Content-Type: multipart/mixed; boundary="-------------0204230537129" This is a multi-part message in MIME format. ---------------0204230537129 Content-Type: text/plain; name="02-194.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-194.keywords" stochastic PDE, stationary solutions, thin film ---------------0204230537129 Content-Type: application/postscript; name="Dirk4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Dirk4.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: Dirk4.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Times-Italic %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D 600 -f Dirk4.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.04.18:1141 %%BeginProcSet: texc.pro %! /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/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/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]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/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: 8r.enc % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "1 July 1998", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "Encoding for TrueType or Type 1 fonts % to be used with TeX." % @} % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both % ASCII and Windows. % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (Dirk4.dvi) @start %DVIPSBitmapFont: Fa cmmi6 6 1 /Fa 1 59 df<127812FCA4127806067A8513>58 D E %EndDVIPSBitmapFont /Fb 200[37 37 37 1[37 37 37 37 48[{TeXBase1Encoding ReEncodeFont}7 74.7198 /Times-Bold rf /Fc 134[33 33 50 33 37 21 29 29 37 37 37 37 54 21 33 1[21 37 37 21 33 37 33 37 37 10[46 1[42 37 46 1[46 54 50 62 42 50 33 25 54 54 46 46 54 50 46 46 6[25 11[19 25 42[37 2[{TeXBase1Encoding ReEncodeFont}48 74.7198 /Times-Italic rf %DVIPSBitmapFont: Fd cmsy5 5 1 /Fd 1 4 df<13E0A438F0E1E0EAF843387E4FC0380F5E00EA01F0A2EA0F5E387E4FC038 F843E0EAF0E13800E000A413127B921F>3 D E %EndDVIPSBitmapFont /Fe 162[14 1[14 91[{TeXBase1Encoding ReEncodeFont}2 41.511 /Times-Roman rf %DVIPSBitmapFont: Ff cmmi5 5 6 /Ff 6 117 df<127012F812FCA2127C120CA31218A2123012601240060D7A8413>59 D78 D<001FB61280A2393E01F00F0038EC03 0012701260EB03E012C0A3C64848C7FCA4495AA449C8FCA4133EA4381FFFF85C211C7C9B 23>84 D107 D<137E3801FF80EA0381380703C0380E0780EB0300EA0F80EA07F86CB4FC6C 1380EA000FEA3003127812F8EB0700EAF00EEA7FFCEA1FF012127C911C>115 D<13C0EA01E0A3EA03C0A4EAFFFCA2EA0780A2EA0F00A4121EA31304EA3C0CA213181370 EA1FE0EA0F800E1A7D9917>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr5 5 4 /Fg 4 53 df48 D<1360EA01E0120F12FF12F11201B3A3387FFF80A2111C7B9B1C>II52 D E %EndDVIPSBitmapFont /Fh 173[42 3[42 78[{TeXBase1Encoding ReEncodeFont}2 58.1154 /Times-Bold rf %DVIPSBitmapFont: Fi cmsy7 7 7 /Fi 7 68 df0 D<1338A50060130C00F8133E00FC137E00FE13 FE383FBBF83807FFC000011300EA007C48B4FC000713C0383FBBF838FE38FE00FC137E00 F8133E0060130C00001300A517197B9A22>3 D<12E012F812FEEA3F80EA0FE0EA03F8EA 00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03F8ED 00FE163E16FEED03F8ED0FE0ED3F80EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03F8EB0F E0EB3F8001FEC8FCEA03F8EA0FE0EA3F80007EC9FC12F812E0CAFCAB007FB612FCB712FE A227357AA734>21 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F13C0A3EA1F80A21300 A25A123EA35AA3127812F8A25A12100D1E7D9F13>48 D<017F157F2601FFE0903803FFC0 000701F890380FF1F0260F83FC90381F0038261E00FF013C7F001890263F8078130C4890 261FC0E07F007090260FE1C07F0060EB07E3913803F780486DB4C7EA01806E5A157E157F 81824B7E0060DAF7E0EB0300913801E3F0DBC3F85B6C90260381FC13066C90260F00FE5B 001C011E90387F803C6C017C90381FE0F82607C7F86DB45A2601FFE0010313C06C6CC86C C7FC391B7C9942>I<49B5FC130F133F01FFC7FCEA01F8EA03E0EA078048C8FC121E121C 123C123812781270A212F05AA2B7FCA300E0C8FCA27E1270A212781238123C121C121E7E 6C7EEA03E0EA01F86CB4FC013FB5FC130F130120277AA12D>I67 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi8 8 1 /Fj 1 89 df<90260FFFFCEB7FFFA29026007FC0EB0FF06E48148018006E6C131E171802 0F5C6F5B02075C6F485A020349C7FCEDF8065E6E6C5A5E6E6C5A5EED7F8093C8FC6F7EA2 6F7E153F156FEDCFE0EC018791380307F0EC0703020E7F141C4A6C7E14704A6C7E495A49 48137F49C7FC010E6E7E5B496E7E5BD801F081D807F8143FD8FFFE0103B5FCA2382D7EAC 3A>88 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy8 8 1 /Fk 1 77 df76 D E %EndDVIPSBitmapFont /Fl 206[25 49[{TeXBase1Encoding ReEncodeFont}1 49.8132 /Times-Roman rf /Fm 144[29 2[16 8[26 5[19 1[19 14[36 26[29 7[19 19 40[{TeXBase1Encoding ReEncodeFont}9 58.1154 /Times-Roman rf %DVIPSBitmapFont: Fn cmex10 10 12 /Fn 12 91 df<1430147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B 1203A2485AA3485AA3121F90C7FCA25AA3123EA2127EA6127C12FCB3A2127C127EA6123E A2123FA37EA27F120FA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C013 01EB00E0147014301462738226>0 D<12C07E12707E123C7E7EA26C7E6C7EA26C7E7F12 007F1378137CA27FA37FA31480130FA214C0A31307A214E0A6130314F0B3A214E01307A6 14C0A2130FA31480A2131F1400A3133EA35BA2137813F85B12015B485AA2485A48C7FCA2 121E5A12385A5A5A14627C8226>I<12F0B3B3B2043674811C>12 D<151E153E157C15F8EC01F0EC03E01407EC0FC0EC1F8015005C147E5CA2495A495AA249 5AA2495AA2495AA249C7FCA2137EA213FE5B12015BA212035BA21207A25B120FA35B121F