Content-Type: multipart/mixed; boundary="-------------0608021225647" This is a multi-part message in MIME format. ---------------0608021225647 Content-Type: text/plain; name="06-215.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-215.comments" E-mail: Armen.Shirikyan@math.u-psud.fr ---------------0608021225647 Content-Type: text/plain; name="06-215.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-215.keywords" Stationary solutions, irreducibility, controllability, 3D Navier-Stokes system. ---------------0608021225647 Content-Type: application/postscript; name="article.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="article.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: jdde2006.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMR9 CMBX9 CMBX12 CMBX10 CMR10 CMMI10 CMMI7 %%+ CMSY10 MSBM10 CMR7 CMSY7 CMEX10 CMTI10 CMMI5 CMBXTI10 CMR6 CMR8 %%+ CMMI8 CMSY8 EUFM10 CMSS10 MSAM10 CMR5 CMMI6 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o article.ps jdde2006.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2006.08.02:1856 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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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 (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 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 80 /P put dup 81 /Q put readonly def /FontBBox{-61 -250 999 759}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BDCEEA888D07B439DBD4E8B4C9 D198C03874B5E6F8FBF4922065A92BC3E66D05DE53971CB1424510E892442858 D69CE1F76E4DA76C87C763A4B2FE36321E54B1328C9155B8ED6361855A151723 3386AEA3D042B8D89C8C0E9A33E5DF3B466F7BB8C2C8A4ED4CDAFF55FC6D3EE6 0AF2CEBFC1AC3A6E6692F8BB81F82D86BAE85016AD62FCB05467082C2E5AD348 44D1439C2B59F65590E57CA0DE481A7A34E79931B1513C4C30156170409A4BB8 46D412D1DAF88AD30722F12DBCA1CCC6B4BCC28D06B0D29149DDEC520C8FBA13 6B82E2E1790F00B216282FF122EF0D47B70A1B29514DDF7C0435ED238C14BDF5 6DA243117FBEF7398F97EB95597707ED63C6797EBA1B46EA19ABB1DABDA171B3 16CD500F5D64CBFBE4F9CBC3E66A34427D3C4D0C432710289381F9BFD91B4FF4 1E3A896C3EEA2F3105C218877D6C0C6B763760FA364D00065E1CAE9DCB5676ED 286A9ED0D1C946DCA6A2A670EE0936FB4706CC62E234CFEED34AA615C48D2872 A087F30990C85E64BA68F3D5C117123467DB411C9F2D6F6858CC70C1E352C477 713097321B4C4FD4C5CDE305415F998E7245908EEDE6E056A736EA77BD8C639C 3A79FFD0B74B3D28F0494A115F2841CF8A8827AB5608F96FD8998A5F40FB3DFE 3AA0C7696DE4E1D18DC0D6E84B943175FC38FFC42A9C0CBB13A908978C98BFE5 034F88480F32B9DEB2FD228FF6CB0B89B045AB02020C82E3F5716DC640613185 9F597CE262729BC52132F43922B9E28BB71A30AC8709634561B22D13C4FAFE0A 12C4451969226B220038AD8DDA990A4E2CAD53DBEAB698898BBD3046234EB4EA 901287E71CB41296C431383AB85F18882F65BE36923F6C0FD6FADAC5B42FDB68 64C06E047434FA7A659EF7F3F1B8ABABD083F8A6581E430B838BCC6625F517E5 2814A27C3488EEE385C04810F5EBDDE2211187687F6A12D0AAA4D26FF25F5638 9F932FE6F4C906185484E06C166AB63F720C4DAD3CFE8BBB7A7BDEFA689AF22A 7403C12697478D87D37C743E9705BDC007166C4A38C14F8F8D172C663C51CDE6 AD30E6315A2F25B6BF62CF269B27C3C12065C2A521B8A5314A865D2F7C7F3705 E0F748810F0654D5A6534BDAF4349F2F765377077FE0B54B075BBEDB8B06BAD1 9FC22CCD1916A0D08A468113A252D910BD0D868E637CE183F19D470605E663B8 46664568434A0857D60F023636281A72332E23B7F987084C7B0696180D60F234 2E5F4FE3ABD35A872EF2BF7D3B456F70344775706B290BFEBE87498AB5DB2FB8 8AB5E9EB27C83F4946EAC4796DA104E25E243EBADA22435ABE98F0439B71EF14 8F57063BF9D9317C551019C4EECA282D145BFD5807059AC54D245EB9C780D876 393E9EA3D3F0431DFF81449D0B0A434A8995EFF7845A83E93696E71D238C5C93 87F3B1C5AF023F3602 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 109 /m put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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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 2 /multiply put dup 20 /lessequal put dup 50 /element put dup 79 /O put dup 84 /T put dup 89 /Y put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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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 59 /comma put dup 68 /D put dup 70 /F put dup 86 /V put dup 102 /f put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 73 /I put dup 82 /R put dup 84 /T put dup 94 /circumflex 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 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 readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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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 readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % 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 (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 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 110 /n put readonly def /FontBBox{-29 -250 1274 754}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 106 /j put dup 109 /m put dup 112 /p put dup 121 /y put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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 12 /fi put dup 45 /hyphen put dup 46 /period 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 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W 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 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 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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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 /ff put dup 12 /fi put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 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 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T 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 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 75 /K put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 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 61 /equal put dup 91 /bracketleft put dup 99 /c put dup 108 /l put dup 111 /o put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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cleartomark %%EndFont %%BeginFont: 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 69 /E put dup 80 /P put dup 82 /R put dup 84 /T put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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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 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 75 /K put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /union put dup 92 /intersection 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 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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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 17 /eta put dup 23 /nu put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 74 /J put dup 76 /L put dup 78 /N put dup 81 /Q put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z put dup 105 /i 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 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 8 /Phi put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 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 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 98 /b 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 109 /m put dup 110 /n 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 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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%%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 2 /Theta put dup 5 /Pi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 24 /cedilla put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal 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 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft 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 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec 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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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period 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 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 84 /T 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 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 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % 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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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 12 /fi put dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 55 /seven put dup 57 /nine put dup 58 /colon put dup 64 /at 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 72 /H put dup 73 /I 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 83 /S put dup 84 /T put dup 85 /U put dup 88 /X put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex 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 123 /endash put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F D1F017CE45884D76EF2CB9BC5821FD25365DDEA6E45F332B5F68A44AD8A530F0 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % 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 (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 65 /A put dup 83 /S put dup 97 /a put dup 101 /e put dup 104 /h put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 121 /y put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % 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 (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 45 /hyphen put dup 78 /N put dup 81 /Q put dup 83 /S put dup 97 /a put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h 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 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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600 (jdde2006.dvi) @start /Fa 133[30 47[34 74[{}2 49.8132 /CMMI6 rf /Fb 204[28 28 28 28 48[{}4 41.511 /CMR5 rf /Fc 138[65 117[{}1 83.022 /MSAM10 rf /Fd 174[61 53 80[{}2 83.022 /CMSS10 rf /Fe 146[64 109[{}1 83.022 /EUFM10 rf /Ff 166[47 4[39 4[56 28[47 29[55 17[55 2[{}6 66.4176 /CMSY8 rf /Fg 133[33 35 2[34 41 14[34 15[41 15[45 1[58 8[20 59[{}9 66.4176 /CMMI8 rf /Fh 134[37 37 51 37 39 27 28 28 37 39 35 39 59 20 37 1[20 39 35 22 31 39 31 39 35 2[35 9[51 1[52 8[25 11[55 8[35 35 35 35 2[20 24 20 2[27 27 27[39 12[{}39 66.4176 /CMR8 rf /Fi 204[30 30 30 30 48[{}4 49.8132 /CMR6 rf /Fj 145[54 110[{}1 83.022 /CMBXTI10 rf /Fk 134[30 8[31 2[51 2[24 106[{}4 41.511 /CMMI5 rf /Fl 132[56 1[59 1[81 59 62 44 44 46 59 62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 9[116 2[78 62 2[77 1[88 1[67 4[88 70 1[86 81 1[85 6[31 4[56 56 56 56 56 2[31 37 32[62 12[{}44 99.6264 /CMBX12 rf /Fm 132[42 1[40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 12[59 47 61 1[56 64 62 74 52 2[32 62 64 54 56 63 59 58 62 10[42 42 42 42 42 42 2[25 30 25 4[25 26[47 51 11[{}55 83.022 /CMTI10 rf /Fn 154[46 9[92 46 1[120 60[62 62 6[61 61 50 50 3[28 2[48 48 6[38 38{}15 83.022 /CMEX10 rf /Fo 180[51 24[45 66 19 26[52 52 16[34 52 1[52{}9 58.1154 /CMSY7 rf /Fp 144[33 2[19 8[30 7[19 29[51 8[33 33 33 33 33 4[51 1[26 26 39[41{}14 58.1154 /CMR7 rf /Fq 171[55 1[60 1[51 10[55 69[{}4 83.022 /MSBM10 rf /Fr 141[69 3[42 2[42 23 32 32 42 42 9[55 55 1[55 59 82 2[45 50 70 1[58 66 1[100 1[63 4[60 1[64 44 55 66 10[0 3[55 83 15[83 5[65 65 4[65 65 2[65 2[42 3[65 7[65 23 65{}39 83.022 /CMSY10 rf /Fs 133[32 34 2[33 39 25 31 31 1[34 1[41 59 1[35 27 23 14[45 1[54 1[39 45 39 2[52 2[53 1[46 1[37 1[54 1[42 1[54 48 50 1[35 2[34 1[20 35[33 5[34 17[{}32 58.1154 /CMMI7 rf /Ft 133[39 41 47 59 40 48 30 39 37 1[42 1[50 73 1[43 34 29 48 40 41 39 43 1[36 1[35 5[57 48 69 78 48 57 49 51 63 66 2[67 81 57 71 46 36 69 65 53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 4[39 52 54 52 4[43 1[36 41 50 48 3[41 36 1[37 1[47 3[55 8[{}67 83.022 /CMMI10 rf /Fu 128[42 42 2[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 23 23 42 23 42 23 51 62 1[85 62 62 60 46 61 1[57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 3[65 1[23 23 42 42 42 42 42 42 42 42 42 42 1[23 28 23 65 1[32 32 23 4[42 9[37 4[42 4[69 46 46 48 60 4[62 2[65 69 52{}91 83.022 /CMR10 rf /Fv 134[50 50 69 1[53 37 38 39 1[53 48 53 80 27 50 1[27 53 48 29 44 53 42 53 46 12[66 3[65 2[91 57 2[36 4[73 69 68 72 7[48 48 48 48 48 48 48 48 48 48 1[27 4[37 37 27[53 12[{}45 83.022 /CMBX10 rf /Fw 135[71 2[75 52 53 55 1[75 67 75 112 37 2[37 75 1[41 61 75 60 1[65 14[100 1[92 2[128 3[50 5[97 1[102 12[67 67 67 67 67 48[{}28 119.552 /CMBX12 rf /Fx 134[47 1[64 1[49 34 35 36 2[44 49 1[25 1[27 25 3[41 49 39 49 43 13[49 5[84 1[69 9[67 6[25 45[49 12[{}22 74.7198 /CMBX9 rf /Fy 132[38 1[41 41 55 41 43 30 30 30 41 43 38 43 64 21 41 23 21 43 38 23 34 43 34 43 38 2[38 21 1[21 2[58 2[58 55 43 1[60 52 60 58 70 48 2[28 58 1[50 52 59 55 54 58 60 5[21 38 1[38 38 38 38 38 38 38 38 1[21 26 21 24[38 6[43 12[{}63 74.7198 /CMR9 rf /Fz 134[51 6[38 3[54 81 1[51 1[27 54 2[43 3[49 13[54 17[73 65[{}11 99.6264 /CMR12 rf /FA 132[66 1[70 2[70 73 51 52 51 70 73 66 73 111 36 70 1[36 73 1[40 58 73 2[66 13[73 1[103 2[99 32[43 45[{}24 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 551 872 a FA(Qualitativ)l(e)42 b(prop)t(erties)h(of)g(stationary)e(measures)i(for)721 1054 y(three-dimensional)g(Na)l(vier{Stok)l(es)g(equations)1584 1295 y Fz(Armen)33 b(Shiriky)m(an)1429 1469 y Fy(Lab)r(oratoire)27 b(de)e(Math)n(\023)-36 b(ematiques)1245 1561 y(Univ)n(ersit)n(\023)g(e) 26 b(de)f(P)n(aris-Sud)h(XI,)f(B^)-38 b(atimen)n(t)26 b(425)1488 1652 y(91405)i(Orsa)n(y)d(Cedex,)h(F)-6 b(rance)1245 1752 y(E-mail:)35 b(Armen.Shiriky)n(an@math.u-psud.fr)1778 2051 y Fx(Abstract)837 2181 y Fy(The)29 b(pap)r(er)f(is)h(dev)n(oted)f 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b(of)h(viscosit)n(y)e(can)h(b)r(e)g(used)h(to)f(describ)r(e) g(turbulen)n(t)515 2906 y(b)r(eha)n(viour)g(of)h(solutions.)71 b(And,)43 b(second,)f(under)d(some)f(additional)h(assumptions,)i(a)515 3005 y(large)27 b(class)g(of)h(w)n(eak)g(solutions)f(for)h(\(0.1\))g (con)n(v)n(erge)e(to)i(a)g(stationary)f(solution)h(as)g(time)515 3105 y(go)r(es)j(to)h(in\014nit)n(y)-7 b(.)51 b(Before)31 b(turning)h(to)g(a)f(description)h(of)g(the)h(con)n(ten)n(ts)e(of)h (this)g(pap)r(er,)515 3204 y(let)c(us)f(recall)g(some)g(earlier)f (results)h(on)g(3D)h(sto)r(c)n(hastic)f(NS)h(equations.)639 3304 y(Existence)g(of)g(w)n(eak)f(solutions)g(for)h(the)g(Cauc)n(h)n(y) f(problem)g(and)h(of)g(stationary)f(solu-)515 3404 y(tions,)21 b(as)d(w)n(ell)h(as)g(some)g(a)g(priori)f(estimates)h(for)g(them,)i(w)n (as)e(established)g(b)n(y)g(Bensoussan,)515 3503 y(T)-7 b(emam)25 b([BT73)o(],)g(Vishik,)h(Komec)n(h,)f(F)-7 b(ursik)n(o)n(v)23 b([VKF79)o(],)j(Capi)r(\023)-44 b(nski,)25 b(G)r(\030)-39 b(atarek)24 b([CG94)o(],)515 3603 y(Flandoli,)c(G)r (\030)-39 b(atarek)18 b([F)n(G95)o(])i(and)f(others.)33 b(A)20 b(\014rst)f(result)g(sho)n(wing)f(the)h(mixing)g(c)n(haracter) 515 3703 y(of)i(3D)h(NS)g(dynamics)f(under)g(non-degenerate)f(random)h (forcing)f(w)n(as)h(obtained)g(b)n(y)g(Flan-)515 3802 y(doli)f([Fla97)o(].)35 b(He)21 b(pro)n(v)n(ed)e(that)i(if)g(the)h (noise)e(is)g(e\013ectiv)n(e)h(in)g(all)g(F)-7 b(ourier)19 b(mo)r(des,)j(then)g(the)515 3902 y(supp)r(ort)f(of)h(an)n(y)f (\\admissible")f(w)n(eak)g(solution)h(coincides)g(with)i(the)f(en)n (tire)f(phase)g(space.)515 4001 y(In)j(the)g(case)f(of)h(a)f(rough)g (white)h(noise,)g(Da)g(Prato)e(and)i(Debussc)n(he)g([DD03])g (constructed)515 4101 y(a)c(Mark)n(o)n(v)f(semigroup)h(concen)n(trated) g(on)h(w)n(eak)f(solutions)h(of)g(the)g(3D)g(NS)h(equations)e(and)515 4201 y(established)27 b(a)g(mixing)h(prop)r(ert)n(y)e(for)h(it.)38 b(Under)28 b(similar)e(conditions,)i(Odasso)e([Oda06)n(])515 4300 y(pro)n(v)n(ed)31 b(that)j(an)n(y)e(solution)g(obtained)h(as)f(a)h (limit)h(of)f(Galerkin)f(appro)n(ximations)f(con-)515 4400 y(v)n(erges)22 b(exp)r(onen)n(tially)i(to)g(a)g(stationary)f (solution.)36 b(Flandoli)24 b(and)g(Romito)g([FR06])g(ha)n(v)n(e)515 4500 y(constructed)36 b(a)f(Mark)n(o)n(v)f(selection)i(of)h(w)n(eak)e (solutions)g(and)h(pro)n(v)n(ed)f(the)i(irreducibil-)515 4599 y(it)n(y)28 b(and)h(strong)e(F)-7 b(eller)28 b(prop)r(ert)n(y)g (for)g(it,)h(pro)n(vided)f(that)h(the)g(random)e(p)r(erturbation)h(is) 515 4699 y(su\016cien)n(tly)i(rough.)45 b(The)31 b(results)f(of)h(this) g(pap)r(er)f(sho)n(w)g(that,)i(in)f(the)g(case)f(of)g(p)r(erio)r(dic) 515 4798 y(b)r(oundary)f(conditions,)h(non-degeneracy)d(of)j(the)g (noise)f(with)i(resp)r(ect)e(to)h(the)g(\014rst)g(few)515 4898 y(F)-7 b(ourier)26 b(mo)r(des)i(ensures)e(mixing)i(c)n(haracter)d (of)j(the)g(dynamics.)1926 5255 y(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 639 523 a Fu(W)-7 b(e)22 b(no)n(w)f(describ)r(e)g 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1995 y Fv(Main)36 b(Theorem.)44 b Fm(Ther)l(e)35 b(is)f(an)g(inte)l (ger)g Ft(N)40 b Fr(\025)30 b Fu(1)j Fm(not)h(dep)l(ending)h(on)f Ft(h)g Fm(and)h Ft(\027)k Fm(such)515 2095 y(that)29 b(if)1489 2194 y Ft(b)1525 2206 y Fs(j)1583 2194 y Fr(6)p Fu(=)23 b(0)84 b Fm(for)31 b Ft(j)d Fu(=)22 b(1)p Ft(;)14 b(:)g(:)g(:)g(;)g(N)8 b(;)515 2340 y Fm(then)29 b(the)h(fol)t(lowing)i (assertions)f(hold.)580 2502 y Fv(\(i\))41 b Fm(A)n(ny)29 b(b)l(al)t(l)g(in)g(the)g(sp)l(ac)l(e)g Ft(H)1578 2471 y Fp(1)1632 2502 y Fr(\\)16 b Ft(H)36 b Fm(has)29 b(a)g(p)l(ositive)h Ft(\026)p Fm(-me)l(asur)l(e.)38 b(In)28 b(p)l(articular,)j(the)722 2601 y(supp)l(ort)f(of)h Ft(\026)e Fm(c)l(oincides)j(with)e Ft(H)7 b Fm(.)553 2765 y Fv(\(ii\))41 b Fm(L)l(et)27 b Ft(F)39 b Fm(b)l(e)27 b(a)g(\014nite-dimensional)i(subsp)l(ac)l(e.)38 b(Then)27 b(the)h(pr)l(oje)l(ction)g(of)g Ft(\026)f Fm(to)g Ft(F)39 b Fm(c)l(an)722 2864 y(b)l(e)26 b(minorise)l(d)h(by)f(a)g(me)l (asur)l(e)g(of)g(the)g(form)h Ft(\032)2156 2876 y Fs(F)2211 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b(urthermore,)28 b(the)515 3623 y(pro)r(of)c(giv)n(en)h(in)g(Section)h(3)f(do)r(es)f (not)i(use)f(the)h(Gaussian)e(structure)h(of)g(the)g(noise)g Ft(\021)s Fu(,)h(and)515 3723 y(therefore)i(a)h(similar)f(result)h(is)g (true)g(for)f(other)h(t)n(yp)r(es)g(of)g(random)f(p)r(erturbations,)g (suc)n(h)515 3823 y(as)f(random)f(kic)n(k)h(forces,)1463 4061 y Ft(\021)s Fu(\()p Ft(t;)14 b(x)p Fu(\))24 b(=)1824 3958 y Fo(1)1797 3982 y Fn(X)1797 4161 y Fs(k)q Fp(=1)1932 4061 y Ft(\021)1973 4073 y Fs(k)2014 4061 y Fu(\()p Ft(x)p Fu(\))p Ft(\016)s Fu(\()p Ft(t)19 b Fr(\000)f Ft(k)s Fu(\))p Ft(;)778 b Fu(\(0.3\))515 4313 y(or)26 b(piecewise-constan)n(t) h(sto)r(c)n(hastic)f(pro)r(cesses,)1425 4551 y Ft(\021)s Fu(\()p Ft(t;)14 b(x)p Fu(\))24 b(=)1786 4447 y Fo(1)1759 4472 y Fn(X)1759 4651 y Fs(k)q Fp(=1)1893 4551 y Ft(\021)1934 4563 y Fs(k)1976 4551 y Fu(\()p Ft(x)p Fu(\))p Ft(I)2123 4566 y Fp([)p Fs(k)q Fo(\000)p Fp(1)p Fs(;k)q Fp(\))2351 4551 y Fu(\()p Ft(t)p Fu(\))p Ft(:)740 b Fu(\(0.4\))515 4807 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b Fr(kr)p Ft(u)p Fr(k)27 b Fu(the)h(norm)f(in)h(the)g(space)e Ft(V)19 b Fu(.)1926 5255 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 515 523 a Fv(De\014nition)33 b(1.1.)41 b Fu(A)30 b(function)g Ft(v)f Fr(2)d(X)1782 535 y Fs(J)1855 523 y Fu(:=)g Ft(L)2026 493 y Fo(1)2026 547 y Fp(lo)r(c)2113 523 y Fu(\()p Ft(J)o(;)14 b(H)7 b Fu(\))19 b Fr(\\)h Ft(L)2486 493 y Fp(2)2486 547 y(lo)r(c)2573 523 y Fu(\()p Ft(J)o(;)14 b(V)19 b Fu(\))30 b(is)f(called)g(a)g Fm(we)l(ak)515 623 y(solution)34 b Fu(for)27 b(\(1.3\))g(if)i(it)f(p)r(ossesses)e(the) i(follo)n(wing)e(prop)r(erties.)580 777 y Fv(\(i\))41 b Fu(Equation)18 b(\(1.3\))g(holds)g(in)g(the)h(sense)f(of)g (distributions,)j(that)d(is,)i(for)e(an)n(y)g(div)n(ergence-)722 877 y(free)28 b(v)n(ector)e(\014eld)i Ft(')23 b Fr(2)h Ft(C)1532 847 y Fo(1)1526 898 y Fp(0)1602 877 y Fu(\()p Ft(J)j Fr(\002)18 b Ft(D)r(;)c Fq(R)1958 847 y Fp(3)1995 877 y Fu(\),)28 b(w)n(e)f(ha)n(v)n(e)971 975 y Fn(Z)1018 1164 y Fs(J)1064 1021 y Fn(\000)1102 1088 y 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Ft(;)g(v)s Fu(\))g Ft(ds)22 b Fu(=)2498 3052 y(1)p 2498 3089 V 2498 3165 a(2)2550 3108 y Fr(k)p Ft(v)s Fu(\(0\))p Fr(k)2783 3073 y Fp(2)2838 3108 y Fu(+)2921 2995 y Fn(Z)3004 3015 y Fs(t)2967 3183 y Fp(0)3033 3108 y Fu(\()p Ft(h;)14 b(v)s Fu(\))g Ft(ds:)3208 3254 y Fu(\(1.6\))515 3354 y(Indeed,)36 b(a)e(standard)g(limiting)h(argumen)n(t)e(sho)n(ws)h(that) h(relation)e(\(1.4\))h(remains)g(v)-5 b(alid)515 3453 y(for)33 b(an)n(y)h(function)g Ft(')h Fr(2)f(Y)1376 3465 y Fs(J)1457 3453 y Fu(suc)n(h)g(that)57 b(_)-46 b Ft(')34 b Fr(2)h Ft(L)2072 3423 y Fp(2)2108 3453 y Fu(\()p Ft(J)o(;)14 b(H)7 b Fu(\))35 b(and)f Ft(')p Fu(\(0\))g(=)f Ft(')p Fu(\()p Ft(T)12 b Fu(\))34 b(=)g(0.)56 b(Let)515 3553 y(us)31 b(\014x)g(an)n(y)f Ft(t)f(>)g Fu(0)i(and)g(consider)f(the)h (sequence)g Ft(')2172 3565 y Fs(k)2242 3553 y Fu(=)e Ft(\037)2388 3565 y Fs(k)2429 3553 y Ft(v)s Fu(,)j(where)f Ft(\037)2823 3565 y Fs(k)2892 3553 y Fr(2)f Ft(C)3042 3523 y Fo(1)3112 3553 y Fu(\()p Fq(R)p Fu(\))i(are)515 3652 y(arbitrary)25 b(functions)j(suc)n(h)f(that)h(0)23 b Fr(\024)g Ft(\037)1798 3664 y Fs(k)1861 3652 y Fr(\024)g Fu(1,)773 3821 y Ft(\037)825 3833 y Fs(k)866 3821 y Fu(\()p Ft(s)p Fu(\))g(=)g(1)82 b(for)1343 3788 y Fp(1)p 1341 3802 37 4 v 1341 3850 a Fs(k)1411 3821 y Fr(\024)23 b Ft(s)g Fr(\024)f Ft(t)d Fr(\000)1791 3788 y Fp(1)p 1790 3802 V 1790 3850 a Fs(k)1836 3821 y Ft(;)180 b(\037)2091 3833 y Fs(k)2132 3821 y Fu(\()p Ft(s)p Fu(\))24 b(=)e(0)83 b(for)27 b Ft(s)c Fr(\024)f Fu(0)28 b(or)e Ft(s)d Fr(\025)g Ft(t:)515 3990 y Fu(W)-7 b(riting)29 b(iden)n(tit)n(y)i(\(1.4\))e(with) h Ft(')d Fu(=)e Ft(')1747 4002 y Fs(k)1818 3990 y Fu(and)k(using)g(the) h(relation)f(\()p Ft(B)t Fu(\()p Ft(v)23 b Fu(+)c Ft(z)t(;)14 b(v)s Fu(\))p Ft(;)g(v)s Fu(\))27 b(=)e(0,)515 4089 y(w)n(e)i(obtain) 556 4295 y Fr(\000)635 4182 y Fn(Z)718 4202 y Fs(t)681 4370 y Fp(0)747 4295 y Fu(\()p Ft(v)s(;)14 b(@)903 4307 y Fs(s)939 4295 y Fu(\()p Ft(\037)1023 4307 y Fs(k)1064 4295 y Ft(v)s Fu(\)\))g Ft(ds)f Fu(+)1358 4182 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4727 y Fr(k)p Ft(v)s Fr(k)2331 4693 y Fp(2)2367 4660 y Fn(\001)2419 4727 y Ft(ds)23 b Fr(!)2640 4671 y Fu(1)p 2640 4708 V 2640 4784 a(2)2692 4660 y Fn(\000)2730 4727 y Fr(k)p Ft(v)s Fu(\()p Ft(t)p Fu(\))p Fr(k)2951 4693 y Fp(2)2997 4727 y Fr(\000)9 b(k)p Ft(v)s Fu(\(0\))p Fr(k)3304 4693 y Fp(2)3341 4660 y Fn(\001)p 515 4850 1146 4 v 607 4904 a Fi(1)642 4927 y Fh(This)28 b(inequalit)n(y)i(is)e(obtained)i(formally)d(b)n(y)j (taking)f(the)h(scalar)e(pro)r(duct)i(of)k(\(1.3\))c(with)f Fg(v)r Fh(,)h(in)n(te-)515 5006 y(grating)24 b(in)f(time,)h(and)g (replacing)g(=)f(b)n(y)h Ff(\024)p Fh(.)1926 5255 y Fu(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 515 523 a Fu(as)27 b Ft(k)f Fr(!)d(1)p Fu(.)37 b(P)n(assing)25 b(to)j(the)g(limit)g(in)g(\(1.7\))f(as)g Ft(k)f Fr(!)d(1)p Fu(,)28 b(w)n(e)f(arriv)n(e)f(at)h(\(1.6\).)639 623 y(The)e(follo)n(wing)e(prop)r(osition)g(establishes)h(a)g(w)n (eak-strong)e(uniqueness)i(for)g(solutions)515 722 y(of)34 b(\(1.3\))o(.)515 888 y Fv(Prop)s(osition)39 b(1.3.)45 b Fm(L)l(et)37 b Ft(v)i Fr(2)e(X)1616 900 y Fs(J)1700 888 y Fm(and)k Fu(~)-46 b Ft(v)40 b Fr(2)d(Y)2095 900 y Fs(J)2179 888 y Fm(b)l(e,)i(r)l(esp)l(e)l(ctively,)i(we)l(ak)d(and)f (str)l(ong)515 988 y(solutions)30 b(for)39 b Fu(\(1.3\))30 b Fm(such)f(that)h Ft(v)s Fu(\(0\))24 b(=)h(~)-45 b Ft(v)s Fu(\(0\))p Fm(.)39 b(Then)31 b Ft(v)26 b Fu(=)g(~)-46 b Ft(v)t Fm(.)639 1154 y Fu(In)23 b(the)g(case)f Ft(z)k Fu(=)d(0,)g(a)f(pro)r(of)g(of)g(Prop)r(osition)f(1.3)h(can)g(b)r(e)h (found)g(in)g([Soh01)n(].)36 b(F)-7 b(or)22 b(the)515 1254 y(reader's)d(con)n(v)n(enience,)i(w)n(e)g(outlined)h(the)f(pro)r (of)g(for)f(the)i(general)e(case)g(in)i(the)f(App)r(endix.)515 1486 y Fl(1.2)112 b(Admissible)26 b(w)m(eak)g(solutions)g(for)f(sto)s (c)m(hastic)g(Na)m(vier{Stok)m(es)770 1602 y(equations)515 1756 y Fu(Let)i(us)h(consider)f(3D)g(Na)n(vier{Stok)n(es)e(equations)i (p)r(erturb)r(ed)h(b)n(y)f(a)g(random)g(force:)1460 1938 y(_)-37 b Ft(u)18 b Fu(+)g Ft(\027)5 b(Lu)18 b Fu(+)g Ft(B)t Fu(\()p 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Fr(g)f Fm(is)g(an)h(orthonormal)g(b)l(asis)g(in) g Ft(V)c Fm(,)24 b Fr(f)p Ft(\014)2081 4015 y Fs(j)2115 4003 y Fr(g)e Fm(is)h(a)g(se)l(quenc)l(e)f(of)h(indep)l(endent)g(stan-) 515 4102 y(dar)l(d)28 b(Br)l(ownian)h(motions)f(on)g Fq(R)1567 4114 y Fp(+)1622 4102 y Fm(,)h(and)f Fr(f)p Ft(b)1913 4114 y Fs(j)s(k)1983 4102 y Fr(g)g Fm(is)g(a)g(family)h(of)g (r)l(e)l(al)f(numb)l(ers)f(satisfying)515 4202 y(the)j(c)l(ondition) 1745 4243 y Fo(1)1718 4268 y Fn(X)1694 4447 y Fs(j;k)q Fp(=1)1876 4347 y Ft(b)1912 4313 y Fp(2)1912 4368 y Fs(j)s(k)2006 4347 y Ft(<)23 b Fr(1)p Ft(:)639 4604 y Fu(W)-7 b(e)29 b(no)n(w)e(recall)g(the)i(concept)f(of)g(an)f(admissible)h(w)n(eak)f (solution)h(for)f(\(1.8\).)38 b(T)-7 b(o)28 b(this)515 4703 y(end,)g(w)n(e)f(\014rst)g(de\014ne)h(the)g(Ornstein{Uhlen)n(b)r (ec)n(k)e(pro)r(cess)1457 4939 y Ft(z)t Fu(\()p Ft(t)p Fu(\))d(=)1704 4826 y Fn(Z)1787 4847 y Fs(t)1750 5015 y Fp(0)1830 4939 y Ft(e)1869 4905 y Fo(\000)p Fs(\027)t Fp(\()p Fs(t)p Fo(\000)p Fs(s)p Fp(\))p Fs(L)2168 4939 y Ft(Qd\020)6 b Fu(\()p Ft(t)p Fu(\))p Ft(:)731 b Fu(\(1.11\))1926 5255 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 515 523 a Fu(Relation)38 b(\(1.11\))g(implies)g (that)h Ft(z)j Fu(is)d(a)f Ft(V)19 b Fu(-v)-5 b(alued)38 b(Gaussian)f(pro)r(cess)h(whose)g(almost)515 623 y(ev)n(ery)25 b(tra)5 b(jectory)25 b(b)r(elongs)g(to)i(the)f(space)g Fr(Y)k Fu(=)23 b Ft(C)6 b Fu(\()p Fq(R)2209 635 y Fp(+)2265 623 y Ft(;)14 b(V)k Fu(\))f Fr(\\)f Ft(L)2545 593 y Fp(2)2545 646 y(lo)r(c)2632 623 y Fu(\()p Fq(R)2724 635 y Fp(+)2779 623 y Ft(;)e(U)9 b Fu(\))27 b(and)f(satis\014es)515 722 y(the)i(Stok)n(es)e(equation)1675 822 y(_)-37 b Ft(u)18 b Fu(+)g Ft(\027)5 b(Lu)22 b Fu(=)h Ft(\021)s Fu(\()p Ft(t)p Fu(\))p Ft(:)935 b Fu(\(1.12\))515 983 y Fv(De\014nition)33 b(1.6.)42 b Fu(An)30 b Ft(H)7 b Fu(-v)-5 b(alued)30 b(random)f(pro)r (cess)g Ft(u)p Fu(\()p Ft(t)p Fu(\))h(is)g(called)f(an)h Fm(admissible)k(so-)515 1083 y(lution)g Fu(for)27 b(\(1.8\))g(if)h(it)g 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(existence)f(of)h(a)f(suitable)g(w)n(eak)g(solution\).)515 3030 y Fv(Prop)s(osition)40 b(1.9.)46 b Fm(Supp)l(ose)38 b(that)g Ft(h)g Fr(2)g Ft(H)45 b Fm(and)38 b(Condition)h(1.4)h(is)e (ful\014l)t(le)l(d.)64 b(Then)515 3130 y(Eq.)30 b Fu(\(1.8\))g Fm(has)g(at)g(le)l(ast)g(one)g(stationary)g(me)l(asur)l(e)g Ft(\026)23 b Fr(2)g(P)7 b Fu(\()p Ft(H)g Fu(\))30 b Fm(such)g(that)1429 3349 y Fe(m)p Fu(\()p Ft(\026)p Fu(\))23 b(:=)1741 3236 y Fn(Z)1787 3424 y Fs(H)1863 3349 y Fr(k)p Ft(v)s Fr(k)1990 3314 y Fp(2)1990 3369 y Fs(V)2047 3349 y Ft(\026)p Fu(\()p Ft(dv)s Fu(\))h Ft(<)f Fr(1)p Ft(:)702 b Fu(\(1.14\))639 3571 y(Note)32 b(that,)h(in)e([VF88,)g(CG94,)g(F)n(G95)o(],)i(the)f (authors)e(do)h(not)h(state)f(explicitly)h(the)515 3671 y(fact)19 b(that)h(the)g(energy)f(inequalit)n(y)h(\(1.5\))f(holds.)34 b(Ho)n(w)n(ev)n(er,)20 b(they)f(construct)g(a)h(solution)f(as)515 3771 y(a)j(p)r(oin)n(t)n(wise)g(limit)h(of)g(Galerkin)f(appro)n 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Fu(\(1.15\))515 5006 y(where)27 b Ft(u)803 5018 y Fp(0)863 5006 y Fr(2)c Ft(V)c Fu(.)37 b(W)-7 b(e)28 b(\014rst)f(in)n(tro)r(duce)g(some)g (notations.)1926 5255 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 639 523 a Fu(Let)23 b Ft(h)g Fr(2)h Ft(H)29 b Fu(b)r(e)24 b(a)e(function,)j(let)e Ft(T)34 b(>)23 b Fu(0)f(b)r(e)h(a)g(constan)n(t,)g(and)g(let)g Ft(J)31 b Fu(=)23 b([0)p Ft(;)14 b(T)e Fu(].)34 b(F)-7 b(or)22 b(an)n(y)515 623 y Ft(u)563 635 y Fp(0)629 623 y Fr(2)30 b Ft(V)19 b Fu(,)33 b(w)n(e)e(denote)h(b)n(y)f(\002)1419 635 y Fs(T)1471 623 y Fu(\()p Ft(h;)14 b(u)1636 635 y Fp(0)1673 623 y Fu(\))32 b(the)g(set)f(of)h(functions)g Ft(\021)h Fr(2)d Ft(L)2694 593 y Fp(2)2731 623 y Fu(\()p Ft(J)2809 635 y Fs(T)2861 623 y Ft(;)14 b(H)7 b Fu(\))32 b(for)f(whic)n(h)515 722 y(problem)26 b(\(1.8\))o(,)h(\(1.15\))f(has)g (a)g(unique)h(solution)f Ft(u)d Fr(2)g(Y)2297 734 y Fs(J)2344 722 y Fu(.)37 b(Using)26 b(the)h(implicit)g(function)515 822 y(theorem,)g(it)h(can)f(b)r(e)h(sho)n(wn)f(that)h(\(see)f(Theorem)g (1.8)g(in)h([Shi06a)o(]\))1006 994 y Fr(D)1070 1006 y Fs(T)1123 994 y Fu(\()p Ft(h)p Fu(\))23 b(:=)g Fr(f)p Fu(\()p Ft(\021)s(;)14 b(u)1572 1006 y Fp(0)1609 994 y Fu(\))23 b Fr(2)g Ft(L)1799 960 y Fp(2)1836 994 y Fu(\()p Ft(J)1914 1006 y Fs(T)1967 994 y Ft(;)14 b(H)7 b Fu(\))18 b Fr(\002)g Ft(V)42 b Fu(:)23 b Ft(\021)j Fr(2)e Fu(\002)2560 1006 y Fs(T)2612 994 y Fu(\()p Ft(h;)14 b(u)2777 1006 y Fp(0)2814 994 y Fu(\))p Fr(g)279 b Fu(\(1.16\))515 1166 y(is)26 b(an)h(op)r(en)f(subset)h(of)f Ft(L)1315 1136 y Fp(2)1352 1166 y Fu(\()p Ft(J)1430 1178 y Fs(T)1483 1166 y Ft(;)14 b(H)7 b Fu(\))16 b Fr(\002)g Ft(V)j Fu(,)27 b(and)f(the)h(op)r(erator)e Fr(R)i Fu(taking)f(\()p Ft(\021)s(;)14 b(u)2991 1178 y Fp(0)3028 1166 y Fu(\))24 b Fr(2)f(D)3226 1178 y Fs(T)3305 1166 y Fu(to)515 1266 y(the)30 b(solution)f Ft(u)d Fr(2)h(Y)1187 1278 y Fs(J)1264 1266 y Fu(of)36 b(\(1.8\))o(,)31 b(\(1.15\))e(is)g(lo)r(cally)g(Lipsc)n(hitz)h(con)n (tin)n(uous.)42 b(W)-7 b(e)30 b(denote)515 1365 y(b)n(y)d Fr(R)700 1377 y Fs(t)757 1365 y Fu(the)h(restriction)f(of)g Fr(R)h Fu(to)g(the)g(time)g Ft(t)23 b Fr(2)g Ft(J)8 b Fu(.)639 1488 y(Let)26 b Ft(E)i Fr(\032)22 b Ft(U)35 b Fu(and)25 b Ft(F)35 b Fr(\032)22 b Ft(H)32 b Fu(b)r(e)26 b(\014nite-dimensional)f(subspaces)f(and)h(let)h Fd(P)2974 1500 y Fs(F)3052 1488 y Fu(:)d Ft(H)30 b Fr(!)23 b Ft(H)515 1587 y Fu(b)r(e)30 b(the)f(orthogonal)f(pro)5 b(jection)28 b(on)n(to)h Ft(F)12 b Fu(.)42 b(In)30 b(the)f(next)h(de\014nition,)g(w) n(e)f(assume)g(that)h Ft(\021)515 1687 y Fu(is)d(an)h Ft(E)5 b Fu(-v)-5 b(alued)27 b(con)n(trol)f(function.)515 1845 y Fv(De\014nition)44 b(1.10.)i Fu(Equation)39 b(\(1.8\))f(is)h (said)g(to)g(b)r(e)h Fm(appr)l(oximately)i(c)l(ontr)l(ol)t(lable)g(in) 515 1944 y(time)33 b Ft(T)43 b Fu(if)33 b(for)e(an)n(y)g Ft(u)1220 1956 y Fp(0)1257 1944 y Ft(;)19 b Fu(^)-47 b Ft(u)29 b Fr(2)h Ft(V)51 b Fu(and)32 b(an)n(y)f Ft(")e(>)h Fu(0)h(there)h(is)g Ft(\021)h Fr(2)d Fu(\002)2647 1956 y Fs(T)2699 1944 y Fu(\()p Ft(h;)14 b(u)2864 1956 y Fp(0)2901 1944 y Fu(\))21 b Fr(\\)h Ft(C)3096 1914 y Fo(1)3167 1944 y Fu(\()p Ft(J)o(;)14 b(E)5 b Fu(\))515 2044 y(suc)n(h)27 b(that)1539 2144 y Fr(kR)1651 2156 y Fs(T)1703 2144 y Fu(\()p Ft(\021)s(;)14 b(u)1864 2156 y Fp(0)1902 2144 y Fu(\))k Fr(\000)24 b Fu(^)-48 b Ft(u)p Fr(k)2125 2156 y Fs(V)2205 2144 y Ft(<)23 b(":)812 b Fu(\(1.17\))515 2287 y(Equation)29 b(\(1.8\))i(is)f(said)g(to)h(b)r(e)g Fm(solid)t(ly)j Ft(F)12 b Fm(-c)l(ontr)l(ol)t(lable)34 b(in)f(time)f Ft(T)42 b Fu(if)31 b(for)f(an)n(y)g Ft(u)3164 2299 y Fp(0)3229 2287 y Fr(2)e Ft(V)515 2386 y Fu(and)37 b(an)n(y)g Ft(R)k(>)f Fu(0)d(there)g(is)h(a)f(constan)n(t)g Ft(\016)43 b(>)d Fu(0)d(and)h(a)f(compact)g(set)h Fr(K)h Fu(in)f(a)g(\014nite-)515 2486 y(dimensional)e(subspace)g Ft(X)45 b Fr(\032)38 b Ft(C)1619 2456 y Fo(1)1689 2486 y Fu(\()p Ft(J)o(;)14 b(E)5 b Fu(\))37 b(suc)n(h)g(that)g Fr(K)j(\032)e Fu(\002)2595 2498 y Fs(T)2647 2486 y Fu(\()p Ft(h;)14 b(u)2812 2498 y Fp(0)2848 2486 y Fu(\),)40 b(and)d(for)f(an)n (y)515 2586 y(con)n(tin)n(uous)26 b(mapping)i Ft(\010)23 b Fu(:)g Fr(K)i(!)e Ft(F)39 b Fu(satisfying)27 b(the)h(inequalit)n(y) 1360 2758 y(sup)1357 2827 y Fs(\021)r Fo(2K)1503 2758 y Fr(k)p Ft(\010)p Fu(\()p Ft(\021)s Fu(\))19 b Fr(\000)f Fd(P)1863 2770 y Fs(F)1918 2758 y Fr(R)1988 2770 y Fs(T)2041 2758 y Fu(\()p Ft(\021)s(;)c(u)2202 2770 y Fp(0)2239 2758 y Fu(\))p Fr(k)2313 2770 y Fs(F)2391 2758 y Fr(\024)22 b Ft(\016)o(;)630 b Fu(\(1.18\))515 2990 y(w)n(e)27 b(ha)n(v)n(e)f Ft(\010)p Fu(\()p Fr(K)q Fu(\))f Fr(\033)d Ft(B)1186 3002 y Fs(F)1242 2990 y Fu(\()p Ft(R)q Fu(\).)639 3148 y(F)-7 b(or)28 b(an)n(y)g(\014nite-dimensional)g(subspace)g Ft(G)c Fr(\032)g Ft(U)9 b Fu(,)29 b(w)n(e)f(denote)g(b)n(y)g Fr(F)8 b Fu(\()p Ft(G)p Fu(\))30 b(the)e(largest)515 3247 y(v)n(ector)22 b(space)g Ft(G)1042 3259 y Fp(1)1103 3247 y Fr(\032)g Ft(U)32 b Fu(suc)n(h)23 b(that)h(an)n(y)e(elemen)n(t)i Ft(\021)2134 3259 y Fp(1)2194 3247 y Fr(2)g Ft(G)2338 3259 y Fp(1)2399 3247 y Fu(is)f(represen)n(table)e(in)j(the)f(form)1555 3497 y Ft(\021)1596 3509 y Fp(1)1656 3497 y Fu(=)g Ft(\021)f Fr(\000)1931 3393 y Fs(k)1890 3418 y Fn(X)1892 3595 y Fs(j)s Fp(=1)2023 3497 y Ft(\025)2071 3509 y Fs(j)2107 3497 y Ft(B)t Fu(\()p Ft(\020)2248 3462 y Fs(j)2284 3497 y Fu(\))p Ft(;)515 3766 y Fu(where)30 b Ft(\021)s(;)14 b(\020)881 3735 y Fp(1)919 3766 y Ft(;)g(:)g(:)g(:)g(;)g(\020)1146 3735 y Fs(k)1216 3766 y Fr(2)29 b Ft(G)j Fu(are)e(some)h(v)n(ectors)e (and)i Ft(\025)2249 3778 y Fp(1)2287 3766 y Ft(;)14 b(:)g(:)g(:)g(;)g (\025)2520 3778 y Fs(k)2592 3766 y Fu(are)30 b(non-negativ)n(e)g(con-) 515 3865 y(stan)n(ts.)54 b(Since)33 b Ft(B)38 