.)0 2802 y(It)26 b(means)g(that)g(in)g(Theorem)f(7.1)g (the)i(constan)n(t)e FH(C)32 b FI(and)26 b(the)g(exp)r(onen)n(t)g FH(\024)g FI(are)f(indep)r(enden)n(t)h(of)g FH(N)9 b FI(,)0 2901 y(while)31 b(in)h(Theorem)e(10.1)g(the)i(constan)n(ts)e(in) i(the)f(estimates)g(for)g(the)h(limiting)f(pro)r(cess)f FH(U)38 b FG(=)29 b FH(U)3126 2871 y FA(N)0 3001 y FI(ma)n(y)e(b)r(e)h (c)n(hosen)e(indep)r(enden)n(t)j(of)e FH(N)9 b FI(.)37 b(The)27 b(pro)r(cess)g FH(U)1785 2971 y FA(N)1875 3001 y FI(satis\034es)g(the)g(Galerkin)g(appro)n(ximation)0 3100 y(to)g(the)h(Euler)g(equation:)1094 3200 y FG(_)-37 b FH(u)18 b FG(+)g(\005)1291 3212 y FA(N)1354 3200 y FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g(0)p FH(;)96 b(u)23 b FF(2)g FH(H)2023 3212 y FA(N)2086 3200 y FH(:)826 b FI(\(10.19\))0 3349 y(This)33 b(is)g(a)f(\034nite\025dimensional)g (Hamiltonian)h(system)f(with)i(t)n(w)n(o)e(in)n(tegrals)f(of)i(motion:) 47 b(the)33 b(en-)0 3449 y(ergy)i FH(E)5 b FG(\()p FH(u)p FG(\))39 b(=)520 3416 y FD(1)p 520 3430 34 4 v 520 3478 a(2)563 3449 y FF(j)p FH(u)p FF(j)657 3419 y FD(2)731 3449 y FI(and)d(the)h(enstroph)n(y)f FG(\012\()p FH(u)p FG(\))i(=)1768 3416 y FD(1)p 1768 3430 V 1768 3478 a(2)1811 3449 y FF(k)p FH(u)p FF(k)1943 3419 y FD(2)1943 3470 y(1)1979 3449 y FI(,)h(see)d([AK01)o(].)64 b(The)37 b(former)e(is)i (the)0 3549 y(hamiltonian)27 b(of)h(the)g(system)f(and)g(the)h(latter)f (is)h(its)g(additional)f(in)n(tegral)f(of)h(motion.)2806 3519 y Fo(12)0 3715 y FJ(Exercise)38 b(10.7.)44 b FI(F)-7 b(or)33 b FH(H)851 3727 y FA(N)947 3715 y FF(3)g FH(u)p FG(\()p FH(t)p FG(\))h(=)1308 3653 y Fz(P)1396 3740 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1595 3715 y FH(u)1643 3727 y FA(s)1678 3715 y FG(\()p FH(t)p FG(\))h FI(write)e(\(10.19\))g(as)g (an)g(explicit)h(system)f(of)0 3825 y(di\033eren)n(tial)c(equations)f (for)h FF(f)p FH(u)1009 3837 y FA(s)1043 3825 y FG(\()p FH(t)p FG(\))p FF(g)p FI(.)42 b(Pro)n(v)n(e)28 b(that)i(the)f(\035o)n (w\025maps)f(of)h(this)g(system)g(preserv)n(e)f(the)0 3924 y(Euclidean)g(v)n(olume)f(in)h FH(H)836 3936 y FA(N)899 3924 y FI(.)959 3894 y Fo(13)125 4090 y FI(In)i(statistical)f(ph)n (ysics)h(the)g(in)n(v)-5 b(arian)n(t)29 b(measure)g(V)-7 b(ol)14 b FG(=)1950 4028 y Fz(Q)2029 4115 y Fw(j)p FA(s)p Fw(j\024)p FA(N)2228 4090 y FH(du)2319 4102 y FA(s)2384 4090 y FI(is)31 b(called)e(the)i FC(c)l(anonic)l(al)p 0 4183 1276 4 v 62 4237 a Fn(11)127 4260 y Fm(If)22 b Fi(b)226 4268 y Ff(s)279 4260 y Fh(\021)d Fi(b)383 4269 y Fb(\000)p Ff(s)486 4260 y Fm(and)j Fi(B)674 4269 y Fc(6)729 4260 y Fi(<)d Fh(1)p Fm(,)i(then)i(the)g(corollary)g(remains)e (true)i(if)e(the)h(set)h Fi(K)i Fm(is)d(mo)r(di\034ed)f(\(i.e.,)g (decreased\))j(to)0 4339 y(satisfy)g(\(10.11\))62 4395 y Fn(12)127 4418 y Fm(Apart)h(from)e Fi(E)k Fm(and)e Fg(\012)f Fm(the)g(Euler)h(equation)g(has)f(in\034nitely)h(man)n(y)e (in)n(tegrals)i(\(10.9\))q(,)f(but)g(non)h(of)f(them)f(is)h(an)0 4497 y(in)n(tegral)j(of)e(motion)g(for)h(\(10.19\))h(\(apart)h(from)d (\(10.9\))q(,)h(where)h Fi(g)g Fm(is)e(an)h(a\036ne)g(function,)h(and)f (whic)n(h)g(equals)g Fg(\012)g Fm(up)0 4576 y(to)e(an)g(a\036ne)g (transformation\),)h(cf.)31 b(Exercise)25 b(2.12.)62 4631 y Fn(13)127 4655 y Fm(The)k(system)f(\(10.19\))i(is)e(Hamiltonian) g(with)g(resp)r(ect)j(to)e(the)g(symplectic)g(structure,)j(giv)n(en)d (b)n(y)g(a)g(t)n(w)n(o-form)0 4734 y(with)24 b(v)l(ariable)g(co)r (e\036cien)n(ts.)33 b(So)24 b(this)f(assertion)i(is)e(not)i(ob)n (vious.)1553 5059 y FI(77)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 78 77 bop 0 523 a FC(me)l(asur)l(e)p FI(.)37 b(F)-7 b(or)27 b(an)n(y)f FG(0)d FH(<)g(a)f(<)h(b)k FI(the)h(surface) 895 703 y FH(S)951 669 y FA(N)946 723 y(a;b)1058 703 y FG(=)22 b FF(f)p FH(u)g FF(2)i FH(H)1405 715 y FA(N)1491 703 y FF(j)f FH(E)5 b FG(\()p FH(u)p FG(\))23 b(=)g FH(a;)41 b FG(\012\()p FH(u)p FG(\))23 b(=)g FH(b)p FF(g)0 883 y FI(is)k(in)n(v)-5 b(arian)n(t)27 b(for)g(the)h(system)f(\(10.19\))o (.)37 b(Therefore)26 b(the)i(normalised)e FH(\016)s FI(-measure)g(of)i (this)g(surface)1158 1104 y FH(\016)1198 1070 y FA(N)1195 1125 y(a;b)1307 1104 y FG(=)1395 1012 y Fz(\020)1445 1104 y FH(Z)1508 1070 y FA(N)1502 1125 y(a;b)1590 1012 y Fz(\021)1640 1029 y Fw(\000)p FD(1)1729 1104 y FI(V)p FH(ol)r FF(j)1881 1124 y FA(S)1925 1104 y Fu(N)1922 1144 y(a;b)2008 1104 y FH(;)0 1316 y FI(is)j(in)n(v)-5 b(arian)n(t)29 b(for)i(the)g(\035o)n(w-maps)e(\(here)i FH(Z)1387 1286 y FA(N)1381 1340 y(a;b)1501 1316 y FI(stands)f(for)g(the)i(co)r (dimension\025t)n(w)n(o)d(v)n(olume)h(of)h(the)0 1429 y(surface\).)54 b(The)33 b(measure)f FH(\016)907 1399 y FA(N)904 1453 y(a;b)1027 1429 y FI(is)h(called)g(the)h FC(micr)l(o)l(c)l(anonic)l(al)i(me)l(asur)l(e)f(for)h(a)f(system)g (with)h(the)0 1540 y(two)31 b(ener)l(gies)p FI(,)e FH(E)34 b FI(and)29 b FG(\012)p FI(.)40 b(Let)28 b(us)h(denote)f FG(\006)1463 1510 y FD(2)1525 1540 y FG(=)d FF(f)p FG(\()p FH(a;)14 b(b)p FG(\))24 b FF(j)h FG(0)f FH(<)h(a)f(<)h(b)p FF(g)p FI(.)39 b(F)-7 b(or)28 b(an)n(y)g(Borel)f(measure)0 1640 y FH(\032)c FF(2)g(P)7 b FG(\(\006)301 1610 y FD(2)338 1640 y FG(\))28 b FI(w)n(e)g(set)1198 1782 y FH(\026)1248 1794 y FA(\032)1309 1782 y FG(=)1397 1669 y Fz(Z)1443 1857 y FD(\006)1490 1841 y Ft(2)1541 1782 y FH(\016)1581 1748 y FA(N)1578 1802 y(a;b)1681 1782 y FH(\032)p FG(\()p FH(da)14 b(db)p FG(\))p FH(:)944 b FI(\(10.20\))0 1976 y(The)29 b(measures)f FH(\026)580 1988 y FA(\032)647 1976 y FI(are)g(in)n(v)-5 b(arian)n(t)28 b(for)g(\(10.19\))o(.)41 b(F)-7 b(rom)28 b(the)i(p)r(oin)n(t)f(of)g(view)g(of)f(statistical)h (ph)n(ysics)0 2076 y(they)f(are)e(the)i(most)g(natural)e(in)n(v)-5 b(arian)n(t)27 b(measures.)35 b(So)28 b(it)g(is)f(natural)g(to)g(ask)g (the)h(follo)n(wing)125 2175 y FJ(Question.)34 b FI(Can)28 b(the)g(measure)e FH(\026)1250 2145 y FA(N)1250 2196 y FD(0)1336 2175 y FG(=)d FF(D)r FH(U)1556 2145 y FA(N)1619 2175 y FG(\()p FH(t)p FG(\))28 b FI(b)r(e)g(represen)n(ted)e(in)i(the)g (form)f(\(10.20\))o(?)125 2324 y(This)19 b(question)h(b)r(ecomes)f(in)n (teresting)g(when)h FH(N)32 b FF(\035)23 b FG(1)p FI(,)e(i.e.)34 b(when)20 b(the)g(Galerkin)f(system)g(\(10.19\))0 2423 y(appro)n(ximates)27 b(w)n(ell)i(the)g(Euler)g(equation.)40 b(F)-7 b(or)29 b(suc)n(h)f FH(N)38 b FI(the)29 b(answ)n(er)f(is)h (negativ)n(e,)f(if)h(w)n(e)g(assume)0 2523 y(some)20 b(natural)f(additional)h(regularit)n(y)e(of)i(the)h(measures)e FH(\032)1855 2535 y FA(N)1918 2523 y FI(.)34 b(Namely)-7 b(,)22 b(that)e FH(\032)2505 2535 y FA(N)2591 2523 y FG(=)j FH(\036)2728 2535 y FA(N)2791 2523 y FG(\()p FH(a;)14 b(b)p FG(\))g FH(da)g(db)p FI(,)0 2623 y(where)29 b FF(j)p FH(\036)314 2635 y FA(N)378 2623 y FF(j)e(\024)f FH(C)37 b FI(and)29 b(the)i(constan)n(t)e FH(C)36 b FI(is)30 b(indep)r(enden)n(t)g(of)g FH(N)9 b FI(.)2154 2592 y Fo(14)2268 2623 y FI(Indeed,)31 b(let)f FH(\026)2736 2592 y FA(N)2736 2643 y FD(0)2826 2623 y FG(=)d FH(\026)2968 2635 y FA(\032)3002 2643 y Fu(N)3090 2623 y FI(for)0 2722 y(all)g FH(N)37 b FI(large)26 b(enough.)36 b(By)27 b(the)h(recen)n(t)f(results)g(of)h(A.)g(Biriuk)f([Bir06)n(],)957 2831 y Fz(Z)1054 2944 y FF(k)p FH(u)p FF(k)1186 2910 y FD(2)1186 2965 y(2)1236 2944 y FH(\016)1276 2910 y FA(N)1273 2965 y(a;b)1362 2944 y FG(\()p FH(du)p FG(\))c FF(!)g(1)83 b FI(a)p FH(s)28 b(N)j FF(!)23 b(1)p FH(;)704 b FI(\(10.21\))0 3172 y(for)27 b(eac)n(h)g FG(\()p FH(a;)14 b(b)p FG(\))23 b FF(2)g FG(\006)656 3142 y FD(2)693 3172 y FI(,)28 b(uniformly)g(on)f(compact)g(subsets)g(of)h FG(\006)2012 3142 y FD(2)2049 3172 y FI(.)37 b(Using)28 b(\(10.10\))f(w)n(e)g(get)g(that)811 3338 y FG(1)p 811 3375 42 4 v 811 3451 a(4)862 3394 y FH(B)925 3406 y FD(0)986 3394 y FG(=)1073 3281 y Fz(Z)1180 3338 y FG(1)p 1180 3375 V 1180 3451 a(2)1232 3394 y FF(k)p FH(u)p FF(k)1364 3360 y FD(2)1364 3415 y(1)1413 3394 y FH(\026)1463 3406 y FA(\032)1497 3414 y Fu(N)1555 3394 y FG(\()p FH(du)p FG(\))d(=)1821 3281 y Fz(Z)1867 3470 y FD(\006)1914 3453 y Ft(2)1965 3394 y FH(b)14 b(\032)2058 3406 y FA(N)2121 3394 y FG(\()p FH(da)g(db)p FG(\))p FH(:)0 3615 y FI(This)28 b(relation)e(and)h(the)h(regularit)n(y)e(assumption)h(imply)h(that)689 3830 y FH(\032)732 3842 y FA(N)795 3762 y Fz(\000)833 3830 y FF(f)p FG(\()p FH(a;)14 b(b)p FG(\))22 b FF(2)i FG(\006)1217 3795 y FD(2)1277 3830 y FF(j)f FH(\015)1371 3795 y Fw(\000)p FD(1)1483 3830 y FF(\025)g FH(b)f FF(\025)h FH(a)18 b FG(+)g FH(\015)5 b FF(g)1952 3762 y Fz(\001)2012 3830 y FF(\025)2110 3773 y FG(1)p 2110 3811 V 2110 3887 a(2)2328 3830 y FF(8)14 b FH(N)22 b(;)0 4037 y FI(for)27 b(a)g(suitable)h FH(\015)f(>)c FG(0)p FI(.)36 b(No)n(w)29 b(\(10.21\))d(implies)i(that)1182 4147 y Fz(Z)1279 4260 y FF(k)p FH(u)p FF(k)1411 4225 y FD(2)1411 4280 y(2)1461 4260 y FH(\026)1511 4272 y FA(\032)1545 4280 y Fu(N)1603 4260 y FG(\()p FH(du)p FG(\))23 b FF(!)g(1)14 b FH(;)0 4482 y FI(in)30 b(con)n(tradiction)f(with)h(the)g(\034rst)g(estimate)g (in)g(the)g(item)g(3\))g(of)g(Theorem)f(10.1)g(\(more)g(precisely)-7 b(,)0 4581 y(with)28 b(an)f(analogy)f(of)i(this)f(estimate)h(for)f(the) h(system)f(\(10.19\))o(\).)p 0 4654 1276 4 v 62 4708 a Fn(14)127 4731 y Fm(It)j(su\036ce)g(to)h(assume)e(that)h(for)g(some)f Fi(\015)34 b(>)c Fg(0)f Fm(the)i Fi(\032)1608 4742 y Ff(N)1666 4731 y Fm(-measure)e(of)h(the)g(set)g Fh(f)p Fg(\()p Fi(a;)13 b(b)p Fg(\))30 b Fh(2)g Fg(\006)2647 4708 y Fc(2)2711 4731 y Fh(j)f Fi(b)h Fh(\025)g Fi(a)20 b Fg(+)f Fi(\015)t Fh(g)31 b Fm(is)0 4810 y Fh(\025)19 b Fi(c)h(>)f Fg(0)24 b Fm(for)g(all)f Fi(N)7 b Fm(.)1553 5059 y FI(78)p eop end PStoPSsaved restore %%Page: (78,79) 40 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 79 78 bop 0 523 a Fy(10.3)112 b(Other)37 b(scalings)0 676 y FI(No)n(w)27 b(let)h(us)f(scale)g(the)h(2D)g(NSE)g (di\033eren)n(tly)-7 b(,)28 b(and)f(consider)g(the)h(equation)1084 859 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(\027)1806 825 y FA(a)1860 859 y FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)816 b FI(\(10.22\))0 1042 y(where)26 b(the)i(random)e (force)g FH(\021)31 b FI(is)c(as)f(in)h(\(10.1\))f(and)h FH(a)c FF(6)p FG(=)g(1)p FH(=)p FG(2)p FI(.)35 b(Let)27 b FH(\026)2175 1054 y FA(\027)2244 1042 y FI(b)r(e)g(the)h(stationary)d (measure)0 1141 y(for)i(this)h(equation,)f(and)g FH(u)860 1153 y FA(\027)901 1141 y FG(\()p FH(t)p FG(\))h FI(b)r(e)g(the)g (corresp)r(onding)e(stationary)g(solution.)0 1241 y(Applying)g(Ito's)g (form)n(ula)e(to)i(the)g(functions)h FF(j)p FH(u)p FF(j)e FI(and)h FF(k)p FH(u)p FF(k)1855 1211 y FD(2)1855 1261 y(1)1917 1241 y FI(as)f(at)h(the)g(b)r(eginning)g(of)g(Section)f(10.1,) 0 1340 y(w)n(e)i(see)g(that)549 1523 y Fk(E)14 b FF(k)p FH(u)716 1535 y FA(\027)756 1523 y FG(\()p FH(t)p FG(\))p FF(k)892 1489 y FD(2)892 1544 y(1)952 1523 y FG(=)23 b FH(\027)1086 1489 y FD(2)p FA(a)p Fw(\000)p FD(1)1268 1490 y(1)p 1268 1504 34 4 v 1268 1552 a(2)1325 1523 y FH(B)1388 1535 y FD(0)1425 1523 y FH(;)106 b Fk(E)14 b FF(k)p FH(u)1721 1535 y FA(\027)1761 1523 y FG(\()p FH(t)p FG(\))p FF(k)1897 1489 y FD(2)1897 1544 y(2)1957 1523 y FG(=)23 b FH(\027)2091 1489 y FD(2)p FA(a)p Fw(\000)p FD(1)2273 1490 y(1)p 2273 1504 V 2273 1552 a(2)2330 1523 y FH(B)2393 1535 y FD(1)2514 1523 y FF(8)14 b FH(t)g(:)294 b FI(\(10.23\))0 1706 y FJ(Theorem)30 b(10.8.)40 b FI(1\))30 b FC(If)g FH(a)23 b(>)998 1673 y FD(1)p 998 1687 V 998 1734 a(2)1041 1706 y FC(,)30 b(then)g FG(lim)14 b FH(\026)1460 1718 y FA(\027)1493 1726 y Fu(j)1552 1706 y FG(=)22 b FH(\016)1676 1718 y FD(0)1713 1706 y FC(.)0 1805 y FI(2\))29 b FC(If)i FH(a)23 b(<)355 1773 y FD(1)p 355 1787 V 355 1834 a(2)398 1805 y FC(,)30 b(then)g FG(lim)14 b FH(\026)817 1817 y FA(\027)850 1825 y Fu(j)915 1805 y FC(do)l(es)31 b(not)e(exist)g(in)h FF(P)7 b FG(\()p FH(H)g FG(\))p FC(.)0 1971 y(Pr)l(o)l(of.)43 b FI(If)28 b FH(')g FI(is)g(a)f(b)r (ounded)h(Lipsc)n(hitz)f(functional)h(on)f FH(H)34 b FI(suc)n(h)28 b(that)f(Lip)p FG(\()p FH(')p FG(\))d FF(\024)f FG(1)p FI(,)k(then)646 2107 y Fz(\014)646 2156 y(\014)673 2177 y Fk(E)750 2110 y Fz(\000)788 2177 y FH(')p FG(\()p FH(u)922 2189 y FA(\027)963 2177 y FG(\(0\)\))19 b FF(\000)f FH(')p FG(\(0\))1363 2110 y Fz(\001)1402 2107 y(\014)1402 2156 y(\014)1452 2177 y FF(\024)23 b Fk(E)14 b FF(j)p FH(u)1688 2189 y FA(\027)1728 2177 y FG(\(0\))p FF(j)24 b(\024)1968 2110 y Fz(\000)2006 2177 y FH(\027)2052 2143 y FD(2)p FA(a)p Fw(\000)p FD(1)2235 2144 y(1)p 2235 2158 V 2235 2206 a(2)2301 2177 y FH(B)2364 2189 y FD(0)2401 2110 y Fz(\001)2439 2127 y FD(1)p FA(=)p FD(2)0 2360 y FI(b)n(y)k(\(10.23\))o(.)37 b(Therefore,)1127 2459 y FF(D)16 b FH(u)1255 2471 y FA(\027)1296 2459 y FG(\(0\))23 b FH(*)g(\016)1568 2471 y FD(0)1651 2459 y FI(if)47 b FH(a)23 b(>)g FG(1)p FH(=)p FG(2)14 b FH(;)871 b FI(\(10.24\))0 2609 y(and)27 b(the)h(\034rst)g(assertion)e(follo)n(ws.)125 2708 y(T)-7 b(o)27 b(pro)n(v)n(e)f(the)i(second,)f(w)n(e)g(mak)n(e)f (in)i(\(10.22\))f(the)h(follo)n(wing)e(substitution:)1126 2891 y FH(u)p FG(\()p FH(t)p FG(\))d(=)f FH(\027)1424 2857 y FA(b)1472 2891 y FH(v)s FG(\()p FH(\034)9 b FG(\))p FH(;)61 b(t)24 b FG(=)e FH(\027)1895 2857 y Fw(\000)p FA(b)1981 2891 y FH(\034)h(:)872 b FI(\(10.25\))0 3117 y(Then)227 3084 y FA(@)t(u)p 227 3098 79 4 v 234 3146 a(@)t(t)338 3117 y FG(=)23 b FH(\027)472 3087 y FA(b)519 3117 y FH(v)562 3087 y Fw(0)559 3138 y FA(\034)601 3117 y FI(,)28 b FH(\021)e FG(=)831 3061 y FH(d)p 816 3098 74 4 v 816 3174 a(dt)923 3117 y(\020)6 b FG(\()p FH(t)p FG(\))24 b(=)e FH(\027)1216 3083 y FA(b)1297 3061 y FH(d)p 1274 3098 89 4 v 1274 3174 a(d\034)1396 3117 y(\020)6 b FG(\()p FH(\027)1516 3083 y Fw(\000)p FA(b)1602 3117 y FH(\034)j FG(\))p FI(,)29 b(and)e(\(10.22\))f(reads)h(as)888 3379 y FH(v)931 3345 y Fw(0)928 3400 y FA(\034)988 3379 y FG(+)18 b FH(\027)1117 3345 y FD(1)p Fw(\000)p FA(b)1235 3379 y FH(Lv)k FG(+)c FH(B)t FG(\()p FH(v)s FG(\))24 b(=)e FH(\027)1768 3345 y FA(a)p Fw(\000)1866 3323 y Ft(3)p 1866 3332 29 3 v 1866 3365 a(2)1923 3345 y FA(b)2013 3323 y FH(d)p 1990 3360 89 4 v 1990 3436 a(d\034)2117 3357 y Fz(b)2112 3379 y FH(\020)6 b FG(\()p FH(\034)j FG(\))14 b FH(;)0 3609 y FI(where)242 3587 y Fz(b)237 3609 y FH(\020)6 b FG(\()p FH(\034)j FG(\))24 b(=)f FH(\027)546 3579 y FA(b=)p FD(2)661 3609 y FH(\020)6 b FG(\()p FH(\027)781 3579 y Fw(\000)p FA(b)867 3609 y FH(\034)j FG(\))p FI(.)37 b(Clearly)23 b FF(D)1355 3542 y Fz(\000)1398 3587 y(b)1393 3609 y FH(\020)6 b FG(\()p FF(\001)p FG(\))1522 3542 y Fz(\001)1584 3609 y FG(=)23 b FF(D)1738 3542 y Fz(\000)1776 3609 y FH(\020)6 b FG(\()p FF(\001)p FG(\))1905 3542 y Fz(\001)1944 3609 y FI(.)36 b(Cho)r(osing)23 b FH(b)g FG(=)g FH(a)12 b FF(\000)2650 3577 y FD(1)p 2650 3591 34 4 v 2650 3638 a(2)2717 3609 y FI(and)24 b(denoting)2 3726 y Fz(e)-48 b