A45B123FA548C8FCA912FEB3A8127FA96C7EA5121F7FA4120F7FA312077FA21203A27F12 01A27F12007F137EA27FA26D7EA26D7EA26D7EA26D7EA26D7E6D7EA2147E80801580EC0F C0EC07E01403EC01F0EC00F8157C153E151E1F94718232>16 D<12F07E127C7E7E6C7E7F 6C7E6C7E12017F6C7E137EA27F6D7EA26D7EA26D7EA26D7EA26D7EA26D7EA280147E147F 80A21580141FA215C0A2140F15E0A3140715F0A4140315F8A5EC01FCA9EC00FEB3A8EC01 FCA9EC03F8A515F01407A415E0140FA315C0141FA21580A2143F1500A25C147E14FE5CA2 495AA2495AA2495AA2495AA2495AA249C7FC137EA25B485A5B1203485A485A5B48C8FC12 3E5A5A5A1F947D8232>I<160F161F163E167C16F8ED01F0ED03E0ED07C0150FED1F8016 00153E157E5D4A5A5D14034A5A5D140F4A5AA24AC7FC143E147E5CA2495AA2495AA2495A A2130F5CA2495AA2133F91C8FCA25B137E13FEA25B1201A25B1203A35B1207A35B120FA3 5BA2121FA45B123FA690C9FC5AAA12FEB3AC127FAA7E7FA6121F7FA4120FA27FA312077F A312037FA312017FA212007FA2137E137F7FA280131FA26D7EA2801307A26D7EA26D7EA2 6D7EA2147E143E143F6E7EA26E7E1407816E7E1401816E7E157E153E811680ED0FC01507 ED03E0ED01F0ED00F8167C163E161F160F28C66E823D>I<12F07E127C7E7E6C7E6C7E6C 7E7F6C7E1200137C137E7F6D7E130F806D7E1303806D7EA26D7E147C147E80A26E7EA26E 7EA26E7EA2811403A26E7EA2811400A281157E157FA2811680A2151F16C0A3150F16E0A3 150716F0A31503A216F8A4150116FCA6150016FEAA167FB3AC16FEAA16FC1501A616F815 03A416F0A21507A316E0150FA316C0151FA31680153FA216005DA2157E15FE5DA214015D A24A5AA214075DA24A5AA24A5AA24AC7FCA2147E147C14FC495AA2495A5C1307495A5C13 1F49C8FC137E137C5B1201485A5B485A485A48C9FC123E5A5A5A28C67E823D>I<161E16 7EED01FE1507ED0FF8ED3FE0ED7FC0EDFF80913801FE004A5A4A5A5D140F4A5A5D143F5D 147F92C7FCA25C5CB3B3B3A313015CA3495AA213075C495AA2495A495A137F49C8FC485A 485AEA07F0EA1FE0485AB4C9FC12FCA2B4FCEA3FC06C7EEA07F0EA03FC6C7E6C7E6D7E13 3F6D7E6D7EA26D7E801303A26D7EA3801300B3B3B3A38080A281143F81141F816E7E1407 816E7E6E7E913800FF80ED7FC0ED3FE0ED0FF8ED07FE1501ED007E161E27C675823E>26 D80 D<167F923801FFC0923803C0F0923807803892380F007892381F01FC15 1E153EA2157E92387C0070170015FCA44A5AA81403A45DA41407A94A5AAA4A5AA95DA414 3FA492C8FCA7143E147EA4147C123800FE13FC5CA2495A5CEA7803387007C0383C0F80D8 0FFEC9FCEA03F82E5C7C7F27>82 D88 D90 D E %EndDVIPSBitmapFont /Fo 134[33 1[48 33 1[18 26 22 2[33 33 52 18 2[18 33 1[22 29 33 1[33 29 11[48 41 37 44 2[48 48 59 41 2[22 48 1[37 41 48 44 44 48 18[17 4[22 22 40[{TeXBase1Encoding ReEncodeFont}36 66.4176 /Times-Roman rf /Fp 134[37 37 55 37 42 23 32 32 42 42 42 42 60 23 37 1[23 42 42 23 37 42 37 42 42 9[69 2[46 3[51 1[55 1[46 4[60 51 51 60 55 1[51 10[42 42 1[42 42 42 2[21 28 21 2[28 28 37[42 2[{TeXBase1Encoding ReEncodeFont} 46 83.022 /Times-Italic rf %DVIPSBitmapFont: Fq cmr7 7 11 /Fq 11 62 df<90383FFFFCA2010090C7FC147EA5903803FF80013F13F89038FC7E7ED8 03E0EB0F80D80FC0EB07E0D81F00EB01F04815F8007EEC00FCA248157EA6007E15FCA26C EC01F86C15F0D80FC0EB07E0D803E0EB0F80D800FCEB7E0090383FFFF801031380D9007E C7FCA514FF013F13FCA227287DA72F>8 D<140EB3A2B812E0A3C7000EC8FCB3A22B2B7D A333>43 D48 D<13381378EA01F8121F12FE12E01200B3AB487EB512F8A215 267BA521>I<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A4 127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA01 80390300030012065A001FB5FC5A485BB5FCA219267DA521>I<13FF000313E0380F01F8 381C007C0030137E003C133E007E133FA4123CC7123E147E147C5C495AEB07E03801FF80 91C7FC380001E06D7E147C80143F801580A21238127C12FEA21500485B0078133E00705B 6C5B381F01F03807FFC0C690C7FC19277DA521>I<1438A2147814F81301A21303130713 06130C131C131813301370136013C012011380EA03005A120E120C121C5A12305A12E0B6 12E0A2C7EAF800A7497E90383FFFE0A21B277EA621>I<0018130C001F137CEBFFF85C5C 1480D819FCC7FC0018C8FCA7137F3819FFE0381F81F0381E0078001C7F0018133EC7FC80 A21580A21230127C12FCA3150012F00060133E127000305B001C5B380F03E03803FFC0C6 48C7FC19277DA521>II<137F3803FFE0380781F8380E007C48131E5A801278A3127C007E131EEA3F 80EBE03C6C6C5A380FFCF03807FFC06C5BC613E0487F38079FFC380F07FEEA1E0348C67E 48133FEC1F8048130FA21407A315001278140E6C5B6C5B380F80F03803FFE0C66CC7FC19 277DA521>56 D61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmsy10 10 23 /Fr 23 108 df<007FB81280B912C0A26C17803204799641>0 D<121C127FEAFF80A5EA 7F00121C0909799917>I<0060150600F8150F6C151F007E153F6C157E6C6C14FC6C6CEB 01F86C6CEB03F06C6CEB07E06C6CEB0FC06C6CEB1F80017EEB3F006D137E6D6C5A90380F C1F8903807E3F0903803F7E06DB45A6D5B6EC7FCA24A7E497F903803F7E0903807E3F090 380FC1F890381F80FC90383F007E017E7F49EB1F804848EB0FC04848EB07E04848EB03F0 4848EB01F84848EB00FC48C8127E007E153F48151F48150F00601506282874A841>I<00 7FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912 FCA26C17F836287BA841>17 D20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC6 6C7EEB3FE0EB0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8 ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE 07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948 C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007F B81280B912C0A26C1780324479B441>I25 D<020FB6128091B712C0130301 0F1680D91FF8C9FCEB7F8001FECAFCEA01F8485A485A485A5B48CBFCA2123EA25AA21278 12F8A25AA87EA21278127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF86DB7 1280010316C01300020F1580323279AD41>I<181EA4181F84A285180785727EA2727E72 7E85197E85F11F80F10FC0F107F0007FBA12FCBCFCA26C19FCCCEA07F0F10FC0F11F80F1 