b Fu(is)33 b(a)g(quadratic)g(op)r(erator)e(con)n(tin)n(uous)i(from)g Ft(U)42 b Fu(to)34 b Ft(H)2920 3835 y Fp(1)2957 3865 y Fu(\()p Ft(D)r(;)14 b Fq(R)3157 3835 y Fp(3)3194 3865 y Fu(\),)35 b(w)n(e)515 3965 y(see)26 b(that)i Fr(F)8 b Fu(\()p Ft(G)p Fu(\))24 b Fr(\032)e Ft(U)36 b Fu(is)27 b(a)g(w)n(ell-de\014ned)g(v)n(ector)f(space)g(of)h(\014nite)h (dimension.)36 b(Also)27 b(note)515 4064 y(that)h Fr(F)8 b Fu(\()p Ft(G)p Fu(\))23 b Fr(\033)g Ft(G)p Fu(.)639 4164 y(W)-7 b(e)28 b(no)n(w)f(de\014ne)h(a)f(sequence)g(of)h(subspaces) e Ft(E)2143 4176 y Fs(k)2208 4164 y Fr(\032)c Ft(U)37 b Fu(b)n(y)27 b(the)h(rule)968 4398 y Ft(E)1029 4410 y Fp(0)1090 4398 y Fu(=)22 b Ft(E)5 b(;)97 b(E)1424 4410 y Fs(k)1488 4398 y Fu(=)23 b Fr(F)8 b Fu(\()p Ft(E)1737 4410 y Fs(k)q Fo(\000)p Fp(1)1863 4398 y Fu(\))84 b(for)27 b Ft(k)f Fr(\025)c Fu(1)p Ft(;)97 b(E)2485 4410 y Fo(1)2578 4398 y Fu(=)2693 4294 y Fo(1)2680 4319 y Fn([)2666 4498 y Fs(k)q Fp(=1)2801 4398 y Ft(E)2862 4410 y Fs(k)2903 4398 y Ft(:)241 b Fu(\(1.19\))515 4649 y(The)27 b(follo)n(wing)g (result)g(is)h(established)f(in)h([Shi06a)o(,)f(Shi06c].)515 4807 y Fv(Prop)s(osition)32 b(1.11.)41 b Fm(L)l(et)30 b Ft(E)h Fr(\032)26 b Ft(U)40 b Fm(b)l(e)31 b(a)h(\014nite-dimensional) g(subsp)l(ac)l(e)g(such)f(that)g Ft(E)3308 4819 y Fo(1)515 4907 y Fm(is)g(dense)f(in)h Ft(H)7 b Fm(.)40 b(Then)31 b(the)g(fol)t(lowing)i(assertions)e(take)g(plac)l(e)g(for)g(any)g Ft(h)24 b Fr(2)h Ft(H)7 b Fm(,)31 b Ft(T)k(>)24 b Fu(0)p Fm(,)515 5006 y(and)30 b Ft(\027)e(>)23 b Fu(0)p Fm(.)1926 5255 y Fu(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 580 523 a Fv(\(i\))41 b Fm(Equation)48 b Fu(\(1.8\))40 b Fm(is)h(appr)l(oximately)i(c)l(ontr)l(ol)t(lable)f (in)f(time)g Ft(T)52 b Fm(by)41 b(an)f Ft(E)5 b Fm(-value)l(d)722 623 y(c)l(ontr)l(ol.)553 789 y Fv(\(ii\))41 b Fm(F)-6 b(or)22 b(any)f(\014nite-dimensional)i(subsp)l(ac)l(e)e Ft(F)35 b Fr(\032)23 b Ft(H)7 b Fm(,)23 b(Eq.)f Fu(\(1.8\))f Fm(is)h(solid)t(ly)h Ft(F)12 b Fm(-c)l(ontr)l(ol)t(lable)722 888 y(in)30 b(time)g Ft(T)41 b Fm(by)30 b(an)g Ft(E)5 b Fm(-value)l(d)30 b(c)l(ontr)l(ol.)515 1121 y Fl(1.4)112 b(Analyticit)m(y)37 b(of)h(the)f(resolving)i(op)s(erator)515 1274 y Fu(Let)31 b Ft(X)37 b Fu(and)31 b Ft(Y)49 b Fu(b)r(e)31 b(Banac)n(h)f(spaces)g(and)g(let)i Fr(D)e(\032)e Ft(X)38 b Fu(b)r(e)31 b(an)f(op)r(en)h(set.)47 b(Recall)30 b(that)h(a)515 1374 y(con)n(tin)n(uous)c(function)i Ft(f)k Fu(:)25 b Fr(D)i(!)e Ft(Y)47 b Fu(is)28 b(said)g(to)h(b)r(e)f Fm(analytic)35 b Fu(if)29 b(for)f(an)n(y)g Ft(x)2873 1386 y Fp(0)2935 1374 y Fr(2)d(D)31 b Fu(there)d(is)515 1473 y(a)f(constan)n(t)g Ft(r)f(>)c Fu(0)27 b(suc)n(h)h(that)1006 1705 y Ft(f)9 b Fu(\()p Ft(x)p Fu(\))24 b(=)f Ft(f)9 b Fu(\()p Ft(x)1408 1717 y Fp(0)1445 1705 y Fu(\))19 b(+)1617 1601 y Fo(1)1590 1626 y Fn(X)1579 1802 y Fs(m)p Fp(=1)1736 1705 y Ft(L)1793 1717 y Fs(m)1855 1705 y Fu(\()p Ft(x)g Fr(\000)f Ft(x)2083 1717 y Fp(0)2121 1705 y Fu(\))83 b(for)28 b Ft(x)23 b Fr(2)g Ft(B)2575 1717 y Fs(X)2639 1705 y Fu(\()p Ft(x)2718 1717 y Fp(0)2755 1705 y Ft(;)14 b(r)r Fu(\))q Ft(;)280 b Fu(\(1.20\))515 1964 y(where)28 b Ft(L)813 1976 y Fs(m)901 1964 y Fu(:)e Ft(X)32 b Fr(!)25 b Ft(Y)48 b Fu(is)29 b(an)f Ft(m)p Fu(-linear)g(op)r(erator)g(dep)r(ending)h(on)g Ft(x)2690 1976 y Fp(0)2727 1964 y Fu(,)h(and)f(series)f(\(1.20\))515 2063 y(con)n(v)n(erges)d(regularly)-7 b(.)35 b(The)27 b(latter)h(means)f(that)1646 2204 y Fo(1)1619 2229 y Fn(X)1607 2405 y Fs(m)p Fp(=1)1764 2308 y Fr(j)-14 b(j)g(j)p Ft(L)1862 2320 y Fs(m)1925 2308 y Fr(j)g(j)g(j)p Ft(r)2005 2274 y Fs(m)2093 2308 y Ft(<)22 b Fr(1)p Ft(;)515 2567 y Fu(where)33 b Fr(j)-14 b(j)g(j)23 b(\001)f(j)-14 b(j)g(j)34 b Fu(stands)f(for)g(the)h(norm)f(of)g(an)g Ft(m)p Fu(-linear)g(op)r (erator)e(\(see)j([VF88)o(])g(for)f(more)515 2666 y(details\).)639 2766 y(Let)j(us)g(\014x)g(an)g(in)n(terv)-5 b(al)35 b Ft(J)45 b Fu(=)37 b([0)p Ft(;)14 b(T)e Fu(])34 b(and)i(consider)f(Eq.)g (\(1.3\),)j(in)e(whic)n(h)g Ft(h)h Fr(2)g Ft(H)515 2866 y Fu(and)j Ft(z)48 b Fr(2)d(Y)931 2878 y Fs(J)1019 2866 y Fu(are)39 b(giv)n(en)h(functions.)76 b(W)-7 b(e)41 b(denote)f(b)n(y)h Fr(C)2438 2878 y Fs(T)2490 2866 y Fu(\()p Ft(h)p Fu(\))g(the)g(set)f(of)h(functions)515 2965 y(\()p Ft(z)t(;)14 b(v)667 2977 y Fp(0)704 2965 y Fu(\))23 b Fr(2)g(Y)892 2977 y Fs(J)954 2965 y Fr(\002)14 b Ft(V)44 b Fu(for)25 b(whic)n(h)g(Eq.)g(\(1.3\))h(has)f(a)g(unique)h (solution)f Ft(v)h Fr(2)d(Y)2823 2977 y Fs(J)2896 2965 y Fu(satisfying)i(the)515 3065 y(initial)j(condition)1767 3165 y Ft(v)s Fu(\(0\))23 b(=)g Ft(v)2067 3177 y Fp(0)2104 3165 y Ft(:)1040 b Fu(\(1.21\))515 3314 y(Theorem)33 b(1.8)g(in)h([Shi06a)o(])g(implies)g(that)h Fr(C)1954 3326 y Fs(T)2006 3314 y Fu(\()p Ft(h)p Fu(\))f(is)g(an)g(op)r(en)g (subset)g(of)g Fr(Y)2988 3326 y Fs(J)3057 3314 y Fr(\002)22 b Ft(V)53 b Fu(and)515 3414 y(the)35 b(op)r(erator)e Fr(S)41 b Fu(taking)34 b(\()p Ft(z)t(;)14 b(v)1513 3426 y Fp(0)1550 3414 y Fu(\))36 b Fr(2)f(C)1752 3426 y Fs(T)1804 3414 y Fu(\()p Ft(h)p Fu(\))g(to)g(the)g(solution)f Ft(v)39 b Fr(2)c(Y)2755 3426 y Fs(J)2837 3414 y Fu(is)f(lo)r(cally)g(Lips-)515 3513 y(c)n(hitz)24 b(con)n(tin)n(uous.)35 b(The)25 b(follo)n(wing)e (prop)r(osition)g(can)i(b)r(e)f(pro)n(v)n(ed)f(b)n(y)i(the)f(metho)r (ds)h(used)515 3613 y(in)j([Kuk82)n(])g(\(see)f(also)g([Brz91)n(]\).) 515 3779 y Fv(Prop)s(osition)33 b(1.12.)41 b Fm(F)-6 b(or)33 b(any)g Ft(h)27 b Fr(2)h Ft(H)7 b Fm(,)33 b Ft(T)39 b(>)27 b Fu(0)p Fm(,)33 b(and)f Ft(\027)h(>)27 b Fu(0)p Fm(,)33 b(the)g(r)l(esolving)g(op)l(er)l(ator)515 3879 y Fr(S)c Fu(:)23 b Fr(C)684 3891 y Fs(T)736 3879 y Fu(\()p Ft(h)p Fu(\))h Fr(!)f(Y)1033 3891 y Fs(T)1115 3879 y Fm(is)31 b(analytic.)515 4111 y Fl(1.5)112 b(Decomp)s(osable)39 b(measures)515 4264 y Fu(Let)27 b Ft(Z)34 b Fu(b)r(e)28 b(a)f(separable)f(Banac)n(h)g(space)h(and)h(let)g Ft(\025)23 b Fr(2)g(P)7 b Fu(\()p Ft(Z)f Fu(\).)515 4430 y Fv(De\014nition)37 b(1.13.)44 b Fu(The)33 b(measure)g Ft(\025)h Fu(is)f(said)g(to)h(b)r(e) g Fm(de)l(c)l(omp)l(osable)42 b Fu(if)34 b(there)f(are)g(t)n(w)n(o)515 4530 y(sequences)22 b(of)i(closed)e(subspaces)g Fr(f)p Ft(F)1688 4542 y Fs(n)1734 4530 y Fr(g)g Fu(and)i Fr(f)p Ft(G)2063 4542 y Fs(n)2108 4530 y Fr(g)e Fu(suc)n(h)h(that)h(the)g (follo)n(wing)e(prop)r(erties)515 4629 y(hold.)580 4796 y Fv(\(i\))41 b Fu(F)-7 b(or)27 b(an)n(y)g Ft(n)c Fr(\025)g Fu(1,)k(w)n(e)g(ha)n(v)n(e)g Ft(F)1648 4808 y Fs(n)1716 4796 y Fr(\032)c Ft(F)1857 4808 y Fs(n)p Fp(+1)1986 4796 y Fu(,)28 b(and)g(the)g(union)f Fr([)2627 4808 y Fs(n)2673 4796 y Ft(F)2726 4808 y Fs(n)2799 4796 y Fu(is)g(dense)h(in)g Ft(Z)6 b Fu(.)1905 5255 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 553 523 a Fv(\(ii\))41 b Fu(F)-7 b(or)27 b(an)n(y)g Ft(n)c Fr(\025)f Fu(1,)27 b(the)h(space)f Ft(Z)33 b Fu(can)27 b(b)r(e)h(decomp)r(osed)f(in)n(to)g(the)h(direct)f (sum)h(of)f Ft(F)3333 535 y Fs(n)722 623 y Fu(and)h Ft(G)949 635 y Fs(n)994 623 y Fu(,)1797 722 y Ft(Z)h Fu(=)23 b Ft(F)2024 734 y Fs(n)2088 722 y Fc(u)18 b Ft(G)2236 734 y Fs(n)2281 722 y Ft(;)863 b Fu(\(1.22\))722 862 y(and)28 b(the)g(measure)e Ft(\025)i Fu(is)g(represen)n(table)e(as)1758 1027 y Ft(\025)e Fu(=)e Fd(P)1970 1039 y Fs(n)2015 1027 y Ft(\025)d Fr(\012)f Fd(Q)2226 1039 y Fs(n)2272 1027 y Ft(\025;)824 b Fu(\(1.23\))722 1193 y(where)19 b Fd(P)1007 1205 y Fs(n)1072 1193 y Fu(and)h Fd(Q)1287 1205 y Fs(n)1352 1193 y Fu(are)e(the)i(pro)5 b(jections)18 b(asso)r(ciated)h(with)h (decomp)r(osition)f(\(1.22\))o(.)515 1345 y Fm(Example)41 b Fu(1.14)p Fm(.)j Fu(Let)33 b Ft(\025)f Fr(2)g(P)7 b Fu(\()p Ft(Z)f Fu(\))33 b(b)r(e)g(a)f(non-degenerate)f(cen)n(tred)h (Gaussian)g(measure,)515 1445 y(that)i(is,)i(a)d(probabilit)n(y)g (Borel)g(measure)g(on)h Ft(Z)40 b Fu(suc)n(h)34 b(that)g(for)f(an)n(y)h (con)n(tin)n(uous)f(func-)515 1544 y(tional)27 b Ft(`)22 b Fr(2)i Ft(Z)949 1514 y Fo(\003)1014 1544 y Fu(the)j(image)g(of)g Ft(\026)g Fu(under)g Ft(`)g Fu(is)g(a)g(cen)n(tred)f(Gaussian)g (measure)h(on)f(the)i(real)515 1644 y(line,)g(and)h(the)f(supp)r(ort)g (of)h Ft(\025)f Fu(coincides)g(with)h Ft(Z)6 b Fu(.)39 b(W)-7 b(e)28 b(claim)g(that)h Ft(\025)g Fu(is)f(decomp)r(osable.)515 1744 y(Indeed,)21 b(let)e Ft(H)7 b Fu(\()p Ft(\025)p Fu(\))20 b(b)r(e)g(the)f(Cameron{Martin)e(space)i(for)f Ft(\025)i Fu(\(see)f(Section)g(2.1)f(in)i([Bog98)n(]\),)515 1843 y(let)29 b Fr(f)p Ft(e)717 1855 y Fs(j)751 1843 y Fr(g)g Fu(b)r(e)h(an)e(orthonormal)g(basis)g(in)h Ft(H)7 b Fu(\()p Ft(\025)p Fu(\),)31 b(and)e(let)g Fr(f)p Ft(\030)2438 1855 y Fs(j)2473 1843 y Fr(g)g Fu(b)r(e)g(a)g(sequence)g(of)g(scalar) 515 1943 y(i.i.d.)d(random)f(v)-5 b(ariables)24 b(on)h(the)h(same)f (probabilit)n(y)g(space)g(suc)n(h)g(that)h(the)g(distribution)515 2043 y(of)e Ft(\030)642 2055 y Fs(j)702 2043 y Fu(is)h(a)f(standard)g (Gaussian)f(measure)h(on)g Fq(R)p Fu(.)36 b(Then,)26 b(b)n(y)e(Theorem)g(3.4.4)f(in)i([Bog98)n(],)515 2142 y(the)j(series)1775 2184 y Fo(1)1748 2209 y Fn(X)1750 2386 y Fs(j)s Fp(=1)1882 2288 y Ft(\030)1918 2300 y Fs(j)1953 2288 y Fu(\()p Ft(!)s Fu(\))p Ft(e)2111 2300 y Fs(j)515 2515 y Fu(con)n(v)n(erges)34 b(almost)j(surely)f(in)i Ft(Z)6 b Fu(,)39 b(and)e(the)h(distribution)f(of)g(its)h(sum)f Ft(\030)t Fu(\()p Ft(!)s Fu(\))h(coincides)515 2615 y(with)28 b Ft(\025)p Fu(.)37 b(Let)28 b(us)g(set)953 2781 y Ft(F)1006 2793 y Fs(n)1075 2781 y Fu(=)22 b(span)p Fr(f)p Ft(e)1410 2793 y Fs(j)1444 2781 y Ft(;)14 b Fu(1)22 b Fr(\024)h Ft(j)28 b Fr(\024)23 b Ft(n)p Fr(g)p Ft(;)96 b(G)2059 2793 y Fs(n)2127 2781 y Fu(=)p 2215 2708 703 4 v 23 w(span)p Fr(f)p Ft(e)2463 2793 y Fs(j)2497 2781 y Ft(;)14 b(j)28 b Fr(\025)22 b Ft(n)d Fu(+)f(1)p Fr(g)o Ft(;)515 2946 y Fu(where)p 749 2880 68 4 v 21 w Ft(B)27 b Fu(denotes)21 b(the)i(closure)e(of)h Ft(B)k Fu(in)d(the)f(space)f Ft(Z)6 b Fu(.)35 b(Let)23 b(us)f(sho)n(w)f(that)h(prop)r(erties)f(\(i\))515 3046 y(and)27 b(\(ii\))i(of)e(De\014nition)h(1.13)f(hold.)639 3146 y(By)35 b(Theorem)f(3.5.1)f(in)i([Bog98)n(],)i(the)e(supp)r(ort)g (of)f Ft(\025)i Fu(coincides)e(with)h(the)g(closure)515 3245 y(of)e Ft(H)7 b Fu(\()p Ft(\025)p Fu(\))33 b(in)h Ft(Z)6 b Fu(.)52 b(By)33 b(assumption,)h(w)n(e)e(ha)n(v)n(e)g(supp)14 b Ft(\025)33 b Fu(=)e Ft(Z)6 b Fu(,)34 b(and)f(therefore)f(the)h(v)n (ector)515 3345 y(space)h Fr([)799 3357 y Fs(n)845 3345 y Ft(F)898 3357 y Fs(n)979 3345 y Fu(=)h(span)o Fr(f)p Ft(e)1326 3357 y Fs(j)1360 3345 y Ft(;)14 b(j)41 b Fr(\025)35 b Fu(1)p Fr(g)f Fu(is)h(dense)g(in)g Ft(Z)6 b Fu(.)59 b(T)-7 b(o)35 b(pro)n(v)n(e)e(\(ii\),)38 b(w)n(e)d(\014x)g(an)n(y)f(in) n(te-)515 3444 y(ger)d Ft(n)f Fr(\025)f Fu(1)j(and)g(note)f(that)i Ft(F)1498 3456 y Fs(n)1564 3444 y Fr(\\)22 b Ft(G)1706 3456 y Fs(n)1782 3444 y Fu(=)29 b Fr(f)p Fu(0)p Fr(g)p Fu(.)49 b(Therefore)30 b(decomp)r(osition)i(\(1.22\))f(will)515 3544 y(b)r(e)j(established)g(if)g(w)n(e)g(sho)n(w)f(that)h(an)n(y)f(v)n (ector)g Ft(z)k Fr(2)d Ft(Z)40 b Fu(is)33 b(represen)n(table)g(in)h (the)g(form)515 3644 y Ft(z)i Fu(=)c Ft(y)728 3656 y Fs(n)795 3644 y Fu(+)22 b Ft(z)921 3656 y Fs(n)965 3644 y Fu(,)35 b(where)e Ft(u)1317 3656 y Fs(n)1394 3644 y Fr(2)g Ft(F)1535 3656 y Fs(n)1614 3644 y Fu(and)g Ft(v)1821 3656 y Fs(n)1899 3644 y Fr(2)g Ft(G)2052 3656 y Fs(n)2098 3644 y Fu(.)54 b(This)33 b(fact)g(is)g(ob)n(vious)f(for)h(elemen)n(ts) 515 3743 y(of)25 b Ft(H)7 b Fu(\()p Ft(\025)p Fu(\))27 b(and)e(can)h(b)r(e)g(pro)n(v)n(ed)e(b)n(y)h(a)g(simple)h(appro)n (ximation)e(argumen)n(t)h(for)g(an)n(y)g Ft(z)h Fr(2)d Ft(Z)6 b Fu(.)515 3843 y(F)-7 b(urthermore,)26 b(to)i(pro)n(v)n(e)e (\(1.23\))o(,)i(w)n(e)f(write)987 4075 y Ft(\030)t Fu(\()p Ft(!)s Fu(\))d(=)1297 3971 y Fs(n)1257 3996 y Fn(X)1260 4173 y Fs(j)s Fp(=1)1391 4075 y Ft(\030)1427 4087 y Fs(j)1462 4075 y Fu(\()p Ft(!)s Fu(\))p Ft(e)1620 4087 y Fs(j)1674 4075 y Fu(+)1827 3971 y Fo(1)1800 3996 y Fn(X)1757 4173 y Fs(j)s Fp(=)p Fs(n)p Fp(+1)1978 4075 y Ft(\030)2014 4087 y Fs(j)2049 4075 y Fu(\()p Ft(!)s Fu(\))p Ft(e)2207 4087 y Fs(j)2265 4075 y Fu(=:)e Ft(\021)2416 4087 y Fs(n)2462 4075 y Fu(\()p Ft(!)s Fu(\))d(+)f Ft(\020)2719 4087 y Fs(n)2764 4075 y Fu(\()p Ft(!)s Fu(\))p Ft(:)515 4329 y Fu(The)25 b(construction)g(implies)h(that)g Fr(D)r Fu(\()p Ft(\021)1754 4341 y Fs(n)1800 4329 y Fu(\))d(=)g Fd(P)1996 4341 y Fs(n)2041 4329 y Ft(\025)j Fu(and)f Fr(D)r Fu(\()p Ft(\020)2408 4341 y Fs(n)2454 4329 y Fu(\))f(=)e Fd(Q)2658 4341 y Fs(n)2704 4329 y Ft(\025)p Fu(.)36 b(Since)26 b Ft(\021)3067 4341 y Fs(n)3138 4329 y Fu(and)f Ft(\020)3333 4341 y Fs(n)515 4429 y Fu(are)h(indep)r(enden)n(t,)j(w)n(e)e(obtain)g (\(1.23\))639 4555 y(In)f(what)f(follo)n(ws,)f(w)n(e)h(shall)g(deal)g (with)g(measures)f Ft(\025)g Fr(2)f(P)7 b Fu(\()p Ft(Z)f Fu(\))25 b(satisfying)g(the)g(follo)n(w-)515 4655 y(ing)i(condition:) 515 4807 y Fv(Condition)20 b(1.15.)32 b Fu(The)19 b(measure)f Ft(\025)h Fu(is)g(decomp)r(osable)e(in)j(the)f(sense)f(of)h (De\014nition)g(1.13,)515 4907 y(its)28 b(supp)r(ort)f(coincides)h (with)g Ft(Z)6 b Fu(,)28 b(and)f(for)h(an)n(y)f Ft(n)c Fr(\025)g Fu(1)k(the)i(pro)5 b(jection)27 b Fd(P)2861 4919 y Fs(n)2906 4907 y Ft(\025)h Fu(p)r(ossesses)e(a)515 5006 y(p)r(ositiv)n(e)h(con)n(tin)n(uous)f(densit)n(y)i(with)g(resp)r (ect)f(to)h(the)g(Leb)r(esgue)f(measure)f(on)i Ft(F)3093 5018 y Fs(n)3138 5006 y Fu(.)1905 5255 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 515 523 a Fm(Example)41 b Fu(1.16)p Fm(.)i Fu(Let)32 b Ft(\025)g Fu(b)r(e)g(a)g(Gaussian)f(measure)g(on)g (a)h(separable)e(Banac)n(h)h(space)g Ft(X)7 b Fu(.)515 623 y(Denote)31 b(b)n(y)f Ft(Z)37 b Fu(the)31 b(supp)r(ort)f(of)h Ft(\025)p Fu(.)47 b(Then)31 b Ft(Z)36 b Fu(is)31 b(also)e(a)i (separable)e(Banac)n(h)g(space,)i(and)515 722 y(the)g(restriction)f(of) h Ft(\025)g Fu(to)g Ft(Z)36 b Fu(is)31 b(a)f(non-degenerate)g(Gaussian) f(measure.)46 b(F)-7 b(urthermore,)515 822 y(an)n(y)27 b(\014nite-dimensional)g(pro)5 b(jection)26 b(of)i Ft(\025)g Fu(p)r(ossesses)e(a)i(p)r(ositiv)n(e)f(smo)r(oth)g(densit)n(y)h(with) 515 922 y(resp)r(ect)f(to)h(the)g(Leb)r(esgue)f(measure.)35 b(Th)n(us,)28 b(an)n(y)f(Gaussian)f(measure)h(satis\014es)g(Condi-)515 1021 y(tion)g(1.15.)515 1296 y Fw(2)134 b(Main)45 b(results)515 1494 y Fl(2.1)112 b(F)-9 b(orm)m(ulations)515 1647 y Fu(Let)32 b(us)g(consider)f(the)h(NS)g(system)g(\(1.8\),)h(where)e Ft(h)g Fr(2)f Ft(H)39 b Fu(is)32 b(a)f(deterministic)h(function)515 1747 y(and)25 b Ft(\021)j Fu(is)e(a)f(random)f(pro)r(cess)g(satisfying) h(Condition)g(1.4.)35 b(In)26 b(what)f(follo)n(ws,)g(w)n(e)g(assume)515 1847 y(that)d Ft(h)g Fu(and)h Ft(\021)i Fu(are)c(\014xed)i(and)f(do)g (not)g(trace)f(the)i(dep)r(endence)g(of)f(v)-5 b(arious)21 b(parameters)f(on)515 1946 y(them.)45 b(Recall)30 b(that)g(the)h (concept)f(of)g(stationary)e(measure)h(for)h(\(1.8\))g(is)g(in)n(tro)r (duced)g(in)515 2046 y(De\014nition)19 b(1.8.)33 b(F)-7 b(or)19 b(an)n(y)f(\014nite-dimensional)g(space)g Ft(F)12 b Fu(,)21 b(w)n(e)d(denote)h(b)n(y)g Ft(`)2840 2058 y Fs(F)2913 2046 y Fu(the)h(Leb)r(esgue)515 2146 y(measure)27 b(on)h(it.)40 b(If)29 b Ft(\026)1207 2158 y Fp(1)1272 2146 y Fu(and)f Ft(\026)1484 2158 y Fp(2)1550 2146 y Fu(are)g(t)n(w)n(o)f(measures)g(suc)n(h)h(that)h Ft(\026)2623 2158 y Fp(1)2660 2146 y Fu(\(\000\))c Fr(\025)f Ft(\026)2940 2158 y Fp(2)2977 2146 y Fu(\(\000\))29 b(for)f(an)n(y)515 2245 y(measurable)i(set)i(\000,)i(then)e(w)n(e)g(write)g Ft(\026)1783 2257 y Fp(1)1850 2245 y Fr(\025)f Ft(\026)1996 2257 y Fp(2)2033 2245 y Fu(.)50 b(The)32 b(follo)n(wing)f(theorem)h(is) g(the)g(main)515 2345 y(result)27 b(of)h(this)f(pap)r(er.)515 2511 y Fv(Theorem)46 b(2.1.)i Fm(Supp)l(ose)43 b(that)f(the)g(image)h (of)g(the)f(op)l(er)l(ator)h Ft(Q)e Fm(in)h(Condition)i(1.4)515 2610 y(c)l(ontains)36 b(a)g(\014nite-dimensional)h(subsp)l(ac)l(e)f Ft(E)k Fr(\032)34 b Ft(U)44 b Fm(for)37 b(which)g(the)g(ve)l(ctor)f(sp) l(ac)l(e)g Ft(E)3308 2622 y Fo(1)515 2710 y Fm(de\014ne)l(d)26 b(in)32 b Fu(\(1.19\))24 b Fm(is)i(dense)g(in)g Ft(H)7 b Fm(.)37 b(L)l(et)25 b Ft(\026)e Fr(2)g(P)7 b Fu(\()p Ft(H)g Fu(\))26 b Fm(b)l(e)f(a)h(stationary)h(me)l(asur)l(e)e(for)35 b Fu(\(1.8\))515 2810 y Fm(such)30 b(that)1431 2931 y Fe(m)p Fu(\()p Ft(\026)p Fu(\))23 b(=)1720 2818 y Fn(Z)1766 3007 y Fs(H)1843 2931 y Fr(k)p Ft(v)s Fr(k)1970 2897 y Fp(2)1970 2952 y Fs(V)2027 2931 y Ft(\026)p Fu(\()p Ft(dv)s Fu(\))h Fr(\024)f Fe(m)2403 2943 y Fp(0)2440 2931 y Ft(;)745 b Fu(\(2.1\))515 3122 y Fm(wher)l(e)30 b Fe(m)813 3134 y Fp(0)873 3122 y Ft(>)23 