FH(\027)28 b FG(=)23 b FH(\027)203 3696 y FD(1)p Fw(\000)p FA(b)345 3726 y FG(=)f FH(\027)489 3673 y Ft(3)p 489 3682 29 3 v 489 3716 a(2)527 3696 y Fw(\000)p FA(a)619 3726 y FI(,)27 b(w)n(e)f(see)g(that)g FH(\027)1147 3696 y FA(a)p Fw(\000)1245 3673 y Ft(3)p 1245 3682 V 1245 3716 a(2)1295 3696 y FA(b)1351 3726 y FG(=)f Fz(e)-48 b FH(\027)1495 3673 y Ft(1)p 1495 3682 V 1495 3716 a(2)1538 3726 y FI(.)36 b(Therefore)25 b FH(v)s FG(\()p FH(\034)9 b FG(\))28 b FI(is)f(a)f(w)n(eak)f(stationary) g(solution)0 3825 y(of)j(the)f(equation)1042 3925 y FH(v)1085 3891 y Fw(0)1082 3946 y FA(\034)1142 3925 y FG(+)c(~)-47 b FH(\027)5 b(Lv)21 b FG(+)d FH(B)t FG(\()p FH(v)s FG(\))24 b(=)1758 3851 y FF(p)p 1827 3851 47 4 v 1832 3925 a FG(~)-47 b FH(\027)19 b(\021)s FG(\()p FH(t;)14 b(x)p FG(\))g FH(:)789 b FI(\(10.26\))0 4074 y(No)n(w)23 b(let)h(us)f(assume)g(that)h (the)g(sequence)f FH(\026)1391 4086 y FA(\027)1424 4094 y Fu(j)1482 4074 y FG(=)g FF(D)16 b FH(u)1698 4086 y FA(\027)1731 4094 y Fu(j)1766 4074 y FG(\(0\))24 b FI(con)n(v)n(erges)d (to)i(a)g(limit)i(in)e FF(P)7 b FG(\()p FH(H)g FG(\))p FI(.)36 b(Then,)0 4174 y(b)n(y)27 b(Prokhoro)n(v's)e(criterion,)1167 4274 y Fk(P)1232 4206 y Fz(\000)1271 4274 y FF(j)p FH(u)1342 4286 y FA(\027)1375 4294 y Fu(j)1410 4274 y FG(\(0\))p FF(j)e FH(>)g(R)1714 4206 y Fz(\001)1799 4274 y FF(\000)-14 b(!)1774 4327 y FA(R)p Fw(!1)1980 4274 y FG(0)0 4467 y FI(uniformly)37 b(in)h FH(j)5 b FI(.)66 b(Since)37 b FH(u)897 4479 y FA(\027)930 4487 y Fu(j)965 4467 y FG(\(0\))j(=)f FH(\027)1261 4437 y FA(a)p Fw(\000)p FD(1)p FA(=)p FD(2)1467 4467 y FH(v)1507 4479 y FA(\027)1540 4487 y Fu(j)1576 4467 y FG(\(0\))e FI(b)n(y)h(\(10.25\),)g(then)g(the)g (con)n(v)n(ergence)c(ab)r(o)n(v)n(e)0 4567 y(implies)c(that)g Fk(P)p FG(\()p FF(j)p FH(v)626 4579 y FA(\027)659 4587 y Fu(j)696 4567 y FG(\(0\))p FF(j)d FH(>)g(")p FG(\))g FF(!)g FG(0)j FI(as)f FH(\027)1369 4579 y FA(j)1431 4567 y FF(!)e FG(0)p FI(,)j(for)g(eac)n(h)f FH(")e(>)g FG(0)p FI(.)43 b(So)30 b FF(D)r FH(v)2445 4579 y FA(\027)2478 4587 y Fu(j)2514 4567 y FG(\(0\))d FH(*)h(\016)2795 4579 y FD(0)2862 4567 y FI(in)i FF(P)7 b FG(\()p FH(H)g FG(\))p FI(.)0 4667 y(But)28 b(b)n(y)f(Theorem)g(10.1,)f(along)h(a)g(suitable)h (subsequence)f(the)h(measures)e FF(D)r FH(v)2504 4679 y FA(\027)2537 4687 y Fu(j)2573 4667 y FG(\(0\))i FI(con)n(v)n(erge)d (to)i(a)0 4766 y(non-zero)f(limit.)37 b(Con)n(tradiction.)p 3128 4766 4 57 v 3132 4714 50 4 v 3132 4766 V 3182 4766 4 57 v 1553 5059 a(79)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 80 79 bop 0 523 a Fy(10.4)112 b(Discussion)0 676 y FI(Consider)27 b(the)g(NSE)i(\(7.1\),)e(where)g FG(0)c FH(<)g(\027)28 b FF(\034)23 b FG(1)p FI(.)36 b(Assume)28 b(that)1468 841 y FH(B)1531 853 y FD(1)p 1468 878 101 4 v 1468 954 a FH(B)1531 966 y FD(0)1602 897 y FF(\030)22 b FG(1)1204 b FI(\(10.27\))0 1116 y(\(that)35 b(is,)i(the)e(force)f(is) h(essen)n(tially)e(supp)r(orted)i(b)n(y)f(lo)n(w)g(mo)r(des\).)59 b(Let)35 b FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))35 b FI(b)r(e)g(a)g (stationary)0 1215 y(solution)27 b(of)h(the)g(equation)e(and)i(assume)f (that)g(its)h(energy)f(is)g(of)h(order)e(one:)1363 1394 y Fk(E)p FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)1614 1359 y FD(2)1674 1394 y FF(\030)d FG(1)p FH(:)1108 b FI(\(10.28\))0 1572 y(Let)33 b(us)g(\034nd)h(a)f(real)f(n)n(um)n(b)r (er)h FH(a)g FI(suc)n(h)g(that)g FH(B)1506 1584 y FD(0)1575 1572 y FG(=)f FH(\027)1718 1542 y FD(2)p FA(a)1792 1572 y FI(,)j(and)e(set)g FH(\021)i FG(=)d FH(\027)2371 1542 y Fw(\000)p FA(a)2486 1539 y(d)p 2473 1553 60 4 v 2473 1601 a(dt)2543 1572 y FH(\020)6 b FI(.)54 b(Then)33 b(for)g FH(\021)j FI(w)n(e)0 1683 y(ha)n(v)n(e)c FH(B)260 1695 y FD(0)329 1683 y FG(=)f(1)p FI(,)j FH(B)587 1695 y FD(1)656 1683 y FF(\030)e FG(1)p FI(.)52 b(By)34 b(\(10.25\))e FH(u)p FG(\()p FH(t)p FG(\))g(=)f FH(\027)1610 1653 y FA(a)p Fw(\000)p FD(1)p FA(=)p FD(2)1803 1683 y FH(v)s FG(\()p FH(\034)9 b FG(\))p FI(,)36 b(where)c FH(v)s FG(\()p FH(\034)9 b FG(\))35 b FI(is)d(a)h(stationary)e(w)n(eak)0 1782 y(solution)f(of)37 b(\(10.26\))o(.)46 b(By)30 b(Corollary)e(10.6,) i(the)h(measure)f FF(D)r FH(v)s FG(\(0\))h FI(is)g(close)f(to)g(the)h (set)g FH(K)36 b FI(\(whic)n(h)0 1882 y(dep)r(ends)f(on)f(the)g (constan)n(ts)g FH(B)1037 1894 y FD(0)1074 1882 y FH(;)14 b(B)1174 1894 y FD(1)1245 1882 y FI(as)34 b(ab)r(o)n(v)n(e\).)56 b(Since)34 b(the)h(measures)e(in)h FH(K)40 b FI(satisfy)34 b(ii\),)j(then)0 1982 y Fk(E)p FF(j)p FH(v)s FG(\(0\))p FF(j)258 1951 y FD(2)318 1982 y FF(\030)416 1949 y FD(1)p 416 1963 34 4 v 416 2010 a(2)473 1982 y FH(B)536 1994 y FD(0)573 1982 y FI(,)28 b(and)f(b)n(y)i(\(10.28\))d(w)n(e)h(m)n(ust)h (ha)n(v)n(e)e FH(a)d FF(\031)g FG(1)p FH(=)p FG(2)p FI(.)125 2081 y(W)-7 b(e)23 b(ha)n(v)n(e)g(seen)g(that)h(if)f(eq.)36 b(\(7.1\))23 b(with)h FG(0)e FH(<)h(\027)28 b FF(\034)23 b FG(1)h FI(has)e(a)i(stationary)d(solution)i FH(u)g FI(with)h(energy)0 2181 y(of)30 b(order)f(one,)h(then)g(the)h(force)e (is)h FF(\030)1213 2121 y(p)p 1282 2121 47 4 v 60 x FH(\027)35 b FI(and)30 b FF(D)r FH(u)p FG(\()p FH(t)p FG(\))g FI(is)g(close)g(to)f (the)i(set)f FH(K)6 b FI(.)43 b(Arguing)30 b(as)f(when)0 2280 y(pro)n(ving)24 b(Corollary)g(10.6)h(w)n(e)g(see)h(that)g(in)h (this)f(case)f(the)i(pro)r(cess)d FH(u)p FG(\()p FF(\001)p FG(\))j FI(is)f(close)f(in)h(distribution)g(to)0 2380 y(an)h(Eulerian)h(limit.)125 2480 y(In)d(the)h(3D)f(case)g(the)g (situation,)h(presumably)-7 b(,)25 b(is)g(v)n(ery)f(di\033eren)n(t.)37 b(Indeed,)26 b(let)f(us)h(assume)e(\(for)0 2579 y(the)i(purp)r(oses)f (of)h(this)h(section\))e(that)h(the)h(3D)f(NSE,)h(written)f(in)g(the)g (form)g(\(10.22\))o(,)g(has)f(a)h(unique)0 2679 y(stationary)k (measure,)i(that)f(the)h(corresp)r(onding)e(stationary)g(solution)h(is) g FH(H)2499 2649 y FD(1)2536 2679 y FI(\025smo)r(oth)g(in)h FH(x)g FI(and)0 2779 y(its)d(energy)g(sta)n(ys)f(of)h(order)f(one)h(as) f FH(\027)k FF(!)26 b FG(0)p FI(.)41 b(Applying)30 b(Ito's)f(form)n (ula)f(to)h(the)h(energy)e(functional)10 2845 y FD(1)p 10 2859 34 4 v 10 2907 a(2)67 2878 y FF(j)p FH(u)p FG(\()p FH(t)p FG(\))p FF(j)255 2890 y FA(L)301 2898 y Ft(2)366 2878 y FI(w)n(e)h(get,)f(as)h(in)g(the)g(2D)g(case,)f(that)h(the)g (rate)f(of)h(dissipation)f(of)h(energy)f FH(")d FG(:=)f Fk(E)p FF(jr)p FH(u)p FF(j)3107 2848 y FD(2)3107 2901 y FA(L)3153 2909 y Ft(2)0 2988 y FI(equals)265 2955 y FD(1)p 265 2969 V 265 3016 a(2)322 2988 y FH(\027)368 2958 y FD(2)p FA(a)441 2988 y FH(B)504 3000 y FD(0)541 2988 y FI(.)46 b(By)31 b(the)g(K)n(olmogoro)n(v's)26 b(theory)k(of)h(turbulence)f(\(see)h(in)g([F)-7 b(ri95)n(]\),)32 b FH(")f FI(con)n(v)n(erges)0 3087 y(to)c(a)h(p)r(ositiv)n(e)f(limit)h (as)f FH(\027)h FF(!)23 b FG(0)p FI(.)1055 3057 y Fo(15)1162 3087 y FI(So)k(no)n(w)g(w)n(e)g(m)n(ust)h(ha)n(v)n(e)e FH(a)d FG(=)g(0)p FI(.)0 3319 y Fy(10.5)112 b(Kic)m(k)m(ed)37 b(equations)0 3472 y FI(Let)28 b(us)f(consider)g(the)h(kic)n(k)n(ed)e (equation)663 3663 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)h(Lu)f FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)1353 3599 y FF(p)p 1422 3599 47 4 v 64 x FH(\027)c(\021)s FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)1816 3599 y FF(p)p 1885 3599 V 64 x FH(\027)1968 3584 y Fz(X)2010 3762 y FA(k)2116 3663 y FH(\021)2160 3628 y FA(!)2157 3683 y(k)2222 3663 y FH(\016)s FG(\()p FH(t)19 b FF(\000)f FH(k)s FG(\))c FH(;)394 b FI(\(10.29\))0 3945 y(where)37 b FH(\021)291 3957 y FA(k)371 3945 y FG(=)474 3866 y Fz(X)622 3945 y FH(b)658 3957 y FA(s)707 3945 y FH(\030)743 3957 y FA(sk)829 3945 y FH(e)868 3957 y FA(s)903 3945 y FI(.)66 b(Here)37 b(the)g(random)f(v)-5 b(ariables)36 b FH(\030)2054 3957 y FA(sk)2164 3945 y FI(are)g(as)g(in)i(Section)f(3,)i FH(b)2970 3957 y FA(s)3044 3945 y FF(6)p FG(=)f(0)0 4067 y FI(for)33 b(all)f FH(s)i FI(and)e FH(B)555 4079 y FD(1)625 4067 y FG(=)722 4004 y Fz(P)837 4067 y FF(j)p FH(s)p FF(j)922 4036 y FD(2)973 4067 y FH(b)1009 4036 y FD(2)1009 4087 y FA(s)1078 4067 y FH(<)g FF(1)p FI(.)54 b(By)33 b(Theorem)f(6.4)g(and)h(its)g(corollaries,)f(for)h(an)n(y)f FH(\027)38 b(>)31 b FG(0)0 4166 y FI(the)g(corresp)r(onding)e(Mark)n(o) n(v)f(c)n(hain)i(has)g(a)h(unique)f(stationary)f(measure)h FH(\026)2481 4178 y FA(\027)2522 4166 y FI(.)47 b(Let)30 b FH(u)2791 4178 y FA(\027)2832 4166 y FG(\()p FH(t)p FG(\))i FI(b)r(e)f(the)0 4266 y(corresp)r(onding)24 b(1\025p)r(erio)r (dic)h(solution)h(of)g(\(10.29\),)f(i.e.)36 b FF(D)r FH(u)1897 4278 y FA(\027)1939 4266 y FG(\()p FH(k)s FG(\))23 b(=)g FH(\026)2210 4278 y FA(\027)2277 4266 y FI(for)j(eac)n(h)f FH(k)k FI(\(see)d(the)h(end)f(of)0 4365 y(Section)i(3.3\).)36 b(Due)28 b(to)f(\(3.12\))o(,)536 4493 y Fz(Z)619 4514 y FA(k)q FD(+1)582 4682 y FA(k)771 4606 y Fk(E)14 b FF(k)p FH(u)938 4618 y FA(\027)978 4606 y FG(\()p FH(s)p FG(\))p FF(k)1123 4572 y FD(2)1123 4627 y(1)1174 4606 y FH(ds)23 b FG(=)1377 4550 y(1)p 1377 4587 42 4 v 1377 4663 a(2)1442 4606 y FH(D)1511 4618 y FD(0)1548 4606 y FH(;)1631 4493 y Fz(Z)1714 4514 y FA(k)q FD(+1)1678 4682 y FA(k)1867 4606 y Fk(E)14 b FF(k)p FH(u)2034 4618 y FA(\027)2074 4606 y FG(\()p FH(s)p FG(\))p FF(k)2219 4572 y FD(2)2219 4627 y(2)2270 4606 y FH(ds)23 b FG(=)2473 4550 y(1)p 2473 4587 V 2473 4663 a(2)2547 4606 y FH(D)2616 4618 y FD(1)2935 4606 y FI(\(10.30\))p 0 4733 1276 4 v 62 4787 a Fn(15)127 4810 y Fm(W)-6 b(e)24 b(assume)f(that)i(the)g(theory)g (applies)f(to)g(the)g(space\025p)r(erio)r(dic)i(turbulence,)g(whic)n(h) e(is)e(not)j(at)f(all)f(ob)n(vious.)1553 5059 y FI(80)p eop end PStoPSsaved restore %%Page: (80,81) 41 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 81 80 bop 0 523 a FI(for)27 b(eac)n(h)g FH(\027)h(>)23 b FG(0)k FI(and)g(eac)n(h)g FH(k)s FI(,)h(where)825 718 y FH(D)894 730 y FD(0)954 718 y FG(=)1041 639 y Fz(X)1189 718 y FH(b)1225 683 y FD(2)1225 738 y FA(s)1285 718 y Fk(E)14 b FH(\030)1402 683 y FD(2)1398 738 y FA(s)p FD(1)1466 718 y FH(;)60 b(D)1618 730 y FD(1)1679 718 y FG(=)1766 639 y Fz(X)1914 718 y FH(b)1950 683 y FD(2)1950 738 y FA(s)2000 718 y FF(j)p FH(s)p FF(j)2085 683 y FD(2)2146 718 y Fk(E)14 b FH(\030)2263 683 y FD(2)2259 738 y FA(s)p FD(1)2341 718 y FH(:)0 917 y FI(The)44 b(constan)n(ts)e FH(D)639 929 y FD(0)720 917 y FI(and)h FH(D)966 929 y FD(1)1047 917 y FI(w)n(ere)f(de\034ned)i(in)g(\(3.1\))o(,)k(and)43 b FH(D)2156 929 y FD(0)2243 917 y FF(2)50 b FG(\(0)p FH(;)14 b(B)2522 929 y FD(0)2559 917 y FG(\))p FI(,)48 b FH(D)2731 929 y FD(1)2818 917 y FF(2)i FG(\(0)p FH(;)14 b(B)3097 929 y FD(1)3134 917 y FG(\))p FI(.)0 1017 y(Moreo)n(v)n(er,)44 b(b)n(y)f(Lemma)f(3.9)h(the)g(solution)f FH(u)p FG(\()p FH(t)p FG(\))h FI(with)h(initial)f(condition)g FH(u)p FG(\(0\))48 b(=)g(0)43 b FI(satis\034es)0 1116 y Fk(E)14 b FF(k)p FH(u)p FG(\()p FH(k)s FG(\))p FF(k)319 1086 y FD(2)p FA(m)319 1137 y FD(1)448 1116 y FF(\024)33 b FH(C)605 1128 y FA(m)703 1116 y FI(for)g(all)h FH(k)s FI(,)i(uniformly)e(in)g FH(\027)5 b FI(.)57 b(As)34 b(at)g(the)g(b)r (eginning)g(of)g(Section)g(10.1,)h(this)0 1216 y(implies)28 b(that)g(the)g(p)r(erio)r(dic)f(solution)g FH(u)1285 1228 y FA(\027)1326 1216 y FG(\()p FH(t)p FG(\))h FI(meets)g(the)g (follo)n(wing)e(estimates:)766 1398 y Fk(E)14 b FF(k)p FH(u)933 1410 y FA(\027)973 1398 y FG(\()p FH(k)s FG(\))p FF(k)1125 1364 y FA(m)1125 1419 y FD(1)1211 1398 y FF(\024)22 b FH(C)1357 1410 y FA(m)1449 1398 y FI(for)27 b(all)g FH(k)s(;)14 b(m)23 b FF(2)g FE(N)p FH(;)41 b FI(all)28 b FH(\027)g(>)23 b FG(0)14 b FH(:)0 1581 y FI(Using)36 b(\(3.5\))f(and)g(the)h(prop)r(ert)n(y)e(\(NS)p FG(1)1297 1551 y Fw(0)1320 1581 y FI(\))i(from)e(Section)i(2.5)e(w)n(e)h(see)f (that)i(this)f(estimate)g(holds)0 1681 y(with)28 b FH(k)j FI(replaced)26 b(b)n(y)h(an)n(y)g FH(t)c FF(\025)g FG(0)p FI(.)125 1780 y(Rep)r(eating)k(the)h(pro)r(of)f(of)g(Theorem)g(10.1)f (w)n(e)h(get)0 1946 y FJ(Theorem)j(10.9.)40 b FC(A)n(ny)29 b(se)l(quenc)l(e)j Fz(e)-48 b FH(\027)1198 1958 y FA(j)1256 1946 y FF(!)23 b FG(0)29 b FC(c)l(ontains)h(a)g(subse)l(quenc)l(e)f FH(\027)2329 1958 y FA(j)2387 1946 y FF(!)23 b FG(0)29 b FC(such)h(that)889 2129 y FF(D)16 b FH(u)1017 2141 y FA(\027)1050 2149 y Fu(j)1085 2129 y FG(\()p FF(\001)p FG(\))24 b FH(*)f FF(D)16 b FH(U)9 b FG(\()p FF(\001)p FG(\))37 b FC(in)g FF(P)1746 2062 y Fz(\000)1784 2129 y FH(C)6 b FG(\(0)p FH(;)14 b FF(1)p FG(;)g FH(H)2156 2095 y FD(1)2193 2129 y FG(\))2225 2062 y Fz(\001)2277 2129 y FH(:)0 2312 y FC(The)34 b(limiting)g(pr)l(o)l(c)l(ess)g FH(U)9 b FG(\()p FF(\001)p FG(\))33 b FC(is)h(1-p)l(erio)l(dic)h(and)f (p)l(ossesses)g(the)g(fol)t(lowing)h(pr)l(op)l(erties:)47 b(every)34 b(its)0 2411 y(tr)l(aje)l(ctory)h FH(U)9 b FG(\()p FH(t)p FG(\))33 b FF(2)f FH(L)715 2423 y FD(2)p FA(;)11 b FD(lo)r(c)866 2411 y FG(\(0)p FH(;)j FF(1)p FG(;)g FH(H)1173 2381 y FD(2)1210 2411 y FG(\))35 b FC(satis\034es)f (the)h(Euler)g(e)l(quation,)i(and)e FF(j)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(j)p FC(,)36 b FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)3006 2423 y FD(1)3078 2411 y FC(ar)l(e)0 2511 y(time-indep)l(endent)40 b(c)l(onstants,)i(as)e(wel)t(l)g(as)g (the)g(quantities)47 b FI(\(10.9\))o FC(.)68 b(Mor)l(e)l(over,)44 b Fk(E)14 b FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)3047 2481 y FD(2)3047 2532 y(1)3124 2511 y FG(=)10 2578 y FD(1)p 10 2592 34 4 v 10 2639 a(2)76 2611 y FH(D)145 2623 y FD(0)212 2611 y FC(for)30 b(e)l(ach)h FH(t)f FC(and)215 2750 y Fz(Z)298 2771 y FA(k)q FD(+1)262 2939 y FA(k)451 2863 y Fk(E)14 b FF(k)p FH(U)9 b FG(\()p FH(s)p FG(\))p FF(k)781 2829 y FD(2)781 2884 y(2)831 2863 y FH(ds)23 b FF(\024)1034 2807 y FG(1)p 1034 2844 42 4 v 1034 2920 a(2)1108 2863 y FH(D)1177 2875 y FD(1)1244 2863 y FC(for)30 b(al)t(l)h FH(k)17 b(;)99 b Fk(E)14 b FF(k)p FH(U)9 b FG(\()p FH(t)p FG(\))p FF(k)1998 2829 y FA(m)1998 2884 y FD(1)2083 2863 y FF(\024)22 b FH(C)2229 2875 y FA(m)2392 2863 y FC(for)30 b(al)t(l)h FH(t)f FC(and)g FH(m)14 b(:)1553 5059 y FI(81)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 82 81 bop 0 523 a FJ(The)32 b(ph)m(ysical)g(relev)-5 b(ance)31 b(of)h(the)f(results.)36 b FI(The)27 b(results)g(on)g(the)h (Eulerian)f(limit)h(agree)e(with)0 623 y(ph)n(ysical)i(heuristics)h (since)g(they)h(sho)n(w)e(that)h(the)h(2D)f(Euler)h(equation)e(describ) r(es)h(certain)f(classes)0 722 y(of)34 b(the)g(homogeneous)e(2D)i (turbulence.)55 b(This)34 b(claim)g(w)n(as)f(often)h(maid)f(b)n(y)h(ph) n(ysicists)f(\(e.g.,)i(b)n(y)0 822 y(Onsager)25 b(in)j([Ons49)o(]\).)37 b(F)-7 b(rom)27 b(other)g(hand,)h(the)g(fact)f(that)h(solutions)f(of)g (the)h(2D)g(NSE,)h(scaled)d(as)1069 1005 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))24 b(=)1745 941 y FF(p)p 1814 941 47 4 v 64 x