3F00197E61614E5A4E5AA24E5A61180F96C7FCA260181EA4482C7BAA53>33 D<91381FFFFE91B6FC1303010F14FED91FF0C7FCEB7F8001FEC8FCEA01F8485A485A485A 5B48C9FCA2123EA25AA2127812F8A25AA2B712FE16FFA216FE00F0C9FCA27EA21278127C A27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF06DB512FE010314FF1300021F13 FE283279AD37>50 D67 D<0307B612FE033FEDFF804AB812C0140791260F807EC7FC91 263C00FEEC3F004A161E4A491418010194C7FC495A01071301A2D90FC05B148014000118 130390C75BA34B5AA3150F5EA34B5AA293B512FC4B5C604B14C0037ECAFCA25DA25D1401 A24A5AA25D14075D140F5D141F92CBFC5C0006133E003E137E007E137CB413FC6D5AEBC1 F0EBF1E06CB45A6C90CCFC6C5AEA07F0423C7EB83C>70 D72 D76 D<0203B512F8027FECFF8049B712F0010F8290273FC3F00313FED978039038003FFF2601 E00702071380D803C06F13C0D807801500000F177FD81F00EE3FE0484A141F123E5A0078 010F150F12C0C7FC4B15C0A3021FED1F80A24B1500183EA2023F5D6092C85A4D5A4D5A4A 4A5A027E020EC7FC173C17F84AEB03E0EE3F80DB1FFEC8FC0101EB7FF89138F8FFC0DAF9 FCC9FC02F8CAFC495AA3495AA3495AA3495AA291CBFC5BA2137EA35B13F013C03B3D7FB8 3A>80 D<923801FFC0031F13F8037F13FE0203B6FC91260FE01F138091261E000313C002 78010013E04A147FD903C0EC3FF04948141F49C8EA0FF8131E491507137C49ED03FC485A A2485A48481501A2120F485AA290C9FC5AA24817F8127EA2170312FE18F0A3EF07E0A26C 17C0170F18806DED1F00127F6D153E6D5D6C6C130F01FC013E5B3B1FFF01F801F06CD9FF E05B6C91388003C000014948485A26007FE049C7FC90C8121E163816F0ED03E0ED078003 3EC8FCEC0FFC0003B500E0140E000F0280143E4801FCC8127C48D9FF8014FC000102F014 F8D8000F01FEEB01F00101D9FFC013E0D9003F9038FC03C0020790B5120002005C031F13 F8030113C0374577BA44>I92 D 102 D<12FCEAFFC0EA07F0EA01FCEA007E7F80131F80130FB3A7801307806D7E6D7EEB00 7EEC1FF0EC07F8EC1FF0EC7E00495A495A495A5C130F5CB3A7131F5C133F91C7FC137E48 5AEA07F0EAFFC000FCC8FC1D537ABD2A>I<14C0EB01E01303A214C01307A21480130FA2 EB1F00A2131E133EA25BA2137813F8A2485AA25B1203A25B1207A2485AA290C7FC5AA212 3EA2123C127CA2127812F8A41278127CA2123C123EA27EA27E7FA26C7EA212037FA21201 7FA26C7EA21378137CA27FA2131E131FA2EB0F80A2130714C0A2130314E0A21301EB00C0 135278BD20>I<126012F07EA21278127CA2123C123EA27EA27E7FA26C7EA212037FA26C 7EA212007FA21378137CA27FA2131E131FA2EB0F80A2130714C0A2130314E0A414C01307 A21480130FA2EB1F00A2131E133EA25BA2137813F8A25B1201A2485AA25B1207A2485AA2 90C7FC5AA2123EA2123C127CA2127812F8A25A126013527CBD20>I<126012F0B3B3B3B3 A91260045377BD17>I<0070131C00F0131EB3B3B3B3A80070131C175277BD2A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr10 10 18 /Fs 18 127 df0 D5 D<011FB512FEA39026001FFEC8FC EC07F8A8EC3FFE0103B512E0D91FF713FC90397F07F87F01FCEC1F80D803F8EC0FE0D807 F06E7ED80FE06E7E001F82D83FC06E7EA2007F8201808000FF1780A7007F170001C05C00 3F5EA2D81FE04A5A000F5ED807F04A5AD803F84A5AD800FCEC1F80017F027FC7FC90391F F7FFFC0103B512E09026003FFEC8FCEC07F8A8EC1FFE011FB512FEA331397BB83C>8 D10 D<146014E0EB01C0EB0380EB0700130E131E5B5BA2 5B485AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3121EA212 1F7EA27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E014601352 78BD20>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C133E13 1EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C 137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<15301578B3 A6007FB812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41>43 D48 DIII<1538A21578 15F8A2140114031407A2140F141F141B14331473146314C313011483EB03031307130613 0C131C131813301370136013C01201EA038013005A120E120C5A123812305A12E0B712F8 A3C73803F800AB4A7E0103B512F8A325397EB82A>I 54 D56 D<121C127FEAFF80A5EA7F00121CC7FCB2121C127FEA FF80A5EA7F00121C092479A317>58 D<007FB812F8B912FCA26C17F8CCFCAE007FB812F8 B912FCA26C17F836167B9F41>61 D<121E123FEA7F80A2EAFFC0EA7F80A2EA3F00121E0A 097AB717>95 D126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmmi7 7 26 /Ft 26 122 df11 D21 D34 D<1238127C12FE12FFA2127F123B1203A31206A3120C12181238127012 2008127A8614>59 D61 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00 FEEC3F80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03F8ED00FE163E16FEED03F8ED0FE0ED 3F80EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03F8EB0FE0EB3F8001FEC8FCEA03F8EA0F E0EA3F8000FEC9FC12F812E027277AA134>I<4B7E1503150782150F151FA2153FA2156F 15CF82EC0187140315071406140E140C02187FA2EC30031460A214C013011480D903007F 91B5FC5B90380C0001A25B13380130805B01E013005B12011203000F4A7ED8FFF890381F FFE0A22B2A7DA932>65 D<903B3FFFF01FFFF8A2D901FCC7EAFE004A5CA2010314015F5C A2010714035F5CA2010F14075F5CA2011F140F91B65AA2913880000F013F141F5F91C7FC A249143F94C7FC137EA201FE5C167E5BA2000115FE5E5BA200031401B539C07FFFE0A235 287DA736>72 D<90263FFFF0EB7FF8A2D901FCC7EA1FC04AEC1E005F010315704C5A4AEB 03804CC7FC0107141C5E4A13E04B5A010FEB0780030EC8FC4A5A157C011F13FE14C3EC87 7F149E90393FB83F8014F09138C01FC0148049486C7EA2017E6D7EA201FE6D7EA2496D7E A200016E7EA249147FA2000382B539C007FFF8A235287DA738>75 D<90383FFFF8A2D901FCC7FC5CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133F A291C8FCA249141C1618137E163801FE1430167049146016E000011401ED03C0491307ED 0F800003147FB7FC160026287DA72E>I78 D<013FB512E016FC903901FC007F4AEB0F80EE07C0010315E016 035C17F01307EE07E05CA2010FEC0FC017804AEB1F00163E011F14F8ED07F091B51280A2 90393F800FE0ED03F002007F15015BA2137EA201FE1303A2495CA20001160817184914E0 