b Fu(0)29 b Fm(is)h(a)g(c)l(onstant.)38 b(Then)31 b(the)e(fol)t(lowing)k (assertions)d(take)g(plac)l(e.)580 3288 y Fv(\(i\))41 b Fm(F)-6 b(or)30 b(any)g(b)l(al)t(l)h Ft(B)c Fr(\032)c Ft(V)49 b Fm(ther)l(e)29 b(is)i(a)f(c)l(onstant)f Ft(p)p Fu(\()p Ft(B)t(;)14 b Fe(m)2405 3300 y Fp(0)2442 3288 y Fu(\))23 b Ft(>)g Fu(0)29 b Fm(such)h(that)1737 3471 y Ft(\026)p Fu(\()p Ft(B)t Fu(\))24 b Fr(\025)f Ft(p)p Fu(\()p Ft(B)t(;)14 b Fe(m)2272 3483 y Fp(0)2309 3471 y Fu(\))p Ft(:)844 b Fu(\(2.2\))722 3654 y Fm(In)30 b(p)l(articular,)h (the)f(supp)l(ort)g(of)g Ft(\026)g Fm(c)l(oincides)h(with)g Ft(H)7 b Fm(.)553 3820 y Fv(\(ii\))41 b Fm(L)l(et)32 b Ft(F)40 b Fr(\032)28 b Ft(H)39 b Fm(b)l(e)33 b(a)g (\014nite-dimensional)g(subsp)l(ac)l(e)g(and)g(let)g Ft(\026)2691 3832 y Fs(F)2778 3820 y Fm(b)l(e)g(the)g(pr)l(oje)l(ction) 722 3919 y(of)h Ft(\026)f Fm(to)g Ft(F)12 b Fm(.)48 b(Then)34 b(ther)l(e)f(is)g(a)g(function)g Ft(\032)2117 3931 y Fs(F)2201 3919 y Fr(2)c Ft(C)2350 3889 y Fo(1)2421 3919 y Fu(\()p Ft(F)12 b Fu(\))33 b Fm(dep)l(ending)i(only)e(on)g Fe(m)3342 3931 y Fp(0)722 4019 y Fm(such)d(that)g Ft(\026)1131 4031 y Fs(F)1209 4019 y Fr(\025)23 b Ft(\032)1340 4031 y Fs(F)1395 4019 y Ft(`)1430 4031 y Fs(F)1514 4019 y Fm(and)31 b Ft(\032)1719 4031 y Fs(F)1774 4019 y Fu(\()p Ft(y)s Fu(\))23 b Ft(>)g Fu(0)29 b Fm(for)i Ft(`)2232 4031 y Fs(F)2286 4019 y Fm(-almost)f(every)h Ft(y)26 b Fr(2)d Ft(F)12 b Fm(.)639 4185 y Fu(A)41 b(pro)r(of)e(of)h(Theorem)f (2.1)g(is)h(giv)n(en)f(in)h(the)h(next)f(section.)73 b(It)40 b(is)g(based)g(on)f(an)515 4285 y(auxiliary)e(result,)k(whic)n (h)e(is)g(of)f(indep)r(enden)n(t)i(in)n(terest.)70 b(Before)38 b(form)n(ulating)f(it,)42 b(w)n(e)515 4384 y(mak)n(e)27 b(t)n(w)n(o)f(remarks.)515 4517 y Fm(R)l(emark)49 b Fu(2.2)p Fm(.)f Fu(Analysing)38 b(the)h(pro)r(of)g(giv)n(en)f(b)r(elo)n(w,)j(it) f(is)f(not)g(di\016cult)g(to)g(see)g(that)515 4617 y(Theorem)25 b(2.1)h(remains)f(v)-5 b(alid)26 b(for)g(an)n(y)g(admissible)g(w)n(eak) f(solution)h Ft(u)f Fu(of)i(Eq.)e(\(1.8\))h(suc)n(h)515 4716 y(that)36 b Fq(P)p Fr(f)p Ft(u)p Fu(\(0\))f Fr(2)i Ft(V)19 b Fr(g)36 b Ft(>)g Fu(0.)61 b(In)36 b(particular,)g(w)n(e)g (can)f(tak)n(e)g(a)g(solution)g(of)h(the)g(Cauc)n(h)n(y)515 4816 y(problem)27 b(\(1.8\))o(,)h(\(1.15\))f(with)h(an)n(y)f (deterministic)h(initial)g(function)g Ft(u)2757 4828 y Fp(0)2817 4816 y Fr(2)23 b Ft(V)c Fu(.)1905 5255 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 515 523 a Fm(R)l(emark)41 b Fu(2.3)p Fm(.)i Fu(F)-7 b(or)31 b(a)f(general)f(b)r(ounded)j(domain)e Ft(D)h Fr(\032)d Fq(R)2417 493 y Fp(3)2454 523 y Fu(,)k(the)f (condition)f(of)h(densit)n(y)515 623 y(of)h Ft(E)675 635 y Fo(1)777 623 y Fu(in)g(the)h(space)e Ft(H)39 b Fu(is)31 b(di\016cult)i(to)f(c)n(hec)n(k.)49 b(Ho)n(w)n(ev)n(er,)31 b(Theorem)g(2.1)g(is)h(v)-5 b(alid)32 b(for)515 722 y(an)n(y)c (three-dimensional)g(torus,)h(and)g(it)h(is)f(sho)n(wn)f(in)i([Shi06a)o (])f(that)g(if)h Ft(E)k Fu(con)n(tains)29 b(the)515 822 y(\014rst)22 b Ft(N)32 b Fu(eigen)n(v)-5 b(alues)21 b(of)i(the)g(Stok)n (es)f(op)r(erator,)g(then)h Ft(E)2285 834 y Fo(1)2378 822 y Fu(is)g(dense)f(in)h Ft(H)30 b Fu(for)22 b(su\016cien)n(tly)515 922 y(large)31 b Ft(N)9 b Fu(.)51 b(W)-7 b(e)33 b(th)n(us)g(obtain)f(a) g(\\uniform")g(v)n(ersion)f(of)h(the)h(Main)g(Theorem)e(stated)i(in)515 1021 y(the)28 b(In)n(tro)r(duction.)639 1150 y(W)-7 b(e)37 b(no)n(w)e(turn)h(to)g(the)g(auxiliary)f(result)h(needed)g(in)g(the)g (pro)r(of)g(of)g(Theorem)f(2.1.)515 1249 y(Let)f Ft(Z)40 b Fu(b)r(e)35 b(a)f(separable)f(Banac)n(h)g(space,)i(let)g Ft(V)53 b Fu(b)r(e)35 b(a)f(P)n(olish)f(space,)i(and)g(let)f Ft(F)47 b Fu(b)r(e)34 b(a)515 1349 y(\014nite-dimensional)25 b(v)n(ector)g(space.)36 b(F)-7 b(or)25 b(an)n(y)g(p)r(oin)n(ts)h Ft(z)2273 1361 y Fp(0)2333 1349 y Fr(2)e Ft(Z)31 b Fu(and)26 b Ft(u)2708 1361 y Fp(0)2768 1349 y Fr(2)e Ft(V)45 b Fu(and)25 b(p)r(ositiv)n(e)515 1448 y(constan)n(ts)h Ft(r)919 1460 y Fp(1)985 1448 y Fu(and)h Ft(r)1183 1460 y Fp(2)1221 1448 y Fu(,)h(w)n(e)f(set)1288 1620 y Ft(D)r Fu(\()p Ft(z)1430 1632 y Fp(0)1467 1620 y Ft(;)14 b(u)1552 1632 y Fp(0)1589 1620 y Fu(\))23 b(=)1760 1599 y(_)1732 1620 y Ft(B)1795 1632 y Fs(Z)1848 1620 y Fu(\()p Ft(z)1919 1632 y Fp(0)1956 1620 y Ft(;)14 b(r)2030 1632 y Fp(1)2068 1620 y Fu(\))k Fr(\002)2230 1599 y Fu(_)2201 1620 y Ft(B)2264 1632 y Fs(V)2322 1620 y Fu(\()p Ft(u)2402 1632 y Fp(0)2439 1620 y Ft(;)c(r)2513 1632 y Fp(2)2551 1620 y Fu(\))p Ft(:)515 1792 y Fu(Let)40 b Ft(f)52 b Fu(:)44 b Ft(D)r Fu(\()p Ft(z)978 1804 y Fp(0)1015 1792 y Ft(;)14 b(u)1100 1804 y Fp(0)1137 1792 y Fu(\))44 b Fr(!)g Ft(F)52 b Fu(b)r(e)41 b(a)e(con)n(tin)n(uous)g(mapping.)74 b(F)-7 b(or)40 b(an)n(y)f Ft(\025)44 b Fr(2)g(P)7 b Fu(\()p Ft(Z)f Fu(\))40 b(and)515 1891 y Ft(\026)23 b Fr(2)g(P)7 b Fu(\()p Ft(V)19 b Fu(\),)k(denote)f(b) n(y)f Ft(f)1320 1903 y Fo(\003)1358 1891 y Fu(\()p Ft(\025;)14 b(\026)p Fu(\))23 b(the)f(image)f(under)h Ft(f)30 b Fu(of)22 b(the)g(restriction)e(of)i(the)g(pro)r(duct)515 1991 y(measure)30 b Ft(\025)22 b Fr(\012)f Ft(\026)32 b Fu(to)g Ft(D)r Fu(\()p Ft(z)1328 2003 y Fp(0)1365 1991 y Ft(;)14 b(u)1450 2003 y Fp(0)1486 1991 y Fu(\).)50 b(The)32 b(follo)n(wing)f (theorem)g(is)h(a)f(mo)r(di\014ed)h(v)n(ersion)f(of)g(a)515 2091 y(result)c(established)g(in)h([AKSS06)o(];)g(its)g(pro)r(of)f(is)g (giv)n(en)g(in)h(the)g(next)g(subsection.)515 2248 y Fv(Theorem)44 b(2.4.)j Fm(L)l(et)40 b Ft(f)50 b Fu(:)42 b Ft(D)r Fu(\()p Ft(z)1582 2260 y Fp(0)1619 2248 y Ft(;)14 b(u)1704 2260 y Fp(0)1741 2248 y Fu(\))42 b Fr(!)g Ft(F)52 b Fm(b)l(e)41 b(a)f(c)l(ontinuous)g(mapping)h(such)g(that,)515 2348 y(for)36 b(any)g Ft(u)c Fr(2)1016 2327 y Fu(_)987 2348 y Ft(B)1050 2360 y Fs(V)1107 2348 y Fu(\()p Ft(u)1187 2360 y Fp(0)1224 2348 y Ft(;)14 b(r)1298 2360 y Fp(2)1336 2348 y Fu(\))p Fm(,)37 b(the)f(function)f Ft(f)9 b Fu(\()p Fr(\001)p Ft(;)14 b(u)p Fu(\))33 b(:)2246 2327 y(_)2217 2348 y Ft(B)2280 2360 y Fs(Z)2333 2348 y Fu(\()p Ft(z)2404 2360 y Fp(0)2441 2348 y Ft(;)14 b(r)2515 2360 y Fp(1)2553 2348 y Fu(\))33 b Fr(!)g Ft(F)48 b Fm(is)35 b(c)l(ontinuously)515 2447 y(di\013er)l(entiable,)45 b(the)40 b(derivative)i Fu(\()p Ft(D)1689 2459 y Fs(z)1727 2447 y Ft(f)9 b Fu(\)\()p Ft(z)t(;)14 b(u)p Fu(\))40 b Fm(is)g(c)l(ontinuous)g(on)g Ft(D)r Fu(\()p Ft(z)2840 2459 y Fp(0)2877 2447 y Ft(;)14 b(u)2962 2459 y Fp(0)2999 2447 y Fu(\))p Fm(,)43 b(and)e(the)515 2547 y(image)e(of)g(the)g(line)l(ar)g(op)l(er)l(ator)g Fu(\()p Ft(D)1697 2559 y Fs(z)1736 2547 y Ft(f)9 b Fu(\)\()p Ft(z)1889 2559 y Fp(0)1926 2547 y Ft(;)14 b(u)2011 2559 y Fp(0)2047 2547 y Fu(\))39 b Fm(c)l(oincides)h(with)f(the)f(entir)l(e) h(sp)l(ac)l(e)g Ft(F)12 b Fm(.)515 2646 y(L)l(et)34 b Ft(\025)e Fr(2)g(P)7 b Fu(\()p Ft(Z)f Fu(\))34 b Fm(and)h Ft(\026)d Fr(2)g(P)7 b Fu(\()p Ft(V)19 b Fu(\))35 b Fm(b)l(e)f(two)h (me)l(asur)l(es)f(such)h(that)f(Condition)i(1.15)h(holds)515 2746 y(for)29 b Ft(\025)p Fm(,)h(and)f Fu(supp)13 b Ft(\026)24 b Fu(=)e Ft(V)d Fm(.)38 b(Then)29 b(ther)l(e)g(is)g(a)g(function)f Ft(\032)23 b Fr(2)g Ft(C)6 b Fu(\()p Ft(F)12 b Fu(\))30 b Fm(such)e(that)h Ft(\032)22 b(>)h Fu(0)28 b Fm(in)h(a)515 2846 y(neighb)l(ourho)l(o)l(d)j(of)e Ft(y)1191 2858 y Fp(0)1251 2846 y Fu(=)23 b Ft(f)9 b Fu(\()p Ft(z)1460 2858 y Fp(0)1497 2846 y Ft(;)14 b(u)1582 2858 y Fp(0)1618 2846 y Fu(\))30 b Fm(and)1674 3017 y Ft(f)1715 3029 y Fo(\003)1753 3017 y Fu(\()p Ft(\025;)14 b(\026)p Fu(\))24 b Fr(\025)f Ft(\032`)2142 3029 y Fs(F)2196 3017 y Ft(:)989 b Fu(\(2.3\))515 3189 y Fm(F)-6 b(urthermor)l(e,)34 b(ther)l(e)f(is)h (an)f(op)l(en)h(b)l(al)t(l)g Ft(B)g Fr(\032)29 b Ft(V)52 b Fm(c)l(entr)l(e)l(d)32 b(at)i Ft(u)2533 3201 y Fp(0)2603 3189 y Fm(and)f(a)h(c)l(omp)l(actly)g(sup-)515 3289 y(p)l(orte)l(d)c (function)g Ft( )c Fr(2)d Ft(C)6 b Fu(\()p Ft(F)31 b Fr(\002)18 b Ft(B)t Fu(\))p Fm(,)31 b(b)l(oth)f(of)h(them)e(not)h(dep)l (ending)h(on)f Ft(\026)p Fm(,)g(such)g(that)1298 3503 y Ft(\032)p Fu(\()p Ft(y)s Fu(\))23 b(=)1559 3390 y Fn(Z)1606 3578 y Fs(B)1677 3503 y Ft( )s Fu(\()p Ft(y)s(;)14 b(u)p Fu(\))g Ft(\026)p Fu(\()p Ft(du)p Fu(\))84 b Fm(for)31 b Ft(y)26 b Fr(2)d Ft(F)12 b(;)612 b Fu(\(2.4\))1699 3681 y Ft( )s Fu(\()p Ft(y)1829 3693 y Fp(0)1866 3681 y Ft(;)14 b(u)1951 3693 y Fp(0)1988 3681 y Fu(\))23 b Ft(>)g Fu(0)p Ft(:)1012 b Fu(\(2.5\))515 3912 y Fl(2.2)112 b(Pro)s(of)37 b(of)h(Theorem)g(2.4)515 4065 y Fu(W)-7 b(e)30 b(rep)r(eat)f(the)h(sc)n(heme)g(used)f(in)h([AKSS06)o(])g(for)f (the)i(case)d(of)i(analytic)f(functions)h(and)515 4165 y(measures)h(on)h(Hilb)r(ert)h(spaces.)50 b(By)33 b(assumption,)g(the)g (image)e(of)i Ft(A)e Fu(:=)g(\()p Ft(D)2997 4177 y Fs(z)3035 4165 y Ft(f)9 b Fu(\)\()p Ft(z)3188 4177 y Fp(0)3225 4165 y Ft(;)14 b(u)3310 4177 y Fp(0)3347 4165 y Fu(\))515 4264 y(coincides)19 b(with)g Ft(F)32 b Fu(and)19 b Fr([)1331 4276 y Fs(n)1376 4264 y Ft(F)1429 4276 y Fs(n)1495 4264 y Fu(is)g(dense)g(in)h Ft(Z)6 b Fu(.)33 b(Therefore)19 b(w)n(e)g(can)g(\014nd)g(an)g(in)n(teger)g Ft(m)k Fr(\025)f Fu(1)515 4364 y(and)33 b(a)g(subspace)f Ft(F)1174 4334 y Fp(1)1162 4385 y Fs(m)1258 4364 y Fr(\032)g Ft(F)1408 4376 y Fs(m)1505 4364 y Fu(of)h(dimension)g(dim)14 b Ft(F)45 b Fu(suc)n(h)33 b(that)g Ft(A)p Fu(\()p Ft(F)2790 4334 y Fp(1)2778 4385 y Fs(m)2842 4364 y Fu(\))g(=)f Ft(F)12 b Fu(.)54 b(Let)33 b(us)515 4464 y(denote)d(b)n(y)h Ft(F)969 4434 y Fp(2)957 4484 y Fs(m)1049 4464 y Fr(\032)d Ft(F)1195 4476 y Fs(m)1289 4464 y Fu(an)n(y)i(subspace)h(suc)n(h)f (that)h Ft(F)2226 4476 y Fs(m)2318 4464 y Fu(=)d Ft(F)2476 4434 y Fp(1)2464 4484 y Fs(m)2548 4464 y Fc(u)20 b Ft(F)2698 4434 y Fp(2)2686 4484 y Fs(m)2750 4464 y Fu(.)47 b(Com)n(bining)30 b(this)515 4563 y(with)e(\(1.22\))o(,)g(w)n(e)f(obtain)g(the)h(direct)g (decomp)r(osition)1567 4735 y Ft(Z)g Fu(=)23 b Ft(F)1805 4701 y Fp(1)1793 4755 y Fs(m)1875 4735 y Fc(u)18 b Ft(F)2023 4701 y Fp(2)2011 4755 y Fs(m)2093 4735 y Fc(u)g Ft(G)2241 4747 y Fs(m)2304 4735 y Ft(:)515 4907 y Fu(F)-7 b(or)40 b Ft(z)49 b Fr(2)e Ft(Z)6 b Fu(,)44 b(w)n(e)d(shall)g(write)g Ft(z)49 b Fu(=)c(\()p Ft(z)1836 4919 y Fs(m)1899 4907 y Ft(;)14 b(z)1979 4876 y Fo(0)1975 4927 y Fs(m)2038 4907 y Fu(\))46 b(=)f(\()p Ft(z)2301 4876 y Fp(1)2297 4927 y Fs(m)2360 4907 y Ft(;)14 b(z)2440 4876 y Fp(2)2436 4927 y Fs(m)2499 4907 y Ft(;)g(z)2579 4876 y Fo(0)2575 4927 y Fs(m)2637 4907 y Fu(\),)45 b(where)c Ft(z)3034 4876 y Fp(1)3030 4927 y Fs(m)3138 4907 y Fr(2)46 b Ft(F)3304 4876 y Fp(1)3292 4927 y Fs(m)3356 4907 y Fu(,)515 5006 y Ft(z)558 4976 y Fp(2)554 5027 y Fs(m)643 5006 y Fr(2)26 b Ft(F)789 4976 y Fp(2)777 5027 y Fs(m)841 5006 y Fu(,)k Ft(z)933 5018 y Fs(m)1022 5006 y Fr(2)c Ft(F)1156 5018 y Fs(m)1220 5006 y Fu(,)k(and)f Ft(z)1479 4976 y Fo(0)1475 5027 y Fs(m)1564 5006 y Fr(2)e Ft(G)1711 5018 y Fs(m)1774 5006 y Fu(.)43 b(Applying)30 b(the)g(implicit)g(function)g(theorem)f (to)1905 5255 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 515 523 a Fu(the)28 b(function)g Ft(f)9 b Fu(\()p Ft(z)t(;)14 b(u)p Fu(\))26 b(in)i(the)g(neigh)n(b)r (ourho)r(o)r(d)f(of)g(\()p Ft(z)2224 535 y Fp(0)2261 523 y Ft(;)14 b(u)2346 535 y Fp(0)2383 523 y Fu(\),)28 b(w)n(e)f(can)h(\014nd)g(op)r(en)f(balls)1169 676 y Ft(V)1217 688 y Fp(1)1278 676 y Fr(\032)c Ft(F)1431 642 y Fp(1)1419 697 y Fs(m)1482 676 y Ft(;)97 b(V)1650 688 y Fp(2)1711 676 y Fr(\032)23 b Ft(F)1864 642 y Fp(2)1852 697 y Fs(m)1915 676 y Ft(;)97 b(V)2083 688 y Fp(3)2144 676 y Fr(\032)22 b Ft(G)2296 688 y Fs(m)2360 676 y Ft(;)97 b(B)27 b Fr(\032)22 b Ft(V)515 829 y Fu(suc)n(h)28 b(that,)h(for)e(an)n(y)h Ft(z)1235 799 y Fp(2)1231 850 y Fs(m)1318 829 y Fr(2)d Ft(V)1446 841 y Fp(2)1483 829 y Fu(,)k Ft(z)1578 799 y Fo(0)1574 850 y Fs(m)1661 829 y Fr(2)24 b Ft(V)1788 841 y Fp(3)1826 829 y Fu(,)29 b(and)f Ft(u)c Fr(2)g Ft(B)t Fu(,)29 b(the)g(mapping)f Ft(f)9 b Fu(\()p Fr(\001)p Ft(;)14 b(z)2981 799 y Fp(2)2977 850 y Fs(m)3039 829 y Ft(;)g(z)3119 799 y Fo(0)3115 850 y Fs(m)3178 829 y Ft(;)g(u)p Fu(\))28 b(is)515 929 y(a)h(di\013eomorphism)g(of)g Ft(V)1316 941 y Fp(1)1384 929 y Fu(on)n(to)f(its)i(image)f Ft(W)12 b Fu(\()p Ft(z)2097 899 y Fp(2)2093 950 y Fs(m)2156 929 y Ft(;)i(z)2236 899 y Fo(0)2232 950 y Fs(m)2294 929 y Ft(;)g(u)p Fu(\).)43 b(Let)29 b Ft(g)s Fu(\()p Fr(\001)p Ft(;)14 b(z)2805 899 y Fp(2)2801 950 y Fs(m)2864 929 y Ft(;)g(z)2944 899 y Fo(0)2940 950 y Fs(m)3002 929 y Ft(;)g(u)p Fu(\))30 b(b)r(e)g(the)515 1029 y(in)n(v)n(erse)c(mapping,)h (so)g(that)h(for)f Ft(z)1606 999 y Fp(1)1602 1049 y Fs(m)1688 1029 y Fr(2)c Ft(V)1814 1041 y Fp(1)1880 1029 y Fu(and)k Ft(y)f Fr(2)d Ft(W)12 b Fu(\()p Ft(z)2351 999 y Fp(2)2347 1049 y Fs(m)2410 1029 y Ft(;)i(z)2490 999 y Fo(0)2486 1049 y Fs(m)2549 1029 y Ft(;)g(u)p Fu(\),)27 b(w)n(e)g(ha)n(v)n(e)890 1182 y Ft(y)f Fu(=)c Ft(f)9 b Fu(\()p Ft(z)1169 1148 y Fp(1)1165 1202 y Fs(m)1228 1182 y Ft(;)14 b(z)1308 1148 y Fp(2)1304 1202 y Fs(m)1366 1182 y Ft(;)g(z)1446 1148 y Fo(0)1442 1202 y Fs(m)1505 1182 y Ft(;)g(u)p Fu(\))83 b(if)28 b(and)f(only)g(if)84 b Ft(z)2299 1148 y Fp(1)2295 1202 y Fs(m)2381 1182 y Fu(=)22 b Ft(g)s Fu(\()p Ft(y)s(;)14 b(z)2667 1148 y Fp(2)2663 1202 y Fs(m)2725 1182 y Ft(;)g(z)2805 1148 y Fo(0)2801 1202 y Fs(m)2864 1182 y Ft(;)g(u)p Fu(\))p Ft(:)204 b Fu(\(2.6\))515 1335 y(Let)26 b(us)g(\014x)g(some)g(bases)f (in)i Ft(F)1467 1305 y Fp(1)1455 1356 y Fs(m)1544 1335 y Fu(and)f Ft(F)39 b Fu(and)26 b(denote)g(b)n(y)g Ft(d)p Fu(\()p Ft(z)2454 1305 y Fp(1)2450 1356 y Fs(m)2513 1335 y Ft(;)14 b(z)2593 1305 y Fp(2)2589 1356 y Fs(m)2651 1335 y Ft(;)g(z)2731 1305 y Fo(0)2727 1356 y Fs(m)2790 1335 y Ft(;)g(u)p Fu(\))26 b(the)g(determi-)515 1435 y(nan)n(t)g(of)h(the)g(deriv)-5 b(ativ)n(e)26 b(\()p Ft(D)1422 1450 y Fs(z)1456 1433 y Fb(1)1454 1466 y Fk(m)1513 1435 y Ft(f)9 b Fu(\)\()p Ft(z)1670 1405 y Fp(1)1666 1455 y Fs(m)1729 1435 y Ft(;)14 b(z)1809 1405 y Fp(2)1805 1455 y Fs(m)1868 1435 y Ft(;)g(z)1948 1405 y Fo(0)1944 1455 y Fs(m)2006 1435 y Ft(;)g(u)p Fu(\).)37 b(No)n(w)26 b(note)h(that,)g(in)g(view)g(of)33 b(\(1.23\))o(,)515 1534 y(the)28 b(pro)r(duct)f(measure)g Ft(\025)19 b Fr(\012)f Ft(\026)27 b Fu(on)h(the)g(space)f Ft(Z)d Fr(\002)18 b Ft(V)46 b Fu(can)27 b(b)r(e)h(written)g(as)1128 1688 y(\()p Ft(\025)19 b Fr(\012)f Ft(\026)p Fu(\)\()p Ft(dz)t(;)c(du)p Fu(\))23 b(=)g Ft(\032)1824 1700 y Fs(m)1887 1688 y Fu(\()p Ft(z)1958 1700 y Fs(m)2021 1688 y Fu(\))14 b Ft(dz)2149 1700 y Fs(m)2226 1688 y Ft(\025)2274 1653 y Fo(0)2274 1708 y Fs(m)2337 1688 y Fu(\()p Ft(dz)2455 1653 y Fo(0)2451 1708 y Fs(m)2514 1688 y Fu(\))g Ft(\026)p Fu(\()p Ft(du)p Fu(\))515 1841 y(where)37 b Ft(dz)847 1853 y Fs(m)948 1841 y Fu(denotes)h(the)g(Leb)r(esgue)g(measure)f(on)g Ft(F)2294 1853 y Fs(m)2358 1841 y Fu(,)j Ft(\032)2464 1853 y Fs(m)2565 1841 y Fu(is)e(the)h(densit)n(y)f(of)f Fd(P)3267 1853 y Fs(m)3331 1841 y Ft(\025)515 1940 y Fu(with)29 b(resp)r(ect)f(to)h Ft(dz)1175 1952 y Fs(m)1238 1940 y Fu(,)g(and)f Ft(\025)1500 1910 y Fo(0)1500 1961 y Fs(m)1588 1940 y Fu(=)d Fd(Q)1739 1952 y Fs(m)1802 1940 y Ft(\025)k Fu(.)40 b(It)29 b(follo)n(ws)e(that)i(if)h Ft(\037)p Fu(\()p Ft(z)t(;)14 b(u)p Fu(\))27 b(is)i(a)f(con)n(tin)n (uous)515 2040 y(function)35 b(on)f Ft(Z)28 b Fr(\002)23 b Ft(V)53 b Fu(with)34 b(supp)r(ort)h(in)f(the)h(set)f Ft(S)39 b Fu(:=)34 b Ft(V)2402 2052 y Fp(1)2462 2040 y Fr(\002)23 b Ft(V)2598 2052 y Fp(2)2658 2040 y Fr(\002)g Ft(V)2794 2052 y Fp(3)2854 2040 y Fr(\002)g Ft(B)t Fu(,)36 b(then)f(the)515 2140 y(image)23 b(of)i(the)g(truncated)f(measure)f Ft(\037)p Fu(\()p Ft(\025)12 b Fr(\012)g Ft(\026)p Fu(\))25 b(under)f Ft(f)34 b Fu(admits)24 b(the)h(represen)n(tation)d(\(cf.)515 2239 y(Sections)27 b(2.2)g(and)g(2.3)g(in)h([AKSS06)o(]\))560 2446 y Ft(f)601 2458 y Fo(\003)639 2378 y Fn(\000)677 2446 y Ft(\037)p Fu(\()p Ft(\025)19 b Fr(\012)f Ft(\026)p Fu(\))993 2378 y Fn(\001)1032 2446 y Fu(\()p Ft(dy)s Fu(\))23 b(=)1294 2329 y Fn(\032)1356 2333 y(Z)1463 2390 y Fu(\()8 b Fn(e)-54 b Ft(\037)8 b Fu(~)-50 b Ft(\032)1590 2402 y Fs(m)1653 2390 y Fu(\)\()p Ft(y)s(;)14 b(z)1841 2359 y Fp(2)1837 2410 y Fs(m)1900 2390 y Ft(;)g(z)1980 2359 y Fo(0)1976 2410 y Fs(m)2038 2390 y Ft(;)g(u)p Fu(\))p 1463 2427 693 4 v 1529 2514 a Fr(j)1567 2492 y Fu(~)1552 2514 y Ft(d)q Fu(\()p Ft(y)s(;)g(z)1752 2490 y Fp(2)1748 2534 y Fs(m)1810 2514 y Ft(;)g(z)1890 2490 y Fo(0)1886 2534 y Fs(m)1949 2514 y Ft(;)g(u)p Fu(\))p Fr(j)2165 2446 y Ft(dz)2251 2411 y Fp(2)2247 2466 y Fs(m)2310 2446 y Ft(\025)2358 2411 y Fo(0)2358 2466 y