FH(\027)c(\021)s FG(\()p FH(t;)14 b(x)p FG(\))g FH(;)801 b FI(\(10.31\))0 1187 y(under)29 b(the)h(double)f(limit)h FH(t)d FF(!)f(1)p FI(,)k FH(\027)h FF(!)26 b FG(0)k FI(con)n(v)n(erge)d (\(in)j(distribution\))f(to)h(solutions)e(of)i(the)f(free)0 1287 y(Euler)37 b(equation)f(w)n(as)g(not)h(kno)n(wn)g(earlier.)63 b(Moreo)n(v)n(er,)37 b(the)g(results)g(obtained)f(sho)n(w)g(that)i(the) 0 1386 y(Kraic)n(hnan)28 b(theory)h(of)g(2D)h(turbulence)g(do)r(es)f (not)h(apply)f(to)h(the)g(stationary)e(space\025p)r(erio)r(dic)g(2D)0 1486 y(turbulence.)49 b(Indeed,)33 b(the)f(theory)f(is)g(based)g(on)g (the)h(assumption)f(that)h(the)g(rate)f(of)h(dissipation)0 1586 y(of)g(enstroph)n(y)f FH(\027)5 b Fk(E)p FF(j)p FG(\001)p FH(u)736 1598 y FA(\027)778 1586 y FF(j)801 1556 y FD(2)869 1586 y FG(=)30 b FH(\027)5 b Fk(E)p FF(jr)14 b FG(curl)g FH(u)p FF(j)1403 1556 y FD(2)1472 1586 y FI(con)n(v)n(erges)30 b(\(as)i FH(\027)k FF(!)31 b FG(0)p FI(\))h(to)g(a)g(non-zero)f(limit.)52 b(But)0 1685 y(due)27 b(to)g(the)h(explicit)f(form)n(ula)f(for)h Fk(E)p FF(j)p FG(\001)p FH(u)1326 1697 y FA(\027)1367 1685 y FF(j)1390 1655 y FD(2)1454 1685 y FI(in)g(Section)g(10.1,)f(this)i(limit)g(v)-5 b(anishes)26 b(if)i(the)f(NSE)h(is)0 1785 y(scaled)k(as)g(\(10.31\))n (.)52 b(In)33 b(the)g(same)f(time,)i(the)f(results)f(of)h(Sections)f (10.3)f(and)i(10.4)e(sho)n(w)h(that)h(if)0 1885 y(the)27 b(randomly)e(forced)g(2D)i(NSE)g(is)f(scaled)g(di\033eren)n(tly)g(from) g(\(10.31\))n(,)h(then)g(its)f(solutions)g(cannot)0 1984 y(con)n(v)n(erge)e(to)i(a)f(non-trivial)g(limit)i(with)f(energy)f(of)h (order)f(one.)36 b(This)26 b(fact)g(is)g(not)g(surprising)e(since)0 2084 y(for)g(the)h FG(2)p FH(\031)s FI(\025p)r(erio)r(dic)f(b)r (oundary)f(conditions)h(the)h(smallest)f(w)n(a)n(v)n(e\025n)n(um)n(b)r (er)e(is)j(one,)g(so)f(the)h(in)n(v)n(erse)0 2183 y(cascade)38 b(of)h(energy)f(\(to)h(the)g(lo)n(w)f(frequencies\),)k(on)d(whic)n(h)g (the)g(Kraic)n(hnan)e(theory)i(is)g(based,)0 2283 y(cannot)30 b(dev)n(elop)g(prop)r(erly)-7 b(.)45 b(It)31 b(is)g(an)f(exciting)h(op) r(en)g(problem)f(to)g(study)h(the)g(limit)h FH(t)c FF(!)g(1)j FI(and)0 2383 y(the)j(double)g(limit)h FH(t)e FF(!)h(1)p FI(,)h FH(\027)k FF(!)34 b FG(0)f FI(for)g(the)i(2D)e(NSE)i(with)g FH(x)f FF(2)f FE(R)2266 2353 y FD(2)2304 2383 y FI(,)i(p)r(erturb)r(ed) f(b)n(y)g(a)f(space\025)0 2482 y(homogeneous)f(\(non-p)r(erio)r(dic\))h (random)f(force,)i(to)g(\034nd)g(under)f(what)g(scaling)g(of)g(the)h (force)f(the)0 2582 y(limit)28 b(is)g(of)f(order)f(one)i(and)f(to)g (compare)g(the)h(result)f(obtained)g(with)h(the)g(Kraic)n(hnan)e (theory)-7 b(.)3078 2552 y Fo(16)p 0 4654 1276 4 v 62 4708 a Fn(16)127 4731 y Fm(The)24 b(theory)h(predicts)g(that)g(in)e (this)g(case)h(the)g(double)h(limit)c(of)i(order)i(one)f(exists)g(if)e (the)j(force)f(is)f(indep)r(enden)n(t)0 4810 y(of)g(the)i(viscosit)n(y) f Fi(\027)t Fm(.)31 b(This)23 b(is)g(a)h(remark)l(able)g(conjecture)i (whic)n(h)e(has)g(to)g(b)r(e)g(c)n(hec)n(k)n(ed.)1553 5059 y FI(82)p eop end PStoPSsaved restore %%Page: (82,83) 42 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 83 82 bop 0 523 a FK(11)131 b(Balance)44 b(relations)h(for)f(the)g(white\025forced)g(NSE)0 721 y Fy(11.1)112 b(The)38 b(balance)g(relations)0 875 y FI(Let)28 b(us)f(consider)g(again)f(the)i(NSE)h(\(10.1\))o(,)e(where)g FH(B)1701 887 y FD(1)1762 875 y FH(<)22 b FF(1)28 b FI(and)1264 1057 y FH(b)1300 1069 y FA(s)1358 1057 y FG(=)22 b FH(b)1481 1069 y Fw(\000)p FA(s)1591 1057 y FF(6)p FG(=)h(0)83 b FF(8)14 b FH(s:)1051 b FI(\(11.1\))0 1240 y(No)n(w)34 b(the)h(scaling)e(factor)870 1180 y FF(p)p 939 1180 47 4 v 60 x FH(\027)40 b FI(in)35 b(the)f(r.h.s.)58 b(is)34 b(not)g(imp)r(ortan)n(t)g(for)g(us)h(\(i.e.,)h(the)f(force)f FH(\021)k FI(ma)n(y)0 1340 y(dep)r(end)25 b(on)f FH(\027)5 b FI(\).)37 b(W)-7 b(e)24 b(in)n(tro)r(duce)g(it)h(to)f(mak)n(e)g(the)h (form)n(ulas)e(whic)n(h)h(w)n(e)g(obtain)g(b)r(elo)n(w)g(indep)r(enden) n(t)0 1439 y(of)k FH(\027)5 b FI(.)125 1539 y(Denoting)22 b(b)n(y)h FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\))24 b FI(the)f(v)n(orticit)n(y)f FH(\030)27 b FG(=)22 b(curl)14 b FH(u)p FI(,)23 b(w)n(e)g(get)f(for)h FH(\030)k FI(the)c(equation)f (\(see)h(Section)g(2.7\))1029 1700 y FG(_)1011 1722 y FH(\030)f FF(\000)c FH(\027)5 b FG(\001)p FH(\030)23 b FG(+)18 b(\()p FH(u)h FF(\001)f(r)p FG(\))p FH(\030)28 b FG(=)1802 1658 y FF(p)p 1871 1658 V 64 x FH(\027)33 b FG(curl)13 b FH(\021)k(;)799 b FI(\(11.2\))0 1904 y(where)1082 2026 y FG(curl)13 b FH(\021)27 b FG(=)1414 1970 y FH(d)p 1399 2007 74 4 v 1399 2083 a(dt)1510 1947 y Fz(X)1496 2133 y FA(s)p Fw(2)p Fv(Z)1611 2113 y Ft(2)1611 2151 y(0)1657 2026 y FH(b)1693 2038 y FA(s)1728 2026 y FH(\014)1775 2038 y FA(s)1811 2026 y FG(\()p FH(t)p FG(\))p FH(')1959 2038 y FA(s)1995 2026 y FG(\()p FH(x)p FG(\))0 2274 y FI(with)768 2401 y FH(')822 2413 y FA(s)881 2401 y FG(=)1024 2345 y FF(j)p FH(s)p FF(j)p 979 2382 175 4 v 979 2398 a(p)p 1048 2398 42 4 v 69 x FG(2)13 b FH(\031)1177 2401 y FG(cos)g FH(s)19 b FF(\001)f FH(x)c(;)125 b(')1664 2413 y Fw(\000)p FA(s)1775 2401 y FG(=)22 b FF(\000)1982 2345 y(j)p FH(s)p FF(j)p 1937 2382 175 4 v 1937 2398 a(p)p 2006 2398 42 4 v 69 x FG(2)13 b FH(\031)2136 2401 y FG(sin)g FH(s)19 b FF(\001)f FH(x;)0 2604 y FI(for)27 b(all)g FH(s)c FF(2)h FE(Z)438 2574 y FD(2)438 2624 y(+)493 2604 y FI(.)125 2703 y(F)-7 b(or)26 b(a)i(scalar)e(function)i FH(g)s FG(\()p FH(x)p FG(\))g FI(w)n(e)f(set)h FF(r)1407 2673 y Fw(?)1463 2703 y FH(g)s FG(\()p FH(x)p FG(\))c(=)e(\()p FH(@)5 b(g)s(=@)g(x)1990 2673 y FD(2)2027 2703 y FH(;)14 b FF(\000)p FH(@)5 b(g)s(=@)g(x)2359 2673 y FD(1)2394 2703 y FG(\))2426 2673 y FA(t)2456 2703 y FI(.)37 b(Then)1279 2886 y FG(curl)14 b FF(r)1501 2852 y Fw(?)1557 2886 y FH(g)26 b FG(=)c FF(\000)p FG(\001)p FH(g)s(:)0 3069 y FI(F)-7 b(or)30 b(an)n(y)f FH(l)g FF(2)f FE(R)i FI(the)h(op)r(erator) e FF(\000)p FG(\001)h FI(de\034nes)g(an)g(isomorphism)f(of)h(the)h (zero\025mean)n(v)-5 b(alue)28 b(Sob)r(olev)0 3168 y(spaces)f FH(H)331 3138 y FA(l)324 3189 y FD(0)361 3168 y FG(\()p FE(T)448 3138 y FD(2)486 3168 y FG(\))h FI(and)f FH(H)783 3133 y FA(l)p Fw(\000)p FD(2)776 3191 y(0)893 3168 y FG(\()p FE(T)980 3138 y FD(2)1018 3168 y FG(\))14 b FH(:)0 3334 y FJ(Exercise)37 b(11.1.)42 b FI(Pro)n(v)n(e)31 b(that)h(for)g(an)n(y)f FH(r)j FF(2)d FE(R)h FI(the)h(di\033eren)n (tial)f(op)r(erator)e FG(curl)i FI(de\034nes)g(an)g(iso-)0 3434 y(morphism)1174 3534 y FG(curl)23 b(:)g FH(H)1458 3499 y FA(r)1517 3534 y FF(!)g FH(H)1699 3498 y FA(r)r Fw(\000)p FD(1)1692 3556 y(0)1821 3534 y FG(\()p FE(T)1908 3499 y FD(2)1946 3534 y FG(\))14 b FH(;)0 3683 y FI(and)27 b(that)h(its)g(in)n(v)n(erse)e(can)h(b)r(e)h(written)g(as)54 b FG(curl)1556 3646 y Fw(\000)p FD(1)1668 3683 y FG(=)23 b FF(r)1825 3653 y Fw(?)1900 3683 y FF(\016)18 b FG(\()p FF(\000)p FG(\001\))2158 3653 y Fw(\000)p FD(1)2247 3683 y FH(:)125 3783 y FI(A)n(ccordingly)-7 b(,)36 b(for)g(an)n(y)f(in)n (teger)h FH(r)k FF(\025)d(\000)p FG(1)e FI(eq)h(\(11.2\))f(with)i FH(u)g FG(=)g(curl)2405 3746 y Fw(\000)p FD(1)2508 3783 y FH(\030)j FI(is)c(a)g(w)n(ell)g(de\034ned)0 3882 y(equation)27 b(for)g FH(\030)t FG(\()p FH(t)p FG(\))d FF(2)f FH(H)778 3852 y FA(r)771 3903 y FD(0)815 3882 y FG(\()p FE(T)902 3852 y FD(2)940 3882 y FG(\))p FI(,)28 b(equiv)-5 b(alen)n(t)27 b(to)g(eq.)37 b(\(10.1\))o(.)125 4048 y(Let)29 b FH(u)323 4060 y FA(\027)393 4048 y FI(b)r(e)g(a)g(stationary)e(solution)i(of)35 b(\(10.1\))o(.)42 b(Then)29 b FH(\030)1923 4060 y FA(\027)1990 4048 y FG(=)c(curl)14 b FH(u)2281 4060 y FA(\027)2351 4048 y FI(is)29 b(a)f(stationary)g(solution)0 4148 y(of)41 b(\(11.2\))o(.)59 b(Due)35 b(to)g(\(11.1\))f(and)g(Theorem)g(7.1)g(the) h(random)f(\034elds)h FH(u)2336 4160 y FA(\027)2377 4148 y FG(\()p FH(t;)28 b(x)p FG(\))35 b FI(and)g FH(\030)2809 4160 y FA(\027)2850 4148 y FG(\()p FH(t;)28 b(x)p FG(\))36 b FI(are)0 4248 y(homogeneous)26 b(in)i FH(x)p FI(.)125 4347 y(Let)j FH(g)i FG(:)d FE(R)g FF(!)g FE(R)i FI(b)r(e)g(a)g(con)n (tin)n(uous)f(function)h(with)g(compact)f(supp)r(ort.)50 b(De\034ne)32 b FH(G)g FI(to)g(b)r(e)g(its)0 4447 y(second)27 b(in)n(tegral,)f(so)h(that)1037 4547 y FH(G)1102 4512 y Fw(00)1167 4547 y FG(=)c FH(g)s(;)179 b(G)p FG(\(0\))24 b(=)e FH(G)1847 4512 y Fw(0)1871 4547 y FG(\(0\))h(=)g(0)p FH(:)0 4696 y FI(Then)895 4796 y FF(j)p FH(G)p FG(\()p FH(r)r FG(\))p FF(j)i(\024)e FH(C)6 b FF(j)p FH(r)r FF(j)p FH(;)98 b FF(j)p FH(G)1581 4761 y Fw(0)1605 4796 y FG(\()p FH(r)r FG(\))p FF(j)24 b(\024)f FH(C)q(;)97 b FF(8)p FH(r)26 b FF(2)e FE(R)p FH(:)683 b FI(\(11.3\))1553 5059 y(83)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 84 83 bop 0 523 a FI(Let)28 b(us)f(consider)g(the)h (stationary)e(pro)r(cess)1183 744 y FH(t)d FF(7!)1343 631 y Fz(Z)1389 819 y Fv(T)1428 803 y Ft(2)1478 744 y FH(G)p FG(\()p FH(\030)1611 756 y FA(\027)1653 744 y FG(\()p FH(t;)28 b(x)p FG(\)\))14 b FH(dx:)0 973 y FI(The)39 b(estimates)h(\(11.3\))e(allo)n(w)g(us)i(to)f(apply)g(to)h(this)f(pro)r (cess)g(the)g(Ito)h(form)n(ula)e(\(see)h([DZ92]\).)0 1072 y(T)-7 b(aking)30 b(the)i(exp)r(ectation)f(and)g(arguing)f(as)g (when)i(deriving)f(\(10.3\))o(,)i(w)n(e)e(\034nd)g(that)h(for)f(all)f FH(t)g FF(\025)e FG(0)0 1172 y FI(the)g(random)f(\034eld)g FH(\030)663 1184 y FA(\027)705 1172 y FG(\()p FH(t;)h(x)p FG(\))g FI(satis\034es)290 1397 y FH(\027)5 b Fk(E)413 1284 y Fz(Z)459 1473 y Fv(T)498 1456 y Ft(2)548 1397 y FH(G)613 1363 y Fw(0)637 1397 y FG(\()p FH(\030)705 1409 y FA(\027)747 1397 y FG(\()p FH(t;)28 b(x)p FG(\)\)\()p FF(\000)p FG(\001\))p FH(\030)1205 1409 y FA(\027)1248 1397 y FG(\()p FH(t;)g(x)p FG(\))14 b FH(dx)24 b FG(=)1666 1341 y(1)p 1666 1378 42 4 v 1666 1454 a(2)1731 1397 y FH(\027)5 b Fk(E)1868 1318 y Fz(X)1854 1503 y FA(s)p Fw(2)p Fv(Z)1969 1483 y Ft(2)1969 1521 y(0)2015 1397 y FH(b)2051 1363 y FD(2)2051 1418 y FA(s)2102 1284 y Fz(Z)2148 1473 y Fv(T)2187 1456 y Ft(2)2237 1397 y FH(G)2302 1363 y Fw(00)2345 1397 y FG(\()p FH(\030)2413 1409 y FA(\027)2455 1397 y FG(\()p FH(t;)28 b(x)p FG(\)\))p FH(')2733 1363 y FD(2)2733 1418 y FA(s)2785 1397 y FH(dx:)125 1678 y FI(In)n(tegrating)e(b)n(y)h(parts,)g(w)n(e)g(see)g(that)h(the)g (l.h.s.)37 b(equals)1213 1899 y FH(\027)5 b Fk(E)1336 1786 y Fz(Z)1433 1899 y FH(g)s FG(\()p FH(\030)1544 1911 y FA(\027)1586 1899 y FG(\))p FF(jr)p FH(\030)1746 1911 y FA(\027)1788 1899 y FF(j)1811 1864 y FD(2)1862 1899 y FH(dx:)0 2129 y FI(Since)28 b FH(b)253 2141 y FA(s)311 2129 y FG(=)23 b FH(b)435 2141 y Fw(\000)p FA(s)549 2129 y FI(and)28 b FH(')765 2099 y FD(2)765 2150 y FA(s)821 2129 y FG(+)18 b FH(')958 2099 y FD(2)958 2150 y Fw(\000)p FA(s)1068 2129 y FG(=)23 b FF(j)p FH(s)p FF(j)1241 2099 y FD(2)1278 2129 y FH(=)p FG(2)p FH(\031)1412 2099 y FD(2)1449 2129 y FI(,)28 b(then)g(the)g(r.h.s.)36 b(is)456 2305 y FG(1)p 456 2342 V 456 2418 a(4)521 2361 y FH(\027)5 b Fk(E)644 2248 y Fz(Z)741 2269 y(\020)791 2361 y FH(g)s FG(\()p FH(\030)t FG(\))966 2282 y Fz(X)952 2467 y FA(s)p Fw(2)p Fv(Z)1067 2447 y Ft(2)1067 2485 y(0)1113 2361 y FH(b)1149 2327 y FD(2)1149 2381 y FA(s)1186 2361 y FG(\()p FH(')1272 2327 y FD(2)1272 2381 y FA(s)1328 2361 y FG(+)18 b FH(')1465 2327 y FD(2)1465 2381 y Fw(\000)p FA(s)1553 2361 y FG(\))1585 2269 y Fz(\021)1649 2361 y FH(dx)23 b FG(=)1860 2305 y(1)p 1860 2342 V 1860 2418 a(2)1911 2361 y(\(2)p FH(\031)s FG(\))2067 2327 y Fw(\000)p FD(2)2157 2361 y FH(B)2220 2373 y FD(1)2257 2361 y Fk(E)2334 2248 y Fz(Z)2430 2361 y FH(g)s FG(\()p FH(\030)2541 2373 y FA(\027)2583 2361 y FG(\))14 b FH(dx:)0 2642 y FI(Th)n(us)27 b(w)n(e)g(ha)n(v)n(e)746 2763 y Fk(E)823 2650 y Fz(Z)920 2763 y FH(g)s FG(\()p FH(\030)1031 2775 y FA(\027)1072 2763 y FG(\))p FF(jr)p FH(\030)1232 2775 y FA(\027)1274 2763 y FF(j)1297 2729 y FD(2)1348 2763 y FH(dx)d FG(=)1560 2707 y(1)p 1560 2744 V 1560 2820 a(2)1611 2763 y(\(2)p FH(\031)s FG(\))1767 2729 y Fw(\000)p FD(2)1857 2763 y FH(B)1920 2775 y FD(1)1957 2763 y Fk(E)2034 2650 y Fz(Z)2130 2763 y FH(g)s FG(\()p FH(\030)2241 2775 y FA(\027)2283 2763 y FG(\))14 b FH(dx:)0 2950 y FI(Since)28 b(the)g(random)e(\034eld) i FH(\030)880 2962 y FA(\027)949 2950 y FI(is)g(homogeneous)e(in)h FH(x)p FI(,)i(then)684 3168 y Fk(E)p FH(g)s FG(\()p FH(\030)858 3180 y FA(\027)899 3168 y FG(\()p FH(t;)f(x)p FG(\)\))p FF(jr)p FH(\030)1251 3180 y FA(\027)1294 3168 y FG(\()p FH(t;)g(x)p FG(\))p FF(j)1509 3133 y FD(2)1570 3168 y FG(=)1668 3111 y(1)p 1668 3148 V 1668 3225 a(2)1719 3168 y(\(2)p FH(\031)s FG(\))1875 3133 y Fw(\000)p FD(2)1965 3168 y FH(B)2028 3180 y FD(1)2065 3168 y Fk(E)p FH(g)s FG(\()p FH(\030)2239 3180 y FA(\027)2280 3168 y FG(\()p FH(t;)g(x)p FG(\)\))473 b FI(\(11.4\))0 3378 y(for)38 b(an)n(y)h FG(\()p FH(t;)28 b(x)p FG(\))p FI(,)42 b(pro)n(vided)c(that) h FH(g)j FI(is)d(a)f(con)n(tin)n(uous)g(function)i(with)f(compact)f (supp)r(ort.)71 b(The)0 3478 y(equalit)n(y)28 b(\(11.4\))f(is)g(called) g(a)h FC(b)l(alanc)l(e)i(r)l(elation)p FI(.)0 3644 y FJ(Exercise)40 b(11.2.)k FI(Use)35 b(Exercise)f(4.12)f(to)i(sho)n(w)f (that)h(if)g FH(B)1967 3656 y FD(3)2040 3644 y FH(<)g FF(1)g FI(then)g(\(11.4\))f(holds)h(for)f(an)n(y)0 3743 y(con)n(tin)n(uous)26 b(function)j FH(g)s FG(\()p FH(r)r FG(\))p FI(,)f(satisfying)1235 3926 y FF(j)p FH(g)s FG(\()p FH(r)r FG(\))p FF(j)c(\024)f FH(C)6 b FG(\(1)18 b(+)g FF(j)p FH(r)r FF(j)p FG(\))1896 3892 y FA(c)1931 3926 y FH(;)1023 b FI(\(11.5\))0 4109 y(for)27 b(some)g FH(C)6 b FI(,)28 b FH(c)23 b(>)g FG(0)p FI(.)125 4275 y(Hin)n(t:)52 b(appro)n(ximate)33 b FH(g)38 b FI(as)d(ab)r(o)n(v)n(e)e(b)n(y)i (functions)h(with)f(compact)g(supp)r(ort)g FH(g)2658 4287 y FA(n)2702 4275 y FG(\()p FH(r)r FG(\))i FI(suc)n(h)e(that)0 4374 y FH(g)40 4386 y FA(n)85 4374 y FG(\()p FH(r)r FG(\))26 b FF(!)f FH(g)s FG(\()p FH(r)r FG(\))k FI(for)f(eac)n(h)g FH(r)k FI(and)c(eac)n(h)g FH(g)1272 4386 y FA(n)1345 4374 y FI(meets)h(\(11.5\))o(.)40 b(Next)29 b(pass)f(to)h(the)g(limit)g (in)g(\(11.4\))f(with)0 4474 y FH(g)i FI(replaced)d(b)n(y)g FH(g)553 4486 y FA(n)598 4474 y FI(.)37 b(\(F)-7 b(or)27 b(a)g(complete)h(pro)r(of)f(see)g([KP05)o(]\).)0 4640 y FJ(Theorem)j(11.3.)39 b FC(L)l(et)29 b FH(B)848 4652 y FD(3)908 4640 y FH(<)23 b FF(1)29 b FC(and)h(let)f FH(\030)1421 4652 y FA(\027)1492 4640 y FC(b)l(e)g(a)g(stationary)h (solution)g(of)g(e)l(q.)g FI(\(11.2\))o FC(.)38 b(Then)30 b(for)0 4740 y(any)g(c)l(ontinuous)f(function)h FH(g)s FC(,)f(satisfying)39 b FI(\(11.5\))o FC(,)30 b(the)g(b)l(alanc)l(e)g(r) l(elation)37 b FI(\(11.4\))29 b FC(holds.)1553 5059 y FI(84)p eop end PStoPSsaved restore %%Page: (84,85) 43 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 85 84 bop 125 523 a FI(T)-7 b(aking)26 b(in)i(\(11.4\))f FH(g)e FG(=)e(1)k FI(\(or)g(using)i(\(10.4\))d(and)i (Exercise)f(2.22\))f(w)n(e)h(see)h(that)1081 711 y Fk(E)p FF(jr)p FH(\030)1272 723 y FA(\027)1314 711 y FG(\()p FH(t;)14 b(x)p FG(\))p FF(j)1515 677 y FD(2)1576 711 y FG(=)1674 655 y(1)p 1674 692 42 4 v 1674 768 a(2)1739 711 y(\(2)p FH(\031)s FG(\))1895 677 y Fw(\000)p FD(2)1984 711 y FH(B)2047 723 y FD(1)2085 711 y FH(:)0 882 y FI(So)38 b(another)f(w)n(a)n(y)h(of)g(stating)g(the)h(assertion)d(of)j(the)g (theorem)e(is)i(that)f(the)h(random)e(v)-5 b(ariables)0 982 y FH(g)s FG(\()p FH(\030)111 994 y FA(\027)152 982 y FG(\()p FH(t;)28 b(x)p FG(\)\))23 b FI(and)e FF(jr)p FH(\030)682 994 y FA(\027)724 982 y FG(\()p FH(t;)28 b(x)p FG(\))p FF(j)939 952 y FD(2)998 982 y FI(are)20 b(uncorrelated)g(for)g(eac)n(h)h FG(\()p FH(t;)14 b(x)p FG(\))22 b FI(and)f(eac)n(h)f(function)i FH(g)s FI(,)g(satisfying)0 1082 y(\(11.5\))o(.)125 1181 y(It)37 b(is)h(straigh)n(tforw)n(ard)c (that)k(the)f(relations)g(\(11.4\))o(,)j(v)-5 b(alid)37 b(for)g(all)g(b)r(ounded)h(con)n(tin)n(uous)e FH(g)s FI(,)0 1281 y(imply)28 b(that)862 1415 y Fk(E)p FG(\()p FF(jr)p FH(\030)1085 1427 y FA(\027)1127 1415 y FG(\()p FH(t;)g(x)p FG(\))p FF(j)1342 1381 y FD(2)1403 1415 y FF(j)23 b(F)1509 1430 y FA(\030)1539 1438 y Fu(\027)1576 1430 y FD(\()p FA(t;)11 b(x)p FD(\))1726 1415 y FG(\))23 b(=)1879 1359 y(1)p 1879 1396 V 1879 1472 a(2)1944 1415 y(\(2)p FH(\031)s FG(\))2100 1381 y Fw(\000)p FD(2)2190 1415 y FH(B)2253 1427 y FD(1)2304 1415 y FH(;)650 b FI(\(11.6\))0 1577 y(where)34 b(the)i(l.h.s.)59 b(is)35 b(a)f(conditional)h(exp)r (ectation)f(of)h(the)h(random)e(v)-5 b(ariable)34 b FF(jr)p FH(\030)2698 1589 y FA(\027)2739 1577 y FG(\()p FH(t;)28 b(x)p FG(\))p FF(j)2954 1547 y FD(2)3028 1577 y FI(with)0 1677 y(resp)r(ect)41 b(to)f(the)i FH(\033)s FI(-algebra,)g(generated)e (b)n(y)g(the)i(random)e(v)-5 b(ariable)39 b FH(\030)2332 1689 y FA(\027)2374 1677 y FG(\()p FH(t;)28 b(x)p FG(\))p FI(.)78 b(Note)41 b(that)g(the)0 1777 y(balance)27 b(relations)f(in)i (the)g(form)f(\(11.6\))g(hold)g(without)h(assuming)f(that)h FH(B)2410 1789 y FD(3)2470 1777 y FH(<)23 b FF(1)p FI(.)125 1876 y(The)k(balance)g(relations)g(\(11.4\))g(and)g(their)h(pro)r(of)f (are)g(related)g(to)g(the)i(classical)d(fact)i(that)g(the)0 1976 y(quan)n(tities)19 b(\(10.9\))g(are)f(in)n(tegrals)g(of)h(motion)h (for)e(the)i(Euler)g(equation.)33 b(Indeed,)22 b(to)d(pro)n(v)n(e)f (the)h(latter)0 2076 y(w)n(e)j(di\033eren)n(tiate)h(\(10.9\))f(along)f (tra)5 b(jectories)21 b(of)29 b(\(10.8\),)23 b(and)g(to)g(establish)f (\(11.4\))g(w)n(e)g(di\033eren)n(tiate)0 2175 y(the)34 b(same)e(quan)n(tit)n(y)-7 b(,)34 b(ev)-5 b(aluated)33 b(for)g FH(U)41 b FG(:=)32 b FH(\030)1481 2187 y FA(\027)1522 2175 y FI(,)j(using)e(Ito's)g(form)n(ula.)52 b(A)34 b(signi\034can)n(t) e(di\033erence)0 2275 y(b)r(et)n(w)n(een)k(the)g(t)n(w)n(o)e(results)h (is)h(that)g(\(10.9\))f(is)g(an)h(in)n(tegral)e(of)h(motion)h(for)f (the)h(Euler)g(equation)0 2374 y(in)j(an)n(y)e(closed)h(Riemann)g (surface)g(\(as)g(w)n(ell)g(as)g(in)g(a)g(domain)g(with)h(the)g(b)r (oundary)e(condition)0 2474 y FH(u)23 b FF(\001)h FH(n)36 b FG(=)g(0)g FI(on)f(the)h(b)r(oundary\).)60 b(On)36 b(the)g(con)n(trary)-7 b(,)35 b(the)h FC(b)l(alanc)l(e)i(r)l(elations)g (hold)g(only)g(for)g(the)0 2574 y(p)l(erio)l(dic)32 b(b)l(oundary)e(c)l (onditions)f FI(since)f(their)f(pro)r(of)h(hea)n(vily)e(uses)i(that)g FH(')2377 2544 y FD(2)2377 2594 y FA(s)2433 2574 y FG(+)18 b FH(')2570 2544 y FD(2)2570 2594 y Fw(\000)p FA(s)2681 2574 y FF(\021)23 b(j)p FH(s)p FF(j)2854 2544 y FD(2)2891 2574 y FH(=)p FG(2)p FH(\031)3025 2544 y FD(2)3090 2574 y FI(for)0 2673 y(eac)n(h)k FH(s)p FI(.)125 2773 y(No)n(w)g(let)g FH(g)s FG(\()p FH(r)r FG(\))i FI(b)r(e)f(a)f(smo)r(oth)g(function)i (suc)n(h)e(that)996 2922 y FF(j)p FH(g)s FG(\()p FH(r)r FG(\))p FF(j)20 b FG(+)e FF(j)p FH(g)1357 2887 y Fw(0)1380 2922 y FG(\()p FH(r)r FG(\))p FF(j)24 b(\024)f FH(C)6 b FG(\(1)18 b(+)g FF(j)p FH(r)r FF(j)1943 2887 y FA(c)1978 2922 y FG(\))83 b FF(8)14 b FH(r)787 b FI(\(11.7\))0 3070 y(with)28 b(some)f FH(C)q(;)h(c)23 b(>)g FG(0)p FI(.)37 b(Since)27 b(for)g(an)n(y)g FH(x)d FF(2)f FE(T)1461 3040 y FD(2)1526 3070 y FI(the)28 b(map)775 3219 y FH(H)851 3185 y FD(4)911 3219 y FF(!)23 b FE(R)1077 3185 y FD(2)1115 3219 y FH(;)97 b(u)p FG(\()p FF(\001)p FG(\))23 b FF(7!)g FG(\(curl)14 b FH(u)p FG(\()p FH(x)p FG(\))p FH(;)g FF(jr)g FG(curl)g FH(u)p FG(\()p FH(x)p FG(\))p FF(j)2321 3185 y FD(2)2358 3219 y FG(\))p FH(;)0 3368 y FI(is)27 b(con)n(tin)n(uous)g (\(see)g(\(S3\))h(in)g(Section)g(1.1\),)f(then)h(the)g(map)784 3517 y FH(H)860 3482 y FD(4)920 3517 y FF(!)23 b FE(R)p FH(;)97 b(u)p FG(\()p FF(\001)p FG(\))23 b FF(7!)g FH(g)s FG(\(curl)14 b FH(u)p FG(\()p FH(x)p FG(\)\))g FF(jr)g FG(curl)g FH(u)p FG(\()p FH(x)p FG(\))p FF(j)2344 3482 y FD(2)2382 3517 y FH(;)0 3666 y FI(satis\034es)28 b(\(7.4\))o(.)39 b(Therefore,)28 b(if)h FH(B)1080 3678 y FD(5)1141 3666 y FH(<)24 b FF(1)p FI(,)29 b(then)g(b)n(y)f(the)g(Strong)g(La)n(w)f(of) i(Large)d(Num)n(b)r(ers)j(\(Theo-)0 3765 y(rem)e(8.4\),)g(for)g(an)n(y) g(solution)g FH(\030)32 b FI(of)i(\(11.2\))27 b(suc)n(h)g(that)h Fk(E)p FF(j)p FH(\030)t FG(\(0\))p FF(j)1984 3735 y FD(2)2044 3765 y FH(<)23 b FF(1)k FI(w)n(e)h(ha)n(v)n(e)372 3919 y FG(1)p 363 3956 61 4 v 363 4032 a FH(T)447 3863 y Fz(Z)530 3883 y FA(T)9 b FD(+1)493 4051 y(1)680 3908 y Fz(\000)718 3976 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\)\))g FF(jr)p FH(\030)t FG(\()p FH(t;)g(x)p FG(\))p FF(j)1390 3941 y FD(2)1430 3908 y Fz(\001)1482 3976 y FH(dt)23 b FF(!)g Fk(E)1747 3908 y Fz(\000)1785 3976 y FH(g)s FG(\()p FH(\030)1896 3988 y FA(\027)1937 3976 y FG(\(0)p FH(;)14 b(x)p FG(\)\))g FF(jr)p FH(\030)2301 3988 y FA(\027)2344 3976 y FG(\(0)p FH(;)g(x)p FG(\))p FF(j)2557 3941 y FD(2)2594 3908 y Fz(\001)2716 3976 y FI(a.s)o FH(:)0 4171 y FI(Since)28 b(a)f(similar)g(relation)f(holds)h (for)g FH(g)s FG(\()p FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\)\))p FI(,)29 b(then)f(w)n(e)g(ha)n(v)n(e)0 4310 y FJ(Corollary)f(11.4.)34 b FC(L)l(et)25 b FH(B)854 4322 y FD(5)914 4310 y FH(<)e FF(1)h FC(and)i FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\))25 b FC(b)l(e)g(a)g(solution)g(of)43 b FI(\(11.2\))24 b FC(such)h(that)f Fk(E)p FF(j)p FH(\030)t FG(\(0\))p FF(j)2932 4280 y FD(2)2993 4310 y FH(<)f FF(1)p FC(.)0 4409 y(Then)30 b(for)h(any)f(smo)l(oth)g(function)g FH(g)s FC(,)g(satisfying)38 b FI(\(11.7\))o FC(,)30 b(and)g(any)g FH(x)24 b FF(2)f FE(T)2382 4379 y FD(2)2450 4409 y FC(we)30 b(have)252 4564 y FG(1)p 243 4601 V 243 4677 a FH(T)326 4507 y Fz(Z)409 4527 y FA(T)9 b FD(+1)373 4695 y FA(T)560 4553 y Fz(\000)598 4620 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\)\))g FF(jr)p FH(\030)t FG(\()p FH(t;)g(x)p FG(\))p FF(j)1270 4586 y FD(2)1328 4620 y FF(\000)1421 4564 y FG(1)p 1421 4601 42 4 v 1421 4677 a(2)1486 4620 y(\(2)p FH(\031)s FG(\))1642 4586 y Fw(\000)p FD(2)1732 4620 y FH(B)1795 4632 y FD(1)1832 4620 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(t;)g(x)p FG(\)\))2157 4553 y Fz(\001)2210 4620 y FH(dt)23 b FF(!)g FG(0)85 b FC(as)46 b FH(T)34 b FF(!)23 b(1)p FH(;)0 4810 y FC(almost)30 b(sur)l(ely.)1553 5059 y FI(85)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 86 85 bop 125 523 a FI(W)-7 b(e)29 b(rep)r(eat)f(that)h (the)g(results)f(of)h(this)g(section)f(also)f(apply)i(to)f(the)h(NSE)h (without)f(the)g(scaling)0 623 y(factor)238 563 y FF(p)p 307 563 47 4 v 60 x FH(\027)k FI(in)28 b(the)g(r.h.s.:)1141 722 y FG(_)-38 b FH(u)18 b FG(+)g FH(\027)5 b(Lu)18 b FG(+)g FH(B)t FG(\()p FH(u)p FG(\))23 b(=)g FH(\021)s FG(\()p FH(t;)14 b(x)p FG(\))p FH(;)915 b FI(\(11.8\))0 864 y(where)34 b(the)h(force)f FH(\021)k FI(has)c(the)h(form)f (\(10.2\))g(with)h FH(B)1698 876 y FD(1)1770 864 y FH(<)g FF(1)f FI(and)h FH(b)2192 876 y FA(s)2262 864 y FF(\021)f FH(b)2397 876 y Fw(\000)p FA(s)2519 864 y FF(6)p FG(=)h(0)p FI(.)57 b(Substituting)0 964 y FH(\021)26 b FG(=)155 904 y FF(p)p 224 904 V 60 x FH(\027)f FG(~)-48 b FH(\021)s FI(,)27 b(w)n(e)f(get)g(eq.)36 b(\(10.1\))25 b(with)i FH(\021)i FI(replaced)c(b)n(y)32 b FG(~)-48 b FH(\021)t FI(.)36 b(Hence,)27 b(the)f(v)n(orticit)n(y)f FH(\030)2593 976 y FA(\027)2661 964 y FI(of)h(a)g(stationary)0 1063 y(solution)h(of)34 b(\(11.8\))27 b(satis\034es)605 1272 y Fk(E)p FH(g)s FG(\()p FH(\030)779 1284 y FA(\027)820 1272 y FG(\()p FH(t;)h(x)p FG(\)\))p FF(jr)p FH(\030)1172 1284 y FA(\027)1215 1272 y FG(\()p FH(t;)g(x)p FG(\)\))1439 1238 y FD(2)1500 1272 y FG(=)1598 1216 y(1)p 1598 1253 42 4 v 1598 1329 a(2)1663 1272 y FH(\027)1709 1238 y Fw(\000)p FD(1)1798 1272 y FG(\(2)p FH(\031)s FG(\))1954 1238 y Fw(\000)p FD(2)2044 1272 y FH(B)2107 1284 y FD(1)2144 1272 y Fk(E)p FH(g)s FG(\()p FH(\030)2318 1284 y FA(\027)2359 1272 y FG(\()p FH(t;)g(x)p FG(\)\))0 1470 y FI(for)f(an)n(y)g FG(\()p FH(t;)h(x)p FG(\))p FI(,)g(if)g FH(g)j FI(satis\034es)26 b(\(11.5\))o(.)0 1700 y Fy(11.2)112 b(The)38 b(co-area)g(form)f(of)h (the)f(balance)h(relations)0 1854 y FI(Let)31 b(us)f(tak)n(e)g(the)h (balance)f(relation)g(\(11.4\))o(,)h(where)f FH(g)k FI(is)c(a)g(b)r (ounded)h(con)n(tin)n(uous)f(function,)i FG(0)27 b FF(\024)0 1953 y FH(g)f FF(\024)c FG(1)p FI(.)37 b(In)n(tegrating)26 b(this)i(relation)e(in)i FH(dx)g FI(w)n(e)g(get:)634 2161 y Fk(E)711 2048 y Fz(Z)808 2161 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))p FF(jr)p FH(\030)t FG(\()p FH(x)p FG(\))p FF(j)1332 2127 y FD(2)1386 2161 y FH(dx)23 b FG(=)1597 2105 y(1)p 1597 2142 V 1597 2218 a(2)1649 2161 y(\(2)p FH(\031)s FG(\))1805 2127 y Fw(\000)p FD(2)1894 2161 y FH(B)1957 2173 y FD(1)1994 2161 y Fk(E)2071 2048 y Fz(Z)2168 2161 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FH(dx:)424 b FI(\(11.9\))0 2373 y(Here)27 b(w)n(e)g(abbreviate)g FH(\030)763 2385 y FA(\027)804 2373 y FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)f FH(\030)t FG(\()p FH(x)p FG(\))29 b FI(\()p FH(t)23 b FF(\025)g FG(0)k FI(is)g(\034xed\).)125 2473 y(Assume)32 b(that)g FH(B)684 2485 y FD(6)752 2473 y FH(<)e FF(1)p FI(.)50 b(Then)33 b FH(\030)t FG(\()p FH(x)p FG(\))e FF(2)g FH(C)1558 2443 y FD(3)1628 2473 y FI(a.s.)50 b(\(see)32 b(Exercise)f(4.12\).)49 b(Mo)r(difying)32 b FH(\030)37 b FI(on)31 b(a)0 2573 y(n)n(ull-set)c(w)n(e)g(ac)n(hiev)n(e)1203 2672 y FH(\030)1243 2638 y FA(!)1292 2672 y FG(\()p FH(x)p FG(\))d FF(2)f FH(C)1570 2638 y FD(3)1608 2672 y FG(\()p FE(T)1695 2638 y FD(2)1733 2672 y FG(\))83 b FF(8)14 b FH(!)s(:)0 2814 y FI(F)-7 b(or)27 b(an)n(y)g FH(")22 b(>)h FG(0)k FI(and)h FH(\034)k FF(2)24 b FE(R)j FI(w)n(e)h(set)270 2984 y FH(K)341 2996 y FA(")376 2984 y FG(\()p FH(!)s FG(\))c(=)e FF(f)p FH(x)h FF(2)h FE(T)852 2950 y FD(2)912 2984 y FF(j)g(jr)p FH(\030)1091 2950 y FA(!)1139 2984 y FG(\()p FH(x)p FG(\)\))p FF(j)g(\025)f FH(")p FF(g)p FH(;)96 b FG(\000)1669 2996 y FA(")1705 2984 y FG(\()p FH(\034)5 b(;)14 b(!)s FG(\))23 b(=)g FF(f)p FH(x)g FF(2)g FH(K)2274 2996 y FA(")2309 2984 y FG(\()p FH(!)s FG(\))g FF(j)g FH(\030)2537 2950 y FA(!)2586 2984 y FG(\()p FH(x)p FG(\))h(=)e FH(\034)9 b FF(g)p FH(:)0 3154 y FI(The)30 b(random)e(set)i FG(\000)663 3166 y FA(")728 3154 y FI(is)f(a)g(\034nite)h(union)g(of)f FH(C)1492 3124 y FD(3)1530 3154 y FI(-smo)r(oth)g(curv)n(es.)41 b(W)-7 b(e)30 b(denote)f(b)n(y)g FH(\015)35 b FI(p)r(oin)n(ts)29 b(of)h FG(\000)3154 3166 y FA(")0 3254 y FI(and)d(denote)h(b)n(y)f FH(d\015)33 b FI(the)28 b(length\025elemen)n(t.)36 b(P)n(erforming)26 b(the)i(co-area)d(c)n(hange)i(of)g(v)-5 b(ariables)751 3424 y FH(K)822 3436 y FA(")857 3424 y FG(\()p FH(!)s FG(\))23 b FF(3)h FH(x)f FF(7!)g FG(\()p FH(\034)5 b(;)14 b(\015)5 b FG(\))p FH(;)98 b(\034)32 b FG(=)23 b FH(\030)1761 3389 y FA(!)1809 3424 y FG(\()p FH(x)p FG(\))p FH(;)38 b(\015)28 b FF(2)23 b FG(\000)2182 3436 y FA(")2218 3424 y FG(\()p FH(\034)5 b(;)14 b(!)s FG(\))p FH(;)0 3593 y FI(w)n(e)27 b(ha)n(v)n(e)g FH(d\034)c(d\015)28 b FG(=)23 b FF(jr)p FH(\030)t FF(j)14 b FH(dx)877 3605 y FD(1)929 3593 y FH(dx)1019 3605 y FD(2)1084 3593 y FI(\(e.g.,)28 b(see)f([Cha84)o(]\).)37 b(So)642 3693 y Fz(Z)688 3881 y FA(K)744 3889 y Fu(")794 3806 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FF(jr)p FH(\030)t FG(\()p FH(x)p FG(\))p FF(j)1332 3772 y FD(2)1386 3806 y FH(dx)24 b FG(=)1587 3693 y Fz(Z)1634 3881 y Fv(R)1694 3806 y FH(g)s FG(\()p FH(\034)9 b FG(\))1860 3693 y Fz(Z)1906 3881 y FD(\000)1947 3889 y Fu(e)1980 3881 y FD(\()p FA(\034)s(;!)r FD(\))2147 3806 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)19 b(d\034)24 b(;)388 b FI(\(11.10\))0 4032 y(and)757 4040 y Fz(Z)803 4229 y FA(K)859 4237 y Fu(")909 4153 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FH(dx)25 b FG(=)1384 4040 y Fz(Z)1430 4229 y Fv(R)1490 4153 y FH(g)s FG(\()p FH(\034)9 b FG(\))1656 4040 y Fz(Z)1702 4229 y FD(\000)1743 4237 y Fu(e)1776 4229 y FD(\()p FA(\034)s(;!)r FD(\))1943 4153 y FF(jr)p FH(\030)t FF(j)2098 4119 y Fw(\000)p FD(1)2201 4153 y FH(d\015)19 b(d\034)24 b(:)503 b FI(\(11.11\))125 4356 y(No)n(w)28 b(w)n(e)h(study)g(the)h(b)r(eha)n (viour)e(of)h(exp)r(ectations)g(of)g(the)g(four)g(in)n(tegrals)f(in)h (\(11.10\))o(,)g(\(11.11\))0 4456 y(as)e FH(")c FF(!)g FG(0)p FI(.)125 4555 y(i\))28 b(Ob)n(viously)-7 b(,)609 4768 y Fk(E)686 4655 y Fz(Z)732 4843 y FA(K)788 4851 y Fu(")837 4768 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FF(jr)p FH(\030)t FG(\()p FH(x)p FG(\))p FF(j)1375 4733 y FD(2)1429 4768 y FH(dx)24 b FF(!)f Fk(E)1726 4655 y Fz(Z)1772 4843 y Fv(T)1811 4827 y Ft(2)1861 4768 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FF(jr)p FH(\030)t FG(\()p FH(x)p FG(\))p FF(j)2399 4733 y FD(2)2453 4768 y FH(dx)g(:)1553 5059 y FI(86)p eop end PStoPSsaved restore %%Page: (86,87) 44 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 87 86 bop 125 523 a FI(ii\))28 b(W)-7 b(e)28 b(ha)n(v)n(e)725 570 y Fz(\014)725 619 y(\014)725 669 y(\014)752 665 y Fk(E)829 552 y Fz(Z)875 741 y Fv(T)914 724 y Ft(2)946 741 y Fw(n)p FA(K)1036 749 y Fu(")1086 665 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FH(dx)1448 570 y Fz(\014)1448 619 y(\014)1448 669 y(\014)1501 665 y FF(\024)22 b Fk(E)1665 552 y Fz(Z)1711 741 y Fv(T)1750 724 y Ft(2)1800 665 y FH(I)1836 680 y Fw(fjr)p FA(\030)r Fw(j)p FA(<")p Fw(g)2119 665 y FG(\()p FH(x;)14 b(!)s FG(\))g FH(dx)g(:)0 864 y FI(When)28 b FH(")23 b FF(!)g FG(0)p FI(,)k(the)h(r.h.s.)37 b(con)n(v)n(erges)24 b(to)743 1073 y Fk(E)820 960 y Fz(Z)866 1148 y Fv(T)905 1132 y Ft(2)955 1073 y FH(I)991 1088 y Fw(fjr)p FA(\030)r Fw(j)p FD(=0)p Fw(g)1275 1073 y FG(\()p FH(x;)14 b(!)s FG(\))g FH(dx)24 b FG(=)e(\(2)p FH(\031)s FG(\))1849 1038 y FD(2)1887 1073 y Fk(P)p FF(fr)p FH(\030)t FG(\()p FH(x)2182 1085 y FD(0)2220 1073 y FG(\))h(=)g(0)p FF(g)0 1290 y FI(\(w)n(e)18 