17380003EDF070B5D8C00113E0923800FFC0C9EA3F002D297DA732>82 D<000FB712E05A9039800FE007D81E009038C001C05A0038011F1300123000705C006015 01023F148012E0481400A2C74890C7FCA2147EA214FEA25CA21301A25CA21303A25CA213 07A25CA2130FA25CA2131F001FB57EA22B287DA727>84 D87 D99 D<130E131F5BA2133E131C90C7FCA7EA03E0 487EEA0C78EA187C1230A212605B12C0A2EA01F0A3485AA2485AA2EBC180EA0F81A2381F 0300A213066C5A131CEA07F06C5A11287DA617>105 D<133EEA07FEA2EA007CA213FCA2 5BA21201A25BA21203EC07809038E01FC0EC38600007EB61E014C3EBC187EBC307D80FC6 13C09038CC038001B8C7FC13E0487E13FEEB3F80EB0FC0486C7E1303003E1460A2127EEC C0C0127CECC18012FC903801E30038F800FE0070137C1B297CA723>107 D<3B07801FC007E03B0FE07FF01FF83B18F0E0F8783C3B30F1807CE03E903AFB007D801E D860FEEB3F005B49133E00C14A133E5B1201A24848495BA35F4848485A1830EE01F0A23C 0F8003E003E060A218C0933801E180271F0007C013E3933800FF00000E6D48137C341B7D 993B>109 D<3907801FC0390FE07FF03918F0E0F83930F1807CEBFB00D860FE133C5B5B 00C1147C5B1201A248485BA34A5AEA07C01660EC03E0A23A0F8007C0C0A2EDC180913803 C300D81F0013C7EC01FE000EEB00F8231B7D9929>I<9038F007C03901FC1FF039031E78 780006EBE03C90381FC01C000CEB801E14005B0018141F133E1200137E153E137CA213FC 157C5B1578000114F0A2EC01E0EC03C03903FC07809038FE1F00EBE7FCEBE1F0D807E0C7 FCA25BA2120FA25B121FEAFFF8A22025809922>112 DI115 D<131C133EA25BA45BA4485AB512 E0A23801F000485AA4485AA4485AA448C7FC1460A214C0123EEB0180EB0300EA1E06EA1F 1CEA0FF8EA03E013267EA419>I<3903E001C03907F003E0380C7807EA187C0030130314 011260EBF80000C014C0A2EA01F0A2EC0180EA03E0A2EC0300EA07C0A21406A25CA20003 5B6D5A3801F0E06CB45A013FC7FC1B1B7D9921>118 D<90387C03C03901FF0FF0390707 9C30390E03B078000CEBF0F8001813E1123015F0396007C0E015001200A2495AA449C7FC 15301238007C1460EAFC3E15C0EAF87E39F06F03803970C70700383F83FE381F01F81D1B 7D9926>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmmi10 10 49 /Fu 49 122 df11 D<1403EC3FF891387FFF80D901E313C014800103133F9138001F80ED07 0092C7FC80A280A2808013018080130080147F81143F8149B47E130790380F8FF0EB3E0F 496C7E13F83801F003D803E07F1207380FC0011380121FEA3F0014005A127EA212FE5D48 1301A35DA24813035D6C13075D127C4A5A6C91C7FC5C6C133E6C6C5A3807C0F03801FFE0 D8003FC8FC223D7DBB25>14 D<133F14C0EB07F06D7E801301A26D7EA3147FA36E7EA36E 7EA36E7EA36E7EA36E7EA36E7EA26E7EA214014A7E5C4A7E91381E3F80143C14784A6C7E 1301EB03E049486C7EEB0F80EB1F00496D7E137E5B48486D7E485A485A000F6E7E485A48 5A48C87E12FE167F4816800070151F293B7CB930>21 D<017E1438D83FFE147E16FEA2D8 01FC14FC12000001140116F85BED03F0120315074914E0150F000715C0ED1F805BED3F00 000F147EA2495B4A5A001F495A5D49485A4A5A003F49C7FC143EEB00F8495A48485AEB0F 80D87E3EC8FC13F8EAFFE0138000F8C9FC27257CA429>23 D<1406A6913807FFC04A13E0 91383F80609138FDFFE0903903F87F804948C7FC495A495A495A137F91C8FC5B5B1201A2 5BA512007F137E90383F3FF090381FFFFC90380FC01C90381FFFF890383C7FE001F0C8FC 485A485A485AA248C9FC121EA25AA2127C1278A312F87EA2127E127F7FEA3FE013FC6CB4 FC6C13E06C13F8000113FF6C6C13C0010F13F001037FEB007F140F14031400A4010C5BEB 0E0190380783E0903801FF80D9007EC7FC234B7EB924>I<013FB612E090B712F05A1207 17E0270F807006C7FC391E00600E48140C003813E04813C048141CEAC001120014800103 5BA213071400A25B1578011E137CA3133E133C137C157E13FC5B1201157F1203497FA3D8 01C0131C2C257EA32F>I<160C161C1618A316381630A316701660A316E05EA315015EA3 01F80103130FD803FE9138001F80D8070F153F000E018015C0001C5C001814060038161F 0030160FD8701F010E13070060140C1703D8E03F168000C0EB001C491318EA007E180001 FE13384913305F000116064913700360130E5F000316184901E013384B133017705F0201 495AD801F849485A4CC7FC160E2600FC035B017EEB0078013FEB01E090390FE30F809026 03FFFEC8FC9038003FF00206C9FCA2140E140CA3141C1418A314381430A314701460324B 7EB936>32 D34 D39 D<121C127FEAFF80A5EA7F00121C0909798817>58 D<121C127FEAFF80A213C0A3127F12 1C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>II<150C151E153EA2153C157CA21578 15F8A215F01401A215E01403A215C01407A21580140FA215005CA2141E143EA2143C147C A2147814F8A25C1301A25C1303A2495AA25C130FA291C7FC5BA2131E133EA2133C137CA2 137813F8A25B1201A25B1203A25B1207A25B120FA290C8FC5AA2121E123EA2123C127CA2 127812F8A25A12601F537BBD2A>I<126012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007F C0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED 01FF9238007FC0EE1FF0EE07FCEE01FF9338007F80EF1FC0A2EF7F80933801FF00EE07FC EE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FC EB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0EA7FC048CBFC12FC1270323279AD41>I< EC03FC91381FFF80027F7F903901F807F0903903C001F890390780007C91C7127E010E80 4980011F1580D93FC0130F17C01607A24A14E0A2011EC7FC90C8FCA5EC0FF0ECFFFC9038 03F00E903907C0078F90381F8001D93E0013CF491300484814FF0003ED7FC05B485A120F 48481580A2485AA2007F160090C8FC167E16FE5A485D15015E1503485D15075E4B5AA200 7E4A5A4BC7FC003E147E003F5C6C6C485A390FC007F03907F01FC06CB5C8FCC613FCEB1F E02B3E7DBB2C>64 D<1760177017F01601A21603A21607160FA24C7EA216331673166316 C3A2ED0183A2ED0303150683150C160115181530A21560A215C014011580DA03007FA202 061300140E140C5C021FB5FC5CA20260C7FC5C83495A8349C8FC1306A25BA25B13385B01 F01680487E000716FFB56C013F13FF5EA2383C7DBB3E>I<9339FF8001C0030F13E0037F 9038F80380913A01FF807E07913A07F8000F0FDA1FE0EB079FDA3F80903803BF0002FFC7 6CB4FCD901FC80495A4948157E495A495A4948153E017F163C49C9FC5B1201484816385B 