Fs(m)2421 2446 y Fu(\()p Ft(dz)2539 2411 y Fo(0)2535 2466 y Fs(m)2599 2446 y Fu(\))p Ft(\026)p Fu(\()p Ft(du)p Fu(\))2836 2329 y Fn(\033)2899 2446 y Ft(`)2934 2458 y Fs(F)2988 2446 y Fu(\()p Ft(dy)s Fu(\))p Ft(;)46 b Fu(\(2.7\))515 2654 y(where)27 b(the)h(in)n(tegral)e(is)i(tak)n(en)f(o)n(v)n(er)e Ft(V)1735 2666 y Fp(2)1792 2654 y Fr(\002)18 b Ft(V)1923 2666 y Fp(3)1979 2654 y Fr(\002)g Ft(B)t Fu(,)1189 2807 y(~)-50 b Ft(\032)1224 2819 y Fs(m)1287 2807 y Fu(\()p Ft(y)s(;)14 b(z)1443 2773 y Fp(2)1439 2828 y Fs(m)1502 2807 y Ft(;)g(z)1582 2773 y Fo(0)1578 2828 y Fs(m)1640 2807 y Ft(;)g(u)p Fu(\))23 b(=)g Ft(\032)1911 2819 y Fs(m)1974 2807 y Fu(\()p Ft(g)s Fu(\()p Ft(y)s(;)14 b(z)2205 2773 y Fp(2)2201 2828 y Fs(m)2263 2807 y Ft(;)g(z)2343 2773 y Fo(0)2339 2828 y Fs(m)2402 2807 y Ft(;)g(u)p Fu(\))p Ft(;)g(z)2599 2773 y Fp(2)2595 2828 y Fs(m)2657 2807 y Fu(\))p Ft(;)515 2961 y Fu(and)28 b(the)g(functions)36 b Fn(e)-53 b Ft(\037;)1282 2939 y Fu(~)1268 2961 y Ft(d)28 b Fu(are)f(de\014ned)i(in)f(a)g(similar)f(w)n(a)n(y)-7 b(.)38 b(Let)28 b(us)g(c)n(ho)r(ose)f(a)h(con)n(tin)n(uous)515 3060 y(function)f Ft(\037)h Fu(supp)r(orted)f(b)n(y)f Ft(S)32 b Fu(and)27 b(equal)g(to)g(1)g(in)g(the)h(neigh)n(b)r(ourho)r (o)r(d)e(of)h(\()p Ft(z)3027 3072 y Fp(0)3064 3060 y Ft(;)14 b(u)3149 3072 y Fp(0)3186 3060 y Fu(\))27 b(and)515 3160 y(denote)626 3346 y Ft( )s Fu(\()p Ft(y)s(;)14 b(u)p Fu(\))23 b(=)987 3233 y Fn(Z)1033 3421 y Fs(V)1072 3429 y Fb(2)1104 3421 y Fo(\002)p Fs(V)1195 3429 y Fb(3)1256 3289 y Fu(\()11 b Fn(e)-57 b Ft(')8 b Fu(~)-50 b Ft(\032)1385 3301 y Fs(m)1448 3289 y Fu(\)\()p Ft(y)s(;)14 b(z)1636 3259 y Fp(2)1632 3310 y Fs(m)1695 3289 y Ft(;)g(z)1775 3259 y Fo(0)1771 3310 y Fs(m)1834 3289 y Ft(;)g(u)p Fu(\))p 1256 3326 695 4 v 1324 3414 a Fr(j)1361 3392 y Fu(~)1347 3414 y Ft(d)p Fu(\()p Ft(y)s(;)g(z)1546 3390 y Fp(2)1542 3434 y Fs(m)1604 3414 y Ft(;)g(z)1684 3390 y Fo(0)1680 3434 y Fs(m)1743 3414 y Ft(;)g(u)p Fu(\))p Fr(j)1960 3346 y Ft(dz)2046 3311 y Fp(2)2042 3366 y Fs(m)2105 3346 y Ft(\025)2153 3311 y Fo(0)2153 3366 y Fs(m)2217 3346 y Fu(\()p Ft(dz)2335 3311 y Fo(0)2331 3366 y Fs(m)2394 3346 y Fu(\))p Ft(;)97 b Fu(\()p Ft(y)s(;)14 b(u)p Fu(\))23 b Fr(2)g Ft(F)31 b Fr(\002)18 b Ft(B)t(:)111 b Fu(\(2.8\))515 3559 y(Then)22 b(it)h(follo)n(ws)e(from)h(\(2.7\))g(that)g(inequalit)n (y)h(\(2.3\))f(holds)g(with)g(the)h(function)g Ft(\032)f Fu(de\014ned)515 3658 y(b)n(y)32 b(\(2.4\))o(.)47 b(The)31 b(con)n(tin)n(uit)n(y)g(of)g(the)g(functions)g Ft( )j Fu(and)d Ft(\032)g Fu(is)g(ob)n(vious)f(from)g(the)i(explicit)515 3758 y(form)n(ulas)e(for)h(them.)50 b(T)-7 b(o)31 b(pro)n(v)n(e)61 b(\(2.5\),)33 b(it)f(su\016ces)f(to)g(note)h(that)g(the)g(supp)r(ort)f (of)h(the)515 3858 y(measure)23 b Ft(dz)921 3827 y Fp(2)917 3878 y Fs(m)993 3858 y Fr(\012)13 b Ft(\025)1119 3827 y Fo(0)1119 3878 y Fs(m)1207 3858 y Fu(is)25 b(the)g(en)n(tire)f(space) g Ft(F)1942 3827 y Fp(2)1930 3878 y Fs(m)2006 3858 y Fr(\002)13 b Ft(G)2149 3870 y Fs(m)2212 3858 y Fu(,)26 b(and)e(the)h(in)n(tegrand)f(in)h(\(2.8\))g(with)515 3957 y Ft(y)36 b Fu(=)d Ft(y)731 3969 y Fp(0)802 3957 y Fu(and)g Ft(u)g Fu(=)g Ft(u)1196 3969 y Fp(0)1267 3957 y Fu(is)h(p)r(ositiv)n(e)f(on)h(an)f(op)r(en)h(set.)56 b(Finally)-7 b(,)35 b(the)f(p)r(ositivit)n(y)g(of)g Ft(\032)g Fu(in)515 4057 y(the)d(neigh)n(b)r(ourho)r(o)r(d)e(of)i Ft(y)1370 4069 y Fp(0)1438 4057 y Fu(follo)n(ws)f(from)g(\(2.5\))g(and) h(the)g(fact)g(that)g(the)g(supp)r(ort)f(of)h Ft(\026)515 4156 y Fu(coincides)c(with)h Ft(V)19 b Fu(.)37 b(The)27 b(pro)r(of)g(is)h(complete.)515 4426 y Fw(3)134 b(Pro)t(of)45 b(of)h(the)f(main)g(theorem)515 4608 y Fu(In)35 b(this)g(section,)i(w)n (e)d(presen)n(t)h(a)f(pro)r(of)g(of)h(Theorem)g(2.1.)58 b(T)-7 b(o)34 b(mak)n(e)g(the)i(main)f(ideas)515 4707 y(more)g(transparen)n(t,)i(w)n(e)e(\014rst)h(pro)n(v)n(e)f(a)g(w)n(eak) n(er)f(v)n(ersion)h(of)h(our)f(result.)62 b(Namely)-7 b(,)38 b(w)n(e)515 4807 y(establish)30 b(assertions)e(\(i\))j(and)f (\(ii\))h(of)f(Theorem)f(2.1)g(for)h(a)g(giv)n(en)f(stationary)g (solution,)515 4907 y(without)e(caring)e(ab)r(out)i(the)g(uniformit)n (y)f(of)h(estimates.)36 b(In)27 b(the)g(second)f(subsection,)g(w)n(e) 515 5006 y(sho)n(w)g(ho)n(w)h(to)h(mo)r(dify)g(the)g(pro)r(of)f(to)g (obtain)h(the)g(result)f(in)h(full)g(generalit)n(y)-7 b(.)1905 5255 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 515 523 a Fl(3.1)112 b(Simpli\014ed)39 b(v)m(ersion:)51 b(non-uniform)39 b(estimates)515 676 y Fu(W)-7 b(e)28 b(\014rst)f(explain)g(the)h(main)f(idea.)37 b(By)27 b(de\014nition,)h(if)g Ft(\026)23 b Fr(2)g(P)7 b Fu(\()p Ft(H)g Fu(\))27 b(is)h(a)f(stationary)f(mea-)515 776 y(sure)h(for)g(\(1.8\))o(,)h(then)g(there)g(is)f(an)h(admissible)f (w)n(eak)f(solution)h Ft(u)p Fu(\()p Ft(t)p Fu(\))h(de\014ned)g(on)g(a) f(com-)515 876 y(plete)35 b(probabilit)n(y)e(space)h(\(\012)p Ft(;)14 b Fr(F)8 b Ft(;)14 b Fq(P)p Fu(\))35 b(with)g(righ)n(t-con)n (tin)n(uous)d(\014ltration)i Fr(F)2969 888 y Fs(t)3033 876 y Fu(suc)n(h)g(that)515 975 y Fr(D)r Fu(\()p Ft(u)p Fu(\()p Ft(t)p Fu(\)\))26 b(=)f Ft(\026)k Fu(for)f(an)n(y)g Ft(t)d Fr(\025)g Fu(0.)40 b(Let)29 b(us)g(represen)n(t)e Ft(u)i Fu(in)g(the)g(form)g(\(1.13\))o(,)g(where)f Ft(z)t Fu(\()p Ft(t)p Fu(\))h(is)515 1075 y(the)20 b(Ornstein{Uhlen)n(b)r(ec)n (k)f(pro)r(cess)f(de\014ned)j(b)n(y)f(\(1.11\))g(and)f Ft(v)s Fu(\()p Ft(t)p Fu(\))i(is)f(an)f Fr(F)2857 1087 y Fs(t)2886 1075 y Fu(-progressiv)n(ely)515 1174 y(measurable)33 b(random)g(pro)r(cess)h(whose)g(almost)g(ev)n(ery)f(tra)5 b(jectory)33 b(is)h(a)g(w)n(eak)g(solution)515 1274 y(of)g(\(1.3\))o(.) j(Let)28 b Ft(J)j Fu(=)23 b([0)p Ft(;)14 b Fu(1])27 b(and)g(let)962 1453 y(\012)1022 1465 y Fp(0)1083 1453 y Fu(=)1170 1386 y Fn(\010)1219 1453 y Ft(!)e Fr(2)f Fu(\012)f(:)g Ft(u)p Fu(\(0\))g Fr(2)g Ft(V)5 b(;)14 b(z)26 b Fr(2)e Ft(C)6 b Fu(\()p Fq(R)2150 1465 y Fp(+)2205 1453 y Ft(;)14 b(V)19 b Fu(\))g Fr(\\)g Ft(L)2491 1419 y Fp(2)2491 1474 y(lo)r(c)2578 1453 y Fu(\()p Fq(R)2670 1465 y Fp(+)2725 1453 y Ft(;)14 b(U)9 b Fu(\))2860 1386 y Fn(\011)2908 1453 y Ft(:)515 1633 y Fu(Then)37 b(\012)801 1645 y Fp(0)877 1633 y Fr(2)i(F)45 b Fu(and)36 b Fq(P)p Fu(\(\012)1389 1645 y Fp(0)1427 1633 y Fu(\))i(=)h(1.)64 b(If)38 b Ft(!)j Fr(2)e Fu(\012)2070 1645 y Fp(0)2107 1633 y Fu(,)g(and)e(Eq.)g(\(1.3\))f(has)h(a)f(strong)g (solu-)515 1732 y(tion)g(^)-45 b Ft(v)35 b Fr(2)e(Y)909 1744 y Fs(J)988 1732 y Fu(satisfying)g(the)g(initial)g(condition)j(^) -45 b Ft(v)s Fu(\(0\))33 b(=)e Ft(u)2453 1744 y Fp(0)2490 1732 y Fu(,)j(where)f Ft(u)2841 1744 y Fp(0)2910 1732 y Fu(=)e Ft(u)p Fu(\(0\),)k(then)515 1832 y(Prop)r(osition)26 b(1.3)g(implies)i(that)1120 2011 y Ft(u)p Fu(\()p Ft(t)p Fu(\))23 b(=)f Ft(z)t Fu(\()p Ft(t)p Fu(\))d(+)f Fr(S)1661 2023 y Fs(t)1690 2011 y Fu(\()p Ft(z)t(;)c(u)1850 2023 y Fp(0)1886 2011 y Fu(\))24 b(=:)e Fr(T)2097 2023 y Fs(t)2127 2011 y Fu(\()p Ft(z)t(;)14 b(u)2287 2023 y Fp(0)2323 2011 y Fu(\))83 b(for)28 b Ft(t)23 b Fr(2)g Ft(J)8 b(:)434 b Fu(\(3.1\))515 2191 y(Here)30 b Fr(S)37 b Fu(denotes)30 b(an)h(op)r(erator)e(taking)h(\()p Ft(z)t(;)14 b(v)1972 2203 y Fp(0)2009 2191 y Fu(\))28 b Fr(2)g(Y)2207 2203 y Fs(J)2275 2191 y Fr(\002)20 b Ft(V)49 b Fu(to)31 b(the)g(solution)f Ft(v)h Fr(2)e(Y)3235 2203 y Fs(J)3312 2191 y Fu(of)515 2290 y(problem)c(\(1.3\),)h(\(1.21\))g(and)f Fr(S)1504 2302 y Fs(t)1560 2290 y Fu(stands)g(for)h(the)g(restriction)f(of)h Fr(S)33 b Fu(to)26 b(the)g(time)g Ft(t)p Fu(.)37 b(What)515 2390 y(has)27 b(b)r(een)h(said)f(implies)h(that)1464 2360 y Fp(2)863 2569 y Ft(\026)p Fu(\(\000\))c(=)e Fq(P)p Fr(f)p Ft(u)p Fu(\(1\))g Fr(2)i Fu(\000)p Fr(g)e(\025)h Fq(P)p Fr(fT)1830 2581 y Fp(1)1866 2569 y Fu(\()p Ft(z)t(;)14 b(u)2026 2581 y Fp(0)2063 2569 y Fu(\))23 b Fr(2)g Fu(\000)p Fr(g)83 b Fu(for)27 b(an)n(y)g(\000)c Fr(2)g(B)s Fu(\()p Ft(H)7 b Fu(\))p Ft(:)177 b Fu(\(3.2\))515 2749 y(Th)n(us,)25 b(Theorem)f(2.1)g(will)h(b)r(e)g(established)g(if)g(w)n(e)f(sho)n(w)g (that)h(assertions)e(\(i\))j(and)e(\(ii\))i(are)515 2848 y(v)-5 b(alid)34 b(for)g(the)h(distribution)f(of)h(the)f(random)g(v)-5 b(ariable)33 b Fr(T)2391 2860 y Fp(1)2429 2848 y Fu(\()p Ft(u)2509 2860 y Fp(0)2546 2848 y Ft(;)14 b(z)t Fu(\).)57 b(The)34 b(\014rst)g(of)h(them)515 2948 y(is)g(a)h(simple)g (consequence)e(of)i(the)g(appro)n(ximate)e(con)n(trollabilit)n(y)g(of)i (the)g(NS)g(system,)515 3047 y(while)g(the)h(other)e(will)i(follo)n(w)e (from)h(the)h(solid)e(con)n(trollabilit)n(y)g(in)h (\014nite-dimensional)515 3147 y(pro)5 b(jections)26 b(and)i(Theorem)e(2.4.)639 3247 y(W)-7 b(e)28 b(no)n(w)f(turn)h(to)f (the)h(accurate)f(pro)r(of.)36 b(It)28 b(is)f(divided)h(in)n(to)g(sev)n (eral)d(steps.)639 3395 y Fm(Step)d(1)p Fu(.)34 b(Let)19 b(us)g(denote)g(b)n(y)g Ft(\025)k Fr(2)h(P)7 b Fu(\()p Fr(Y)1821 3407 y Fs(J)1867 3395 y Fu(\))20 b(the)f(la)n(w)f(of)h(the)g (restriction)f(of)h(the)g(Ornstein{)515 3494 y(Uhlen)n(b)r(ec)n(k)26 b(pro)r(cess)f(\(1.11\))g(to)h(the)g(in)n(terv)-5 b(al)25 b Ft(J)35 b Fu(and)25 b(let)i Ft(Z)h Fu(=)23 b(supp)14 b Ft(\025)p Fu(.)37 b(It)26 b(is)g(w)n(ell)g(kno)n(wn)515 3594 y(that)33 b Ft(\025)h Fu(is)f(a)f(Gaussian)h(measure)f(on)g Fr(Y)1812 3606 y Fs(J)1859 3594 y Fu(.)54 b(As)33 b(w)n(as)f(explained) h(in)g(Example)f(1.16,)i(the)515 3694 y(measure)20 b Ft(\025)k Fr(2)f(P)7 b Fu(\()p Ft(Z)f Fu(\))22 b(satis\014es)e (Condition)i(1.15.)33 b(W)-7 b(e)22 b(claim)g(that,)h(for)e(an)n(y)g (Borel)f(subsets)515 3793 y Ft(B)578 3805 y Fp(1)638 3793 y Fr(\032)j Ft(Z)33 b Fu(and)27 b Ft(B)1040 3805 y Fp(2)1101 3793 y Fr(\032)22 b Ft(V)47 b Fu(satisfying)27 b(the)h(inclusion)1632 3973 y Ft(B)1695 3985 y Fp(1)1751 3973 y Fr(\002)18 b Ft(B)1897 3985 y Fp(2)1957 3973 y Fr(\032)23 b(C)2089 3985 y Fp(1)2126 3973 y Fu(\()p Ft(h)p Fu(\))p Ft(;)947 b Fu(\(3.3\))515 4152 y(w)n(e)27 b(ha)n(v)n(e)1610 4252 y Ft(\026)c Fr(\025)g(T)1816 4264 y Fp(1)p Fo(\003)1887 4252 y Fu(\()p Ft(\025)1967 4264 y Fs(B)2017 4272 y Fb(1)2055 4252 y Ft(;)14 b(\026)2142 4264 y Fs(B)2192 4272 y Fb(2)2228 4252 y Fu(\))p Ft(;)925 b Fu(\(3.4\))515 4399 y(where)19 b(the)g(righ)n(t-hand)g(side)g(of)26 b(\(3.4\))19 b(stands)g(for)g(the) g(image)g(under)g Fr(T)2731 4411 y Fp(1)2788 4399 y Fu(of)g(the)h (restriction)515 4499 y(of)27 b(the)h(pro)r(duct)g(measure)e Ft(\025)19 b Fr(\012)f Ft(\026)28 b Fu(to)f(the)h(set)g Ft(B)2052 4511 y Fp(1)2108 4499 y Fr(\002)18 b Ft(B)2254 4511 y Fp(2)2291 4499 y Fu(.)639 4598 y(T)-7 b(o)27 b(pro)n(v)n(e)f (\(3.4\),)i(let)g(us)f(tak)n(e)g(an)n(y)g(Borel)f(set)i(\000)23 b Fr(\032)g Ft(H)34 b Fu(and)28 b(write)999 4778 y Ft(\026)p Fu(\(\000\))23 b(=)g Fq(P)p Fr(f)p Ft(u)p Fu(\(1\))f Fr(2)h Fu(\000)p Fr(g)g(\025)f Fq(P)p Fr(f)p Ft(z)k Fr(2)d Ft(B)2126 4790 y Fp(1)2163 4778 y Ft(;)14 b(u)2248 4790 y Fp(0)2308 4778 y Fr(2)24 b Ft(B)2450 4790 y Fp(2)2487 4778 y Ft(;)14 b(u)p Fu(\(1\))22 b Fr(2)i Fu(\000)p Fr(g)p Ft(;)312 b Fu(\(3.5\))p 515 4850 1146 4 v 607 4904 a Fi(2)642 4927 y Fh(Inequalit)n(y)31 b(\(3.2\))f(is)e(not)i(en)n(tirely) f(accurate,)j(b)r(ecause)e(the)g(op)r(erator)g Ff(T)2653 4936 y Fi(1)2716 4927 y Fh(is)f(de\014ned)h(only)f(on)h(an)515 5006 y(op)r(en)24 b(subset)h(of)e Ff(Y)1029 5017 y Fa(J)1087 5006 y Ff(\002)16 b Fg(V)g Fh(.)1905 5255 y Fu(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 515 523 a Fu(where)30 b Ft(u)806 535 y Fp(0)871 523 y Fu(=)f Ft(u)p Fu(\(0\).)46 b(In)31 b(view)g(of)g(the)g(w)n(eak-strong)d(uniqueness)j(\(see)f(Prop)r (osition)g(1.3\))515 623 y(and)d(inclusion)h(\(3.3\))o(,)g(w)n(e)f(ha)n (v)n(e)1252 804 y Ft(u)p Fu(\(1\))c(=)f Fr(T)1561 816 y Fp(1)1599 804 y Fu(\()p Ft(z)t(;)14 b(u)1759 816 y Fp(0)1795 804 y Fu(\))83 b(for)27 b Ft(z)g Fr(2)c Ft(B)2244 816 y Fp(1)2281 804 y Fu(,)28 b Ft(u)2380 816 y Fp(0)2440 804 y Fr(2)23 b Ft(B)2581 816 y Fp(2)2619 804 y Ft(:)515 986 y Fu(Substituting)i(this)g(relation)f(in)n(to)h(\(3.5\))f(and)h (using)f(the)h(indep)r(endence)h(of)f Ft(z)j Fu(and)c Ft(u)3199 998 y Fp(0)3236 986 y Fu(,)h(w)n(e)515 1086 y(obtain)1076 1268 y Ft(\026)p Fu(\(\000\))e Fr(\025)g Fq(P)p Fr(f)p Ft(z)i Fr(2)f Ft(B)1652 1280 y Fp(1)1689 1268 y Ft(;)14 b(u)1774 1280 y Fp(0)1834 1268 y Fr(2)23 b Ft(B)1975 1280 y Fp(2)2012 1268 y Ft(;)14 b Fr(T)2094 1280 y Fp(1)2132 1268 y Fu(\()p Ft(z)t(;)g(u)2292 1280 y Fp(0)2328 1268 y Fu(\))23 b Fr(2)h Fu(\000)p Fr(g)1265 1392 y Fu(=)f Fq(E)14 b(P)p Fr(f)p Ft(z)25 b Fr(2)f Ft(B)1721 1404 y Fp(1)1758 1392 y Ft(;)14 b(u)1843 1404 y Fp(0)1903 1392 y Fr(2)23 b Ft(B)2044 1404 y Fp(2)2081 1392 y Ft(;)14 b Fr(T)2163 1404 y Fp(1)2201 1392 y Fu(\()p Ft(z)t(;)g(u)2361 1404 y Fp(0)2397 1392 y Fu(\))23 b Fr(2)h Fu(\000)14 b Fr(j)g(F)2694 1404 y Fp(0)2730 1392 y Fr(g)1265 1551 y Fu(=)23 b Fq(E)1408 1459 y Fn(\020)1458 1551 y Ft(I)1494 1563 y Fs(B)1544 1571 y Fb(2)1581 1551 y Fu(\()p Ft(u)1661 1563 y Fp(0)1698 1551 y Fu(\))14 b Fq(E)1813 1483 y Fn(\010)1862 1551 y Ft(I)1898 1563 y Fs(B)1948 1571 y Fb(1)1985 1551 y Fu(\()p Ft(z)t Fu(\))p Ft(I)2128 1563 y Fp(\000)2173 1551 y Fu(\()p Fr(T)2250 1563 y Fp(1)2288 1551 y Fu(\()p Ft(z)t(;)g(v)s Fu(\)\))2507 1483 y Fn(\011)2556 1480 y(\014)2556 1530 y(\014)2583 1584 y Fs(v)r Fp(=)p Fs(u)2708 1592 y Fb(0)2746 1459 y Fn(\021)2795 1551 y Ft(;)390 b Fu(\(3.6\))515 1753 y(where)27 b Ft(I)791 1765 y Fp(\000)864 1753 y Fu(denotes)g(the)h(indicator)f(function)h(of)g(\000.)37 b(No)n(w)27 b(note)g(that)912 1935 y Fq(E)981 1867 y Fn(\010)1029 1935 y Ft(I)1065 1947 y Fs(B)1115 1955 y Fb(1)1153 1935 y Fu(\()p Ft(z)t Fu(\))p Ft(I)1296 1947 y Fp(\000)1341 1935 y Fu(\()p Fr(T)1418 1947 y Fp(1)1456 1935 y Fu(\()p Ft(z)t(;)14 b(v)s Fu(\)\))1675 1867 y Fn(\011)1747 1935 y Fu(=)22 b Fr(T)1879 1947 y Fp(1)p Fo(\003)1951 1935 y Fu(\()p Ft(\025)2031 1947 y Fs(B)2081 1955 y Fb(1)2118 1935 y Ft(;)14 b(v)s Fu(\)\(\000\))84 b(for)27 b(an)n(y)g Ft(v)f Fr(2)e Ft(B)2922 1947 y Fp(2)2959 1935 y Ft(;)226 b Fu(\(3.7\))515 2116 y(where)37 b Fr(T)810 2128 y Fp(1)p Fo(\003)881 2116 y Fu(\()p Ft(\025)961 2128 y Fs(B)1011 2136 y Fb(1)1049 2116 y Ft(;)14 b(v)s Fu(\))38 b(denotes)f(the)h(image)f(of)h(the)g(restriction)f(of)g Ft(\025)i Fu(to)e Ft(B)2943 2128 y Fp(1)3018 2116 y Fu(under)h(the)515 2216 y(mapping)27 b Fr(T)901 2228 y Fp(1)939 2216 y Fu(\()p Fr(\001)p Ft(;)14 b(v)s Fu(\).)37 b(Substituting)30 b(\(3.7\))d(in)n (to)h(\(3.6\))o(,)g(w)n(e)f(deriv)n(e)949 2398 y Ft(\026)p Fu(\(\000\))c Fr(\025)g Fq(E)1281 2331 y Fn(\010)1330 2398 y Ft(I)1366 2410 y Fs(B)1416 2418 y Fb(2)1453 2398 y Fu(\()p Ft(u)1533 2410 y Fp(0)1570 2398 y Fu(\))p Fr(T)1647 2410 y Fp(1)p Fo(\003)1719 2398 y Fu(\()p Ft(\025)1799 2410 y Fs(B)1849 2418 y Fb(1)1886 2398 y Ft(;)14 b(u)1971 2410 y Fp(0)2008 2398 y Fu(\)\(\000\))2156 2331 y Fn(\011)2228 2398 y Fu(=)23 b Fr(T)2361 2410 y Fp(1)p Fo(\003)2432 2398 y Fu(\()p Ft(\025)2512 2410 y Fs(B)2562 2418 y Fb(1)2600 2398 y Ft(;)14 b(\026)2687 2410 y Fs(B)2737 2418 y Fb(2)2773 2398 y Fu(\)\(\000\))p Ft(:)515 2580 y Fu(Since)28 b(\000)f(w)n(as)g (arbitrary)-7 b(,)25 b(w)n(e)j(arriv)n(e)d(at)j(the)g(required)e (inequalit)n(y)j(\(3.4\))o(.)639 2729 y Fm(Step)37 b(2)p Fu(.)59 b(W)-7 b(e)35 b(no)n(w)g(sho)n(w)f(that)h(for)g(an)n(y)k(^)-47 b Ft(u)2033 2741 y Fp(0)2105 2729 y Fr(2)35 b Ft(V)54 b Fu(and)35 b(an)n(y)f(ball)h Ft(B)40 b Fr(\032)35 b Ft(V)53 b Fu(there)35 b(is)520 2828 y(^)-47 b Ft(z)26 b Fr(2)d Fu(supp)14 b Ft(\025)29 b Fu(suc)n(h)e(that)1696 2928 y Fr(T)1741 2940 y Fp(1)1778 2928 y Fu(\()5 b(^)-47 b Ft(z)t(;)19 b Fu(^)-47 b Ft(u)1938 2940 y Fp(0)1974 2928 y Fu(\))24 b Fr(2)f Ft(B)t(:)1010 b Fu(\(3.8\))515 3077 y(Indeed,)31 b(in)f(view)g(of)g(Prop)r(osition)e(1.11,)h(for)h(an) n(y)k(^)-47 b Ft(u)2188 3089 y Fp(0)2252 3077 y Fr(2)27 b Ft(V)49 b Fu(there)30 b(exists)35 b(^)-48 b Ft(\021)31 b Fr(2)c Ft(C)3096 3047 y Fo(1)3167 3077 y Fu(\()p Ft(J)o(;)14 b(E)5 b Fu(\))515 3176 y(suc)n(h)27 b(that)1682 3276 y Fr(R)1752 3288 y Fp(1)1790 3276 y Fu(\()6 b(^)-48 b Ft(\021)s(;)19 b Fu(^)-47 b Ft(u)1951 3288 y Fp(0)1988 3276 y Fu(\))23 b Fr(2)h Ft(B)t(:)996 b Fu(\(3.9\))515 3425 y(Let)27 b(us)h(set)1463 3562 y(^)-47 b Ft(z)s Fu(\()p Ft(t)p Fu(\))24 b(=)1705 3449 y Fn(Z)1788 3469 y Fs(t)1752 3637 y Fp(0)1832 3562 y Ft(e)1871 3527 y Fo(\000)p Fs(\027)t Fp(\()p Fs(t)p Fo(\000)p Fs(s)p Fp(\))p Fs(L)2175 3562 y Fu(^)-48 b Ft(\021)s Fu(\()p Ft(s)p Fu(\))14 b Ft(ds:)515 3752 y Fu(It)28 b(is)f(a)g(matter)h(of)f(direct)h(v)n(eri\014cation)e (to)h(sho)n(w)g(that)1217 3934 y Fr(R)1287 3946 y Fp(1)1325 3934 y Fu(\()6 