b(use)g(the)h(homogeneit)n(y)e(of)h FH(\030)t FI(\).)35 b(It)18 b(can)g(b)r(e)h(sho)n(wn)f(\(see)g(App)r(endix)h(in)g([Kuk05)n (]\))g(that)g Fk(P)p FF(fr)p FH(\030)t FG(\()p FH(x)3089 1302 y FD(0)3127 1290 y FG(\))k(=)0 1390 y(0)p FF(g)f FG(=)h(0)p FI(.)36 b(Therefore,)927 1532 y Fk(E)1004 1419 y Fz(Z)1050 1608 y FA(K)1106 1616 y Fu(")1156 1532 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FH(dx)25 b FF(!)e Fk(E)1726 1419 y Fz(Z)1772 1608 y Fv(T)1811 1591 y Ft(2)1861 1532 y FH(g)s FG(\()p FH(\030)t FG(\()p FH(x)p FG(\)\))14 b FH(dx)g(:)125 1729 y FI(iii\))28 b(Denote)1045 1829 y FG(\000\()p FH(\034)5 b(;)14 b(!)s FG(\))23 b(=)g FF(f)p FH(x)g FF(2)g FE(T)1650 1795 y FD(2)1711 1829 y FF(j)g FH(\030)t FG(\()p FH(x)p FG(\))h(=)f FH(\034)9 b FF(g)14 b FH(:)791 b FI(\(11.12\))0 1972 y(By)27 b(the)h(Sard)f(lemma,)758 1941 y Fo(17)856 1972 y FI(for)g(eac)n(h)f FH(!)31 b FI(the)d(set)f FG(\000\()p FH(\034)5 b(;)14 b(!)s FG(\))28 b FI(is)f FH(C)1949 1941 y FD(3)1987 1972 y FI(-smo)r(oth)g(for)g(a.e.)36 b FH(\034)d FF(2)24 b FE(R)p FI(.)36 b(So)884 2072 y Fz(Z)930 2261 y FD(\000)971 2269 y Fu(")1003 2261 y FD(\()p FA(\034)s(;!)r FD(\))1171 2185 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)28 b FF(\045)1560 2072 y Fz(Z)1606 2261 y FD(\000\()p FA(\034)s(;!)r FD(\))1814 2185 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)28 b(<)23 b FF(1)14 b FH(;)0 2412 y FI(for)27 b(ev)n(ery)f FH(!)31 b FI(and)c(a.e.)36 b FH(\034)9 b FI(.)38 b(Hence,)547 2621 y Fk(E)624 2508 y Fz(Z)670 2697 y Fv(R)730 2621 y FH(g)s FG(\()p FH(\034)9 b FG(\))896 2508 y Fz(Z)943 2697 y FD(\000)984 2705 y Fu(")1016 2697 y FD(\()p FA(\034)s(;!)r FD(\))1183 2621 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)19 b(d\034)33 b FF(!)23 b Fk(E)1752 2508 y Fz(Z)1798 2697 y Fv(R)1858 2621 y FH(g)s FG(\()p FH(\034)9 b FG(\))2024 2508 y Fz(Z)2071 2697 y FD(\000\()p FA(\034)s(;!)r FD(\))2279 2621 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)19 b(d\034)0 2848 y FI(b)n(y)27 b(the)h(monotone)f(con)n(v)n(ergence)e(theorem.)125 2948 y(iv\))i(Similar,)795 2977 y Fz(Z)841 3166 y FD(\000)882 3174 y Fu(")914 3166 y FD(\()p FA(\034)s(;!)r FD(\))1081 3090 y FF(jr)p FH(\030)t FF(j)1236 3056 y Fw(\000)p FD(1)1340 3090 y FH(d\015)h FF(\045)1560 2977 y Fz(Z)1606 3166 y FD(\000\()p FA(\034)s(;!)r FD(\))1814 3090 y FF(jr)p FH(\030)t FF(j)1969 3056 y Fw(\000)p FD(1)2073 3090 y FH(d\015)g FF(\024)22 b(1)14 b FH(;)0 3299 y FI(for)27 b(ev)n(ery)f FH(!)31 b FI(and)c(a.e.)36 b FH(\034)9 b FI(,)29 b(where)e(w)n(e)g(accepted)g(the)h(follo)n(wing)f(agreemen)n (t:)35 b FG(0)2518 3269 y Fw(\000)p FD(1)2630 3299 y FG(=)22 b(0)p FI(.)37 b(Hence,)458 3508 y Fk(E)535 3395 y Fz(Z)581 3584 y Fv(R)641 3508 y FH(g)s FG(\()p FH(\034)9 b FG(\))807 3395 y Fz(Z)854 3584 y FD(\000)895 3592 y Fu(")927 3584 y FD(\()p FA(\034)s(;!)r FD(\))1094 3508 y FF(jr)p FH(\030)t FF(j)1249 3474 y Fw(\000)p FD(1)1352 3508 y FH(d\015)19 b(d\034)33 b FF(!)23 b Fk(E)1752 3395 y Fz(Z)1798 3584 y Fv(R)1858 3508 y FH(g)s FG(\()p FH(\034)9 b FG(\))2024 3395 y Fz(Z)2071 3584 y FD(\000\()p FA(\034)s(;!)r FD(\))2279 3508 y FF(jr)p FH(\030)t FF(j)2434 3474 y Fw(\000)p FD(1)2537 3508 y FH(d\015)19 b(d\034)125 3740 y FI(No)n(w)28 b(\(11.9\))o(\025\(11.11\))e(and)h(i\)-iv\))h(imply)g (that)177 3954 y Fk(E)254 3841 y Fz(Z)300 4029 y Fv(R)359 3954 y FH(g)s FG(\()p FH(\034)9 b FG(\))525 3841 y Fz(Z)572 4029 y FD(\000\()p FA(\034)s(;!)r FD(\))781 3954 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)19 b(d\034)32 b FG(=)1264 3898 y(1)p 1264 3935 42 4 v 1264 4011 a(2)1329 3954 y(\(2)p FH(\031)s FG(\))1485 3920 y Fw(\000)p FD(2)1575 3954 y FH(B)1638 3966 y FD(1)1689 3954 y Fk(E)1766 3841 y Fz(Z)1812 4029 y Fv(R)1872 3954 y FH(g)s FG(\()p FH(\034)9 b FG(\))2038 3841 y Fz(Z)2084 4029 y FD(\000\()p FA(\034)s(;!)r FD(\))2293 3954 y FF(jr)p FH(\030)t FF(j)2448 3920 y Fw(\000)p FD(1)2551 3954 y FH(d\015)19 b(d\034)200 b FI(\(11.13\))0 4186 y(\(in)32 b(particular,)f(the)g(r.h.s.)47 b(is)32 b FH(<)c FF(1)p FI(\).)48 b(Since)32 b(for)e(eac)n(h)h FH(!)j FI(the)d(set)h(of)f(critical)f(v)-5 b(alues)31 b FH(\034)41 b FI(of)31 b FH(\030)36 b FI(has)0 4286 y(zero)25 b(measure,)g(then)h(w)n(e)g(can)f(arbitrarily)f(de\034ne)i (the)g(in)n(tegrals)f(of)g FF(jr)p FH(\030)t FF(j)2335 4256 y Fw(\000)p FD(1)2451 4286 y FI(o)n(v)n(er)f(critical)h(lev)n(els) g(of)0 4385 y FH(\030)t FI(,)j(without)g(c)n(hanging)e(the)i(r.h.s..)36 b(Belo)n(w)27 b(w)n(e)g(adopt)g(the)h(follo)n(wing)f(natural)f(con)n(v) n(en)n(tion:)663 4481 y Fz(Z)709 4670 y FD(\000\()p FA(\034)s(;!)r FD(\))917 4594 y FF(jr)p FH(\030)t FF(j)1072 4560 y Fw(\000)p FD(1)1175 4594 y FH(d\015)i FG(=)23 b FF(1)46 b FI(if)h FH(\034)55 b FI(is)28 b(a)f(critical)g(v)-5 b(alue)27 b(of)46 b FH(\030)18 b(:)409 b FI(\(11.14\))p 0 4733 1276 4 v 62 4787 a Fn(17)127 4810 y Fm(The)24 b(lemma)d(applies)j (since)g Fi(\030)e Fh(2)e Fi(C)1083 4787 y Fc(3)1141 4810 y Fm(and)k(dim)10 b Fi(x)20 b Fg(=)f(2)p Fm(,)k(dim)11 b Fi(\030)22 b Fg(=)e(1)p Fm(.)31 b(See)24 b([GG73)q(].)1553 5059 y FI(87)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 88 87 bop 0 523 a FI(The)24 b(in)n(tegral)466 456 y Fz(R)505 553 y FD(\000\()p FA(\034)s(;!)r FD(\))714 523 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)28 b FF(\024)23 b(1)h FI(is)g(alw)n(a)n(ys)e(w)n(ell)i(de\034ned)h(as)f(a)g(limit)g(of) h(the)f(in)n(tegrals)f(o)n(v)n(er)g FG(\000)3131 535 y FA(")3166 523 y FI(.)125 638 y(The)34 b(relation)g(\(11.13\))g(holds) g(for)g(eac)n(h)g(con)n(tin)n(uous)g FH(g)s FI(,)i FG(0)f FF(\024)f FH(g)k FF(\024)d FG(1)p FI(.)58 b(Similar,)36 b(it)f(holds)g(for)0 737 y(con)n(tin)n(uous)26 b FH(g)s FI(,)i FF(\000)p FG(1)22 b FF(\024)h FH(g)i FF(\024)e FG(0)p FI(.)36 b(So)28 b(it)g(is)f(satis\034ed)g(b)n(y)g(an)n(y)g FH(g)f FF(2)d FH(C)2056 749 y FA(b)2090 737 y FG(\()p FE(R)p FG(\))p FI(,)28 b(and)g(w)n(e)f(get)0 903 y FJ(Theorem)j(11.5.) 40 b FC(If)30 b FH(B)793 915 y FD(6)853 903 y FH(<)23 b FF(1)p FC(,)30 b(then)g(for)g(any)g FH(\027)f(>)22 b FG(0)30 b FC(and)g FH(t)23 b FF(\025)g FG(0)29 b FC(we)h(have)88 1124 y Fk(E)165 1011 y Fz(Z)211 1199 y Fw(f)p FA(x)p Fw(j)p FA(\030)335 1183 y Fu(!)333 1216 y(\027)377 1199 y FD(\()p FA(t;x)p FD(\)=)p FA(\034)7 b Fw(g)652 1124 y FF(jr)p FH(\030)784 1089 y FA(!)780 1144 y(\027)832 1124 y FG(\()p FH(t;)14 b(x)p FG(\))p FF(j)g FH(d\015)29 b FG(=)1259 1068 y(1)p 1259 1105 42 4 v 1259 1181 a(2)1325 1124 y(\(2)p FH(\031)s FG(\))1481 1089 y Fw(\000)p FD(2)1570 1124 y FH(B)1633 1136 y FD(1)1671 1124 y Fk(E)1748 1011 y Fz(Z)1793 1199 y Fw(f)p FA(x)p Fw(j)p FA(\030)1917 1183 y Fu(!)1915 1216 y(\027)1959 1199 y FD(\()p FA(t;x)p FD(\)=)p FA(\034)7 b Fw(g)2234 1124 y FF(jr)p FH(\030)2366 1089 y FA(!)2362 1144 y(\027)2414 1124 y FG(\()p FH(t;)14 b(x)p FG(\))p FF(j)2615 1089 y Fw(\000)p FD(1)2719 1124 y FH(d\015)19 b(;)88 b FI(\(11.15\))0 1370 y FC(for)31 b(a.a.)40 b FH(\034)32 b FF(2)24 b FE(R)p FC(.)38 b(Her)l(e)30 b(we)g(assume)36 b FI(\(11.14\))n FC(,)30 b(wher)l(e)h FG(\000)23 b(=)f FF(f)p FH(x)h FF(j)h FH(\030)2100 1339 y FA(!)2096 1390 y(\027)2148 1370 y FG(\()p FH(t;)14 b(x)p FG(\))24 b(=)e FH(\034)9 b FF(g)p FC(.)125 1536 y FI(Noting)27 b(that)h(\(11.10\))e(and)h(i\),)h(iii\))h(with)f FH(g)d FG(=)e(1)k FI(imply)821 1761 y Fk(E)912 1648 y Fz(Z)958 1836 y Fv(T)997 1820 y Ft(2)1047 1761 y FF(jr)p FH(\030)t FF(j)1202 1726 y FD(2)1253 1761 y FH(dx)d FG(=)1455 1648 y Fz(Z)1501 1836 y Fv(R)1561 1669 y Fz(\020)1610 1761 y Fk(E)1701 1648 y Fz(Z)1747 1836 y FD(\000\()p FA(\034)s(;!)r FD(\))1955 1761 y FF(jr)p FH(\030)t FF(j)14 b FH(d\015)2215 1669 y Fz(\021)2279 1761 y FH(d\034)0 2004 y FI(and)34 b(using)i(\(11.9\))d(with)i FH(g)j FG(=)c(1)p FI(,)i(w)n(e)e(see)g(that)h(the)g(in)n(tegral)e(in)i FH(d\034)44 b FI(o)n(v)n(er)33 b FE(R)h FI(of)h(the)g(function)g(in)0 2104 y(\(11.15\))26 b(equals)543 2071 y FD(1)p 543 2085 34 4 v 543 2133 a(2)586 2104 y FH(B)649 2116 y FD(1)686 2104 y FI(.)125 2204 y(Similar)20 b(to)h(Section)g(11.1,)g(the)g(v)n (orticit)n(y)f FH(\030)t FG(\()p FH(t;)14 b(x)p FG(\))22 b FI(of)f(a)f(stationary)g(solution)g(for)h(equation)f(\(11.8\))0 2303 y(for)27 b(an)n(y)g FH(t)c FF(\025)f FG(0)28 b FI(satis\034es)275 2524 y Fk(E)352 2411 y Fz(Z)398 2599 y Fw(f)p FA(x)p Fw(j)p FA(\030)522 2583 y Fu(!)564 2599 y FD(=)p FA(\034)7 b Fw(g)704 2524 y FF(jr)p FH(\030)836 2490 y FA(!)884 2524 y FG(\()p FH(t;)14 b(x)p FG(\))p FF(j)g FH(d\015)29 b FG(=)1312 2468 y(1)p 1312 2505 42 4 v 1312 2581 a(2)1377 2524 y FH(\027)1423 2490 y Fw(\000)p FD(1)1526 2524 y FG(\(2)p FH(\031)s FG(\))1682 2490 y Fw(\000)p FD(2)1772 2524 y FH(B)1835 2536 y FD(1)1872 2524 y Fk(E)1949 2411 y Fz(Z)1995 2599 y Fw(f)p FA(x)p Fw(j)p FA(\030)2119 2583 y Fu(!)2161 2599 y FD(=)p FA(\034)7 b Fw(g)2301 2524 y FF(jr)p FH(\030)2433 2490 y FA(!)2481 2524 y FG(\()p FH(t;)14 b(x)p FG(\))p FF(j)2682 2490 y Fw(\000)p FD(1)2786 2524 y FH(d\015)19 b(;)0 2763 y FI(for)27 b(a.a.)36 b FH(\034)9 b FI(.)0 2962 y FJ(The)44 b(ph)m(ysical)f(relev)-5 b(ance)43 b(of)h(the)f(results.)65 b FI(The)38 b(balance)f(relations)f (\(11.4\))h(and)g(\(11.15\))0 3061 y(indicate)30 b(that)g(in)h(some)e (sense)h(`the)h(p)r(erio)r(dic)e(2D)i(turbulence)f(is)g(in)n (tegrable',)f(if)i(the)f(2D)h(\035uid)f(is)0 3161 y(stirred)h(b)n(y)h (a)g(white)h(in)f(time)h(Gaussian)e(random)g(force.)50 b(W)-7 b(e)33 b(are)e(certain)h(that)g(these)g(relations)0 3261 y(mak)n(e)39 b(a)h(p)r(o)n(w)n(erful)f(to)r(ol)h(to)g(study)g(the) g(2D)g(turbulence,)j(but)e(no)n(w)e(w)n(e)h(kno)n(w)f(only)h(one)f (their)0 3360 y(application:)h(in)30 b([Kuk05)o(])g(the)g(balance)f (relations)f(\(11.4\))h(are)g(used)h(to)f(pro)n(v)n(e)f(that)i(a)g (stationary)0 3460 y(solution)f(of)h(the)g(randomly)e(forced)i(2D)f (NSE)i(and)e(its)h(v)n(orticit)n(y)-7 b(,)29 b(ev)-5 b(aluated)30 b(at)f(an)n(y)g(p)r(oin)n(t,)i(b)r(oth)0 3560 y(are)c(random)f(v)-5 b(ariables)26 b(with)i(\034nite)h(exp)r (onen)n(tial)e(momen)n(ts)g(\(see)g(the)h(estimates)f(\(10.11\))o(\).) 1553 5059 y(88)p eop end PStoPSsaved restore %%Page: (88,89) 45 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 89 88 bop 0 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y(kno)n(wn;)g(Theorem)g(3.5)f(and)i(its)f(pro)r(of)g(are)g(due)h (to)f(Bogolyub)r(o)n(v\025Krylo)n(v.)125 1582 y(The)36 b(results)h(of)f(Section)h(3.4)f(are)g(less)g(kno)n(wn,)j(but)e(rather) f(straigh)n(tforw)n(ard.)62 b(They)36 b(form)0 1682 y(a)30 b(`p)r(o)r(or)h(man's)f(Ito)h(form)n(ula')f(for)g(the)h(kic)n (k\025forced)e(NSE)j(\(their)f(coun)n(ter\025parts)e(for)h(the)h (white\025)0 1781 y(forced)k(equation)g(in)h(Sections)f(4.3)g(and)g (10.1)f(are)h(obtained)g(b)n(y)g(direct)h(application)f(of)g(the)h(Ito) 0 1881 y(form)n(ula\).)125 2021 y(The)21 b(results)h(of)f(Sections)h (4.1\0254.3)d(in)j FJ(Chapter)k(4)c FI(go)e(bac)n(k)h(to)h(the)g (thesis)g(of)g(Viot)f([Vio76)o(])h(and)0 2120 y(the)30 b(b)r(o)r(ok)f([VF88)o(].)42 b(Namely)-7 b(,)30 b(it)g(is)f(pro)n(v)n (ed)f(in)h(these)h(t)n(w)n(o)e(references)g(that)i(the)g(white)f (forced)g(2D)0 2220 y(NSE)j(de\034nes)f(a)f(Mark)n(o)n(v)f(pro)r(cess)g (in)i(a)g(suitable)f(function)i(space,)f(that)g(the)g(Ito)g(lemma)g (applies)0 2319 y(to)f(solutions)g(and)g(implies)g(the)h(a)f(priori)f (estimates)h(of)g(Theorem)g(4.7.)44 b(A)31 b(more)e(delicate)h(task)g (is)0 2419 y(to)h(c)n(hec)n(k)f(these)i(prop)r(erties)e(for)g(white)i (forces)e(of)h(lo)n(w)g(smo)r(othness.)47 b(Since)31 b(all)g(forces)f(in)i(nature)0 2519 y(are)g(smo)r(oth,)j(w)n(e)e(a)n(v) n(oid)f(the)h(men)n(tioned)h(di\036cult)n(y)f(b)n(y)g(assuming)g(that)g FH(B)2466 2531 y FD(1)2536 2519 y FG(:=)2656 2456 y Fz(P)2758 2519 y FF(j)p FH(s)p FF(j)2843 2489 y FD(2)2880 2519 y FH(b)2916 2489 y FD(2)2916 2539 y FA(s)2986 2519 y FH(<)f FF(1)p FI(,)0 2618 y(i.e.)54 b(that)33 b(the)g(forces)g(are)f (`smo)r(oth)g(enough'.)53 b(F)-7 b(or)33 b(the)g(case)g(of)g(forces)f (of)h(lo)n(w)f(smo)r(othness)g(see)0 2718 y(e.g.)46 b([Fla94)n(],)32 b([FM95)o(],)g([F)-7 b(er03)o(].)46 b(Also,)31 b(see)f([DD02])h(for)f (a)g(study)h(of)g(the)g(2D)g(NSE,)g(driv)n(en)f(b)n(y)g(a)0 2818 y(`v)n(ery)i(rough')g(force.)53 b(If)34 b(the)f(co)r(e\036cien)n (ts)g FH(b)1424 2830 y FA(s)1492 2818 y FI(deca)n(y)g(exp)r(onen)n (tially)f(with)i FF(j)p FH(s)p FF(j)p FI(,)g(then)g(the)g(force)e(is)0 2917 y(analytic)c(in)h FH(x)p FI(,)g(and)g(solutions)e(are)h(analytic)g (with)h(probabilit)n(y)e(one)h(\(see)h([BKL00)n(],)g([Shi02]\).)40 b(In)0 3017 y(this)28 b(case)g(the)g(equation)f(can)h(b)r(e)h(studied)f (in)g(the)h(space)e(of)h(analytical)f(functions.)39 b(It)28 b(seems)g(that)0 3116 y(this)34 b(option)f(giv)n(es)g(no)g(serious)g (tec)n(hnical)g(adv)-5 b(an)n(tages)32 b(and)h(do)r(es)h(not)f(lead)h (to)f(new)h(in)n(teresting)0 3216 y(results.)i(In)27 b(three)g(space)f(dimensions)g(situation)h(b)r(ecomes)f(signi\034can)n (tly)g(more)g(complicated)h(due)0 3316 y(to)35 b(the)h(luc)n(k)f(of)g (uniqueness.)60 b(F)-7 b(or)35 b(results,)i(a)n(v)-5 b(ailable)34 b(for)h(the)h(randomly)e(forced)h(3D)g(NSE)h(see)0 3415 y([VF88)o(,)28 b(FMR)-7 b(T01,)27 b(MR03],)g(references)g(in)g (these)h(w)n(orks,)e(and)h(the)h(review)f([Fla05)o(].)125 3515 y(The)g(estimates)g(of)g(Exercise)f(4.10)g(are)g(less)h (traditional.)36 b(F)-7 b(or)26 b(their)i(pro)r(ofs)e(see)h([KS03)n(].) 