1207485A1830121F4993C7FCA2485AA3127F5BA312FF90CCFCA41703A25F1706A26C160E 170C171C5F6C7E5F001F5E6D4A5A6C6C4A5A16076C6C020EC8FC6C6C143C6C6C5C6CB449 5A90393FE00FC0010FB5C9FC010313FC9038007FC03A3D7CBA3B>67 D<0103B7FC4916E018F8903B0007F80007FE4BEB00FFF03F80020FED1FC0180F4B15E0F0 07F0021F1503A24B15F81801143F19FC5DA2147FA292C8FCA25C18035CA2130119F84A15 07A2130319F04A150FA2010717E0181F4A16C0A2010FEE3F80A24AED7F00187E011F16FE 4D5A4A5D4D5A013F4B5A4D5A4A4A5A057FC7FC017F15FEEE03FC91C7EA0FF049EC7FC0B8 C8FC16FC16C03E397DB845>I<0103B812F05BA290260007F8C7123F4B1407F003E0020F 150118005DA2141FA25D19C0143FA24B1330A2027F1470190092C7126017E05C16014A49 5A160F49B6FCA25F9138FC000F01031407A24A6DC8FCA201075C18034A130660010F1606 93C7FC4A150E180C011F161C18184A1538A2013F5E18F04A4A5AA2017F15074D5A91C812 3F49913803FF80B9FCA295C7FC3C397DB83D>I<0103B5D8F803B512F8495DA290260007 F8C73807F8004B5DA2020F150F615DA2021F151F615DA2023F153F615DA2027F157F96C7 FC92C8FCA24A5D605CA249B7FC60A202FCC7120101031503605CA201071507605CA2010F 150F605CA2011F151F605CA2013F153F605CA2017F157F95C8FC91C8FC496C4A7EB690B6 FCA345397DB845>72 D<0103B500F8903807FFFC5BA290260007F8C813804BEDFC0019F0 020F4B5AF003804B4AC7FC180E021F1538604B5CEF0380023F4AC8FC170E4B133C177002 7F5C4C5ADB0007C9FC160E4A5B167E4A13FE4B7E01015B92380E7F80ECFC1CED383F0103 01E07FECFDC04A486C7EECFF00D907FC6D7E5C4A130783130F707E5C1601011F81A24A6D 7EA2013F6F7EA24A143F84137F717E91C8123F496C81B60107B512C0A26146397DB847> 75 D<0103B6FC5B5E90260007FCC8FC5D5D140FA25DA2141FA25DA2143FA25DA2147FA2 92C9FCA25CA25CA21301A25CA21303A25CA2130718404A15C0A2010F150118804A1403A2 011F16005F4A1406170E013F151E171C4A143C177C017F5D160391C7120F49EC7FF0B8FC A25F32397DB839>I<902603FFF893383FFF80496081D900079438FF80000206DC01BFC7 FCA2020E4C5A1A7E020C1606190CDA1C7E16FE4F5A02181630A20238166162023016C1F0 0181DA703F158395380303F002601506A202E0ED0C076202C01518183001016D6C140F06 605B028015C0A20103923801801FDD03005B140092380FC00649173F4D91C8FC01065DA2 010E4B5B4D137E130C6F6C5A011C17FEDCE1805B011802E3C7FCA2013802E6130104EC5C 1330ED03F8017016034C5C01F05CD807FC4C7EB500E0D9C007B512F01680150151397CB8 51>I<902603FFF891381FFFF8496D5CA2D90007030113006FEC007C02061678DA0EFF15 7081020C6D1460A2DA1C3F15E0705CEC181F82023815016F6C5C1430150702706D130303 0392C7FC02607FA2DAE0015C701306ECC0008201016E130EEF800C5C163F0103EDC01C04 1F131891C713E0160F49EDF03818300106140717F8010E02031370EFFC60130CEE01FE01 1C16E004005B011815FF177F1338600130153FA20170151F95C8FC01F081EA07FCB512E0 1706A245397DB843>I<4BB4FC031F13F09238FE01FC913903F0007EDA07C0EB1F80DA1F 80EB0FC0023EC7EA07E002FCEC03F0495A4948EC01F8495A4948EC00FC495A013F16FE49 C9FC13FE187F485A12035B12075B120F4916FF121FA2485AA34848ED01FEA448C9EA03FC A3EF07F8A218F0170F18E0171F18C0EF3F807EEF7F0017FEDA07C05B6C90391FF001F890 3980383803001F496C485A9139E00C0FE0260FC0C0EB1F80D807E1D90E3FC7FC0280137E D803F1EB07F8D801F95C3A007FC00FC0903A3FE07F0003903807FFFE0100018F5BDA000F 1306170E171E705A177CEEC1F816FF5FA25F5F6F5B6F48C7FCED00F8384B7CBA42>81 D<0103B612F849EDFF8018E0903B0007F8001FF84BEB03FCEF00FE020F157FA24BEC3F80 A2021F16C0A25DA2143FF07F805DA2027FEDFF006092C7485A4D5A4A4A5A4D5A4AEC1F80 057FC7FC0101EC07F891B612E094C8FC9139FC000FC00103EC03F0707E4A6D7E83130717 7E5C177F010F5D5F5CA2011F1401A25CA2133F16034A4A1360A2017F17E019C091C71401 496C01011480B61503933900FE0700EF7E0ECAEA1FFCEF07F03B3B7DB83F>I<0003B812 FEA25A903AF8003FC00101C0913880007E4848163C90C7007F141C121E001C92C7FCA248 5CA200305C007017180060130112E0485CA21403C716005DA21407A25DA2140FA25DA214 1FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CEB0FFC003FB6FC 5AA237397EB831>84 D<277FFFFC01B500F890B51280B5FC60000390C7D807FCC7380FF8 0001FC4BEC03E000016204035E98C7FC621A0604075DA2040F5DA2041B5D6216336D0273 5D1663000003C34A5A83DB01834AC8FC04815CDB0301140603075D1506030C5DA203185D 1970033015606115606D01E04A5A15C090267F01804AC9FC17FEDA030014060400130E02 06150C020E5D140C4A5DA24A5D18E04A5D715A5C4A92CAFCA26DC85AA2013E157C177813 3C1770133801301560513B7CB84E>87 D<1578EC01FEEC07C6EC0F861507EC1E03143E14 7C1507ECF806A2EB01F00103130EECE00C1307A2ECC01C010F1318153890381F80301570 156090383F00E015C01401017F1380EB7E03EC07001406EBFE0E495A5C14300001137049 5AEBF9C0EBFB8001FFC7FC5B5B485AA25BA4485A120F121DEA39F0127100E1140C008014 3C0000147015E090387801C0EC078090383C1E00EB1FF8EB07E0203C7FBA23>96 D<147E903803FF8090390FC1C38090391F00EFC0017E137F49133F485A4848EB1F801207 5B000F143F48481400A2485A5D007F147E90C7FCA215FE485C5AA214015D48150CA21403 EDF01C16181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0F9C0 3A03FF007F80D800FCEB1F0026267DA42C>I<133FEA1FFFA3C67E137EA313FE5BA31201 5BA312035BA31207EBE0FCEBE3FF9038E707C0390FFE03E09038F801F001F013F8EBE000 485A15FC5BA2123F90C7FCA214015A127EA2140312FE4814F8A2140715F05AEC0FE0A215 C0EC1F80143F00781400007C137E5C383C01F86C485A380F07C06CB4C7FCEA01FC1E3B7C B924>II<163FED1FFFA3ED007F167EA216FEA216FCA21501A216F8A21503A216F0A215 07A2027E13E0903803FF8790380FC1CF90381F00EF017EEB7FC049133F485A4848131F00 0715805B000F143F485A1600485A5D127F90C7127EA215FE5A485CA21401A248ECF80CA2 1403161CEDF0181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0 F9C03A03FF007F80D800FCEB1F00283B7DB92B>II<16F8ED03FEED0F8792381F 