b(^)-48 b Ft(\021)s(;)20 b Fu(^)-48 b Ft(u)1486 3946 y Fp(0)1523 3934 y Fu(\))23 b(=)28 b(^)-47 b Ft(z)s Fu(\(1\))19 b(+)f Fr(S)1966 3946 y Fp(1)2003 3934 y Fu(\()5 b(^)-47 b Ft(z)t(;)19 b Fu(^)-47 b Ft(u)2163 3946 y Fp(0)2200 3934 y Fu(\))23 b(=)f Fr(T)2387 3946 y Fp(1)2425 3934 y Fu(\()5 b(^)-47 b Ft(z)t(;)19 b Fu(^)-47 b Ft(u)2585 3946 y Fp(0)2621 3934 y Fu(\))p Ft(;)491 b Fu(\(3.10\))515 4116 y(and)31 b(therefore)f(\(3.8\))h(follo)n(ws)g (immediately)g(from)g(\(3.9\).)48 b(T)-7 b(o)31 b(pro)n(v)n(e)f(that)36 b(^)-47 b Ft(z)33 b Fr(2)d Fu(supp)14 b Ft(\025)p Fu(,)515 4216 y(\014rst)27 b(note)h(that)1672 4315 y(^)-47 b Ft(z)26 b Fu(=)d(\()p Ft(M)k Fr(\016)18 b Ft(I)7 b Fu(\)\()f(^)-48 b Ft(\021)t Fu(\))p Ft(;)940 b Fu(\(3.11\))515 4464 y(where)27 b(w)n(e)g(set)819 4683 y(\()p Ft(I)7 b(\030)t Fu(\)\()p Ft(t)p Fu(\))24 b(=)1172 4570 y Fn(Z)1255 4590 y Fs(t)1218 4758 y Fp(0)1298 4683 y Ft(\030)t Fu(\()p Ft(s)p Fu(\))14 b Ft(ds;)97 b Fu(\()p Ft(M)9 b(\030)t Fu(\)\()p Ft(t)p Fu(\))24 b(=)2057 4570 y Fn(Z)2140 4590 y Fs(t)2103 4758 y Fp(0)2183 4683 y Ft(e)2222 4649 y Fo(\000)p Fs(\027)t Fp(\()p Fs(t)p Fo(\000)p Fs(s)p Fp(\))p Fs(L)2521 4683 y Ft(@)2565 4695 y Fs(s)2600 4683 y Ft(\030)t Fu(\()p Ft(s)p Fu(\))14 b Ft(ds:)305 b Fu(\(3.12\))515 4907 y(Let)21 b(us)h(denote)f(b)n(y)h Fr(W)28 b Fu(the)22 b(la)n(w)f(of)g(the)h (restriction)f(of)g(the)h(pro)r(cess)f Ft(Q\020)28 b Fu(to)21 b(the)h(in)n(terv)-5 b(al)21 b Ft(J)8 b Fu(.)515 5006 y(Th)n(us,)32 b Fr(W)39 b Fu(is)31 b(a)g(Gaussian)g(measure)f(on)i (the)g(space)e Ft(L)2271 4976 y Fp(2)2308 5006 y Fu(\()p Ft(J)o(;)14 b(V)19 b Fu(\).)49 b(In)32 b(view)f(of)38 b(\(1.11\))31 b(and)1905 5255 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 515 523 a Fu(the)34 b(second)f(relation)h(in)g (\(3.12\))o(,)i(w)n(e)d(ha)n(v)n(e)g Ft(\025)h Fu(=)g Ft(M)2212 535 y Fo(\003)2249 523 y Fu(\()p Fr(W)7 b Fu(\),)36 b(where)e Ft(M)2789 535 y Fo(\003)2827 523 y Fu(\()p Fr(W)7 b Fu(\))34 b(stands)g(for)515 623 y(the)c(image)g(of)g Fr(W)37 b Fu(under)30 b(the)g(linear)g(op)r(erator)e Ft(M)9 b Fu(.)44 b(By)30 b(assumption,)g(the)h(image)e(of)h Ft(Q)515 722 y Fu(con)n(tains)38 b Ft(E)5 b Fu(,)43 b(and)d(therefore)e Ft(I)12 b Fu(^)-47 b Ft(\021)47 b Fr(2)c Fu(supp)14 b Fr(W)7 b Fu(.)73 b(Recalling)40 b(\(3.11\))o(,)j(w)n(e)c(see)g(that)45 b(^)-47 b Ft(z)43 b Fu(is)515 822 y(con)n(tained)27 b(in)h(the)f(supp)r (ort)h(of)f Ft(M)1613 834 y Fo(\003)1651 822 y Fu(\()p Fr(W)7 b Fu(\).)639 971 y Fm(Step)30 b(3)p Fu(.)37 b(Let)28 b(us)g(pro)n(v)n(e)d(that)1382 1154 y Ft(\026)p Fu(\()p Ft(B)t Fu(\))f Ft(>)e Fu(0)83 b(for)27 b(an)n(y)g(ball)g Ft(B)g Fr(\032)c Ft(V)c(:)655 b Fu(\(3.13\))515 1337 y(Fix)25 b(a)f(ball)h Ft(B)k Fu(and)24 b(a)g(p)r(oin)n(t)31 b(^)-48 b Ft(u)1464 1349 y Fp(0)1524 1337 y Fr(2)24 b Fu(supp)14 b Ft(\026)f Fr(\\)g Ft(V)18 b Fu(.)36 b(W)-7 b(e)25 b(c)n(ho)r(ose)k(^)-47 b Ft(z)26 b Fr(2)e Fu(supp)14 b Ft(\025)25 b Fu(suc)n(h)f(that)h(\(3.9\))515 1436 y(holds.)44 b(Since)30 b(the)h(function)g Fr(T)1510 1448 y Fp(1)1575 1436 y Fu(:)c Fr(C)1669 1448 y Fp(1)1706 1436 y Fu(\()p Ft(h)p Fu(\))h Fr(!)f Ft(V)49 b Fu(is)30 b(con)n(tin)n(uous)f(\(see)i (Prop)r(osition)d(1.12\),)515 1536 y(w)n(e)f(can)g(\014nd)h(balls)f Ft(B)1212 1548 y Fp(1)1273 1536 y Fr(\032)22 b(Y)1415 1548 y Fs(J)1490 1536 y Fu(and)27 b Ft(B)1714 1548 y Fp(2)1775 1536 y Fr(\032)22 b Ft(V)47 b Fu(satisfying)28 b(\(3.3\))f(suc)n(h)g(that)1300 1719 y Fr(T)1345 1731 y Fp(1)1382 1719 y Fu(\()p Ft(z)t(;)14 b(u)1542 1731 y Fp(0)1579 1719 y Fu(\))23 b Fr(2)g Ft(B)88 b Fu(for)27 b Ft(z)f Fr(2)d Ft(B)2196 1731 y Fp(1)2234 1719 y Fu(,)k Ft(u)2332 1731 y Fp(0)2392 1719 y Fr(2)d Ft(B)2534 1731 y Fp(2)2571 1719 y Ft(;)573 b Fu(\(3.14\))1509 1843 y Ft(\025)p Fu(\()p Ft(B)1652 1855 y Fp(1)1690 1843 y Fu(\))23 b Ft(>)g Fu(0)p Ft(;)96 b(\026)p Fu(\()p Ft(B)2139 1855 y Fp(2)2177 1843 y Fu(\))23 b Ft(>)g Fu(0)p Ft(:)782 b Fu(\(3.15\))515 2026 y(Com)n(bining)28 b(\(3.4\))f(and)h(\(3.14\))o (,)g(w)n(e)f(obtain)1209 2208 y Ft(\026)p Fu(\()p Ft(B)t Fu(\))d Fr(\025)e(T)1546 2220 y Fp(1)p Fo(\003)1618 2208 y Fu(\()p Ft(\025)1698 2220 y Fs(B)1748 2228 y Fb(1)1785 2208 y Ft(;)14 b(\026)1872 2220 y Fs(B)1922 2228 y Fb(2)1959 2208 y Fu(\)\()p Ft(B)t Fu(\))24 b(=)f Ft(\025)p Fu(\()p Ft(B)2377 2220 y Fp(1)2415 2208 y Fu(\))p Ft(\026)p Fu(\()p Ft(B)2592 2220 y Fp(2)2629 2208 y Fu(\))p Ft(:)515 2391 y Fu(In)28 b(view)f(of)34 b(\(3.15\))o(,)28 b(this)g(implies)g(the)g (required)e(inequalit)n(y)i(\(3.13\).)639 2540 y Fm(Step)33 b(4)p Fu(.)48 b(W)-7 b(e)31 b(no)n(w)f(turn)h(to)g(the)h(pro)r(of)e(of) h(assertion)e(\(ii\).)48 b(Let)31 b(us)g(\014x)g(an)g(arbitrary)515 2640 y(\014nite-dimensional)d(subspace)h Ft(F)37 b Fr(\032)25 b Ft(H)7 b Fu(.)41 b(W)-7 b(e)29 b(claim)g(that)g(there)g(is)g(a)f (sequence)h(of)f(balls)515 2740 y Ft(B)578 2752 y Fs(j)636 2740 y Fu(=)22 b Ft(B)786 2752 y Fs(F)842 2740 y Fu(\()p Ft(y)915 2752 y Fs(j)950 2740 y Ft(;)14 b(r)1024 2752 y Fs(j)1059 2740 y Fu(\))28 b(and)f(functions)h Ft(')1692 2752 y Fs(j)1751 2740 y Fr(2)23 b Ft(C)1894 2710 y Fo(1)1965 2740 y Fu(\()p Ft(F)12 b Fu(\))28 b(suc)n(h)f(that)1665 2922 y Ft(`)1700 2934 y Fs(F)1755 2922 y Fu(\()p Ft(F)k Fr(n)17 b Ft(F)1983 2934 y Fp(0)2021 2922 y Fu(\))23 b(=)g(0)p Ft(;)938 b Fu(\(3.16\))1507 3047 y Ft(\026)23 b Fr(\025)g Ft(')1722 3059 y Fs(j)1757 3047 y Ft(`)1792 3059 y Fs(F)1930 3047 y Fu(for)k(all)g Ft(j)h Fr(\025)23 b Fu(1)p Ft(;)780 b Fu(\(3.17\))1534 3171 y Ft(')1588 3183 y Fs(j)1623 3171 y Fu(\()p Ft(y)s Fu(\))23 b Ft(>)g Fu(0)83 b(for)27 b Ft(y)e Fr(2)f Ft(B)2302 3183 y Fs(j)2337 3171 y Ft(;)807 b Fu(\(3.18\))515 3354 y(where)30 b Ft(F)811 3366 y Fp(0)876 3354 y Fu(=)d Fr([)1023 3366 y Fs(j)1059 3354 y Ft(B)1122 3366 y Fs(j)1157 3354 y Fu(.)45 b(If)31 b(this)g(claim)f(is)g(established,)h(then)g(the)g(required)e(result)i (can)f(b)r(e)515 3454 y(pro)n(v)n(ed)c(b)n(y)h(the)h(follo)n(wing)f (simple)g(argumen)n(t.)639 3553 y(Without)h(loss)f(of)g(generalit)n(y) -7 b(,)25 b(w)n(e)i(assume)g(that)g(0)c Fr(\024)f Ft(')2417 3565 y Fs(j)2476 3553 y Fr(\024)g Fu(1.)37 b(Let)27 b Ft(\037)2865 3565 y Fs(j)2923 3553 y Fr(2)c Ft(C)3066 3523 y Fo(1)3137 3553 y Fu(\()p Ft(F)12 b Fu(\))28 b(b)r(e)515 3653 y(suc)n(h)f(that)h(0)22 b Fr(\024)h Ft(\037)1086 3665 y Fs(j)1144 3653 y Fr(\024)g Fu(1,)798 3836 y Ft(\037)850 3848 y Fs(j)885 3836 y Fu(\()p Ft(y)s Fu(\))g(=)g(0)k(for)g Ft(y)35 b(=)-51 b Fr(2)23 b Ft(B)1508 3848 y Fs(F)1564 3836 y Fu(\()p Ft(y)1637 3848 y Fs(j)1671 3836 y Ft(;)14 b Fu(2)p Ft(r)1787 3848 y Fs(j)1822 3836 y Fu(\))q Ft(;)180 b(\037)2110 3848 y Fs(j)2144 3836 y Fu(\()p Ft(y)s Fu(\))24 b(=)e(1)28 b(for)f Ft(y)e Fr(2)f Ft(B)2768 3848 y Fs(F)2823 3836 y Fu(\()p Ft(y)2896 3848 y Fs(j)2931 3836 y Ft(;)14 b(r)3005 3848 y Fs(j)3040 3836 y Fu(\))q Ft(:)515 4018 y Fu(Let)27 b Fr(f)p Ft(")744 4030 y Fs(j)779 4018 y Fr(g)g Fu(b)r(e)h(a)f(sequence)g(of)h(p)r(ositiv)n(e)f(constan)n(ts)f (suc)n(h)i(that)1349 4167 y Fo(1)1322 4192 y Fn(X)1324 4368 y Fs(j)s Fp(=1)1456 4270 y Ft(")1495 4282 y Fs(j)1552 4270 y Fu(=)23 b(1)p Ft(;)1828 4167 y Fo(1)1801 4192 y Fn(X)1804 4368 y Fs(j)s Fp(=1)1935 4270 y Ft(")1974 4282 y Fs(j)2009 4270 y Fr(k)p Ft(\037)2103 4282 y Fs(j)2137 4270 y Ft(')2191 4282 y Fs(j)2227 4270 y Fr(k)2269 4286 y Fs(C)2321 4270 y Fk(j)33 b Ft(<)23 b Fr(1)p Ft(;)515 4548 y Fu(where)35 b Fr(k)24 b(\001)g(k)918 4564 y Fs(C)970 4547 y Fk(j)1041 4548 y Fu(stands)36 b(for)f(the)i(usual)f(norm)f(in)i (the)f(space)g(of)g(b)r(ounded)g Ft(C)3050 4518 y Fs(j)3086 4548 y Fu(-smo)r(oth)515 4648 y(functions)28 b(on)f Ft(F)40 b Fu(with)28 b(b)r(ounded)g(deriv)-5 b(ativ)n(es)26 b(up)i(to)g(the)g (order)e Ft(j)5 b Fu(.)37 b(Then)27 b(the)h(function)1504 4892 y Ft(\032)p Fu(\()p Ft(y)s Fu(\))c(:=)1816 4788 y Fo(1)1789 4813 y Fn(X)1792 4990 y Fs(j)s Fp(=1)1923 4892 y Ft(")1962 4904 y Fs(j)1997 4892 y Ft(\037)2049 4904 y Fs(j)2084 4892 y Fu(\()p Ft(y)s Fu(\))p Ft(')2246 4904 y Fs(j)2281 4892 y Fu(\()p Ft(y)s Fu(\))1905 5255 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 515 523 a Fu(is)29 b(in\014nitely)i(smo)r(oth)e (and)h Ft(\032)p Fu(\()p Ft(y)s Fu(\))c Ft(>)g Fu(0)k(for)f Ft(y)g Fr(2)e Ft(F)2084 535 y Fp(0)2122 523 y Fu(.)43 b(It)30 b(follo)n(ws)f(from)g(\(3.16\))g(that)h Ft(\032)c(>)g Fu(0)515 623 y(almost)h(ev)n(erywhere)e(on)j Ft(F)12 b Fu(.)37 b(F)-7 b(urthermore,)26 b(inequalit)n(y)i(\(3.17\))f(implies) h(that)1536 848 y Ft(\026)p Fu(\(\000\))c Fr(\025)1813 735 y Fn(Z)1860 923 y Fp(\000)1919 848 y Ft(')1973 860 y Fs(j)2008 848 y Fu(\()p Ft(y)s Fu(\))p Ft(`)2151 860 y Fs(F)2206 848 y Fu(\()p Ft(dy)s Fu(\))810 b(\(3.19\))515 1077 y(for)33 b(an)n(y)h Ft(j)39 b Fr(\025)34 b Fu(1)g(and)g(\000)g Fr(2)h(B)s Fu(\()p Ft(F)12 b Fu(\).)56 b(Multiplying)36 b(\(3.19\))e(b)n(y)g Ft(")2538 1089 y Fs(j)2607 1077 y Fu(and)g(summing)g(up)h(the)515 1177 y(resulting)27 b(inequalities,)g(w)n(e)g(obtain)1188 1397 y Ft(\026)p Fu(\(\000\))c Fr(\025)1465 1284 y Fn(Z)1511 1473 y Fp(\000)1570 1397 y Ft(\032)p Fu(\()p Ft(y)s Fu(\))14 b Ft(`)1770 1409 y Fs(F)1825 1397 y Fu(\()p Ft(dy)s Fu(\))83 b(for)27 b(an)n(y)g(\000)c Fr(2)h(B)s Fu(\()p Ft(F)12 b Fu(\))o Ft(:)515 1622 y Fu(whence)27 b(it)h(follo)n(ws)f(that)h Ft(\026)23 b Fr(\025)g Ft(\032`)1580 1634 y Fs(F)1634 1622 y Fu(.)639 1771 y Fm(Step)37 b(5)p Fu(.)62 b(T)-7 b(o)35 b(construct)g(sequences)g Fr(f)p Ft(B)1951 1783 y Fs(j)1985 1771 y Fr(g)g Fu(and)h Fr(f)p Ft(')2328 1783 y Fs(j)2363 1771 y Fr(g)f Fu(satisfying)h(\(3.16\))f({)g(\(3.18\))o(,) 515 1871 y(it)e(su\016ces)g(to)g(pro)n(v)n(e)e(that)j(for)e(an)n(y)g (in)n(teger)h Ft(r)h(>)e Fu(0)h(there)g(are)f(coun)n(tably)g(man)n(y)g (balls)515 1970 y Ft(B)582 1940 y Fs(r)578 1994 y(k)648 1970 y Fr(\032)d Ft(F)44 b Fu(and)31 b(functions)h Ft(')1420 1940 y Fs(r)1420 1994 y(k)1491 1970 y Fr(2)d Ft(C)1640 1940 y Fo(1)1711 1970 y Fu(\()p Ft(F)12 b Fu(\))32 b(suc)n(h)f(that)h (\(3.17\))f(and)g(\(3.18\))g(hold)g(with)h Ft(B)3344 1982 y Fs(j)515 2070 y Fu(and)27 b Ft(')730 2082 y Fs(j)793 2070 y Fu(replaced)g(b)n(y)g Ft(B)1303 2040 y Fs(r)1299 2094 y(k)1368 2070 y Fu(and)g Ft(')1583 2040 y Fs(r)1583 2094 y(k)1624 2070 y Fu(,)h(resp)r(ectiv)n(ely)-7 b(,)27 b(and)1526 2253 y Ft(`)1561 2265 y Fs(F)1616 2185 y Fn(\000)1654 2253 y Ft(B)1717 2265 y Fs(F)1772 2253 y Fu(\()p Ft(r)r Fu(\))20 b Fr(n)e([)2010 2265 y Fs(k)2051 2253 y Ft(B)2118 2218 y Fs(r)2114 2273 y(k)2155 2185 y Fn(\001)2216 2253 y Fu(=)k(0)p Ft(:)799 b Fu(\(3.20\))515 2435 y(If)31 b(suc)n(h)g(sequences)f(are)f(constructed,)j(then)f(w)n(e)g(can)f(tak)n (e)g(the)h(union)g(with)h(resp)r(ect)e(to)515 2535 y(all)d(p)r(ositiv)n (e)g(in)n(tegers)g Ft(r)j Fu(and)e Ft(k)i Fu(to)d(obtain)h(the)g (required)e(sequences)h Fr(f)p Ft(B)2857 2547 y Fs(j)2892 2535 y Fr(g)g Fu(and)g Fr(f)p Ft(')3218 2547 y Fs(j)3253 2535 y Fr(g)p Fu(.)639 2684 y Fm(Step)j(6)p Fu(.)37 b(Let)28 b(us)g(set)1165 2867 y Ft(f)9 b Fu(\()p Ft(z)t(;)14 b(u)1375 2879 y Fp(0)1410 2867 y Fu(\))24 b(=)e Fd(P)1606 2879 y Fs(F)1662 2867 y Fr(T)1707 2879 y Fp(1)1744 2867 y Fu(\()p Ft(z)t(;)14 b(u)1904 2879 y Fp(0)1940 2867 y Fu(\))84 b(for)27 b(\()p Ft(z)t(;)14 b(u)2343 2879 y Fp(0)2379 2867 y Fu(\))23 b Fr(2)h(C)2557 2879 y Fp(1)2594 2867 y Fu(\()p Ft(h)p Fu(\))p Ft(;)438 b Fu(\(3.21\))515 3050 y(where)32 b Ft(z)j Fu(and)d Ft(u)1048 3062 y Fp(0)1117 3050 y Fu(are)g(regarded)e(as)i(deterministic)g(functions.)52 b(By)32 b(Prop)r(osition)e(1.12,)515 3149 y(the)g(function)h Ft(f)39 b Fu(is)30 b(analytic)g(on)f(its)i(domain)f(of)g(de\014nition)g Fr(C)2519 3161 y Fp(1)2556 3149 y Fu(\()p Ft(h)p Fu(\).)46 b(W)-7 b(e)30 b(wish)g(to)g(apply)515 3249 y(Theorem)d(2.4)f(to)i Ft(f)9 b Fu(.)639 3349 y(Let)34 b(us)g(\014x)g(a)g(constan)n(t)f Ft(r)j(>)d Fu(0)h(and)g(a)f(p)r(oin)n(t)39 b(^)-47 b Ft(u)2209 3361 y Fp(0)2279 3349 y Fr(2)34 b Ft(V)42 b Fr(\\)23 b Fu(supp)14 b Ft(\026)p Fu(.)56 b(W)-7 b(e)34 b(claim)g(that)515 3448 y(there)27 b(is)g(a)g(\014nite-dimensional)g (subspace)f Ft(Z)1951 3460 y Fp(0)2011 3448 y Fr(\032)d Ft(Z)33 b Fu(and)27 b(an)g(op)r(en)g(subset)g Fr(O)f(\032)c Ft(Z)3155 3460 y Fp(0)3219 3448 y Fu(suc)n(h)515 3548 y(that)28 b Fr(O)d(\032)e(C)918 3560 y Fp(1)955 3548 y Fu(\()p Ft(h)p Fu(\))28 b(and)1617 3647 y Ft(f)9 b Fu(\()p Fr(O)r Ft(;)19 b Fu(^)-47 b Ft(u)1852 3659 y Fp(0)1889 3647 y Fu(\))23 b Fr(\033)g Ft(B)2095 3659 y Fs(F)2150 3647 y Fu(\()p Ft(r)r Fu(\))p Ft(:)891 b Fu(\(3.22\))515 3797 y(Indeed,)22 b(b)n(y)d(Prop)r(osition)g(1.11)f (\(ii\),)k(there)e(is)g(a)g(compact)f(subset)h Fr(K)i Fu(in)e(a)f(\014nite-dimensio-)515 3897 y(nal)27 b(space)g Ft(X)944 3909 y Fp(0)1004 3897 y Fr(\032)c Ft(C)1157 3866 y Fo(1)1227 3897 y Fu(\()p Ft(J)o(;)14 b(E)5 b Fu(\))28 b(suc)n(h)g(that)f Fr(K)e(\032)e Fu(\002)2075 3909 y Fp(1)2111 3897 y Fu(\()p Ft(h;)d Fu(^)-48 b Ft(u)2276 3909 y Fp(0)2313 3897 y Fu(\))28 b(and)1535 4079 y Fd(P)1588 4091 y Fs(F)1644 4079 y Fr(R)1714 4091 y Fp(1)1751 4079 y Fu(\()p Fr(K)q Ft(;)20 b Fu(^)-48 b Ft(u)1932 4091 y Fp(0)1970 4079 y Fu(\))23 b Fr(\033)g Ft(B)2176 4091 y Fs(F)2231 4079 y Fu(\()p Ft(r)r Fu(\))p Ft(:)810 b Fu(\(3.23\))515 4262 y(Let)40 b(us)g(denote)g(b)n(y)g Ft(Z)1260 4274 y Fp(0)1337 4262 y Fu(the)g(image)f(of)h Ft(X)1920 4274 y Fp(0)1997 4262 y Fu(under)g(the)g(linear)g(op)r (erator)e Ft(M)49 b Fu(de\014ned)515 4361 y(in)23 b(\(3.12\))o(.)36 b(Then)23 b Ft(Z)1147 4373 y Fp(0)1207 4361 y Fu(is)g(a)f (\014nite-dimensional)h(subspace)f(of)h Ft(Z)29 b Fu(and)22 b Ft(M)9 b Fu(\()p Fr(K)q Fu(\))24 b(is)f(a)g(compact)515 4461 y(subset)k(of)g Ft(Z)920 4473 y Fp(0)985 4461 y Fu(con)n(tained)f(in)i Fr(C)1501 4473 y Fp(1)1538 4461 y Fu(\()p Ft(h)p Fu(\).)37 b(F)-7 b(urthermore,)27 b(it)h(follo)n(ws)e (from)h(\(3.10\))g(and)g(\(3.23\))515 4561 y(that)1542 4660 y Ft(f)9 b Fu(\()p Ft(M)g Fu(\()p Fr(K)q Fu(\))p Ft(;)19 b Fu(^)-47 b Ft(u)1927 4672 y Fp(0)1964 4660 y Fu(\))23 b Fr(\033)g Ft(B)2170 4672 y Fs(F)2225 4660 y Fu(\()p Ft(r)r Fu(\))p Ft(:)515 4810 y Fu(Since)i Fr(C)773 4822 y Fp(1)810 4810 y Fu(\()p Ft(h)p Fu(\))h(is)f(op)r(en,)h(w)n(e)e (conclude)h(that)h(\(3.22\))e(holds)h(for)g(a)g(small)f(neigh)n(b)r (ourho)r(o)r(d)h Fr(O)515 4909 y Fu(of)i(the)h(compact)f(set)h Ft(M)9 b Fu(\()p Fr(K)q Fu(\).)1905 5255 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 639 523 a Fu(Th)n(us,)28 b(the)g(image)e(of)i (the)g(smo)r(oth)f(mapping)1652 697 y Ft(f)9 b Fu(\()p Fr(\001)p Ft(;)19 b Fu(^)-47 b Ft(u)1842 709 y Fp(0)1878 697 y Fu(\))24 b(:)f Fr(O)i(!)e Ft(F)937 b Fu(\(3.24\))515 871 y(con)n(tains)29 b(the)h(ball)f Ft(B)1214 883 y Fs(F)1270 871 y Fu(\()p Ft(r)r Fu(\).)45 b(By)29 b(the)h(Sard)f(theorem,)h (almost)g(ev)n(ery)e(p)r(oin)n(t)i(of)g Ft(B)3134 883 y Fs(F)3189 871 y Fu(\()p Ft(r)r Fu(\))h(is)515 970 y(regular)783 940 y Fp(3)849 970 y Fu(\(see)e([Ste83)o(]\).)42 b(Applying)29 b(Theorem)f(2.4,)h(for)f(almost)h(ev)n(ery)f Ft(y)2890 982 y Fp(0)2952 970 y Fr(2)e Ft(B)3096 982 y Fs(F)3151 970 y Fu(\()p Ft(r)r Fu(\))k(w)n(e)515 1070 y(can)h(construct)f(a)h (function)h Ft(\032)1484 1082 y Fs(y)1518 1090 y Fb(0)1584 1070 y Fr(2)d Ft(C)6 b Fu(\()p Ft(F)12 b Fu(\))32 b(and)g(a)f(closed)f (ball)h Ft(B)2609 1082 y Fs(y)2643 1090 y Fb(0)2709 1070 y Fr(\032)e Ft(F)43 b Fu(cen)n(tred)31 b(at)g Ft(y)3342 1082 y Fp(0)515 1170 y Fu(suc)n(h)c(that)1020 1343 y Ft(f)1061 1355 y Fo(\003)1099 1343 y Fu(\()p Ft(\025)1179 1355 y Fs(B)1229 1363 y Fb(1)1266 1343 y Ft(;)14 b(\026)1353 1355 y Fs(B)1403 1363 y Fb(2)1440 1343 y Fu(\))23 b Fr(\025)g Ft(\032)1626 1355 y Fs(y)1660 1363 y Fb(0)1696 1343 y Ft(`)1731 1355 y Fs(F)1786 1343 y Ft(;)180 b(\032)2032 1355 y Fs(y)2066 1363 y Fb(0)2102 1343 y Fu(\()p Ft(y)s Fu(\))23 b Ft(>)g Fu(0)82 b(for)27 b Ft(y)f Fr(2)e Ft(B)2781 1355 y Fs(y)2815 1363 y Fb(0)2851 1343 y Ft(;)515 1517 y Fu(where)e Ft(B)813 1529 y Fp(1)873 1517 y Fr(\032)h(O)i Fu(and)d Ft(B)1271 1529 y Fp(2)1331 1517 y Fr(\032)h Ft(H)30 b Fu(are)21 b(some)h(balls)g(suc)n(h)h(that)f Ft(B)2463 1529 y Fp(1)2509 1517 y Fr(\002)8 b Ft(B)2645 1529 y Fp(2)2705 1517 y Fr(\032)23 b(C)2837 1529 y Fp(1)2874 1517 y Fu(\()p Ft(h)p Fu(\).)36 b(It)23 b(is)f(clear)515 1617 y(that)29 b Ft(\037)748 1629 y Fs(y)782 1637 y Fb(0)847 1617 y Fu(can)f(b)r(e)i(minorised)e(b)n(y)g(a)h(function)g Ft(')2062 1629 y Fs(y)2096 1637 y Fb(0)2158 1617 y Fr(2)c Ft(C)2303 1587 y Fo(1)2374 1617 y Fu(\()p Ft(F)12 b Fu(\))29 b(that)g(is)g(p)r(ositiv)n(e)f(on)h Ft(B)3286 1629 y Fs(y)3320 1637 y Fb(0)3356 1617 y