37 b(The-)0 3615 y(orem)25 b(4.11)g(is)h(w)n(ell)f(kno)n(wn)h(and)f(ma) n(y)h(b)r(e)g(found,)h(e.g.)36 b(in)26 b([VF88)o(].)37 b(Theorem)25 b(4.13)g(b)r(elongs)g(to)h(the)0 3714 y(large)e(group)f (of)i(results)g(in)g(mo)r(dern)g(sto)r(c)n(hastic)f(PDE)j(\(SPDE\),)g (motiv)-5 b(ated)25 b(b)n(y)g(n)n(umerical)f(meth-)0 3814 y(o)r(ds,)32 b(since)g(a)f(p)r(opular)f(w)n(a)n(y)h(to)g (calculate)g(solutions)f(of)i(a)f(white)h(forced)f(equation)f(is)i(to)f (replace)0 3913 y(it)d(b)n(y)f(a)g(kic)n(k-forced)e(equation.)36 b(In)28 b(n)n(umerical)e(analysis)g(this)h(is)h(called)e(the)i FC(splitting)i(up)f(metho)l(d)p FI(.)0 4013 y(F)-7 b(or)32 b(some)h(linear)f(and)h(quasilinear)e(SPDE)k(the)e(rate)f(of)h(con)n(v) n(ergence)e(for)h(this)i(metho)r(d)f(can)g(b)r(e)0 4113 y(estimated,)28 b(see)f([GK03)o(].)125 4212 y(All)e(results)g(of)g FJ(Chapter)k(5)c FI(are)g(w)n(ell)f(kno)n(wn)h(and)g(can)g(b)r(e)g (found)h(in)f(b)r(o)r(oks)f(\(some)h(references)0 4312 y(are)j(giv)n(en)f(in)i(the)g(main)f(text\).)41 b(The)28 b(statemen)n(t)h(in)f(Section)h(5.3,)f(whic)n(h)g(w)n(e)g(call)g(the)h (Dobrushin)0 4412 y(lemma,)38 b(seems)e(to)g(b)r(e)h(a)f(Mosco)n(w)f (folklore)g(result)g(from)h(1960's.)61 b(Our)36 b(attempts)g(to)g (trace)g(its)0 4511 y(in)n(v)n(en)n(tor)24 b(failed.)36 b(Roland)25 b(Dobrushin)h(w)n(as)e(the)i(\034rst)f(who)g (systematically)g(used)g(this)h(result)f(\(in)h(a)0 4611 y(di\033eren)n(t,)i(but)g(equiv)-5 b(alen)n(t)27 b(form\))h(in)f(his)h (celebrated)f(w)n(orks)f(on)h(the)h(Gibbs)g(systems.)125 4710 y(The)23 b(main)g(result)f(of)h FJ(Chapter)28 b(6)p FI(,)c(Theorem)e(6.4,)h(essen)n(tially)f(\(i.e.,)i(without)g(sp)r (ecifying)f(that)0 4810 y(the)c(rate)f(of)g(con)n(v)n(ergence)e(to)i (the)h(stationary)e(measure)g(is)i(exp)r(onen)n(tial\))f(is)g(pro)n(v)n (ed)f(b)n(y)h(A.)h(Shiriky)n(an)1553 5059 y(89)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 90 89 bop 0 523 a FI(and)34 b(the)h(author)f(in)g([KS00)o (])h(\(see)f(also)f([KS02b)o(]\).)58 b(This)35 b(pap)r(er)f(w)n(as)f (written)i(during)f(the)h(y)n(ear)0 623 y(1999,)25 b(when)h(its)h (approac)n(h)d(w)n(as)h(discussed)h(at)h(a)f(n)n(um)n(b)r(er)g(of)g (informal)g(seminars)f(in)h(Heriot-W)-7 b(att)0 722 y(Univ)n(ersit)n(y) g(.)55 b(A)n(t)34 b(the)h(end)f(of)g(that)g(y)n(ear)e(the)j(\034rst)e (talk)h(of)g(the)g(results)g(obtained)f(w)n(as)g(giv)n(en)g(at)0 822 y(a)e(meeting)g(of)h(Mosco)n(w)e(Mathematical)h(So)r(ciet)n(y)-7 b(,)32 b(and)f(a)g(preprin)n(t)g(of)g(the)h(pap)r(er)f(app)r(eared.)47 b(T)-7 b(o)0 922 y(pro)n(v)n(e)25 b(the)i(results)f(w)n(e)g(in)g([KS00) o(])h(used)f(a)g(F)-7 b(oias\025Pro)r(di)25 b(t)n(yp)r(e)i(reduction)f (of)h(the)f(NSE)i(\(3.3\))e(to)g(an)0 1021 y FH(N)9 b FI(-dimensional)25 b(system)g(with)i(dela)n(y)-7 b(,)25 b(whic)n(h)h(is)f(satis\034ed)h(b)n(y)f(the)i(v)n(ector,)d(formed)i(b)n (y)f(the)i(\034rst)e FH(N)0 1121 y FI(F)-7 b(ourier)30 b(comp)r(onen)n(ts)f(of)i(a)f(solution.)45 b(The)31 b(new)g(system)f (turned)h(out)f(to)h(b)r(e)g(of)f(the)h(Gibbs)g(t)n(yp)r(e)0 1220 y(\(similar)c(systems)g(are)g(considered,)f(sa)n(y)-7 b(,)27 b(in)h([Rue68)o(,)f(Bo)n(w75)n(,)h(Sin72)o(]\).)38 b(Since)1225 1396 y FH(b)1261 1408 y FA(s)1319 1396 y FF(6)p FG(=)23 b(0)165 b FF(8)14 b(j)p FH(s)p FF(j)23 b(\024)g FH(N)t(;)1013 b FI(\(12.1\))0 1571 y(then)22 b(the)g(noise,)h(whic)n(h)e(stirs)h(it,)h(is)f(non-degenerate.)33 b(Therefore,)21 b(due)h(to)g(a)f(Ruelle\025t)n(yp)r(e)h(theorem)0 1670 y(\(pro)n(v)n(ed)28 b(in)i([KS00)o(]\))g(the)g(new)f(system)h(has) f(a)g(unique)h(in)n(v)-5 b(arian)n(t)28 b(\(\020Gibbs\021\))i(measure,) f(and)g(\(3.3\))0 1770 y(has)35 b(a)h(unique)g(stationary)e(measure)h FH(\026)p FI(.)61 b(W)-7 b(e)36 b(note)g(that)g(imp)r(ortance)f(of)h (the)g(Gibbs)g(measures)0 1870 y(for)e(the)h(study)g(of)g(sto)r(c)n (hastic)e(pro)r(cesses,)i(de\034ned)g(b)n(y)g(randomly)e(forced)h (nonlinear)g(PDE,)i(w)n(as)0 1969 y(adv)n(o)r(cated)26 b(b)n(y)i(Y)-7 b(a.)27 b(Sinai)h(in)f(1990's.)35 b(E.g.,)28 b(see)f(his)h(w)n(ork)e([Sin91)o(])i(on)f(the)h(Burgers)e(equation.)125 2069 y(The)34 b(condition)g(\(12.1\))o(,)i(whic)n(h)e(guaran)n(tees)e (the)i(non-degeneracy)e(of)i(the)h(reduced)e(system,)0 2168 y(is)28 b(crucial)f(for)g(all)h(w)n(orks)e(on)i(stationary)f (measures)f(for)i(randomly)e(forced)i(PDE's,)h(written)f(after)0 2268 y([KS00)o(])f(up)h(to)g(no)n(w)f(\(cf.)37 b(b)r(elo)n(w)27 b(discussion)g(of)h(the)g(w)n(ork)e([HM04)o(]\).)125 2368 y(In)21 b([KS02b)o(],)i([KS01a)n(])e(and)g([KPS02])g(w)n(e)g(dev)n (elop)r(ed)g(a)g(coupling-approac)n(h)d(to)j(study)g(the)h(NSE)0 2467 y(and)30 b(related)f(nonlinear)g(PDE.)i(This)f(approac)n(h)e(uses) h(not)h(the)h(F)-7 b(oias\025Pro)r(di)28 b(reduction,)i(but)h(the)0 2567 y(main)f(lemma)h(the)g(reduction)f(is)g(based)g(up)r(on,)h(whic)n (h)g(w)n(as)e(\034nally)h(transformed)f(to)i(Lemma)f(6.1)0 2667 y(of)c(this)h(b)r(o)r(ok.)36 b(It)26 b(giv)n(es)f(a)h(shorter)f (pro)r(of)g(of)i(the)f(uniqueness)g(and)g(implies)h(that)f(the)h(con)n (v)n(ergence)0 2766 y(to)37 b(the)g(stationary)e(measure)h(is)g(exp)r (onen)n(tially)g(fast.)65 b(The)36 b(pro)r(of)h(w)n(as)e(clari\034ed)h (in)h([Kuk02b)o(],)0 2866 y(using)23 b(the)i(tec)n(hnique)f(of)f(the)i (Kan)n(toro)n(vic)n(h)c(functionals.)35 b(The)24 b(pro)r(of)f(of)h (Theorem)f(6.4,)h(presen)n(ted)0 2965 y(in)38 b(Section)f(6.3,)i(is)e (based)g(on)g(that)h(pap)r(er.)66 b(Theorem)36 b(6.4)h(remains)f(true)h (for)g(the)h(NSE,)g(p)r(er-)0 3065 y(turb)r(ed)f(b)n(y)f(un)n(b)r (ounded)g(random)g(kic)n(ks,)i(see)e([KS01b)n(],)j([Shi04)o(].)64 b(A)37 b(bit)f(later,)i(indep)r(enden)n(tly)0 3165 y(from)24 b([KS01a)o(],)h(a)g(similar)e(coupling-approac)n(h)f(to)j(study)g(the)g (randomly)f(kic)n(k)n(ed)f(NSE)j(w)n(as)e(dev)n(el-)0 3264 y(op)r(ed)k(b)n(y)f(Masmoudi\025Y)-7 b(oung)26 b([MY02)o(].)125 3364 y(Theorem)34 b(6.4)h(and)g(its)g(pro)r(of)g(remain)g(v)-5 b(alid)35 b(for)g(a)g(large)f(class)g(of)i(nonlinear)e(PDE,)j(forced)0 3464 y(b)n(y)28 b(random)e(kic)n(ks.)37 b(In)28 b(particular,)f(for)g (the)i(equations)e(\(0.2\))o(\025\(0.4\))g(and)h(for)f(the)i(2D)e(NSE)i (under)0 3563 y(the)h(Diric)n(hlet)g(b)r(oundary)g(conditions.)43 b(See)30 b([KS00)o(],)h([KS01a)n(],)g([KPS02)o(])f(for)g(the)g(corresp) r(onding)0 3663 y(abstract)c(theorems.)125 3762 y(Existence)i(of)g(a)f (random)g(attractor)f(for)i(a)f(randomly)g(forced)g(2D)h(NSE)h(w)n(as)e (\034rst)g(established)0 3862 y(in)h([CF94)o(].)37 b(F)-7 b(or)27 b(more)g(recen)n(t)g(results)g(see)g([Arn98)o(])g(and)h (references)e(therein.)125 3962 y(The)f(main)g(theorem)g(of)h FJ(Chapter)k(7)p FI(,)c(Theorem)e(7.1,)h(has)g(a)g(longer)f(history)-7 b(.)36 b(The)25 b(\034rst)g(result)0 4061 y(on)d(the)h(uniqueness)f(of) g(a)g(stationary)f(measure)g(for)h(the)h(white-forced)f(2D)g(NSE)h(is)g (due)f(to)g(Flandoli)0 4161 y(and)32 b(Maslo)n(wski)f([FM95])h(who)g (established)h(it)g(for)f(equation)g(\(7.1\))o(,)i(where)e(the)h(force) f FH(\020)39 b FI(is)33 b(non-)0 4261 y(smo)r(oth)27 b(in)h(the)g(space)f(v)-5 b(ariable)26 b FH(x)p FI(,)j(i.e.)1019 4436 y FH(C)1084 4401 y Fw(\000)p FD(1)1174 4436 y FF(j)p FH(s)p FF(j)1259 4401 y Fw(\000)p FD(1)1371 4436 y FF(\024)23 b FH(b)1495 4448 y FA(s)1553 4436 y FF(\024)g FH(C)6 b FF(j)p FH(s)p FF(j)1791 4401 y Fw(\000)1853 4379 y Ft(3)p 1853 4388 29 3 v 1853 4421 a(4)1891 4401 y Fw(\000)p FA(")2061 4436 y FF(8)p FH(s;)808 b FI(\(12.2\))0 4611 y(with)40 b(some)f FH(")k(>)g FG(0)p FI(,)f FH(C)49 b FF(\025)43 b FG(1)p FI(.)73 b(The)40 b(result)f(follo)n(ws)f(b)n(y)i (applying)f(a)g(v)n(ersion)f(of)i(the)g(classical)0 4710 y(Do)r(ob)35 b(theorem)f(to)h(a)f(Mark)n(o)n(v)e(pro)r(cess,)k(whic)n (h)f(the)g(equation)f(de\034nes)h(in)g(a)f(Sob)r(olev)g(space)g(of)0 4810 y(lo)n(w)29 b(smo)r(othness.)42 b(The)30 b(assumption)f(\(12.2\))f (w)n(as)h(w)n(eak)n(ened)f(in)i([F)-7 b(er99)o(].)43 b(In)30 b([BKL01)n(])g(a)f(similar)1553 5059 y(90)p eop end PStoPSsaved restore %%Page: (90,91) 46 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 91 90 bop 0 523 a FI(result)31 b(w)n(as)f(obtained)h(for) g(the)g(case)g(when)g(the)h(estimates)f(\(12.2\))f(are)g(replaced)h(b)n (y)g(exp)r(onen)n(tial)0 623 y(b)r(ounds.)37 b(The)26 b(men)n(tioned)g(results)g(are)f(not)h(quite)h(satisfactory)d(since)i (it)h(is)f(unnatural)g(to)g(imp)r(ose)0 722 y(a)h(lo)n(w)n(er)f(b)r (ound)i(on)f(the)h(energy)f(of)g(the)h(noise)f(in)h(eac)n(h)f(F)-7 b(ourier)26 b(mo)r(de.)125 822 y(In)31 b([Mat99)o(])g(the)h(uniqueness) f(of)g(a)g(stationary)e(measure)i(for)f(equation)h(\(7.1\))g(w)n(as)f (pro)n(v)n(ed)g(in)0 922 y(the)e(laminar)f(case)623 857 y FF(p)p 692 857 101 4 v 65 x FH(B)755 934 y FD(0)816 922 y FF(\034)c FH(\027)5 b FI(.)125 1021 y(After)40 b(the)g(w)n(ork)f([KS00)o(])h(on)g(the)g(kic)n(k-forced)e(NSE,)k(E,)e (Mattingly)-7 b(,)43 b(Sinai)d([EMS01])g(and)0 1121 y(Bricmon)n(t,)25 b(Sprained,)g(Lefev)n(ere)f([BKL02)n(])h(applied)g(the)g(F)-7 b(oias\025Pro)r(di)24 b(reduction)h(as)f(in)h([KS00)o(])g(to)0 1220 y(study)j(the)g(equation)f(\(7.1\))g(for)g(the)h(case)f(when)g FH(\020)34 b FI(is)28 b(a)f(\034nite)h(trigonometrical)e(p)r (olynomial,)1180 1409 y FH(\020)k FG(=)1376 1330 y Fz(X)1334 1512 y Fw(j)p FA(s)p Fw(j\024)p FA(N)1516 1496 y Fr(0)1551 1409 y FH(b)1587 1421 y FA(s)1622 1409 y FH(\014)1669 1421 y FA(s)1705 1409 y FG(\()p FH(t)p FG(\))p FH(e)1838 1421 y FA(s)1874 1409 y FG(\()p FH(x)p FG(\))p FH(;)0 1678 y FI(satisfying)c(\(7.2\))f(\()p FH(N)673 1648 y Fw(0)722 1678 y FI(is)g(an)n(y)f(\034nite)i(n)n(um)n(b)r(er,)g(bigger)e (than)h FH(N)9 b FI(\).)36 b(In)26 b([EMS01])f(it)h(w)n(as)e(pro)n(v)n (ed)g(that)0 1778 y(the)30 b(equation)f(has)g(a)g(unique)h(stationary)e (measure,)h(while)h(in)f([BKL02)o(])g(it)h(w)n(as)f(also)f(established) 0 1877 y(that)927 1977 y FF(B)982 1989 y FA(t)1011 1977 y FH(f)9 b FG(\()p FH(u)p FG(\))23 b FF(!)g FG(\()p FH(f)t(;)28 b(\026)p FG(\))83 b FI(exp)r(onen)n(tially)27 b(fast)p FH(;)715 b FI(\(12.3\))0 2120 y(for)28 b FH(\026)p FI(\025a.a.)40 b FH(u)25 b FF(2)h FH(H)36 b FI(\(it)30 b(w)n(as)e(p)r(oin)n(ted)h(out) g(in)g([Bri02)n(])h(that)f(the)g(argumen)n(ts)f(of)h([BKL02)n(],)h(in)f (fact,)0 2220 y(imply)i(the)g(con)n(v)n(ergence)d(for)i(all)h FH(u)p FI(\).)46 b(W)-7 b(e)31 b(note)f(that)h(the)g(uniqueness)f(of)h (a)f(stationary)f(measure)0 2319 y(is)34 b(a)f(prop)r(ert)n(y)f(of)i (the)g(equation,)h(whic)n(h)e(is)h(essen)n(tially)e(w)n(eak)n(er)g (then)i(the)g(exp)r(onen)n(tial)f(mixing)0 2419 y(\(7.3\):)42 b(the)31 b(former)e(follo)n(ws)h(from)g(the)g(latter,)h(but)g(do)r(es)f (not)h(implies)f(it,)i(as)d(w)n(ell)i(as)e(it)i(do)r(es)f(not)0 2519 y(imply)e(the)g(consequences)e(from)h(\(7.3\),)h(discussed)f(in)g (Chapters)g(8\0259.)125 2618 y(In)20 b([Mat02)n(])g(the)h(con)n(v)n (ergence)c(\(12.3\))i(w)n(as)g(pro)n(v)n(ed)f(for)h(all)h FH(u)p FI(.)34 b(The)20 b(pro)r(of)f(uses)g(the)h(F)-7 b(oias\025Pro)r(di)0 2718 y(reduction,)32 b(whic)n(h)f(follo)n(ws)f(b)n (y)h(some)f(coupling)h(argumen)n(ts.)46 b(Unfortunately)-7 b(,)32 b(w)n(e)f(found)h(it)f(v)n(ery)0 2818 y(di\036cult)21 b(to)f(follo)n(w)g(the)h(argumen)n(ts)e(of)h(that)h(w)n(ork.)33 b(In)20 b(the)h(pap)r(ers)f([EH01,)g(Hai02)o(])h(the)f(uniqueness)0 2917 y(of)j(a)h(stationary)d(measure)i(is)g(obtained)g(for)g(a)h(class) e(of)i(randomly)e(p)r(erturb)r(ed)i(parab)r(olic)e(problems)0 3017 y(with)28 b(strong)e(nonlinear)h(dissipation,)g(including)h(the)f (Ginzburg\025Landau)f(equation)h(\(0.3\).)125 3116 y(In)33 b([KS02a)o(])h(it)g(is)f(sho)n(wn)g(that)h(the)g(ideas,)h(dev)n(elop)r (ed)e(earlier)g(in)h([KS00)n(,)g(KS02a)n(])g(to)g(study)0 3216 y(the)k(kic)n(k)n(ed)f(equations,)i(apply)e(as)g(w)n(ell)g(in)h (the)g(white\025forced)f(case)g(\(without)h(the)g(unpleasan)n(t)0 3316 y(restriction)e(that)h FH(b)632 3328 y FA(s)707 3316 y FG(=)h(0)f FI(if)g FF(j)p FH(s)p FF(j)i FH(>)g(N)1278 3286 y Fw(0)1301 3316 y FI(\).)66 b(T)-7 b(o)37 b(implemen)n(t)g(the)h (ideas)e(some)h(tec)n(hnical)f(lemmas)0 3415 y(from)i([EMS01])g(w)n (ere)g(used.)69 b(The)39 b(sc)n(heme)e(of)i(this)g(pro)r(of)e(is)i (presen)n(ted)e(in)i(Section)f(7.1)g(\(and)0 3515 y(Theorem)24 b(7.1)h(is)g(the)g(main)g(result)g(of)g([KS02a)o(]\).)36 b(The)25 b(results)g(of)g([KS02a)n(])h(apply)e(to)h(the)h(2D)f(NSE)0 3615 y(b)r(oth)30 b(under)g(the)h(p)r(erio)r(dic)e(and)h(the)g(Diric)n (hlet)g(b)r(oundary)g(conditions.)43 b(A)30 b(simpli\034ed)h(v)n (ersion)d(of)0 3714 y(the)g(pro)r(of)f(see)g(in)h([Shi05)o(].)125 3814 y(A)d(disadv)-5 b(an)n(tage)23 b(of)j(Theorems)e(7.1)g(and)h(6.4)f (is)h(that)h(the)f(n)n(um)n(b)r(er)g FH(N)34 b FI(of)25 b(activ)n(e)g(mo)r(des)g(of)g(the)0 3913 y(force)30 b FH(\021)k FI(gro)n(ws)29 b(inde\034nitely)j(as)e FH(\027)k FF(!)29 b FG(0)p FI(.)46 b(This)31 b(assumption)g(is)f(not)h(to)r(o)g (restrictiv)n(e)f(for)g(ph)n(ysical)0 4013 y(applications)23 b(since)h(the)g(forces)f(in)h(nature)f(ha)n(v)n(e)g(energy)g(in)h(eac)n (h)f(F)-7 b(ourier)23 b(mo)r(de,)h(so)g FH(b)2767 4025 y FA(s)2825 4013 y FF(6)p FG(=)f(0)g FI(for)g(all)0 4113 y FH(s)i FI(and)h(\(12.1\))e(holds)h(for)g(eac)n(h)g FH(N)9 b FI(.)36 b(But)26 b(the)f(assumption)g(is)h(violated)e(in)i(n)n (umerical)e(exp)r(erimen)n(ts,)0 4212 y(where)34 b(usually)g(the)h (force)f(is)g(a)h(\034nal)f(trigonometric)f(p)r(olynomial.)57 b(So)35 b(it)g(is)f(desirable)g(to)g(ha)n(v)n(e)0 4312 y(a)i(v)n(ersion)e(of)i(the)h(theorem,)h(where)d FH(N)45 b FI(is)36 b(a)g(constan)n(t,)i(indep)r(enden)n(t)e(of)h FH(\027)5 b FI(.)62 b(Recen)n(tly)36 b(serious)0 4412 y(progress)27 b(in)j(this)g(direction)f(w)n(as)g(claimed)g(in)h([HM04)o (]:)41 b(the)30 b(main)g(theorem)f(of)h(this)f(w)n(ork)g(states)0 4511 y(that)39 b(eq.)f(\(7.1\))g(has)g(a)g(unique)h(stationary)e (measure)g(if)i FH(b)1925 4523 y FA(s)2002 4511 y FF(6)p FG(=)i(0)d FI(for)g(all)g FF(j)p FH(s)p FF(j)j(\024)g FG(2)p FI(.)69 b(The)39 b(pro)r(of)0 4611 y(uses)32 b(essen)n(tially)f (the)h(results)g(of)g(the)h(w)n(ork)d([MP04],)k(where)d(it)i(is)f(sho)n (wn)f(\(using)h(the)h(Mallia)n(vin)0 4710 y(calculus\))28 b(that)h(if)h FH(b)642 4722 y FA(s)702 4710 y FF(6)p FG(=)24 b(0)29 b FI(for)f(all)g FF(j)p FH(s)p FF(j)d(\024)g FG(2)p FI(,)k(then)g(an)n(y)f(solution)g FH(u)p FG(\()p FH(t;)14 b(x)p FG(\))29 b FI(of)35 b(\(7.1\))29 b(is)f(suc)n(h)h(that)g (the)0 4810 y(distribution)c(of)f(the)i(v)n(ector,)e(formed)g(b)n(y)h (its)f(F)-7 b(ourier)24 b(harmonics)g FH(u)2181 4822 y FA(s)2216 4810 y FG(\()p FH(t)p FG(\))p FI(,)i FF(j)p FH(s)p FF(j)d(\024)f FH(N)34 b FI(\()p FH(N)g FI(is)25 b(the)g(same)1553 5059 y(91)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 92 91 bop 0 523 a FI(as)34 b(in)g(\(12.1\)\),)i(is)f (absolutely)e(con)n(tin)n(uous)g(with)i(resp)r(ect)g(to)f(the)h(Leb)r (esgue)f(measure.)56 b(F)-7 b(or)34 b(the)0 623 y(pro)r(of)26 b(of)h(the)h(uniqueness)e(this)i(result)e(pla)n(ys)g(a)h(role,)f (similar)g(to)h(that)g(of)g(the)g(assumption)g(\(12.1\))o(.)0 722 y(W)-7 b(e)28 b(did)h(not)f(include)g(the)h(results)e(of)h([HM04)o (])h(in)f(the)g(main)g(text)h(since)e(w)n(e)h(are)f(w)n(aiting)g(till)i (they)0 822 y(are)e(prop)r(erly)f(c)n(hec)n(k)n(ed.)125 922 y(Theorem)g(7.3)h(is)g(pro)n(v)n(ed)f(in)i([Kuk02a)n(],)g(and)f (Theorem)g(7.4)g(\025)g(in)h([KS03)o(].)125 1021 y(Theorem)g(8.2)g(in) 58 b FJ(Chapter)34 b(8)29 b FI(is)g(a)g(classical)e(result.)41 b(Theorems)28 b(8.3)h(and)f(8.4)h(w)n(ere)f(pro)n(v)n(ed)0 1121 y(in)g([Kuk02a)n(])g(and)g(in)g([Shi06].)38 b(W)-7 b(e)28 b(note)g(that)g(the)g(pro)r(of)g(of)f(Theorem)h(8.3,)f(giv)n(en) g(in)h(Chapter)f(8,)0 1220 y(follo)n(ws)f(from)h(Corollary)e(6.5.)35 b(That)27 b(statemen)n(t)h(is)f(equiv)-5 b(alen)n(t)26 b(to)h(the)h(exp)r(onen)n(tial)f(con)n(v)n(ergence)0 1320 y(of)36 b(distributions)f(of)h(solutions)f(to)g(the)i(stationary)d (measure)g(in)i(the)g(Prokhoro)n(v)e(distance)h(\(see)0 1420 y(Exercise)d(6.7\).)49 b(As)33 b(it)f(is)g(p)r(oin)n(ted)h(out)f (at)g(the)g(end)h(of)f(Section)g(6.3,)g(this)h(con)n(v)n(ergence,)d(in) j(fact,)0 1519 y(follo)n(ws)27 b(from)g(\(6.11\))o(.)125 1619 y(The)i(CL)-7 b(T)30 b(from)g FJ(Chapter)35 b(9)30 b FI(w)n(as)e(\034rst)i(pro)n(v)n(ed)e(in)i([Kuk02a)n(])h(b)n(y)e (comparing)f FH(f)9 b FG(\()p FH(u)p FG(\()p FH(s)p FG(\)\))31 b FI(with)0 1719 y FH(f)9 b FG(\()p FH(U)g FG(\()p FH(t)p FG(\)\))29 b FI(where)e FH(U)9 b FG(\()p FH(t)p FG(\))29 b FI(is)f(a)g(\(w)n(eak\))g(stationary)e(solution,)i(and)g(ev)n(oking)f (the)i(CL)-7 b(T)28 b(for)g(stationary)0 1818 y(pro)r(cesses)k(from)g ([Dur91].)53 b(The)34 b(martingale)d(appro)n(ximation)h(\(9.5\))h(is)g (a)f(kno)n(wn)h(to)r(ol)g(to)g(study)0 1918 y(Mark)n(o)n(v)e(pro)r (cesses)g(and)i(RDS)h(\(brough)n(t)e(to)h(our)f(atten)n(tion)h(b)n(y)f (Y)-7 b(u.)34 b(Kifer\).)53 b(The)33 b(martingale\025)0 2017 y(pro)r(of)27 b(of)h(the)g(theorem)g(is)f(tak)n(en)h(from)f ([Shi06],)h(where)f(the)h(rate)f(of)h(con)n(v)n(ergence)e(to)h(the)i (limit)f(is)0 2117 y(giv)n(en.)36 b(W)-7 b(e)28 b(do)f(not)h(doubt)g (that)f(in)h(\(9.2\))k FG(^)-46 b FH(\033)26 b(>)d FG(0)p FI(,)k(but)h(this)g(fact)g(is)f(not)h(pro)n(v)n(ed)e(y)n(et.)125 2217 y(The)35 b(Eulerian)h(limit)g(\(Theorem)f(10.1\))g(from)g FJ(Chapter)42 b(10)35 b FI(w)n(as)g(established)g(in)h([Kuk04)o(].)0 2316 y(The)28 b(results)f(in)i(Section)e(10.2)g(are)g(direct)h (consequence)f(from)g(the)h(con)n(v)n(ergence)e(\(10.21\))n(,)j(pro)n (v)n(ed)0 2416 y(in)j([Bir06)n(].)48 b(It)32 b(is)f(sho)n(wn)f(in)i ([KS04b)o(])f(that)h(an)f(analogy)e(of)i(Theorem)g(10.1)f(holds)h(for)f (the)i(CGL)0 2516 y(equation)27 b(\(0.4\).)37 b(More)27 b(generally)-7 b(,)26 b(similar)h(results)g(ma)n(y)g(b)r(e)h (established)g(for)f(man)n(y)g(other)g(PDEs)0 2615 y(of)h(the)f(form) 528 2798 y FF(h)14 b FI(Hamiltonian)28 b(PDE)15 b FF(i)k FG(+)f FH(\027)5 b FF(h)14 b FI(damping)f FF(i)24 b FG(=)1938 2734 y FF(p)p 2008 2734 47 4 v 2008 2798 a FH(\027)19 b FF(h)14 b FI(random)26 b(force)13 b FF(i)p FH(;)317 b FI(\(12.4\))0 2980 y(where)29 b(the)g(r.h.s.)41 b(is)29 b(a)g(kic)n(k)14 b(-)28 b(or)g(a)h(white\025force,)g(and)g(the)g (Hamiltonian)g(PDE)i(has)d(at)h(least)g(t)n(w)n(o)0 3080 y(`go)r(o)r(d')e(in)n(tegrals)f(of)i(motion.)125 3180 y(The)h(equation)f(\(0.3\))g(has)h(the)g(form)g(\(12.4\))f(\(without)i (the)f(factor)2295 3120 y FF(p)p 2364 3120 V 60 x FH(\027)35 b FI(in)29 b(its)g(r.h.s.