0F80ED3E3F167F157CA215FC1700161C4A48C7FCA414035DA414075DA20107B512F0A390 26000FE0C7FC5DA4141F5DA4143F92C8FCA45C147EA514FE5CA413015CA4495AA45C1307 A25C121E123F387F8F80A200FF90C9FC131E12FEEA7C3CEA7878EA1FF0EA07C0294C7CBA 29>I107 D109 DII<90390F8003F0 90391FE00FFC903939F03C1F903A70F8700F80903AE0FDE007C09038C0FF80030013E000 01491303018015F05CEA038113015CA2D800031407A25CA20107140FA24A14E0A2010F14 1F17C05CEE3F80131FEE7F004A137E16FE013F5C6E485A4B5A6E485A90397F700F80DA38 3FC7FC90387E1FFCEC07E001FEC9FCA25BA21201A25BA21203A25B1207B512C0A32C3583 A42A>I<02FC13C0903803FF0190380F838390383F01C790397E00EF8049137F485A4848 133F000715005B485A001F5C157E485AA2007F14FE90C75AA3481301485CA31403485CA3 14075D140F127C141F007E495A003E137F381F01EF380F839F3903FF1F80EA00FC130014 3F92C7FCA35C147EA314FE5C130190387FFFF0A322357DA425>I<3903E001F83907F807 FE390E3C1E07391C3E381F3A183F703F800038EBE07F0030EBC0FF00705B00601500EC00 7E153CD8E07F90C7FCEAC07EA2120013FE5BA312015BA312035BA312075BA3120F5BA312 1F5B0007C9FC21267EA425>I<14FF010313C090380F80F090383E00380178131C153C49 13FC0001130113E0A33903F000F06D13007F3801FFE014FC14FF6C14806D13C0011F13E0 13039038003FF014071403001E1301127FA24814E0A348EB03C012F800E0EB07800070EB 0F006C133E001E13F83807FFE0000190C7FC1E267CA427>II<13F8D803FE1438D8070F147C000E6D13FC121C1218 003814011230D8701F5C12601503EAE03F00C001005B5BD8007E1307A201FE5C5B150F12 01495CA2151F120349EC80C0A2153F1681EE0180A2ED7F0303FF130012014A5B3A00F807 9F0E90397C0E0F1C90393FFC07F8903907F001F02A267EA430>I<01F8EB03C0D803FEEB 07E0D8070F130F000E018013F0121C12180038140700301403D8701F130112601500D8E0 3F14E000C090C7FC5BEA007E16C013FE5B1501000115805B150316001203495B1506150E 150C151C151815385D00015C6D485A6C6C485AD97E0FC7FCEB1FFEEB07F024267EA428> I<01F816F0D803FE9138E001F8D8070F903801F003000ED9800314FC121C121800380207 13010030EDE000D8701F167C1260030F143CD8E03F163800C001005B5BD8007E131F1830 01FE5C5B033F1470000117604991C7FCA218E000034A14C049137E17011880170318005F 03FE1306170E000101015C01F801BF5B3B00FC039F8070903A7E0F0FC0E0903A1FFC03FF C0902703F0007FC7FC36267EA43B>I<903907E001F090391FF807FC9039783E0E0F9039 E01F1C1FD801C09038383F803A03800FF07F0100EBE0FF5A000E4A1300000C157E021F13 3C001C4AC7FC1218A2C7123FA292C8FCA25CA2147EA214FEA24A130CA20101141C001E15 18003F5BD87F81143801835C00FF1560010714E03AFE0E7C01C0D87C1C495A2778383E0F C7FC391FF00FFC3907C003F029267EA42F>I<13F8D803FE1470D8070F14F8000EEB8001 121C121800381403003015F0EA701F1260013F130700E0010013E012C05BD8007E130F16 C013FE5B151F000115805BA2153F000315005BA25D157EA315FE5D1401000113033800F8 0790387C1FF8EB3FF9EB0FE1EB00035DA2000E1307D83F805B007F495AA24A5A92C7FCEB 003E007C5B00705B6C485A381E07C06CB4C8FCEA01FC25367EA429>I E %EndDVIPSBitmapFont /Fv 134[42 1[60 1[46 28 32 37 1[46 42 46 69 23 46 1[23 46 42 1[37 46 37 46 42 12[55 1[60 1[51 1[60 1[55 6[55 3[60 7[42 42 42 42 42 42 42 42 42 42 1[21 46[{ TeXBase1Encoding ReEncodeFont}38 83.022 /Times-Bold rf %DVIPSBitmapFont: Fw cmtt9 9 25 /Fw 25 122 df<007FB512F8B612FCA46C14F81E067C9927>45 D<121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A0A728927>I64 D<3A7FFE07FFE0B54813F0A36C486C13E03A07E0007E00AF90B512FEA59038E0007EB03A 7FFE07FFE0B54813F0A36C486C13E0242E7FAD27>72 D77 D<3803FFC0000F13F04813FC4813FF811380EC1FC0381F000F0004 80C71207A2EB0FFF137F0003B5FC120F5A383FFC07EA7FC0130012FE5AA46C130F007F13 1FEBC0FF6CB612806C15C07E000313F1C69038807F8022207C9F27>97 DI III<153F90391FC0FF80D97FF313C048B612 E05A4814EF390FF07F873A1FC01FC3C0EDC000EB800F48486C7EA66C6C485AEBC01FA239 0FF07F8090B5C7FC5C485BEB7FF0EB1FC090C9FCA27F6CB5FC15E015F84814FE4880EB80 01007EC7EA3F80007C140F00FC15C0481407A46C140F007C1580007F143F6C6CEB7F0090 38F807FF6CB55A000714F86C5CC614C0D90FFCC7FC23337EA027>103 DI<130F497E497EA46D 5A6DC7FC90C8FCA7383FFF80487FA37EEA000FB3A4007FB512F0B6FC15F815F07E1D2F7B AE27>I107 D<387FFF80B57EA37EEA000FB3B2007FB512F8B612FCA36C14F81E2E7CAD27>I<397F07 C01F3AFF9FF07FC09039FFF9FFE091B57E7E3A0FFC7FF1F89038F03FC001E0138001C013 00A3EB803EB03A7FF0FFC3FF486C01E3138001F913E701F813E36C4801C313002920819F 27>I<387FE07F39FFF1FFC001F713F090B5FC6C80000313C1EC01FCEBFE005B5BA25BB0 3A7FFF83FFE0B500C713F0A36C018313E024207F9F27>II<387FE0FFD8FFF313C090B512F0816C800003EB81FE49C6 7E49EB3F8049131F16C049130FA216E01507A6150F16C07F151F6DEB3F80157F6DEBFF00 9038FF83FEECFFFC5D5D01F313C0D9F0FEC7FC91C8FCAC387FFF80B57EA36C5B23317F9F 27>I<397FFC03FC39FFFE0FFF023F13804A13C0007F90B5FC39007FFE1F14F89138F00F 809138E002004AC7FC5CA291C8FCA2137EAD007FB57EB67EA36C5C22207E9F27>114 D<9038FFF3800007EBFFC0121F5A5AEB803F38FC000F5AA2EC07806C90C7FCEA7F8013FC 383FFFF06C13FC000713FF00011480D8000F13C09038003FE014070078EB03F000FC1301 A27E14036CEB07E0EBE01F90B512C01580150000FB13FC38707FF01C207B9F27>I<133C 137EA8007FB512F0B612F8A36C14F0D8007EC7FCAE1518157EA415FE6D13FC1483ECFFF8 6D13F06D13E0010313C0010013001F297EA827>I<397FE01FF8486C487EA3007F131F00 031300B21401A21403EBFC0F6CB612E016F07EEB3FFE90390FF87FE024207F9F27>I<3A 7FFE07FFE000FF15F06D5A497E007F15E03A0F80001F00A36D5B0007143EA414F0EBC1F8 3903E3FC7CA4EBE79EA200011478A301F713F8A2EBFF0F6C5CA3EBFE0790387C03E02420 7F9F27>119 D<3A7FFC0FFF80486C4813C0A36C486C13803A07E000F800000313015D13 