Fu(.)515 1717 y(Recalling)e (inequalit)n(y)h(\(3.4\),)f(w)n(e)h(see)f(that)1563 1890 y Ft(\026)1613 1902 y Fs(F)1692 1890 y Fr(\025)22 b Ft(\032)1822 1902 y Fs(y)1856 1910 y Fb(0)1892 1890 y Ft(`)1927 1902 y Fs(F)2005 1890 y Fr(\025)h Ft(')2147 1902 y Fs(y)2181 1910 y Fb(0)2217 1890 y Ft(`)2252 1902 y Fs(F)2307 1890 y Ft(:)837 b Fu(\(3.25\))639 2064 y(W)-7 b(e)20 b(can)f(no)n(w)g (complete)g(the)h(construction)f(of)g Ft(B)2181 2034 y Fs(r)2177 2088 y(k)2238 2064 y Fu(and)g Ft(')2445 2034 y Fs(r)2445 2088 y(k)2505 2064 y Fu(b)n(y)h(a)f(standard)f(argumen)n (t.)515 2164 y(Namely)-7 b(,)26 b(let)h Fr(f)p Ft(K)1066 2176 y Fs(n)1110 2164 y Fr(g)1152 2176 y Fs(n)p Fo(\025)p Fp(1)1309 2164 y Fu(b)r(e)g(a)f(sequence)g(of)g(compact)g(subsets)g(of) h Ft(B)2699 2176 y Fs(F)2754 2164 y Fu(\()p Ft(r)r Fu(\))h(suc)n(h)e (that)h(an)n(y)515 2264 y(p)r(oin)n(t)32 b Ft(y)777 2276 y Fp(0)844 2264 y Fr(2)e Ft(K)1000 2276 y Fs(n)1077 2264 y Fu(is)h(regular)f(for)i(\(3.24\))f(and)g Ft(`)2026 2276 y Fs(F)2081 2196 y Fn(\000)2119 2264 y Ft(B)2182 2276 y Fs(F)2237 2264 y Fu(\()p Ft(r)r Fu(\))23 b Fr(n)e([)2481 2276 y Fs(n)2526 2264 y Ft(K)2597 2276 y Fs(n)2642 2196 y Fn(\001)2710 2264 y Fu(=)30 b(0.)49 b(Eac)n(h)30 b(set)i Ft(K)3334 2276 y Fs(n)515 2363 y Fu(can)27 b(b)r(e)h(co)n(v)n(ered)e(b) n(y)h(\014nitely)h(man)n(y)f(closed)g(balls)g Ft(B)2200 2375 y Fs(in)2269 2363 y Fu(,)h Ft(i)22 b Fu(=)h(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(I)2721 2375 y Fs(n)2767 2363 y Fu(,)27 b(suc)n(h)h(that)1392 2537 y Ft(\026)1442 2549 y Fs(F)1521 2537 y Fr(\025)22 b Ft(')1662 2549 y Fs(in)1731 2537 y Ft(`)1766 2549 y Fs(F)1904 2537 y Fu(for)27 b Ft(i)c Fu(=)g(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(I)2433 2549 y Fs(n)2478 2537 y Ft(;)515 2711 y Fu(where)38 b Ft(')820 2723 y Fs(in)930 2711 y Fr(\025)j Fu(0)d(is)g(a)g(con)n(tin)n(uous)g (function)h(on)f Ft(F)50 b Fu(that)39 b(is)f(p)r(ositiv)n(e)h(on)f Ft(B)3075 2723 y Fs(in)3143 2711 y Fu(.)70 b(The)515 2810 y(required)22 b(families)h Ft(B)1201 2780 y Fs(r)1197 2834 y(k)1261 2810 y Fu(and)f Ft(')1471 2780 y Fs(r)1471 2834 y(k)1536 2810 y Fu(can)g(b)r(e)i(obtained)e(b)n(y)h(taking)f(the)i (union)f(of)g Ft(B)3008 2822 y Fs(in)3099 2810 y Fu(and)g Ft(')3310 2822 y Fs(in)515 2910 y Fu(o)n(v)n(er)35 b(all)h Ft(i)i Fu(=)f(1)p Ft(;)14 b(:)g(:)g(:)g(;)g(I)1258 2922 y Fs(n)1340 2910 y Fu(and)36 b Ft(n)i Fr(\025)g Fu(1.)64 b(This)36 b(completes)h(the)g(pro)r(of)f(of)g(assertions)f(\(i\))515 3010 y(and)27 b(\(ii\))i(for)e(a)g(giv)n(en)g(stationary)f(measure.)515 3241 y Fl(3.2)112 b(General)38 b(case:)51 b(uniform)39 b(estimates)515 3394 y Fu(The)29 b(deriv)-5 b(ation)29 b(of)g(uniform)g(estimates)g(is)h(based)e(on)h(the)h(follo)n(wing)e (simple)i(result.)42 b(It)515 3493 y(sho)n(ws)30 b(that)i(a)e(compact)h (subset)h(of)f Ft(V)50 b Fu(carries)30 b(some)g(uniformly)i(p)r(ositiv) n(e)e(parts)h(of)g(all)515 3593 y(stationary)26 b(measures)g Ft(\026)i Fu(satisfying)g(\(2.1\).)515 3752 y Fv(Prop)s(osition)35 b(3.1.)43 b Fm(F)-6 b(or)34 b(any)g Fe(m)1615 3764 y Fp(0)1683 3752 y Ft(>)c Fu(0)j Fm(ther)l(e)h(is)g(a)h(c)l(omp)l(act)f (set)g Fr(A)d(\032)f Ft(V)53 b Fm(and)34 b(a)g(c)l(on-)515 3852 y(stant)29 b Ft(\016)d(>)c Fu(0)30 b Fm(such)f(that)804 4026 y Ft(\026)p Fu(\()p Fr(A)p Fu(\))24 b Fr(\025)f Ft(\016)88 b Fm(for)30 b(any)g(stationary)h(me)l(asur)l(e)e Ft(\026)h Fm(satisfying)39 b Fu(\(2.1\))o Ft(:)290 b Fu(\(3.26\))515 4199 y Fm(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)30 b(\014rst)f(note)g(that)h(inequalit)n(y)g(\(3.4\))f(pro)n(v)n(ed)f(for) h Ft(t)d Fu(=)f(1)k(is)h(true)f(for)g(an)n(y)g(time)515 4299 y Ft(t)37 b Fu(=)h Ft(s)f(>)g Fu(0.)63 b(Namely)-7 b(,)39 b(let)d(us)h(set)f Ft(J)1751 4311 y Fs(s)1824 4299 y Fu(=)i([0)p Ft(;)14 b(s)p Fu(])36 b(and)g(denote)g(b)n(y)g Ft(\025)2745 4269 y Fs(s)2817 4299 y Fu(the)h(la)n(w)f(for)g(the)515 4399 y(restriction)30 b(of)h(the)g(Ornstein{Uhlen)n(b)r(ec)n(k)f(pro)r (cess)g(\(1.11\))g(to)h(the)g(in)n(terv)-5 b(al)31 b Ft(J)3060 4411 y Fs(s)3127 4399 y Fu(and)f(b)n(y)515 4498 y Ft(Z)572 4510 y Fs(s)630 4498 y Fr(\032)23 b Ft(C)6 b Fu(\()p Ft(J)861 4510 y Fs(s)897 4498 y Ft(;)14 b(V)k Fu(\))q Fr(\\)q Ft(L)1146 4468 y Fp(2)1184 4498 y Fu(\()p Ft(J)1262 4510 y Fs(s)1298 4498 y Ft(;)c(U)9 b Fu(\))18 b(the)i(supp)r(ort)e(of)h Ft(\025)2018 4510 y Fs(s)2054 4498 y Fu(.)34 b(Then)19 b(for)g(an)n(y)f(Borel)f(subsets)i Ft(B)3139 4510 y Fp(1)3199 4498 y Fr(\032)k Ft(Z)3344 4510 y Fs(s)515 4598 y Fu(and)k Ft(B)739 4610 y Fp(2)799 4598 y Fr(\032)c Ft(V)47 b Fu(satisfying)27 b(the)h(inclusion)f Ft(B)1904 4610 y Fp(1)1960 4598 y Fr(\002)18 b Ft(B)2106 4610 y Fp(2)2166 4598 y Fr(\032)23 b(C)2298 4610 y Fs(s)2333 4598 y Fu(\()p Ft(h)p Fu(\))28 b(w)n(e)f(ha)n(v)n(e)1611 4772 y Ft(\026)c Fr(\025)g(T)1817 4784 y Fs(s)p Fo(\003)1886 4772 y Fu(\()p Ft(\025)1966 4738 y Fs(s)1966 4792 y(B)2016 4800 y Fb(1)2054 4772 y Ft(;)14 b(\026)2141 4784 y Fs(B)2191 4792 y Fb(2)2227 4772 y Fu(\))p Ft(:)885 b Fu(\(3.27\))p 515 4850 1146 4 v 607 4904 a Fi(3)642 4927 y Fh(Recall)28 b(that)i(a)e(p)r(oin)n(t)h Fg(y)1304 4936 y Fi(0)1366 4927 y Ff(2)e Fg(F)38 b Fh(is)27 b(said)h(to)h(b)r(e)g(regular)f(for)f (\(3.24\))j(if)d(the)i(rank)g(of)f(the)h(deriv)l(ativ)n(e)515 5006 y(\()p Fg(D)600 5014 y Fa(z)636 5006 y Fg(f)7 b Fh(\)\()p Fg(z)764 5015 y Fi(0)799 5006 y Fg(;)17 b Fh(^)-40 b Fg(u)872 5015 y Fi(0)906 5006 y Fh(\))24 b(is)f(maximal)g(for)g(an)n (y)h(p)r(oin)n(t)h Fg(z)1775 5015 y Fi(0)1829 5006 y Ff(2)19 b(O)26 b Fh(suc)n(h)e(that)h Fg(f)7 b Fh(\()p Fg(z)2390 5015 y Fi(0)2425 5006 y Fg(;)16 b Fh(^)-40 b Fg(u)2497 5015 y Fi(0)2532 5006 y Fh(\))20 b(=)f Fg(y)2688 5015 y Fi(0)2723 5006 y Fh(.)1905 5255 y Fu(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 639 523 a Fu(Let)37 b Ft(B)860 535 y Fp(2)936 523 y Fr(\032)h Ft(V)55 b Fu(b)r(e)37 b(a)g(ball)f(suc)n(h)h(that)g Ft(\026)p Fu(\()p Ft(B)2044 535 y Fp(2)2081 523 y Fu(\))i Fr(\025)f Fu(1)p Ft(=)p Fu(2)d(for)h(an)n(y)g Ft(\026)h Fu(satisfying)g(\(2.1\).)515 623 y(Standard)g(lo)r(cal)h(existence)g(results)g(for)f(the)i(NS-t)n (yp)r(e)f(system)g(\(1.3\))g(imply)h(that)f(w)n(e)515 722 y(can)h(c)n(ho)r(ose)f Ft(s)43 b(>)f Fu(0)e(and)f(a)g(compact)g (set)g Ft(B)2027 734 y Fp(1)2108 722 y Fr(\032)j Ft(Z)2272 734 y Fs(s)2347 722 y Fu(of)d(p)r(ositiv)n(e)g Ft(\025)2820 692 y Fs(s)2856 722 y Fu(-measure)f(suc)n(h)515 822 y(that)d Ft(B)765 834 y Fp(1)825 822 y Fr(\002)23 b Ft(B)976 834 y Fp(2)1049 822 y Fr(\032)35 b(C)1193 834 y Fs(s)1228 822 y Fu(\()p Ft(h)p Fu(\))g(\(for)g(instance,)h(see)f([FK64)o(])g(or)f ([T)-7 b(a)n(y97)n(]\).)59 b(F)-7 b(urthermore,)36 b(it)515 922 y(follo)n(ws)c(from)i(the)f(regularising)f(prop)r(ert)n(y)g(of)i (the)g(resolving)e(op)r(erator)f(for)i(\(1.3\))h(that)515 1021 y Fr(T)560 1033 y Fs(s)595 1021 y Fu(\()p Ft(B)690 1033 y Fp(1)750 1021 y Fr(\002)22 b Ft(B)900 1033 y Fp(2)937 1021 y Fu(\))33 b(is)g(con)n(tained)g(in)g(a)g(compact)f(subset)h Fr(A)h Fu(of)f Ft(V)18 b Fu(.)54 b(Inequalit)n(y)33 b(\(3.27\))g(no)n (w)515 1121 y(implies)28 b(that)1347 1255 y Ft(\026)p Fu(\()p Fr(A)p Fu(\))c Fr(\025)f Ft(\025)1687 1221 y Fs(s)1723 1255 y Fu(\()p Ft(B)1818 1267 y Fp(1)1855 1255 y Fu(\))p Ft(\026)p Fu(\()p Ft(B)2032 1267 y Fp(2)2070 1255 y Fu(\))h Fr(\025)2223 1199 y Fu(1)p 2223 1236 42 4 v 2223 1312 a(2)2275 1255 y Ft(\025)2323 1221 y Fs(s)2359 1255 y Fu(\()p Ft(B)2454 1267 y Fp(1)2491 1255 y Fu(\))p Ft(:)515 1415 y Fu(It)f(remains)e(to)h(note)h(that)f(the)h(righ)n (t-hand)e(side)i(of)f(this)h(inequalit)n(y)f(is)g(p)r(ositiv)n(e)g(and) g(do)r(es)515 1515 y(not)27 b(dep)r(end)i(on)e Ft(\026)p Fu(.)p 3318 1515 4 57 v 3322 1462 50 4 v 3322 1515 V 3372 1515 4 57 v 639 1677 a(W)-7 b(e)29 b(no)n(w)g(turn)f(to)h(the)g (pro)r(of)f(of)36 b(\(2.2\))o(.)41 b(Rep)r(eating)29 b(the)g(argumen)n(t)e(used)i(in)g(Step)g(3)515 1776 y(of)37 b(Subsection)f(3.1,)j(for)d(an)n(y)41 b(^)-47 b Ft(u)1562 1788 y Fp(0)1637 1776 y Fr(2)39 b(A)e Fu(w)n(e)g(can)f(\014nd)h(op)r (en)g(balls)g Ft(B)2779 1788 y Fp(1)2816 1776 y Fu(\()5 b(^)-47 b Ft(u)2896 1788 y Fp(0)2933 1776 y Fu(\))38 b Fr(\032)g(Y)3161 1788 y Fs(J)3245 1776 y Fu(and)515 1876 y Ft(B)578 1888 y Fp(2)615 1876 y Fu(\()5 b(^)-47 b Ft(u)695 1888 y Fp(0)732 1876 y Fu(\))23 b Fr(\032)g Ft(V)47 b Fu(suc)n(h)27 b(that)h Ft(B)1400 1888 y Fp(1)1437 1876 y Fu(\()5 b(^)-47 b Ft(u)1517 1888 y Fp(0)1554 1876 y Fu(\))19 b Fr(\002)f Ft(B)1751 1888 y Fp(2)1788 1876 y Fu(\()5 b(^)-47 b Ft(u)1868 1888 y Fp(0)1905 1876 y Fu(\))23 b Fr(\032)g(C)2092 1888 y Fp(1)2129 1876 y Fu(\()p Ft(h)p Fu(\))28 b(and)1151 2038 y Fr(T)1196 2050 y Fp(1)1233 2038 y Fu(\()p Ft(z)t(;)14 b(u)1393 2050 y Fp(0)1429 2038 y Fu(\))24 b Fr(2)f Ft(B)87 b Fu(for)27 b Ft(z)g Fr(2)c Ft(B)2047 2050 y Fp(1)2084 2038 y Fu(\()5 b(^)-47 b Ft(u)2164 2050 y Fp(0)2201 2038 y Fu(\),)28 b Ft(u)2332 2050 y Fp(0)2392 2038 y Fr(2)c Ft(B)2534 2050 y Fp(2)2571 2038 y Fu(\()5 b(^)-47 b Ft(u)2651 2050 y Fp(0)2688 2038 y Fu(\))p Ft(;)424 b Fu(\(3.28\))1678 2162 y Ft(\025)p Fu(\()p Ft(B)1821 2174 y Fp(1)1859 2162 y Fu(\()5 b(^)-47 b Ft(u)1939 2174 y Fp(0)1976 2162 y Fu(\)\))24 b Ft(>)e Fu(0)p Ft(:)951 b Fu(\(3.29\))515 2324 y(The)34 b(family)h Fr(f)p Ft(B)1058 2336 y Fp(2)1095 2324 y Fu(\()5 b(^)-47 b Ft(u)1175 2336 y Fp(0)1212 2324 y Fu(\))p Ft(;)19 b Fu(^)-47 b Ft(u)1329 2336 y Fp(0)1400 2324 y Fr(2)35 b(Ag)f Fu(forms)g(an)h(op)r(en)f(co)n(v)n(ering)e(for)i(the)h(compact)f (set)h Fr(A)p Fu(,)515 2424 y(and)27 b(w)n(e)h(can)f(c)n(ho)r(ose)g(a)g (\014nite)h(sub)r(co)n(v)n(ering)e Fr(f)p Ft(B)2058 2394 y Fs(i)2054 2445 y Fp(2)2091 2424 y Fr(g)2133 2394 y Fs(N)2133 2446 y(i)p Fp(=1)2244 2424 y Fu(.)38 b(Denote)28 b(b)n(y)f Fr(f)p Ft(B)2814 2394 y Fs(i)2810 2445 y Fp(1)2847 2424 y Fr(g)2889 2394 y Fs(N)2889 2446 y(i)p Fp(=1)3028 2424 y Fu(the)h(corre-)515 2524 y(sp)r(onding)f(set)h(of)f(balls)g(in)h (the)g(space)f Fr(Y)1803 2536 y Fs(J)1850 2524 y Fu(.)639 2623 y(No)n(w)37 b(let)h Ft(\026)h Fr(2)h(P)7 b Fu(\()p Ft(H)g Fu(\))38 b(b)r(e)g(a)f(stationary)e(measure)i(for)g(\(1.8\))g (that)h(satis\014es)e(\(2.1\).)515 2723 y(Then)28 b(it)g(follo)n(ws)e (from)h(\(3.26\))g(that)1441 2885 y Ft(\026)p Fu(\()p Ft(B)1590 2845 y Fs(j)1586 2907 y Fp(2)1625 2885 y Fu(\))c Fr(\025)g Ft(N)1844 2850 y Fo(\000)p Fp(1)1933 2885 y Ft(\016)86 b Fu(for)27 b(some)g Ft(j)5 b(:)515 3047 y Fu(Com)n(bining)27 b(this)h(inequalit)n(y)f(with)h(\(3.28\))f(and)g (\(3.29\))o(,)h(w)n(e)f(see)h(that)877 3209 y Ft(\026)p Fu(\()p Ft(B)t Fu(\))c Fr(\025)e(T)1214 3221 y Fp(1)p Fo(\003)1286 3209 y Fu(\()p Ft(\025)1366 3236 y Fs(B)1419 3208 y Fk(j)1416 3254 y Fb(1)1455 3209 y Ft(;)14 b(\026)1542 3236 y Fs(B)1595 3208 y Fk(j)1592 3254 y Fb(2)1630 3209 y Fu(\))23 b Fr(\025)g Ft(\025)p Fu(\()p Ft(B)1920 3169 y Fs(j)1916 3231 y Fp(1)1955 3209 y Fu(\))14 b Ft(\026)p Fu(\()p Ft(B)2150 3169 y Fs(j)2146 3231 y Fp(2)2186 3209 y Fu(\))23 b Fr(\025)g Ft(N)2405 3174 y Fo(\000)p Fp(1)2494 3209 y Ft(\016)57 b Fu(min)2548 3262 y Fp(1)p Fo(\024)p Fs(i)p Fo(\024)p Fs(N)2781 3209 y Ft(\025)p Fu(\()p Ft(B)2928 3174 y Fs(i)2924 3229 y Fp(1)2962 3209 y Fu(\))p Ft(:)515 3409 y Fu(It)23 b(remains)e(to)h(note)h(that)f(the)h(righ)n(t-hand)e (side)i(of)f(this)h(inequalit)n(y)f(is)g(p)r(ositiv)n(e)g(and)g(do)r (es)515 3509 y(not)27 b(dep)r(end)i(on)e Ft(\026)p Fu(.)639 3650 y(Let)35 b(us)g(pro)n(v)n(e)e(assertion)g(\(ii\))j(with)f(a)g (function)g Ft(\032)2288 3662 y Fs(F)2378 3650 y Fu(not)g(dep)r(ending) g(on)g Ft(\026)p Fu(.)58 b(As)35 b(is)515 3750 y(sho)n(wn)f(in)h(Steps) g(4{6)f(of)g(Subsection)h(3.1,)h(it)g(su\016ces)e(to)h(pro)n(v)n(e)e (that)i(for)g(an)n(y)f Ft(r)k(>)c Fu(0)515 3850 y(and)24 b(almost)f(ev)n(ery)g(p)r(oin)n(t)h Ft(y)1407 3862 y Fp(0)1467 3850 y Fr(2)g Ft(B)1609 3862 y Fs(F)1664 3850 y Fu(\()p Ft(r)r Fu(\))h(there)f(is)g(a)g(function)h Ft(')2523 3862 y Fs(y)2557 3870 y Fb(0)2616 3850 y Fr(2)f Ft(C)2760 3819 y Fo(1)2830 3850 y Fu(\()p Ft(F)12 b Fu(\),)26 b(dep)r(ending)515 3949 y(only)h(on)g Fe(m)876 3961 y Fp(0)941 3949 y Fu(and)g(p)r(ositiv)n(e)g(at)h Ft(y)1552 3961 y Fp(0)1589 3949 y Fu(,)g(suc)n(h)f(that)1720 4111 y Ft(\026)1770 4123 y Fs(F)1848 4111 y Fr(\025)c Ft(')1990 4123 y Fs(y)2024 4131 y Fb(0)2061 4111 y Ft(`)2096 4123 y Fs(F)2150 4111 y Ft(:)994 b Fu(\(3.30\))515 4273 y(Rep)r(eating)32 b(the)g(argumen)n(t)f(used)h(in)g(Step)h(6)e(of)h(Subsection)g(3.1)g (and)f(applying)h(Theo-)515 4373 y(rem)e(2.4,)g(for)f(almost)h(ev)n (ery)f Ft(y)1507 4385 y Fp(0)1571 4373 y Fr(2)f Ft(B)1717 4385 y Fs(F)1772 4373 y Fu(\()p Ft(r)r Fu(\))k(w)n(e)e(can)g(construct) f(op)r(en)h(balls)g Ft(B)3017 4385 y Fp(1)3054 4373 y Fu(\()p Ft(y)3127 4385 y Fp(0)3165 4373 y Fu(\))d Fr(\032)g Ft(Z)515 4472 y Fu(and)h Ft(B)740 4484 y Fs(y)774 4492 y Fb(0)836 4472 y Fr(\032)c Ft(B)988 4484 y Fp(2)1025 4472 y Fu(\()p Ft(y)1098 4484 y Fp(0)1136 4472 y Fu(\))h Fr(\032)g Ft(V)19 b Fu(,)29 b(a)f(constan)n(t)g Ft(\016)g(>)d Fu(0,)k(and)f(a)h(compactly)f(supp)r(orted)g(function)515 4572 y Ft( )569 4584 y Fs(y)603 4592 y Fb(0)662 4572 y Fr(2)c Ft(C)6 b Fu(\()p Ft(F)30 b Fr(\002)18 b Ft(B)1067 4584 y Fs(y)1101 4592 y Fb(0)1138 4572 y Fu(\))28 b(suc)n(h)f(that)1432 4734 y Ft( )1486 4746 y Fs(y)1520 4754 y Fb(0)1556 4734 y Fu(\()p Ft(y)1629 4746 y Fp(0)1667 4734 y Ft(;)14 b(u)1752 4746 y Fp(0)1788 4734 y Fu(\))24 b Ft(>)e Fu(0)1194 b(\(3.31\))1132 4910 y Ft(f)1173 4922 y Fo(\003)1211 4910 y Fu(\()p Ft(\025)1291 4925 y Fs(B)1341 4933 y Fb(1)1374 4925 y Fp(\()p Fs(y)1434 4933 y Fb(0)1467 4925 y Fp(\))1497 4910 y Ft(;)14 b(\026)1584 4925 y Fs(B)1634 4933 y Fb(2)1666 4925 y Fp(\()p Fs(y)1726 4933 y Fb(0)1758 4925 y Fp(\))1788 4910 y Fu(\))24 b Fr(\025)e Ft(`)1966 4922 y Fs(F)2035 4797 y Fn(Z)2081 4985 y Fs(B)2131 4993 y Fk(y)2161 5005 y Fb(0)2216 4910 y Ft( )2270 4922 y Fs(y)2304 4930 y Fb(0)2340 4910 y Fu(\()p Ft(y)s(;)14 b(u)p Fu(\))p Ft(\026)p Fu(\()p Ft(du)p Fu(\))p Ft(;)406 b Fu(\(3.32\))1905 5255 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 515 523 a Fu(where)27 b Ft(u)803 535 y Fp(0)869 523 y Fu(denotes)g(the)i(cen)n(tre)f(of)g Ft(B)1719 535 y Fs(y)1753 543 y Fb(0)1789 523 y Fu(.)39 b(Let)28 b(us)h(\014x)f(a)g(closed)f(ball)h Ft(Q)2769 535 y Fs(y)2803 543 y Fb(0)2863 523 y Fr(\032)c Ft(B)3015 535 y Fs(y)3049 543 y Fb(0)3114 523 y Fu(cen)n(tred)515 623 y(at)j Ft(u)664 635 y Fp(0)729 623 y Fu(and)g(set)1513 722 y Ft(\016)s Fu(\()p Ft(y)s Fu(\))d(:=)74 b(inf)1796 776 y Fs(u)p Fo(2)p Fs(Q)1932 784 y Fk(y)1962 796 y Fb(0)2040 722 y Ft( )2094 734 y Fs(y)2128 742 y Fb(0)2165 722 y Fu(\()p Ft(y)s(;)14 b(u)p Fu(\))p Ft(:)515 925 y Fu(It)32 b(is)f(clear)g(that)h Ft(\016)j Fu(is)d(a)f(Borel)f(function.)50 b(In)32 b(view)g(of)38 b(\(3.31\))o(,)33 b(w)n(e)e(can)h(c)n(ho)r(ose)e Ft(Q)3203 937 y Fs(y)3237 945 y Fb(0)3305 925 y Fu(so)515 1025 y(small)d(that)1523 1125 y Ft(\016)s Fu(\()p Ft(y)s Fu(\))c Fr(\025)g Ft(\016)1819 1137 y Fp(0)1939 1125 y Fu(for)k Ft(y)f Fr(2)d(O)2277 1137 y Fs(y)2311 1145 y Fb(0)2348 1125 y Ft(;)796 b Fu(\(3.33\))515 1274 y(where)32 b Ft(\016)797 1286 y Fp(0)865 1274 y Ft(>)f Fu(0)h(is)g(a)g(constan)n (t)g(and)g Fr(O)1769 1286 y Fs(y)1803 1294 y Fb(0)1872 1274 y Fu(is)g(a)g(ball)h(cen)n(tred)e(at)i Ft(y)2646 1286 y Fp(0)2683 1274 y Fu(.)51 b(Com)n(bining)33 b(\(3.4\),)515 1374 y(\(2.2\))o(,)28 b(and)g(\(3.32\))o(,)f(w)n(e)h(obtain)1353 1556 y Ft(\026)1403 1568 y Fs(F)1458 1556 y Fu(\()p Ft(dy)s Fu(\))c Fr(\025)e Ft(p)p Fu(\()p Ft(Q)1860 1568 y Fs(y)1894 1576 y Fb(0)1930 1556 y Ft(;)14 b Fe(m)2031 1568 y Fp(0)2068 1556 y Fu(\))g Ft(\016)s Fu(\()p Ft(y)s Fu(\))g Ft(`)2311 1568 y Fs(F)2366 1556 y Fu(\()p Ft(dy)s Fu(\))p Ft(:)627 b Fu(\(3.34\))515 1739 y(In)25 b(view)g(of)31 b(\(3.33\))o(,)26 b(w)n(e)f(can)f(minorise)h Ft(p)p Fu(\()p Ft(Q)1904 1751 y Fs(y)1938 1759 y Fb(0)1973 1739 y Ft(;)14 b Fe(m)2074 1751 y Fp(0)2111 1739 y Fu(\))p Ft(\016)29 b Fu(b)n(y)24 b(a)h(function)h Ft(')2765 1751 y Fs(y)2799 1759 y Fb(0)2858 1739 y Fr(2)e Ft(C)3002 1709 y Fo(1)3072 1739 y Fu(\()p Ft(F)12 b Fu(\))26 b(that)515 1839 y(is)f(p)r(ositiv)n(e)g(at)g Ft(y)1041 1851 y Fp(0)1078 1839 y Fu(.)36 b(Inequalit)n(y)26 b(\(3.30\))e(is)i(no)n(w)e(implied)i(b)n(y)g(\(3.34\))o(.)37 b(This)25 b(completes)g(the)515 1938 y(pro)r(of)i(of)g(Theorem)g(2.1)g (in)h(the)g(general)e(case.)515 2213 y Fw(4)134 b(App)t(endix)515 2411 y Fl(4.1)112 b(Pro)s(of)37 b(of)h(Prop)s(osition)f(1.3)515 2564 y Fu(A)26 