\),)g(but)h(the)0 3279 y(corresp)r(onding)g (Hamiltonian)i(PDE)i(is)e(degenerate,)g(as)g(w)n(ell)g(as)g(its)g(in)n (tegrals)f(of)h(motion:)46 b(they)0 3379 y(con)n(tain)35 b(no)g(partial)g(deriv)-5 b(ativ)n(es)34 b(of)i FH(u)p FI(.)60 b(The)35 b(limiting)h(b)r(eha)n(viour)f(of)g(solution)g(as)g FH(\027)42 b FF(!)36 b FG(0)f FI(w)n(as)0 3479 y(studies)26 b(in)g([Kuk97)n(,)g(Kuk99)o(],)g(using)f(di\033eren)n(t)h(ideas.)36 b(In)26 b(these)f(w)n(orks)f(lo)n(w)n(er)h(and)g(upp)r(er)h(b)r(ounds)0 3578 y(for)20 b(high)h(Sob)r(olev)f(norms)g(of)h(stationary)e (solutions)h(w)n(ere)g(obtained.)35 b(No)20 b(lo)n(v)n(er)f(b)r(ounds)i (for)g Fk(E)14 b FF(j)p FH(u)p FF(j)3108 3548 y FD(2)3108 3601 y FA(L)3154 3609 y Ft(2)0 3678 y FI(are)30 b(a)n(v)-5 b(ailable,)30 b(and)g(it)h(is)g(not)f(clear)g(ho)n(w)g(one)g(should)h (scale)e(the)i(force)f FH(\021)k FI(to)d(k)n(eep)f(this)h(quan)n(tit)n (y)0 3778 y FF(\030)23 b FG(1)k FI(as)g FH(\027)h FF(!)23 b FG(0)p FI(.)125 3877 y(The)30 b(results)g(of)h FJ(Chapter)36 b(11)30 b FI(are)g(v)n(ery)g(recen)n(t.)45 b(Theorem)30 b(11.3)g(is)g(pro)n(v)n(ed)f(in)i([KP05)o(])g(and)0 3977 y(Theorem)i(11.5)f(\026)i(in)g([Kuk05)n(].)56 b(The)34 b(lev)n(el)f(sets)g FG(\000\()p FH(\034)9 b FG(\))35 b FI(of)f(the)g(v)n(orticit)n(y)e(of)i(a)f(solution)h(for)f(the)0 4076 y FC(deterministic)21 b FI(Na)n(vier\025Stok)n(es)d(equation)i(in) h(the)f(2d)h(and)f(3d)g(cases,)h(and)f(of)g(solutions)g(for)g(equation) 0 4176 y(\(11.2\))27 b(without)g(assuming)g(that)h FH(\030)f FG(=)22 b(curl)14 b FH(u)27 b FI(\(but)h(imp)r(osing)f(certain)g (a-priori)f(b)r(ounds)h(on)g FH(u)g FI(and)0 4276 y FH(\030)t FI(\),)g(w)n(ere)f(studied)h(b)n(y)f(P)-7 b(.)28 b(Constan)n(tin)e(and) g(others,)g(e.g)g(see)h([Con90)n(,)g(CD96)o(].)37 b(There)26 b(the)h(areas)e(of)0 4375 y(the)31 b(sets)f FG(\000\()p FH(\034)9 b FG(\))32 b FI(are)d(estimated)i(\(with)g(and)g(without)f(a) n(v)n(eraging)e(in)j FH(t)f FI(and)g FH(\034)9 b FI(\),)33 b(as)d(w)n(ell)g(as)g(certain)0 4475 y(in)n(tegrals)c(o)n(v)n(er)g (these)h(sets.)1553 5059 y(92)p eop end PStoPSsaved restore %%Page: (92,93) 47 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 93 92 bop 0 523 a FK(References)0 705 y FI([AK01])158 b(V.)30 b(Arnold)g(and)g(B.)g(Khesin,)g FC(T)-6 b(op)l(olo)l(gic)l(al)34 b(Metho)l(ds)f(in)f(Hydr)l(o)l (dynamics)p FI(,)h(Springer-)415 804 y(V)-7 b(erlag,)27 b(Berlin,)g(2001.)0 965 y([Arn98])144 b(L.)28 b(Arnold,)f FC(R)l(andom)j(Dynamic)l(al)g(Systems)p FI(,)e(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1998.)0 1126 y([Bat82])152 b(G.)33 b(K.)g(Batc)n(helor,)g FC(The)j(The)l(ory)g(of)g(Homo)l(gene)l(ous)f(T) -6 b(urbulenc)l(e)p FI(,)34 b(Cam)n(bridge)e(Uni-)415 1226 y(v)n(ersit)n(y)26 b(Press,)h(Cam)n(bridge,)f(1982.)0 1386 y([Bil99])180 b(P)-7 b(.)25 b(Billingsley)-7 b(,)24 b FC(Conver)l(genc)l(e)j(of)g(Pr)l(ob)l(ability)i(Me)l(asur)l(es)p FI(,)c(John)f(Wiley)g(&)g(Sons,)g(New)415 1486 y(Y)-7 b(ork,)27 b(1999.)0 1647 y([Bir06])170 b(A.)27 b(Biryuk,)f FC(On)h(invariant)j(me)l(asur)l(es)e(of)h(the)g(Euler)g(e)l(quation)p FI(,)e(J.)f(Statist.)h(Ph)n(ys.,)f(to)415 1746 y(app)r(ear)h(\(2006\).) 0 1907 y([BKL00])109 b(J.)33 b(Bricmon)n(t,)g(A.)g(Kupiainen,)h(and)f (R.)g(Lefev)n(ere,)g FC(Pr)l(ob)l(abilistic)k(estimates)d(for)i(the)415 2007 y(two-dimensional)44 b(sto)l(chastic)g(Navier\025Stokes)g(e)l (quations)p FI(,)h(J.)c(Statist.)h(Ph)n(ys.)g FJ(100)415 2106 y FI(\(2000\),)26 b(743\025756.)0 2267 y([BKL01])p 415 2267 250 4 v 372 w(,)33 b FC(Er)l(go)l(dicity)j(of)f(the)f(2D)g (Navier\025Stokes)i(e)l(quations)e(with)g(r)l(andom)h(for)l(cing)p FI(,)415 2367 y(Comm.)28 b(Math.)f(Ph)n(ys.)h FJ(224)f FI(\(2001\),)f(65\02581.)0 2528 y([BKL02])p 415 2528 V 372 w(,)36 b FC(Exp)l(onential)h(mixing)f(for)g(the)h(2D)f(sto)l (chastic)g(Navier\025Stokes)i(dynamics)p FI(,)415 2627 y(Comm.)28 b(Math.)f(Ph)n(ys.)h FJ(230)f FI(\(2002\),)f(no.)h(1,)g (87\025132.)0 2788 y([Bog98])142 b(V.)24 b(I.)h(Bogac)n(hev,)d FC(Gaussian)27 b(Me)l(asur)l(es)p FI(,)e(Mathematical)f(Surv)n(eys)e (and)i(Monographs,)415 2888 y(v)n(ol.)j(62,)g(AMS,)h(Pro)n(vidence,)e (1998.)0 3048 y([Bo)n(w75])126 b(R.)29 b(Bo)n(w)n(en,)f FC(Equilibrium)k(States)e(and)h(the)g(Er)l(go)l(dic)h(The)l(ory)g(of)g (Anosov)f(Di\033e)l(omor-)415 3148 y(phisms)p FI(,)e(v)n(ol.)e(470,)f (Springer-V)-7 b(erlag,)25 b(Berlin,)i(1975.)0 3309 y([Bri02])170 b(J.)37 b(Bricmon)n(t,)j FC(Er)l(go)l(dicity)g(and)g(mixing)f(for)g (sto)l(chastic)h(p)l(artial)g(di\033er)l(ential)g(e)l(qua-)415 3408 y(tions)p FI(,)27 b(Pro)r(ceedings)e(of)h(the)g(In)n(ternational)f (Congress)f(of)i(Mathematicians)f(\(Beijing,)415 3508 y(2002\),)h(v)n(ol.)h(1,)g(Higher)g(Ed.)h(Press,)f(Beijing,)h(2002,)e (pp.)i(567\025585.)0 3669 y([BV92])164 b(A.)51 b(Babin)f(and)h(M.)f (Vishik,)57 b FC(A)n(ttr)l(actors)50 b(of)h(Evolutionary)i(Equations)p FI(,)k(North-)415 3768 y(Holland,)27 b(Amsterdam,)h(1992.)0 3929 y([CD96])162 b(P)-7 b(.)28 b(Constan)n(tin)e(and)h(Ch.)g(Do)r (ering,)g FC(He)l(at)i(tr)l(ansfer)g(in)g(c)l(onve)l(ctive)h(turbulenc) l(e)p FI(,)c(Non-)415 4029 y(linearit)n(y)i FJ(9)f FI(\(1996\),)f (1049\0251060.)0 4190 y([CDF97])108 b(H.)38 b(Crauel,)h(A.)f(Debussc)n (he,)i(and)d(F.)h(Flandoli,)i FC(R)l(andom)f(attr)l(actors)p FI(,)h(J.)d(Dynam.)415 4289 y(Di\033eren)n(tial)28 b(Equations)f FJ(9)g FI(\(1997\),)f(no.)i(2,)f(307\025341.)0 4450 y([CF88])171 b(P)-7 b(.)30 b(Constan)n(tin)f(and)g(C.)h(F)-7 b(oias,)29 b FC(Navier-Stokes)j(Equations)p FI(,)f(Univ)n(ersit)n(y)d(of)h (Chicago)415 4550 y(Press,)e(Chicago,)f(1988.)0 4710 y([CF94])171 b(H.)22 b(Crauel)e(and)h(F.)h(Flandoli,)g FC(A)n(ttr)l(actors)h(for)i(r)l(andom)g(dynamic)l(al)g(systems)p FI(,)e(Probab.)415 4810 y(Theory)j(Related)i(Fields)g FJ(100)e FI(\(1994\),)h(no.)g(3,)g(365\025393.)1553 5059 y(93)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 94 93 bop 0 523 a FI([Cha84])137 b(I.)36 b(Cha)n(v)n(el,)h FC(Eigenvalues)i(in)f(Riemannian)g(Ge)l(ometry)p FI(,)h(A)n(cademic)c(Press,)j(Boston,)415 623 y(1984.)0 789 y([Con90])137 b(P)-7 b(.)25 b(Constan)n(tin,)g FC(Navier\025Stokes) j(e)l(quations)f(and)g(ar)l(e)l(a)g(of)h(interfac)l(es)p FI(,)e(Comm.)e(Math.)415 888 y(Ph)n(ys.)k FJ(129)e FI(\(1990\),)h (241\025266.)0 1054 y([Cra91])150 b(H.)48 b(Crauel,)j FC(Markov)e(me)l(asur)l(es)f(for)g(r)l(andom)g(dynamic)l(al)i(systems)p FI(,)i(Sto)r(c)n(hastics)415 1154 y(Sto)r(c)n(hastics)27 b(Rep.)h FJ(37)f FI(\(1991\),)f(no.)h(3,)h(153\025173.)0 1320 y([Cra01])p 415 1320 250 4 v 413 w(,)41 b FC(R)l(andom)e(p)l(oint) h(attr)l(actors)f(versus)g(r)l(andom)h(set)f(attr)l(actors)p FI(,)i(J.)d(London)415 1420 y(Math.)28 b(So)r(c.)f(\(2\))h FJ(63)f FI(\(2001\),)f(no.)i(2,)f(413\025427.)0 1586 y([DD02])159 b(G.)38 b(Da)g(Prato)f(and)g(A.)h(Debussc)n(he,)i FC(Two-dimensional)h(Navier-Stokes)f(e)l(quations)415 1685 y(driven)27 b(by)f(a)h(sp)l(ac)l(e-time)f(white)h(noise)p FI(,)e(J)e(of)h(F)-7 b(unct.)24 b(Analysis)f FJ(196)f FI(\(2002\),)h(180\025210.)0 1851 y([Deb98])139 b(A.)21 b(Debussc)n(he,)g FC(Hausdor\033)i(dimension)i(of)e(a)h(r)l(andom)f (invariant)h(set)p FI(,)e(J.)e(Math.)g(Pures)415 1951 y(Appl.)28 b(\(9\))g FJ(77)f FI(\(1998\),)f(no.)i(10,)e(967\025988.)0 2117 y([Den04])139 b(D.)37 b(Deniso)n(v,)h FC(L)l(aw)g(of)g(iter)l(ate) l(d)g(lo)l(garithm)h(for)g(r)l(andomly)g(for)l(c)l(e)l(d)f(PDEs)p FI(,)h(Preprin)n(t)415 2217 y(\(2004\).)0 2383 y([DG95])157 b(Ch.)27 b(R.)g(Do)r(ering)e(and)i(J.D.)g(Gibb)r(on,)g FC(Applie)l(d)j(analysis)g(of)g(the)f(navier-stokes)g(e)l(qua-)415 2482 y(tions)p FI(,)f(Cam)n(bridge)e(Univ)n(ersit)n(y)h(Press,)g(1995.) 0 2648 y([Dud89])130 b(R.)40 b(M.)f(Dudley)-7 b(,)43 b FC(R)l(e)l(al)e(Analysis)g(and)g(Pr)l(ob)l(ability)p FI(,)k(W)-7 b(adsw)n(orth)38 b(&)h(Bro)r(oks/Cole,)415 2748 y(P)n(aci\034c)27 b(Gro)n(v)n(e,)f(California,)h(1989.)0 2914 y([Dur91])143 b(R.)36 b(Durret,)i FC(Pr)l(ob)l(ability:)56 b(The)l(ory)39 b(and)f(Examples)p FI(,)h(W)-7 b(adsw)n(orth)34 b(&)i(Bro)r(oks/Cole,)415 3014 y(P)n(aci\034c)27 b(Gro)n(v)n(e,)f (California,)h(1991.)0 3180 y([DZ92])171 b(G.)35 b(Da)f(Prato)g(and)g (J.)g(Zab)r(czyk,)h FC(Sto)l(chastic)i(Equations)f(in)g(In\034nite)g (Dimensions)p FI(,)415 3279 y(Cam)n(bridge)26 b(Univ)n(ersit)n(y)h (Press,)g(Cam)n(bridge,)f(1992.)0 3445 y([DZ96])171 b(G.)35 b(Da)f(Prato)g(and)h(J.)f(Zab)r(c)n(h)n(yk,)h FC(Er)l(go)l(dicity)j (for)f(In\034nite)e(Dimensional)i(Systems)p FI(,)415 3545 y(Cam)n(bridge)26 b(Univ)n(ersit)n(y)h(Press,)g(Cam)n(bridge,)f (1996.)0 3711 y([EH01])167 b(J.-P)-7 b(.)39 b(Ec)n(kmann)f(and)h(M.)f (Hairer,)j FC(Uniqueness)e(of)i(the)f(invariant)g(me)l(asur)l(e)g(for)h (a)415 3811 y(sto)l(chastic)30 b(PDE)g(driven)g(by)f(de)l(gener)l(ate)h (noise)p FI(,)e(Comm.)f(Math.)g(Ph)n(ys.)g FJ(219)f FI(\(2001\),)415 3910 y(523\025565.)0 4076 y([EKMS00])42 b(W.)22 b(E,)h(K.)e(Khanin,)i (A.)f(Mazel,)h(and)f(Y)-7 b(a.)22 b(G.)g(Sinai,)h FC(Invariant)i(me)l (asur)l(es)f(for)h(Bur)l(gers)415 4176 y(e)l(quation)30 b(with)g(sto)l(chastic)h(for)l(cing)p FI(,)e(Annals)e(of)h(Math.)f FJ(151)g FI(\(2000\),)f(877\025960.)0 4342 y([EMS01])107 b(W.)32 b(E,)h(J.)e(C.)h(Mattingly)-7 b(,)33 b(and)e(Y)-7 b(a.)32 b(G.)g(Sinai,)g FC(Gibbsian)k(dynamics)e(and)g(er)l(go)l (dicity)415 4442 y(for)26 b(the)g(sto)l(chastic)l(al)t(ly)i(for)l(c)l (e)l(d)e(Navier\025Stokes)h(e)l(quation)p FI(,)e(Comm.)e(Math.)g(Ph)n (ys.)g FJ(224)415 4541 y FI(\(2001\),)j(83\025106.)0 4707 y([F)-7 b(er99])168 b(B.)25 b(F)-7 b(errario,)24 b FC(Sto)l(chastic)k(Navier-Stokes)h(e)l(quations:)37 b(analysis)29 b(of)g(the)f(noise)g(to)g(have)415 4807 y(a)i(unique)f(invariant)h(me)l(asur)l(e)p FI(,)d(Ann.)g(Mat.)g(Pura)g (Appl.)h(\(4\))f FJ(177)g FI(\(1999\),)e(331\025347.)1553 5059 y(94)p eop end PStoPSsaved restore %%Page: (94,95) 48 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 95 94 bop 0 523 a FI([F)-7 b(er03])p 415 523 250 4 v 431 w(,)38 b FC(Uniqueness)f(r)l(esult)f(for)j(the)e(2D)g (Navier-Stokes)h(e)l(quation)g(with)g(additive)415 623 y(noise)p FI(,)28 b(Sto)r(c)n(hastics)f(and)h(Sto)r(c)n(hastics)e(Rep.) i FJ(75)f FI(\(2003\),)g(435\025442.)0 789 y([Fla94])166 b(F.)22 b(Flandoli,)h FC(Dissip)l(ativity)k(and)e(invariant)g(me)l (asur)l(es)f(for)i(sto)l(chastic)f(Navier-Stokes)415 888 y(e)l(quations)p FI(,)j(NoDEA)g FJ(1)g FI(\(1994\),)e(403\025423.)0 1054 y([Fla05])p 415 1054 V 429 w(,)48 b FC(A)n(n)43 b(intr)l(o)l(duction)i(to)f(3D)g(sto)l(chastic)i(\035uid)e(dynamics)p FI(,)49 b(CIME)c(Lecture)415 1154 y(Notes)27 b(\(2005\).)0 1320 y([FM95])155 b(F.)34 b(Flandoli)f(and)g(B.)h(Maslo)n(wski,)f FC(Er)l(go)l(dicity)k(of)f(the)f(2D)g(Navier-Stokes)h(e)l(quation)415 1420 y(under)30 b(r)l(andom)g(p)l(erturb)l(ations)p FI(,)d(Comm.)h (Math.)g(Ph)n(ys.)f FJ(172)g FI(\(1995\),)f(119\025141.)0 1586 y([FMR)-7 b(T01])41 b(C.)29 b(F)-7 b(oias,)29 b(O.)g(Manley)-7 b(,)30 b(R.)g(Rosa,)e(and)h(R.)h(T)-7 b(emam,)30 b FC(Navier-Stokes)i (Equations)f(and)415 1685 y(Turbulenc)l(e)p FI(,)d(Cam)n(bridge)e(Univ) n(ersit)n(y)g(Press,)h(Cam)n(bridge,)g(2001.)0 1851 y([F)-7 b(ri95])182 b(U.)28 b(F)-7 b(risc)n(h,)27 b FC(T)-6 b(urbulenc)l(e:)38 b(the)30 b(Le)l(gacy)g(of)h(A.)f(N.)f(Kolmo)l(gor)l(ov.)p FI(,)h(Cam)n(bridge)c(Univ)n(er-)415 1951 y(sit)n(y)h(Press,)g(Cam)n (bridge,)g(1995.)0 2117 y([Gal01])155 b(G.)24 b(Galla)n(v)n(otti,)g FC(F)-6 b(oundations)27 b(of)g(Fluid)g(Dynamics)p FI(,)e(Springer-V)-7 b(erlag,)23 b(Berlin,)h(2001.)0 2283 y([GG73])155 b(M.)34 b(Golubitsky)g(and)f(V.)i(Guillemin,)h FC(Stable)g(Mappings)h(and)f (Their)h(Singularities)p FI(,)415 2383 y(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1973.)0 2549 y([GK03])155 b(I.)30 b(Gy)n(ongy)f(and)h(N.)g(Krylo)n(v,)f FC(On)i(splitting)i(up)f(metho)l (d)g(and)h(sto)l(chastic)g(p)l(artial)g(dif-)415 2648 y(fer)l(ential)e(e)l(quations)p FI(,)c(Ann.)i(Prob.)e FJ(2)g FI(\(2003\),)g(564\025591.)0 2814 y([Hai02])158 b(M.)24 b(Hairer,)f FC(Exp)l(onential)k(mixing)f(pr)l(op)l(erties)h(of) g(sto)l(chastic)f(PDE's)h(thr)l(ough)f(asymp-)415 2914 y(totic)k(c)l(oupling)p FI(,)e(Probab.)f(Theory)g(Relat.)g(Fields)h FJ(124)f FI(\(2002\),)f(345\025380.)0 3080 y([HH80])161 b(P)-7 b(.)28 b(Hall)e(and)h(C.)g(C.)g(Heyde,)g FC(Martingale)j(Limit)g (The)l(ory)g(and)g(Its)e(Applic)l(ation)p FI(,)h(A)n(ca-)415 3180 y(demic)f(Press,)f(Boston,)g(1980.)0 3346 y([HM04])147 b(M.)21 b(Hairer)e(and)h(J.)h(Mattingly)-7 b(,)22 b FC(Er)l(go)l (dicity)j(of)f(the)f(2D)g(Navier-Stokes)h(e)l(quations)f(with)415 3445 y(de)l(gener)l(ate)30 b(sto)l(chastic)h(for)l(cing)p FI(,)e(Preprin)n(t)e(\(2004\).)0 3611 y([IW89])170 b(N.)34 