F00001130301F85B1200A26D485A137CA290387E0F80133EA2011F90C7FC5CA2130F149E 14BE130714FC1303A25C1301A25CA213035CA213075C1208EA3E0F007F5B131FD87E7FC8 FCEA7FFE6C5A5B6C5AEA07C022317E9F27>121 D E %EndDVIPSBitmapFont /Fx 75[25 11[25 17[37 27[33 37 37 54 37 37 21 29 25 37 37 37 37 58 21 37 21 21 37 37 25 33 37 33 37 33 3[25 1[25 46 54 1[71 54 54 46 42 50 1[42 1[54 66 46 54 29 25 54 54 42 46 54 50 50 54 6[21 37 37 37 37 37 37 37 37 37 37 1[19 25 19 2[25 25 25 58 20[21 25 13[42 2[{ TeXBase1Encoding ReEncodeFont}75 74.7198 /Times-Roman rf %DVIPSBitmapFont: Fy cmr6 6 6 /Fy 6 56 df<1438B2B712FEA3C70038C7FCB227277C9F2F>43 D<13E01201120712FF12 F91201B3A7487EB512C0A212217AA01E>49 D I<00101330381E01F0381FFFE014C01480EBFE00EA1BF00018C7FCA513FE381BFF80381F 03C0381C01E0381800F014F8C71278A2147CA21230127812F8A214784813F8006013F038 7001E01238381E07803807FF00EA01F816227CA01E>53 DI<1230123C003FB5FCA24813FE14FC3860001C143814 704813E014C0EA0001EB0380EB07001306130E5BA25BA21378A35BA41201A76C5A18237C A11E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmr8 8 2 /Fz 2 51 df<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23 >49 DI E %EndDVIPSBitmapFont /FA 87[33 50[55 33 39 44 2[50 55 83 28 55 1[28 55 1[33 44 55 44 55 50 12[66 1[72 1[61 1[72 94 3[39 78 3[72 1[66 11[50 50 50 50 50 50 49[{TeXBase1Encoding ReEncodeFont}33 99.6264 /Times-Bold rf /FB 75[28 11[28 19[37 37 10[28 13[37 42 42 60 42 42 23 32 28 42 42 42 42 65 23 42 23 23 42 42 28 37 42 37 42 37 3[28 1[28 51 60 1[78 60 60 51 46 55 1[46 60 60 74 51 60 32 28 60 60 46 51 60 55 55 60 6[23 42 42 42 42 42 42 42 42 42 42 1[21 28 21 2[28 28 28 35[46 46 2[{TeXBase1Encoding ReEncodeFont}76 83.022 /Times-Roman rf /FC 134[60 1[86 1[66 40 47 53 1[66 60 66 100 33 2[33 66 60 40 53 66 2[60 13[66 37[40 42[66 2[{TeXBase1Encoding ReEncodeFont}21 119.552 /Times-Bold rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 1067 756 a FC(Stationary)30 b(solutions)f(f)m(or)h(a)f(model)h (of)1225 905 y(amor)o(phous)g(thin-\002lm)h(gr)n(o)o(wth)1704 1146 y FB(April)20 b(18,)f(2002)664 1689 y FA(Dirk)25 b(Bl)990 1688 y(\250)982 1689 y(omk)o(er)1257 1653 y Fz(1)1322 1689 y FA(and)h(Martin)f(Hair)n(er)2123 1653 y Fz(2)664 1831 y Fy(1)717 1863 y Fx(Institut)19 b(f)6 b(\250)-31 b(ur)19 b(Mathematik,)g(R)l(WTH)f(Aachen,)h(German)o(y)717 1955 y(Email:)k Fw(bloemker@instmath.rwth-aachen.)q(de)664 2014 y Fy(2)717 2046 y Fx(D)t(\264)-29 b(epartement)20 b(de)f(Physique)h(Th)t(\264)-29 b(eorique,)19 b(Uni)n(v)o(ersity)g(of)g (Gene)n(v)n(a,)h(Switzerland)717 2137 y(Email:)j Fw (Martin.Hairer@physics.unige.ch)664 2344 y Fv(Abstract)664 2465 y Fx(W)-6 b(e)23 b(consider)h(a)f(class)g(of)h(stochastic)f (partial)g(dif)n(ferential)g(equations)i(arising)e(as)g(a)h(model)f (for)664 2556 y(amorphous)17 b(thin)e(\002lm)g(gro)n(wth.)22 b(Using)16 b(a)f(spectral)g(Galerkin)h(method,)g(we)f(v)o(erify)g(the)h (e)o(xistence)664 2648 y(of)23 b(stationary)g(mild)g(solutions,)h (although)g(the)f(speci\002c)g(nature)h(of)f(the)g(nonlinearity)g(pre)n (v)o(ents)664 2739 y(us)16 b(from)g(sho)n(wing)h(the)f(uniqueness)i(of) e(the)g(solutions)h(as)f(well)f(as)h(their)g(boundedness)i(\(in)e (time\).)515 2944 y FA(1)99 b(Intr)n(oduction)515 3119 y FB(This)25 b(paper)f(sho)n(ws)i(the)f(e)o(xistence)f(of)h(a)h (stationary)e(solution)g(for)h(a)h(stochastic)f(partial)g(dif)n(fer)n (-)515 3218 y(ential)20 b(equation)e(\(SPDE\),)i(where)f(the)h (solutions)g(may)f(not)h(form)f(a)h(Mark)o(o)o(v)f(semigroup)f(due)h (to)515 3318 y(the)h(lack)g(of)g(uniqueness.)j(W)-7 b(e)22 b(consider)d(the)h(f)o(amily)g(of)g(equations)998 3500 y Fu(@)1042 3512 y Ft(t)1072 3500 y Fu(u)i Fs(=)h Fr(\000)p Fu(@)1344 3466 y Fq(4)1339 3521 y Ft(x)1381 3500 y Fu(u)18 b Fs(+)g Fu(\027)5 b(@)1625 3466 y Fq(2)1620 3521 y Ft(x)1662 3500 y Fu(u)18 b Fr(\000)g Fu(@)1860 3466 y Fq(2)1855 3521 y Ft(x)1897 3500 y FB(\()p Fu(@)1969 3521 y Ft(x)2010 3500 y Fu(u)p FB(\))2086 3466 y Fq(2)2141 3500 y Fs(+)g Fu(\030)t FB(\()p Fu(x;)c(t)p FB(\))23 b(,)166 b Fu(\027)28 b Fr(2)23 b Fv(R)g FB(,)325 b(\(1.1\))515 3683 y(for)20 b(a)i(real-v)n(alued)d(scalar)i Fu(u)p FB(\()p Fu(t;)14 b(x)p FB(\))21 b(with)g Fu(t)k(>)f Fs(0)e FB(and)e Fu(x)25 b Fr(2)g FB([)p Fs(0)p Fu(;)14 b(L)p FB(],)20 b(subject)h(to)g (suitable)g(boundary)515 3783 y(conditions)28 b(\()p Fp(e)o(.g)o(.)h FB(periodic)f(or)i(Neumann)e(type\).)53 b(The)29 b(symbol)g Fu(\030)35 b FB(denotes)29 b(a)h(noise)g(process) 515 3882 y(which)19 b(should)g(be)g(thought)f(of)i(as)g(the)g (generalized)e(deri)n(v)n(ati)n(v)o(e)f(of)j(some)f(W)m(iener)h (process)f(to)h(be)515 3982 y(speci\002ed)g(later)g(on.)639 4082 y(Equations)27 b(of)g(the)h(type)f(\(1.1\))g(arise)h(in)g(the)g (gro)n(wth)e(of)i(thin)f(\002lms)i(\(see)f Fp(e)o(.g)o(.)f FB([RML)3221 4052 y Fq(+)3275 4082 y FB(00)o(,)515 4181 y(SP94)o(,)i(BS95]\).)51 b(The)29 b(function)e Fu(u)p FB(\()p Fu(t;)14 b Fr(\001)g FB(\))29 b(describes)f(the)h(graph)f(of)h (a)g(surf)o(ace)f(at)i(time)f Fu(t)40 b(>)f Fs(0)p FB(.)515 4281 y(Usually)17 b(these)h(equations)e(are)h(equipped)e(with)j(a)g (lot)g(of)f(physical)f(parameters,)h(which)f(we)i(set)g(to)515 4381 y Fs(1)i FB(for)f(simplicity)-5 b(.)25 b(In)20 