b(standard)f(limiting)i(argumen)n(t)e(sho)n(ws)f(that)j (if)f Ft(v)h Fr(2)c(X)2319 2576 y Fs(J)2392 2564 y Fu(is)j(a)f(w)n(eak) g(solution)h(for)f(\(1.3\),)515 2664 y(then)35 b(relation)f(\(1.4\))h (is)g(true)g(for)f(an)n(y)g(function)i Ft(')f Fr(2)h(Y)2369 2676 y Fs(J)2451 2664 y Fu(suc)n(h)f(that)57 b(_)-45 b Ft(')35 b Fr(2)h Ft(L)3070 2634 y Fp(2)3070 2688 y(lo)r(c)3157 2664 y Fu(\()p Ft(J)o(;)14 b(H)7 b Fu(\))515 2764 y(and)31 b Ft(')p Fu(\(0\))e(=)f Ft(')p Fu(\()p Ft(T)12 b Fu(\))29 b(=)g(0.)47 b(In)31 b(particular,)g(w)n(e)f(can)h(tak)n(e)f Ft(')g Fu(=)e Ft(\037)2587 2776 y Fs(k)2631 2764 y Fu(~)-45 b Ft(v)s Fu(,)32 b(where)f Fr(f)p Ft(\037)3064 2776 y Fs(k)3104 2764 y Fr(g)g Fu(is)g(the)515 2863 y(sequence)c(of)g (functions)h(de\014ned)g(in)g(Section)g(1.1.)36 b(W)-7 b(e)28 b(th)n(us)f(obtain)556 3099 y Fr(\000)635 2986 y Fn(Z)718 3006 y Fs(t)681 3175 y Fp(0)747 3099 y Fu(\()p Ft(v)s(;)14 b(@)903 3111 y Fs(s)939 3099 y Fu(\()p Ft(\037)1023 3111 y Fs(k)1067 3099 y Fu(~)-45 b Ft(v)s Fu(\)\))14 b Ft(ds)i Fu(+)1363 2986 y Fn(Z)1447 3006 y Fs(t)1410 3175 y Fp(0)1490 3099 y Ft(\037)1542 3111 y Fs(k)1582 3099 y Fu(\()p Ft(s)p Fu(\))1685 3032 y Fn(\000)1724 3099 y Ft(\027)5 b Fu(\()p Ft(v)s(;)18 b Fu(~)-46 b Ft(v)t Fu(\))1958 3111 y Fs(V)2031 3099 y Fu(+)16 b(\()p Ft(B)t Fu(\()p Ft(v)j Fu(+)c Ft(z)t Fu(\))p Ft(;)i Fu(~)-45 b Ft(v)s Fu(\))16 b Fr(\000)f Fu(\()p Ft(h;)i Fu(~)-45 b Ft(v)s Fu(\))2857 3032 y Fn(\001)2909 3099 y Ft(ds)24 b Fu(=)e(0)p Ft(:)41 b Fu(\(4.1\))515 3328 y(F)-7 b(urthermore,)38 b(a)e(similar)g(argumen)n(t)g(sho)n(ws)f(that,)40 b(in)d(iden)n(tit)n (y)h(\(1.4\))e(for)g(the)h(strong)515 3428 y(solution)30 b(~)-45 b Ft(v)s Fu(,)28 b(w)n(e)f(can)g(tak)n(e)g Ft(')c Fu(=)g Ft(\037)1594 3440 y Fs(k)1635 3428 y Ft(v)s Fu(.)37 b(This)28 b(results)f(in)556 3664 y Fr(\000)635 3551 y Fn(Z)718 3571 y Fs(t)681 3739 y Fp(0)747 3664 y Fu(\()s(~)-45 b Ft(v)s(;)14 b(@)903 3676 y Fs(s)939 3664 y Fu(\()p Ft(\037)1023 3676 y Fs(k)1064 3664 y Ft(v)s Fu(\)\))g Ft(ds)i Fu(+)1363 3551 y Fn(Z)1447 3571 y Fs(t)1410 3739 y Fp(0)1490 3664 y Ft(\037)1542 3676 y Fs(k)1582 3664 y Fu(\()p Ft(s)p Fu(\))1685 3596 y Fn(\000)1724 3664 y Ft(\027)5 b Fu(\()s(~)-45 b Ft(v)t(;)14 b(v)s Fu(\))1958 3676 y Fs(V)2031 3664 y Fu(+)i(\()p Ft(B)t Fu(\()s(~)-45 b Ft(v)19 b Fu(+)c Ft(z)t Fu(\))p Ft(;)f(v)s Fu(\))i Fr(\000)f Fu(\()p Ft(h;)f(v)s Fu(\))2857 3596 y Fn(\001)2909 3664 y Ft(ds)24 b Fu(=)e(0)p Ft(:)41 b Fu(\(4.2\))515 3893 y(T)-7 b(aking)23 b(the)i(sum)f(of)31 b(\(4.1\))24 b(and)g(\(4.2\))g(and)g(passing)f(to)h(the)h(limit)g(as)e Ft(k)j Fr(!)d(1)p Fu(,)i(w)n(e)f(obtain)598 4158 y(\()p Ft(v)s Fu(\()p Ft(t)p Fu(\))p Ft(;)18 b Fu(~)-46 b Ft(v)t Fu(\()p Ft(t)p Fu(\)\))19 b(+)1076 4045 y Fn(Z)1159 4065 y Fs(t)1122 4234 y Fp(0)1189 4091 y Fn(\000)1227 4158 y Fu(2)p Ft(\027)5 b Fu(\()s(~)-45 b Ft(v)s(;)14 b(v)s Fu(\))1502 4170 y Fs(V)1578 4158 y Fu(+)k(\()p Ft(B)t Fu(\()p Ft(v)23 b Fu(+)18 b Ft(z)t Fu(\))p Ft(;)e Fu(~)-44 b Ft(v)r Fu(\))19 b(+)f(\()p Ft(B)t Fu(\()s(~)-45 b Ft(v)23 b Fu(+)18 b Ft(z)t Fu(\))p Ft(;)c(v)s Fu(\))2690 4091 y Fn(\001)2741 4158 y Ft(ds)1968 4395 y Fu(=)22 b(\()p Ft(v)s Fu(\(0\))p Ft(;)c Fu(~)-46 b Ft(v)t Fu(\(0\)\))19 b(+)2557 4282 y Fn(Z)2640 4303 y Fs(t)2603 4471 y Fp(0)2669 4395 y Fu(\()p Ft(h;)e Fu(~)-45 b Ft(v)22 b Fu(+)c Ft(v)s Fu(\))c Ft(ds:)83 b Fu(\(4.3\))515 4624 y(Adding)42 b(together)e (relations)h(\(1.5\))g(and)h(\(1.6\))f(\(with)i Ft(v)i Fu(replaced)c(b)n(y)j(~)-45 b Ft(v)s Fu(\),)46 b(subtract-)515 4724 y(ing)28 b(\(4.3\),)g(and)f(carrying)f(out)h(some)g(simple)h (transformations,)e(w)n(e)h(deriv)n(e)1089 4908 y(1)p 1089 4945 42 4 v 1089 5021 a(2)1141 4964 y Fr(k)p Ft(w)r Fr(k)1286 4930 y Fp(2)1341 4964 y Fu(+)1424 4851 y Fn(Z)1507 4872 y Fs(t)1470 5040 y Fp(0)1537 4897 y Fn(\000)1575 4964 y Ft(\027)5 b Fr(k)p Ft(w)r Fr(k)1766 4930 y Fp(2)1766 4985 y Fs(V)1842 4964 y Fu(+)18 b(\()p Ft(B)t Fu(\()p Ft(w)r(;)c(z)23 b Fu(+)e(~)-45 b Ft(v)s Fu(\))p Ft(;)14 b(w)r Fu(\))2504 4897 y Fn(\001)2557 4964 y Ft(ds)23 b Fr(\024)g Fu(0)p Ft(;)393 b Fu(\(4.4\))1905 5255 y(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 515 523 a Fu(where)29 b Ft(w)g Fu(=)e Ft(v)c Fr(\000)f Fu(~)-45 b Ft(v)s Fu(.)44 b(In)30 b(view)f(of)h(standard)f(estimates)g(for)h(the)g(nonlinear)f(term)g Ft(B)34 b Fu(and)515 623 y(Sob)r(olev)27 b(em)n(b)r(edding)g(theorems,) g(w)n(e)h(ha)n(v)n(e)982 805 y Fr(j)p Fu(\()p Ft(B)t Fu(\()p Ft(w)r(;)14 b(z)23 b Fu(+)e(~)-45 b Ft(v)t Fu(\))p Ft(;)14 b(w)r Fu(\))p Fr(j)24 b(\024)e Ft(C)1778 817 y Fp(1)1816 805 y Fr(k)p Ft(w)r Fr(k)1961 771 y Fp(2)1961 828 y Fs(L)2007 811 y Fb(3)2043 738 y Fn(\000)2081 805 y Fr(kr)p Ft(z)t Fr(k)2277 820 y Fs(L)2323 803 y Fb(3)2376 805 y Fu(+)c Fr(kr)s Fu(~)-45 b Ft(v)s Fr(k)2655 820 y Fs(L)2701 803 y Fb(3)2736 738 y Fn(\001)1632 939 y Fr(\024)22 b Ft(C)1778 951 y Fp(2)1816 939 y Fr(k)p Ft(w)r Fr(k)1961 951 y Fs(V)2018 939 y Fr(k)p Ft(w)r Fr(k)2163 872 y Fn(\000)2201 939 y Fr(k)p Ft(z)t Fr(k)2328 951 y Fs(U)2400 939 y Fu(+)c Fr(k)s Fu(~)-45 b Ft(v)s Fr(k)2610 951 y Fs(U)2665 872 y Fn(\001)1632 1074 y Fr(\024)22 b Ft(\027)5 b Fr(k)p Ft(w)r Fr(k)1910 1040 y Fp(2)1910 1094 y Fs(V)1987 1074 y Fu(+)18 b Ft(C)2129 1086 y Fp(3)2166 1074 y Fr(k)p Ft(w)r Fr(k)2311 1040 y Fp(2)2348 1007 y Fn(\000)2386 1074 y Fr(k)p Ft(z)t Fr(k)2513 1040 y Fp(2)2513 1094 y Fs(U)2585 1074 y Fu(+)h Fr(k)s Fu(~)-45 b Ft(v)r Fr(k)2795 1040 y Fp(2)2795 1094 y Fs(U)2850 1007 y Fn(\001)2888 1074 y Ft(:)515 1256 y Fu(Substituting)28 b(this)g(inequalit)n(y)f(in)n(to)h(\(4.4\))o(,)g(w)n(e)f(see)g(that) 1059 1497 y Fr(k)p Ft(w)r Fu(\()p Ft(t)p Fu(\))p Fr(k)1298 1462 y Fp(2)1359 1497 y Fr(\024)22 b Ft(C)1505 1509 y Fp(4)1557 1384 y Fn(Z)1640 1404 y Fs(t)1603 1572 y Fp(0)1683 1497 y Fr(k)p Ft(w)r Fu(\()p Ft(s)p Fu(\))p Fr(k)1931 1462 y Fp(2)1968 1430 y Fn(\000)2006 1497 y Fr(k)p Ft(z)t Fu(\()p Ft(s)p Fu(\))p Fr(k)2236 1462 y Fp(2)2236 1517 y Fs(U)2309 1497 y Fu(+)c Fr(k)s Fu(~)-45 b Ft(v)s Fu(\()p Ft(s)p Fu(\))p Fr(k)2622 1462 y Fp(2)2622 1517 y Fs(U)2678 1430 y Fn(\001)2729 1497 y Ft(ds:)515 1721 y Fu(Application)27 b(of)h(the)g(Gron)n(w)n(all)e(inequalit)n(y)h(sho)n(ws)g(that)g Ft(w)f Fu(=)d Ft(v)e Fr(\000)h Fu(~)-46 b Ft(v)27 b Fr(\021)22 b Fu(0.)515 1954 y Fl(4.2)112 b(Pro)s(of)37 b(of)h(Lemma)h(1.5)515 2107 y Fu(Let)g Ft(\021)j Fu(b)r(e)e(a)e(random)h(pro)r(cess)f (satisfying)g(Condition)h(1.4)f(and)h(let)g Ft(\020)46 b Fu(b)r(e)40 b(the)f(corre-)515 2207 y(sp)r(onding)32 b Ft(H)7 b Fu(-v)-5 b(alued)32 b(cylindrical)g(Wiener)h(pro)r(cess.)51 b(Then)33 b(there)g(is)g(an)f(orthonormal)515 2306 y(basis)21 b Fr(f)p Ft(e)794 2318 y Fs(j)828 2306 y Fr(g)h Fu(in)h Ft(H)29 b Fu(and)22 b(a)g(sequence)f(of)h(indep)r(enden)n(t)h(standard) f(Bro)n(wnian)e(motions)i Fr(f)p Ft(\014)3303 2318 y Fs(j)3337 2306 y Fr(g)515 2406 y Fu(suc)n(h)27 b(that)1587 2551 y Ft(\020)6 b Fu(\()p Ft(t)p Fu(\))24 b(=)1861 2448 y Fo(1)1834 2472 y Fn(X)1837 2649 y Fs(j)s Fp(=1)1968 2551 y Ft(\014)2015 2563 y Fs(j)2050 2551 y Fu(\()p Ft(t)p Fu(\))p Ft(Qe)2249 2563 y Fs(j)2284 2551 y Ft(;)515 2795 y Fu(where)k(the)i(series)e(con)n(v)n(erges)e(in)j Ft(L)1653 2764 y Fp(2)1690 2795 y Fu(\(\012)20 b Fr(\002)f Fu([0)p Ft(;)14 b(T)e Fu(])p Ft(;)i(H)7 b Fu(\))27 b(for)i(an)n(y)f Ft(T)37 b(>)25 b Fu(0.)41 b(Using)29 b(the)g(p)r(olar)515 2894 y(decomp)r(osition)e(for)g Ft(Q)g Fu(\(see)h([RS80)o(]\),)g(w)n(e) f(can)g(rewrite)g Ft(\020)34 b Fu(as)1052 3143 y Ft(\020)6 b Fu(\()p Ft(t)p Fu(\))24 b(=)1327 3040 y Fo(1)1300 3065 y Fn(X)1302 3241 y Fs(j)s Fp(=1)1434 3143 y Ft(\014)1481 3155 y Fs(j)1516 3143 y Fu(\()p Ft(t)p Fu(\))p Ft(AU)9 b(e)1777 3155 y Fs(j)1835 3143 y Fu(=)1973 3040 y Fo(1)1946 3065 y Fn(X)1922 3243 y Fs(j;k)q Fp(=1)2104 3143 y Ft(b)2140 3155 y Fs(k)2181 3143 y Ft(\014)2228 3155 y Fs(j)2262 3143 y Fu(\()p Ft(t)p Fu(\)\()p Ft(U)g(e)2493 3155 y Fs(j)2529 3143 y Ft(;)14 b(f)2607 3155 y Fs(k)2647 3143 y Fu(\))2679 3155 y Fs(V)2737 3143 y Ft(f)2778 3155 y Fs(k)2818 3143 y Ft(;)367 b Fu(\(4.5\))515 3412 y(where)37 b Ft(U)49 b Fu(:)40 b Ft(H)47 b Fr(!)40 b Ft(V)57 b Fu(is)38 b(a)g(partial)f(isometry)-7 b(,)39 b Ft(A)g Fu(is)e(a)h(Hilb)r(ert{Sc)n (hmidt)g(selfadjoin)n(t)515 3511 y(op)r(erator)22 b(in)j Ft(V)43 b Fu(with)24 b(eigen)n(basis)f Fr(f)p Ft(f)1682 3523 y Fs(k)1722 3511 y Fr(g)h Fu(and)g(eigen)n(v)-5 b(alues)23 b Fr(f)p Ft(b)2450 3523 y Fs(k)2490 3511 y Fr(g)p Fu(,)h(and)g(\()p Fr(\001)p Ft(;)14 b Fr(\001)p Fu(\))2884 3523 y Fs(V)2967 3511 y Fu(denotes)24 b(the)515 3611 y(scalar)29 b(pro)r(duct)i(in)g Ft(V)19 b Fu(.)47 b(Setting)31 b Ft(b)1631 3623 y Fs(j)s(k)1731 3611 y Fu(=)d Ft(b)1860 3623 y Fs(k)1901 3611 y Fu(\()p Ft(U)9 b(e)2038 3623 y Fs(j)2072 3611 y Ft(;)14 b(f)2150 3623 y Fs(k)2191 3611 y Fu(\))2223 3623 y Fs(V)2311 3611 y Fu(and)31 b(di\013eren)n(tiating)h(\(4.5\))f(with)515 3711 y(resp)r(ect)c(to)h Ft(t)p Fu(,)f(w)n(e)h(obtain)f(\(1.10\))o(.) 639 3810 y(Con)n(v)n(ersely)-7 b(,)32 b(supp)r(ose)g(that)h Ft(\021)j Fu(is)c(represen)n(table)f(in)i(the)g(form)f(\(1.10\))o(.)52 b(Cho)r(ose)31 b(an)515 3910 y(arbitrary)f(orthonormal)g(basis)h Fr(f)p Ft(e)1642 3922 y Fs(j)1676 3910 y Fr(g)h Fu(in)g Ft(H)39 b Fu(and)31 b(de\014ne)i(a)e(Hilb)r(ert{Sc)n(hmidt)i(op)r (erator)515 4010 y Ft(Q)22 b Fu(:)i Ft(H)29 b Fr(!)23 b Ft(V)47 b Fu(b)n(y)27 b(the)h(relations)1493 4254 y Ft(Qe)1598 4266 y Fs(j)1655 4254 y Fu(=)1770 4150 y Fo(1)1744 4175 y Fn(X)1743 4354 y Fs(k)q Fp(=1)1878 4254 y Ft(b)1914 4266 y Fs(j)s(k)1985 4254 y Ft(f)2026 4266 y Fs(k)2066 4254 y Ft(;)97 b(j)28 b Fr(\025)23 b Fu(1)p Ft(:)515 4516 y Fu(W)-7 b(e)28 b(can)f(rewrite)g(\(1.10\))g(as)1586 4682 y Ft(\021)s Fu(\()p Ft(t)p Fu(\))d(=)1862 4578 y Fo(1)1835 4603 y Fn(X)1838 4780 y Fs(j)s Fp(=1)1990 4660 y Fu(_)1969 4682 y Ft(\014)2016 4694 y Fs(j)2051 4682 y Fu(\()p Ft(t)p Fu(\))p Ft(Qe)2250 4694 y Fs(j)2285 4682 y Ft(:)515 4920 y Fu(This)j(is)h(equiv)-5 b(alen)n(t)27 b(to)h(represen)n(tation)d(\(1.9\).)1905 5255 y(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 515 523 a Fw(References)515 705 y Fu([AKSS06])40 b(A.)34 b(Agrac)n(hev,)e(S.)h(Kuksin,)g(A.)g(Saryc)n (hev,)g(and)f(A.)h(Shiriky)n(an,)g Fm(On)h(\014nite-)904 805 y(dimensional)47 b(pr)l(oje)l(ctions)f(of)f(distributions)h(for)f (solutions)g(of)h(r)l(andomly)904 904 y(for)l(c)l(e)l(d)31 b(PDE's)p Fu(,)d(Ann.)g(Inst.)f(H.)h(P)n(oincar)n(\023)-39 b(e)24 b(Probab.)h(Statist.)j(\(2006\),)e(to)h(ap-)904 1004 y(p)r(ear.)515 1170 y([Bog98])116 b(V.)22 b(I.)f(Bogac)n(hev,)g Fm(Gaussian)j(Me)l(asur)l(es)p Fu(,)f(American)e(Mathematical)f(So)r (ciet)n(y)-7 b(,)904 1269 y(Pro)n(vidence,)26 b(RI,)i(1998.)515 1435 y([Brz91])130 b(Z.)33 b(Brze)n(\023)-39 b(zniak,)31 b Fm(On)j(analytic)h(dep)l(endenc)l(e)h(of)f(solutions)f(of)h (Navier-Stokes)904 1535 y(e)l(quations)44 b(with)g(r)l(esp)l(e)l(ct)f (to)g(exterior)h(for)l(c)l(e)g(and)g(initial)g(velo)l(city)p Fu(,)k(Univ.)904 1635 y(Iagel.)27 b(Acta)h(Math.)g(\(1991\),)e(no.)h (28,)g(111{124.)515 1801 y([BT73])140 b(A.)51 b(Bensoussan)e(and)h(R.)h (T)-7 b(emam,)56 b Fm(Equations)51 b(sto)l(chastiques)g(du)g(typ)l(e) 904 1900 y(Navier-Stokes)p Fu(,)29 b(J.)e(F)-7 b(unct.)29 b(Anal.)e Fv(13)h Fu(\(1973\),)e(195{222.)515 2066 y([CG94])134 b(M.)36 b(Capi)r(\023)-44 b(nski)34 b(and)h(D.)h(G)1755 2120 y(`)1746 2066 y(atarek,)g Fm(Sto)l(chastic)h(e)l(quations)g(in)g (Hilb)l(ert)f(sp)l(ac)l(e)904 2166 y(with)47 b(applic)l(ation)h(to)e (Navier-Stokes)g(e)l(quations)g(in)g(any)g(dimension)p Fu(,)k(J.)904 2266 y(F)-7 b(unct.)29 b(Anal.)f Fv(126)f Fu(\(1994\),)f(no.)h(1,)g(26{35.)515 2432 y([DD03])133 b(G.)43 b(Da)g(Prato)d(and)j(A.)g(Debussc)n(he,)j Fm(Er)l(go)l(dicity)f (for)f(the)g(3D)f(sto)l(chastic)904 2531 y(Navier{Stokes)32 b(e)l(quations)p Fu(,)c(J.)f(Math.)h(Pures)e(Appl.)j Fv(82)e Fu(\(2003\),)f(877{947.)515 2697 y([F)n(G95])142 b(F.)28 b(Flandoli)f(and)g(D.)h(G)1684 2751 y(`)1675 2697 y(a)o(tarek,)f Fm(Martingale)k(and)e(stationary)i(solutions)e(for) 904 2797 y(sto)l(chastic)40 b(Navier-Stokes)f(e)l(quations)p Fu(,)h(Probab.)c(Theory)g(Related)h(Fields)904 2897 y Fv(102)27 b Fu(\(1995\),)g(no.)g(3,)g(367{391.)515 3063 y([FK64])140 b(H.)25 b(F)-7 b(ujita)24 b(and)g(T.)g(Kato,)f Fm(On)j(the)g(Navier-Stokes)h(initial)g(value)g(pr)l(oblem.)h(I)p Fu(,)904 3162 y(Arc)n(h.)g(Rational)f(Mec)n(h.)g(Anal.)h Fv(16)f Fu(\(1964\),)f(269{315.)515 3328 y([Fla97])140 b(F.)33 b(Flandoli,)g Fm(Irr)l(e)l(ducibility)j(of)f(the)f(3-D)g(sto)l (chastic)g(Navier-Stokes)h(e)l(qua-)904 3428 y(tion)p Fu(,)28 b(J.)g(F)-7 b(unct.)28 b(Anal.)g Fv(149)f Fu(\(1997\),)f(no.)i (1,)f(160{177.)515 3594 y([FR06])144 b(F.)29 b(Flandoli)g(and)f(M.)h (Romito,)g Fm(Markov)j(sele)l(ctions)g(for)f(the)g(3D)g(sto)l(chastic) 904 3694 y(Navier-Stokes)g(e)l(quations)p Fu(,)d(preprin)n(t)f (\(2006\).)515 3860 y([Kuk82])104 b(S.)42 b(Kuksin,)i Fm(Di\013e)l(omorphisms)g(of)f(function)g(sp)l(ac)l(es)f(that)h(c)l (orr)l(esp)l(ond)g(to)904 3959 y(quasiline)l(ar)g(p)l(ar)l(ab)l(olic)h (e)l(quations)p Fu(,)g(Mat.)d(Sb.)g(\(N.S.\))h Fv(117\(159\))d Fu(\(1982\),)904 4059 y(no.)28 b(3,)f(359{378,)d(431.)515 4225 y([MP06])126 b(J.)21 b(Mattingly)f(and)1533 4204 y(\023)1526 4225 y(E.)g(P)n(ardoux,)g Fm(Mal)t(liavin)26 b(c)l(alculus)d(for)h(the)g(sto)l(chastic)g(2D)904 4325 y(Navier{Stokes)j(e)l(quation)p Fu(,)d(Comm.)f(Pure)f(Appl.)i(Math.)f (\(2006\),)f(to)h(app)r(ear.)515 4491 y([Oda06])106 b(C.)33 b(Odasso,)e Fm(Exp)l(onential)k(mixing)f(for)h(the)f(3D)g(sto)l (chastic)h(Navier{Stokes)904 4590 y(e)l(quations)p Fu(,)28 b(Comm.)g(Math.)g(Ph)n(ys.)e(\(2006\),)h(to)g(app)r(ear.)515 4756 y([Rom01])87 b(M.)44 b(Romito,)j Fm(Existenc)l(e)d(of)h (martingale)g(and)g(stationary)g(suitable)f(we)l(ak)904 4856 y(solutions)36 b(for)h(a)f(sto)l(chastic)g(Navier{Stokes)h(e)l (quations)p Fu(,)f(preprin)n(t)d(\(2001\),)904 4956 y(23)27 b(pages.)1905 5255 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 515 523 a Fu([RS80])152 b(M.)27 b(Reed)g(and)f(B.)g(Simon,)h Fm(Metho)l(ds)j(of)f(Mo)l(dern)h (Mathematic)l(al)g(Physics)h(I.)904 623 y(Functional)g(Analysis)p Fu(,)d(Academic)g(Press,)e(New)i(Y)-7 b(ork,)26 b(1980.)515 789 y([Shi06a])102 b(A.)73 b(Shiriky)n(an,)82 b Fm(Appr)l(oximate)72 b(c)l(ontr)l(ol)t(lability)h(of)f(thr)l(e)l(e-dimensional)904 888 y(Navier{Stokes)36 b(e)l(quations)p Fu(,)d(Comm)n(un.)g(Math.)f(Ph) n(ys.)f Fv(266)g Fu(\(2006\),)h(no.)g(1,)904 988 y(123{151.)515 1154 y([Shi06b])p 904 1154 250 4 v 361 w(,)26 b Fm(Contr)l(ol)t (lability)k(of)f(thr)l(e)l(e-dimensional)g(Navier-Stokes)f(e)l (quations)904 1254 y(and)53 b(applic)l(ations)p Fu(,)61 b(S)n(\023)-39 b(eminaire:)2069 1233 y(\023)2061 1254 y(Equations)51 b(aux)h(D)n(\023)-39 b(eriv)n(\023)g(ees)50 b(P)n(artielles.)904 1353 y(2005{2006,)1336 1332 y(\023)1329 1353 y(Ecole)26 b(P)n(olytec)n(h.,)h(P)n(alaiseau,)e(2006.)515 1519 y([Shi06c])p 904 1519 V 370 w(,)53 b Fm(Exact)c(c)l(ontr)l(ol)t (lability)h(in)f(pr)l(oje)l(ctions)g(for)h(thr)l(e)l(e-dimensional)904 1619 y(Navier{Stokes)32 b(e)l(quations)p Fu(,)d(Ann.)g(Inst.)f(H.)h(P)n (oincar)n(\023)-39 b(e)25 b(Anal.)j(Non)g(Lin)n(\023)-39 b(eaire)904 1719 y(\(2006\),)27 b(to)g(app)r(ear.)515 1885 y([Soh01])125 b(H.)43 b(Sohr,)h Fm(The)g(Navier{Stokes)g (Equations)p Fu(,)i(Birkh\177)-42 b(auser)40 b(V)-7 b(erlag,)44 b(Basel,)904 1984 y(2001.)515 2150 y([Ste83])144 b(S.)29 b(Stern)n(b)r(erg,)e Fm(L)l(e)l(ctur)l(es)i(on)i(Di\013er)l(ential)f (Ge)l(ometry)p Fu(,)f(Chelsea)f(Publishing)904 2250 y(Co.,)g(New)g(Y)-7 b(ork,)26 b(1983.)515 2416 y([T)-7 b(a)n(y97])122 b(M.)31 b(E.)e(T)-7 b(a)n(ylor,)29 b Fm(Partial)34 b(Di\013er)l(ential)e (Equations.)h(I{III)p Fu(,)e(Springer-V)-7 b(erlag,)904 2516 y(New)28 b(Y)-7 b(ork,)27 b(1996-1997.)515 2682 y([T)-7 b(em79])100 b(R.)41 b(T)-7 b(emam,)43 b Fm(Navier{Stokes)g (Equations)p Fu(,)h(North-Holland,)e(Amsterdam,)904 2781 y(1979.)515 2947 y([VF88])143 b(M.)21 b(I.)g(Vishik)g(and)f(A.)h(V.)g (F)-7 b(ursik)n(o)n(v,)21 b Fm(Mathematic)l(al)k(Pr)l(oblems)f(in)g (Statistic)l(al)904 3047 y(Hydr)l(ome)l(chanics)p Fu(,)30 b(Klu)n(w)n(er,)c(Dordrec)n(h)n(t,)h(1988.)515 3213 y([VKF79])78 b(M.)28 b(I.)g(Vishik,)g(A.)f(I.)h(Komec)n(h,)f(and)g(A.)h(V.)g(F)-7 b(ursik)n(o)n(v,)26 b Fm(Some)k(mathematic)l(al)904 3313 y(pr)l(oblems)e(of)g(statistic)l(al)f(hydr)l(ome)l(chanics)p Fu(,)h(Russ.)d(Math.)f(Surv.)g Fv(34)g Fu(\(1979\),)904 3412 y(149{234.)1905 5255 y(24)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0608021225647--