b(Ik)n(eda)e(and)h(S.)h(W)-7 b(atanab)r(e,)35 b FC(Sto)l(chastic)g (Di\033er)l(ential)h(Equations)f(and)h(Di\033usion)415 3711 y(Pr)l(o)l(c)l(esses)p FI(,)28 b(North-Holland,)f(Amsterdam,)g (1989.)0 3877 y([KA77])158 b(L.)20 b(B.)f(Kan)n(toro)n(vic)n(h)e(and)j (G.)g(P)-7 b(.)20 b(Akilo)n(v,)h FC(F)-6 b(unctional)22 b(Analysis)h(\(in)f(Russian\))p FI(,)f(Nauk)-5 b(a,)415 3977 y(Mosco)n(w,)26 b(1977.)0 4143 y([Kha80])132 b(R.)38 b(Khasminski,)i FC(Sto)l(chastic)g(Stability)g(of)g(Di\033er)l(ential)g (Equations)p FI(,)h(Osijho\033)d(and)415 4242 y(No)r(ordho\033,)27 b(1980.)0 4408 y([Kif86])172 b(Y.)40 b(Kifer,)i FC(Er)l(go)l(dic)h(The) l(ory)f(of)g(Random)f(Tr)l(ansformations)p FI(,)j(Birkhauser,)d(Basel,) 415 4508 y(1986.)0 4674 y([KM80])144 b(R.)44 b(H.)f(Kraic)n(hnan)f(and) h(D.)g(Mon)n(tgomery)-7 b(,)46 b FC(Two-dimensional)g(turbulenc)l(e)p FI(,)h(Rep.)415 4774 y(Prog.)27 b(Ph)n(ys.)g FJ(43)g FI(\(1980\),)g(547\025619.)1553 5059 y(95)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 96 95 bop 0 523 a FI([KP05])164 b(S.)42 b(B.)g(Kuksin)f(and)h(O.)g(P)n(enrose,)i FC(A)f(family)h(of)g(b)l (alanc)l(e)g(r)l(elations)f(for)h(the)f(two-)415 623 y(dimensional)c(Navier\025Stokes)f(e)l(quations)f(with)g(r)l(andom)g (for)l(cing)p FI(,)i(J.)c(Stat.)g(Ph)n(ysics)415 722 y FJ(118)27 b FI(\(2005\),)f(437\025449.)0 888 y([KPS02])118 b(S.)33 b(B.)f(Kuksin,)h(A.)g(Piatnitski,)h(and)e(A.)h(Shiriky)n(an,)f FC(A)i(c)l(oupling)h(appr)l(o)l(ach)h(to)e(r)l(an-)415 988 y(domly)d(for)l(c)l(e)l(d)g(nonline)l(ar)f(PDE's.)h(II)p FI(,)c(Comm.)h(Math.)f(Ph)n(ys.)h FJ(230)f FI(\(2002\),)f(81\02585.)0 1154 y([Kry03])143 b(N.)38 b(V.)f(Krylo)n(v,)h FC(Intr)l(o)l(duction)g (to)g(the)h(The)l(ory)h(of)f(Di\033usion)g(Pr)l(o)l(c)l(esses)p FI(,)h(v)n(ol.)c(142,)415 1254 y(AMS)28 b(T)-7 b(ranslations)26 b(of)h(Mathematical)g(Monographs,)f(Pro)n(vidence,)h(RI,)g(2003.)0 1420 y([KS00])174 b(S.)27 b(B.)g(Kuksin)g(and)g(A.)g(Shiriky)n(an,)f FC(Sto)l(chastic)k(dissip)l(ative)h(PDE's)f(and)g(Gibbs)g(me)l(a-)415 1519 y(sur)l(es)p FI(,)d(Comm.)h(Math.)f(Ph)n(ys.)h FJ(213)f FI(\(2000\),)f(291\025330.)0 1685 y([KS01a])p 415 1685 250 4 v 395 w(,)h FC(A)h(c)l(oupling)h(appr)l(o)l(ach)i(to)d(r)l (andomly)i(for)l(c)l(e)l(d)f(nonline)l(ar)g(PDE's.)h(I)p FI(,)c(Comm.)415 1785 y(Math.)i(Ph)n(ys.)f FJ(221)g FI(\(2001\),)f (351\025366.)0 1951 y([KS01b])p 415 1951 V 391 w(,)54 b FC(Er)l(go)l(dicity)d(for)f(the)f(r)l(andomly)h(for)l(c)l(e)l(d)g(2D) f(Navier\025Stokes)i(e)l(quations)p FI(,)415 2051 y(Math.)28 b(Ph)n(ys.)f(Anal.)h(Geom.)f FJ(4)h FI(\(2001\),)e(147\025195.)0 2217 y([KS02a])p 415 2217 V 395 w(,)i FC(Coupling)k(appr)l(o)l(ach)g (to)e(white-for)l(c)l(e)l(d)i(nonline)l(ar)e(PDE's)p FI(,)f(J.)f(Math.)g(Pures)415 2316 y(Appl.)g FJ(81)f FI(\(2002\),)g(567\025602.)0 2482 y([KS02b])p 415 2482 V 391 w(,)f FC(On)i(dissip)l(ative)i(systems)e(p)l(erturb)l(e)l(d)g(by) h(b)l(ounde)l(d)g(r)l(andom)f(kick-for)l(c)l(es)p FI(,)g(Er-)415 2582 y(go)r(dic)f(Theory)f(and)i(Dynamical)f(Systems)g FJ(22)h FI(\(2002\),)e(1487\0251495.)0 2748 y([KS03])p 415 2748 V 437 w(,)g FC(Some)i(limiting)g(pr)l(op)l(erties)h(of)g(r)l (andomly)g(for)l(c)l(e)l(d)f(2D)g(Navier\025Stokes)h(e)l(qua-)415 2848 y(tions)p FI(,)c(Pro)r(ceedings)f(A)h(of)f(the)h(Ro)n(y)n(al)e(So) r(ciet)n(y)h(of)g(Edin)n(burgh)h FJ(133)f FI(\(2003\),)f(875\025891.)0 3014 y([KS04a])p 415 3014 V 395 w(,)35 b FC(On)f(r)l(andom)i(attr)l (actors)e(for)i(mixing-typ)l(e)f(systems)p FI(,)g(F)-7 b(unct.)34 b(Anal.)f(Appl.)415 3113 y FJ(38)27 b FI(\(2004\),)f (28\02537.)0 3279 y([KS04b])p 415 3279 V 391 w(,)21 b FC(R)l(andomly)i(for)l(c)l(e)l(d)h(CGL)f(e)l(quation:)35 b(stationary)23 b(me)l(asur)l(es)f(and)h(the)f(inviscid)415 3379 y(limit)p FI(,)28 b(J.)g(Ph)n(ys.)f(A:)h(Math.)g(Gen.)g FJ(37)f FI(\(2004\),)f(1\02518.)0 3545 y([Kuk97])130 b(S.)36 b(B.)g(Kuksin,)i FC(On)f(turbulenc)l(e)f(in)i(nonline)l(ar)g (Shr)l(\366)l(dinger)g(e)l(quations)p FI(,)h(Geometric)415 3645 y(and)27 b(F)-7 b(unctional)28 b(Analysis)f FJ(7)g FI(\(1997\),)g(338\025363.)0 3811 y([Kuk99])p 415 3811 V 393 w(,)j FC(Sp)l(e)l(ctr)l(al)i(pr)l(op)l(erties)g(of)h(solutions)f (for)g(nonline)l(ar)g(PDEs)g(in)g(the)g(turbulent)415 3910 y(r)l(e)l(gime)p FI(,)c(Geometric)f(and)h(F)-7 b(unctional)27 b(Analysis)g FJ(9)g FI(\(1999\),)g(141\025184.)0 4076 y([Kuk02a])p 415 4076 V 351 w(,)c FC(Er)l(go)l(dic)j(the)l(or)l(ems)f (for)g(2D)g(statistic)l(al)g(hydr)l(o)l(dynamics)p FI(,)h(Rev.)c(Math.) g(Ph)n(ys.)415 4176 y FJ(14)27 b FI(\(2002\),)f(585\025600.)0 4342 y([Kuk02b])p 415 4342 V 347 w(,)38 b FC(On)e(exp)l(onential)i(c)l (onver)l(genc)l(e)g(to)f(a)h(stationary)g(me)l(asur)l(e)e(for)j (nonline)l(ar)415 4442 y(PDEs,)d(p)l(erturb)l(e)l(d)d(by)h(r)l(andom)h (kick-for)l(c)l(es,)i(and)d(the)g(turbulenc)l(e-limit)p FI(,)f(The)f(M.)g(I.)415 4541 y(Vishik)d(Mosco)n(w)f(PDE)i(seminar,)e (Amer.)h(Math.)h(So)r(c.)e(T)-7 b(ransl.,)29 b(Amer.)g(Math.)g(So)r (c.,)415 4641 y(2002,)d(pp.)i(161\025176.)1553 5059 y(96)p eop end PStoPSsaved restore %%Page: (96,97) 49 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 97 96 bop 0 523 a FI([Kuk04])p 415 523 250 4 v 393 w(,)25 b FC(The)i(Eulerian)g(limit)g(for)h(2D)e(statistic)l (al)h(hydr)l(o)l(dynamics)p FI(,)g(J.)d(Stat.)g(Ph)n(ysics)415 623 y FJ(115)j FI(\(2004\),)f(469\025492.)0 789 y([Kuk05])p 415 789 V 393 w(,)37 b FC(R)l(emarks)f(on)h(the)g(b)l(alanc)l(e)g(r)l (elations)g(for)h(the)e(two-dimensional)j(Navier\025)415 888 y(Stokes)30 b(e)l(quation)g(with)g(r)l(andom)g(for)l(cing)p FI(,)f(J.)e(Stat.)h(Ph)n(ysics,)f(to)h(app)r(ear)f(\(2005\).)0 1054 y([Le)g(87])169 b(Y.)34 b(Le)g(Jan,)g FC(\311quilibr)l(e)j (statistique)e(p)l(our)g(les)h(pr)l(o)l(duits)f(de)h(di\033\351)l (omorphismes)i(al\351)l(a-)415 1154 y(toir)l(es)27 b(ind\351p)l (endants)p FI(,)e(Ann.)g(Inst.)f(H.)h(P)n(oincar\351)e(Probab.)g (Statist.)i FJ(23)e FI(\(1987\),)h(no.)f(1,)415 1254 y(111\025120.)0 1420 y([Led86])150 b(F.)22 b(Ledrappier,)g FC(Positivity)k(of)g(the)e(exp)l(onent)g(for)i(stationary)f(se)l(quenc) l(es)f(of)i(matric)l(es)p FI(,)415 1519 y(Ly)n(apuno)n(v)g(Exp)r(onen)n (ts)i(\(Bremen,)f(1984\),)f(Springer,)g(Berlin,)i(1986,)d(pp.)j (56\02573.)0 1685 y([Lin92])164 b(T.)37 b(Lindv)-5 b(all,)39 b FC(L)l(e)l(ctur)l(es)e(on)h(the)g(Coupling)i(Metho)l(ds)p FI(,)g(John)c(Willey)h(&)f(Sons,)j(New)415 1785 y(Y)-7 b(ork,)27 b(1992.)0 1951 y([Lio69])168 b(J.-L.)21 b(Lions,)h FC(Quelques)i(M\351tho)l(des)h(de)g(R\351solution)f(des)h(Pr)l (obl\350mes)g(aux)f(Limites)g(Non)415 2051 y(Lin\351)l(air)l(es)p FI(,)29 b(Gauthier-Villars,)d(P)n(aris,)g(1969.)0 2217 y([LS89])187 b(R.)22 b(S.)g(Liptser)f(and)h(A.)g(N.)g(Shiry)n(a)n(y)n (ev,)f FC(The)l(ory)26 b(of)f(Martingales)p FI(,)g(Klu)n(w)n(er,)c (Dordrec)n(h)n(t,)415 2316 y(1989.)0 2482 y([Mat99])135 b(J.)31 b(Mattingly)-7 b(,)32 b FC(Er)l(go)l(dicity)i(of)g(2D)f (Navier-Stokes)h(e)l(quations)f(with)g(r)l(andom)h(for)l(cing)415 2582 y(and)c(lar)l(ge)h(visc)l(osity)p FI(,)e(Comm.)e(Math.)h(Ph)n(ys.) f FJ(206)g FI(\(1999\),)f(273\025288.)0 2748 y([Mat02])p 415 2748 V 398 w(,)33 b FC(Exp)l(onential)h(c)l(onver)l(genc)l(e)h(for) f(the)g(sto)l(chastic)l(al)t(ly)i(for)l(c)l(e)l(d)f(Navier\025Stokes) 415 2848 y(e)l(quations)c(and)g(other)g(p)l(artial)t(ly)h(dissip)l (ative)h(dynamics)p FI(,)d(Comm.)f(Math.)f(Ph)n(ys.)h FJ(230)415 2947 y FI(\(2002\),)d(421\025462.)0 3113 y([MP04])153 b(J.)24 b(Mattingly)g(and)f(E.)i(P)n(ardoux,)f FC(Mal)t(liavin)29 b(c)l(alculus)d(for)i(the)e(sto)l(chastic)i(2D)e(Navier)415 3213 y(Stokes)k(e)l(quation)p FI(,)e(Preprin)n(t)f(\(2004\).)0 3379 y([MR03])148 b(R.)22 b(Mikulevicius)f(and)g(B.)h(Rozo)n(vskii,)f FC(Sto)l(chastic)j(Navier\025Stokes)i(e)l(quations)e(for)h(tur-)415 3479 y(bulent)k(\035ows)p FI(,)f(SIAM)g(J.)g(Math.)f(Anal.)h FJ(35)f FI(\(2003\),)f(1250\0251310.)0 3645 y([MY02])147 b(N.)22 b(Masmoudi)f(and)g(L.-S.)g(Y)-7 b(oung,)23 b FC(Er)l(go)l(dic)i(the)l(ory)g(of)g(in\034nite)e(dimensional)j(systems) 415 3744 y(with)41 b(applic)l(ations)h(to)e(dissip)l(ative)i(p)l(ar)l (ab)l(olic)g(PDE's)p FI(,)g(Comm.)d(Math.)g(Ph)n(ys.)g FJ(227)415 3844 y FI(\(2002\),)26 b(461\025481.)0 4010 y([\330k03])176 b(B.)26 b(\330ksendal,)g FC(Sto)l(chastic)j(Di\033er)l (ential)h(Equations.)f(an)g(intr)l(o)l(duction)g(with)g(applic)l(a-)415 4110 y(tions.)p FI(,)f(Springer\025V)-7 b(erlag,)25 b(Berlin,)i(2003.)0 4276 y([Ons49])141 b(L.)26 b(Onsager,)e FC(Statistic)l(al)k(hydr)l(o)l (dynamics)p FI(,)h(Nuo)n(v)n(o)24 b(Cimen)n(to)i(Suppl.)g FJ(6)g FI(\(1949\),)f(279\025)415 4375 y(287,)h(also)h(in)h(\020The)f (collected)g(w)n(orks)f(of)h(Lars)g(Onsager",)e(W)-7 b(orld)27 b(Scien)n(ti\034c,)h(1996.)0 4541 y([Ros71])149 b(M.)59 b(Rosen)n(blatt,)65 b FC(Markov)60 b(Pr)l(o)l(c)l(esses,)65 b(structur)l(e)57 b(and)h(asymptotic)h(b)l(ehaviour)p FI(,)415 4641 y(Springer,)27 b(1971.)1553 5059 y(97)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat TeXDict begin 98 97 bop 0 523 a FI([Rue68])141 b(D.)43 b(Ruelle,)k FC(Statistic)l(al)d(me)l(chanics)g(of)h(a)f (one-dimensional)h(lattic)l(e)f(gas)p FI(,)j(Comm.)415 623 y(Math.)28 b(Ph)n(ys.)f FJ(9)h FI(\(1968\),)e(267\025278.)0 789 y([Shi96])170 b(A.)28 b(Shiry)n(aev,)e FC(Pr)l(ob)l(ability)p FI(,)k(second)d(ed.,)h(Springer\025V)-7 b(erlag,)24 b(Berlin,)k(1996.)0 955 y([Shi02])170 b(A.)38 b(Shiriky)n(an,)g FC(A)n(nalyticity)h(of)g (solutions)g(for)g(r)l(andomly)h(for)l(c)l(e)l(d)f(two-dimensional)415 1054 y(Navier\025Stokes)31 b(e)l(quations)p FI(,)d(Russian)f(Math.)h (Surv)n(eys)e FJ(57)h FI(\(2002\),)f(785\025799.)0 1220 y([Shi04])p 415 1220 250 4 v 433 w(,)f FC(Exp)l(onential)i(mixing)g (for)h(2d)f(Navier\025Stokes)h(system)f(p)l(erturb)l(e)l(d)f(by)h(an)g (un-)415 1320 y(b)l(ounde)l(d)j(noise)p FI(,)e(J.)g(Math.)g(Fluid)g (Mec)n(h.)f FJ(6)h FI(\(2004\),)e(169\025193.)0 1486 y([Shi05])p 415 1486 V 433 w(,)33 b FC(Er)l(go)l(dicity)j(for)e(a)h (class)f(of)h(Markov)g(pr)l(o)l(c)l(esses)f(and)h(applic)l(ations)h(to) d(r)l(an-)415 1586 y(domly)38 b(for)l(c)l(e)l(d)f(PDE's)p FI(,)g(Russian)d(Journal)f(for)i(Mathematical)f(Ph)n(ysics)g FJ(12)g FI(\(2005\),)415 1685 y(81\02596.)0 1851 y([Shi06])p 415 1851 V 433 w(,)29 b FC(L)l(aw)i(of)g(lar)l(ge)h(numb)l(ers)d(and)j (c)l(entr)l(al)e(limit)h(the)l(or)l(em)g(for)g(r)l(andomly)h(for)l(c)l (e)l(d)415 1951 y(PDE's)p FI(,)c(Ergo)r(dic)f(Theory)g(and)g(Related)h (Fields)f FJ(134)g FI(\(2006\),)g(215\025247.)0 2117 y([Sin72])170 b(Y)-7 b(a.)19 b(Sinai,)i FC(Gibbs)i(me)l(asur)l(es)f(in) g(er)l(go)l(dic)h(the)l(ory)p FI(,)f(Russian)c(Math.)i(Surv)n(eys)e FJ(27)g FI(\(1972\),)415 2217 y(21\02569.)0 2383 y([Sin91])p 415 2383 V 433 w(,)25 b FC(Two)i(r)l(esults)f(c)l(onc)l(erning)h (asymptotic)g(b)l(ehavior)i(of)e(solutions)g(of)h(the)e(Bur)l(g-)415 2482 y(ers)k(e)l(quation)g(with)g(for)l(c)l(e)p FI(,)f(J.)e(Statist.)h (Ph)n(ys.)f FJ(64)h FI(\(1991\),)e(1\02512.)0 2648 y([VF88])169 b(M.)34 b(I.)h(Vishik)f(and)g(A.)g(V.)h(F)-7 b(ursik)n(o)n(v,)34 b FC(Mathematic)l(al)k(Pr)l(oblems)e(in)g(Statistic)l(al)g(Hy-)415 2748 y(dr)l(ome)l(chanics)p FI(,)29 b(Klu)n(w)n(er,)e(Dordrec)n(h)n(t,) f(1988.)0 2914 y([Vio76])158 b(M.)32 b(Viot,)h FC(Solutions)g(Faibles)i (d'Equations)g(aux)e(D\351riv\351)l(es)h(Partiel)t(les)h(Sto)l (chastique)415 3014 y(Non\025Line)l(air)l(es)p FI(,)29 b(Ph.D.)g(thesis,)e(P)n(aris,)g(1976.)1553 5059 y(98)p eop end PStoPSsaved restore %%Page: (98,99) 50 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat TeXDict begin 99 98 bop 0 648 a FK(Index)0 813 y FI(adapted)27 b(random)g(pro)r(cess,)f(37)0 993 y(balance)h(relation,)f(83,)h(84)0 1093 y(basis)g(of)g FH(H)7 b FI(,)28 b(10)0 1193 y(Bogolyub)r(o)n (v\025Krylo)n(v)23 b(ansatz,)k(32)0 1373 y(canonical)f(measure,)h(78)0 1473 y(Cen)n(tral)g(Limit)h(Theorem,)e(68)0 1572 y(Chapman-K)n (olmogoro)n(v)d(relation,)j(31)0 1672 y(co-area)f(form)i(of)h(balance)f (relations,)f(86)0 1772 y(correlation)f(tensor,)i(56)0 1871 y(coupling,)g(46)0 2052 y(Dobrushin)g(lemma,)h(46,)f(49)0 2151 y(Donsk)n(er)f(principle,)i(42)0 2332 y(energy)-7 b(,)27 b(27)0 2432 y(enstroph)n(y)-7 b(,)27 b(27)0 2531 y(ergo)r(dic)f(transformation,)g(65)0 2631 y(Eulerian)h(limit,)i(71,)d (76)0 2731 y(exp)r(onen)n(tial)h(mixing,)g(48)0 2911 y(F)-7 b(eller)27 b(prop)r(ert)n(y)-7 b(,)27 b(31)0 3011 y(F)-7 b(oias\025Pro)r(di)26 b(reduction,)h(90,)g(91)0 3191 y(Ginsburg\025Landau)f(equation,)h(4)0 3372 y(high-frequency)f (kic)n(k-force,)g(41,)h(63)0 3471 y(homogeneous)f(random)g(\034eld,)i (36)0 3652 y(in)n(terp)r(olation,)f(8)0 3752 y(Ito)g(form)n(ula,)g(38)0 3932 y(Kan)n(toro)n(vic)n(h)e(distance,)i(47)0 4032 y(Kan)n(toro)n(vic) n(h)e(functional,)i(47,)g(52,)g(59)0 4131 y(Kan)n(toro)n(vic)n(h)e (Theorem,)h(47)0 4231 y(kic)n(k)h(n)n(um)n(b)r(er)g FH(k)s FI(,)h(29)0 4331 y(kic)n(k-force,)e(26)0 4430 y(Kraic)n(hnan)g(theory) -7 b(,)27 b(82)0 4611 y(La)n(w)g(of)g(Iterated)g(Logarithm,)g(68)0 4710 y(Lera)n(y)f(pro)5 b(jection,)26 b(11)0 4810 y(Lipsc)n(hitz-dual)h (distance,)g(44)1636 813 y(Lipsc)n(hitz-norm,)f(44)1636 995 y(Mark)n(o)n(v)f(disin)n(tegration,)h(58)1636 1095 y(Mark)n(o)n(v)f(semigroup,)i(31)1636 1195 y(martingale)f(appro)n (ximation,)g(69)1636 1294 y(maximal)h(coupling,)g(46,)g(49)1636 1394 y(micro)r(canonical)f(measure,)g(78)1636 1494 y(mild)i(solution,)f (24)1636 1676 y(Na)n(vier\025Stok)n(es)e(semigroup,)h(23)1636 1859 y(Prokhoro)n(v)f(distance,)i(55)1636 2042 y(Random)g(Dynamical)g (System,)h(30,)f(57)1636 2141 y(random)g(kic)n(k-force,)e(29)1636 2241 y(random)i(p)r(oin)n(t)g(attractor,)f(57)1636 2340 y(random)h(set,)g(57)1636 2440 y(rate)g(of)g(dissipation)g(of)h(energy) -7 b(,)26 b(27)1636 2540 y(rate)h(of)g(dissipation)g(of)h(enstroph)n(y) -7 b(,)26 b(28)1636 2639 y(reaction-di\033usion)g(equation,)h(4)1636 2739 y(Reynolds)g(n)n(um)n(b)r(er,)g(27,)g(72)1636 2922 y(singular)f(force,)h(24)1636 3021 y(Sk)n(orokho)r(d)f(em)n(b)r(edding) h(theorem,)g(42)1636 3121 y(smo)r(othing)g(prop)r(ert)n(y)-7 b(,)26 b(20)1636 3220 y(Sob)r(olev)h(em)n(b)r(edding,)h(7)1636 3320 y(splitting)g(up)g(metho)r(d,)g(89)1636 3420 y(stationary)e (measure,)g(32,)h(41)1636 3519 y(Strong)g(La)n(w)f(of)i(Large)e(Num)n (b)r(ers,)h(65,)g(68)1636 3702 y(transition)g(function,)h(31)1636 3885 y(v)-5 b(ariational)26 b(distance,)h(45)1636 3984 y(v)n(orticit)n(y)-7 b(,)26 b(27)1636 4167 y(w)n(eak)h(con)n(v)n (ergence)e(of)i(measures,)f(44)1636 4267 y(w)n(eak)h(solution)g(\(of)g (a)h(kic)n(k)n(ed)e(equation\),)h(50)1636 4366 y(w)n(eak)g(solution)g (\(of)g(white-forced)g(NSE\),)i(37)1636 4466 y(white-forced)e(NSE,)h (36)1553 5059 y(99)p eop end PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat showpage PStoPSsaved restore %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0606080500408--