b(some)g(models)f (an)h(additional)f(additi)n(v)o(e)g(nonlinear)f(term)i(\()p Fu(@)2930 4393 y Ft(x)2972 4381 y Fu(u)p FB(\))3048 4350 y Fq(2)3105 4381 y FB(appears.)515 4480 y(W)-7 b(e)20 b(can)e(treat)h(that)g(case)h(too,)e(b)n(ut)h(the)g(analysis)g(is)h (more)e(in)m(v)n(olv)o(ed)f(without)h(contrib)n(uting)f(much)515 4580 y(to)j(the)g(general)f(understanding)f(of)h(the)i(situation,)e(so) i(we)f(do)g(not)g(present)f(it.)639 4679 y(It)g(is)g(easy)g(to)g(v)o (erify)e(that)h(there)g(e)o(xists)h(a)g(v)n(alue)f Fu(\027)2079 4691 y Ft(c)2136 4679 y Fu(<)k Fs(0)d FB(such)f(that)h(if)f Fu(\027)29 b(<)22 b(\027)2866 4691 y Ft(c)2900 4679 y FB(,)d(the)g(equations)515 4779 y(under)27 b(consideration)g(present)h (a)i(linear)e(instability)-5 b(.)51 b(W)-7 b(e)30 b(will)f(therefore)f (distinguish)f(in)j(the)515 4879 y(sequel)g(the)g Fp(stable)g(case)g FB(\()p Fu(\027)47 b(>)41 b(\027)1560 4891 y Ft(c)1594 4879 y FB(\))31 b(from)e(the)h Fp(unstable)f(case)h FB(\()p Fu(\027)47 b Fr(\024)41 b Fu(\027)2730 4891 y Ft(c)2764 4879 y FB(\).)55 b(This)31 b(instability)515 4978 y(is)25 b(responsible)e(for)g(the)h(formation)f(of)g(hills,)j(which)e(is)h (frequently)d(seen)i(in)g(e)o(xperiments)e(\(see)p eop %%Page: 2 2 2 1 bop 517 232 a FB(I)t Fo(N)t(T)t(R)q(O)t(D)t(U)t(C)t(T)t(I)t(O)t(N) 2271 b Fv(2)515 523 y Fp(e)o(.g)o(.)30 b FB([MMS99)n(])i(and)f(the)g (references)f(therein\).)57 b(On)32 b(the)f(other)g(hand)f(we)i(ha)n(v) o(e)e(a)i(quadratic)515 623 y(nonlinearity)24 b(that)i(compensates)f (this)i(instability)-5 b(.)42 b(Unfortunately)-5 b(,)25 b(this)i(nonlinearity)d(mak)o(es)515 722 y(it)g(dif)n(\002cult)f(to)g (deri)n(v)o(e)f(uniform)g(bounds)g(on)h(the)g(solution.)34 b(Moreo)o(v)o(er)m(,)21 b(it)j(is)h(an)e(open)f(problem)515 822 y(ho)n(w)15 b(to)h(establish)f(bounds)g(in)g(case)i(of)e(a)h(tw)o (o-dimensional)d(square,)j(which)f(is)i(ob)o(viously)c(a)j(more)515 922 y(realistic)25 b(model)f(than)g(the)h(one-dimensional)d(case)j(we)g (treat)g(in)g(this)g(paper)-5 b(.)39 b(This)25 b(scenario)f(is)515 1021 y(similar)18 b(to)h(the)f(K)o(uramoto-Si)n(v)n(ashinsk)o(y)c (equation,)j(where)h(there)f(are)i(no)f(results)g(for)g(truly)g(tw)o (o-)515 1121 y(dimensional)g(domains.)639 1220 y(One)i(v)o(ery)f (helpful)g(tool)h(in)h(the)f(analysis)g(is)h(the)f(conserv)n(ation)e (of)i(mass:)515 1432 y Fv(Remark)g(1.1)40 b FB(The)24 b(quantity)f Fu(M)9 b FB(\()p Fu(u)p FB(\))29 b Fs(:=)1763 1366 y Fn(R)1818 1386 y Ft(L)1802 1462 y Fq(0)1882 1432 y Fu(u)p FB(\()p Fu(x)p FB(\))14 b Fu(dx)24 b FB(decouples)f(from)g (the)h(rest)h(of)e(the)i(equa-)515 1532 y(tion.)f(Therefore,)17 b(we)j(can)f(assume)g(without)f(loss)i(of)f(generality)f(that)h Fu(M)9 b Fs(\()p Fu(u)p FB(\()p Fu(t)p FB(\))p Fs(\))22 b Fr(\021)h Fs(0)c FB(for)f(a)i(solu-)515 1632 y(tion)g(of)g(\(1.1\).)k (The)d(v)n(arious)e(Sobole)n(v)g(spaces)i(appearing)d(in)j(the)f (sequel)g(should)g(be)g(thought)f(of)515 1731 y(as)i(the)f(orthogonal)d (complement)i(to)h(the)g(constant)g(function)e Fs(1)i FB(of)g(the)g(usual)g(spaces.)639 1897 y(The)27 b(local)f(e)o(xistence) g(of)h(unique)e(solutions)h(to)h(equations)e(of)i(the)g(type)f(\(1.1\)) f(is)j(standard)515 1997 y(for)19 b(suf)n(\002ciently)f(smooth)h (initial)g(conditions.)k(But)d(the)g(e)o(xistence)f(of)g(global)f (solutions)h(is)i(much)515 2097 y(more)27 b(complicated,)h(and)f(w)o (as)h(sho)n(wn)g(in)f([BG02)o(])i(or)e([BGR02)o(])h(using)g(a)g (spectral)g(Galerkin)515 2196 y(method.)c(Ne)n(v)o(ertheless)c(the)g (question)g(of)g(uniqueness)f(of)h(global)g(solutions)g(is)h(still)h (open,)d Fp(e)o(ven)515 2296 y(in)h(the)h(deterministic)g(case)p FB(,)f(as)i(it)f(is)g(out)g(of)f(reach)g(to)g(sho)n(w)h(enough)d(re)o (gularity)-5 b(.)24 b(Therefore)19 b(the)515 2395 y(equation)24 b(does)h(not)g(necessarily)f(generate)h(a)g(Mark)o(o)o(v)f(semigroup,)h (and)f(the)i(standard)e(theory)515 2495 y(for)19 b(in)m(v)n(ariant)g (measures)g(\(cf.)25 b([DPZ96)o(]\))20 b(does)g(not)f(apply)-5 b(.)639 2595 y(W)e(e)33 b(sho)n(w)f(in)g(this)g(paper)f(that)h(there)f (e)o(xists)h(ne)n(v)o(ertheless)f(an)g(\223in)m(v)n(ariant)f (measure\224)h(for)515 2694 y(\(1.1\).)j(T)-7 b(o)23 b(be)h(more)f(precise,)h(we)g(construct)e(a)i(stationary)f(solution)g Fr(f)p Fu(u)p FB(\()p Fu(t)p FB(\))p Fu(;)36 b(t)30 b Fr(2)f Fv(R)p Fr(g)24 b FB(such)f(that)515 2794 y(the)d(distrib)n (ution)f Fr(P)1104 2764 y Ft(t)1156 2794 y Fs(:=)j Fr(L)p Fs(\()q Fu(u)p FB(\()p Fu(t)p FB(\))p Fs(\))1521 2764 y Fm(1)1575 2794 y FB(of)e Fu(u)p FB(\()p Fu(t)p FB(\))g(is)h(constant) e(in)h(time.)639 2894 y(Our)28 b(concept)g(of)g(solutions)g(is)h(a)g (martingale)e(solution)g(of)h(the)h(corresponding)c(mild)j(for)n(-)515 2993 y(mulation.)i(Hence,)22 b(we)h(allo)n(w)f(a)h(change)e(of)h(the)g (underlying)d(probability)i(space)h(and)g(consider)515 3093 y(solutions)29 b(not)g(of)g(the)g(SPDE,)h(b)n(ut)f(of)g(the)h (corresponding)25 b(v)n(ariation)j(of)h(constants)g(formula.)515 3192 y(Since)g(we)g(use)g(spectral)g(Galerkin)f(methods,)h(our)f (approach)f(is)j(similar)f(to)g(pre)n(vious)e(results)515 3292 y(\(see)f([FG95)n(])g(or)g([CG94)n(]\))g(for)f(the)h(stochastic)f (Na)n(vier)n(-Stok)o(es)g(equation.)40 b(One)25 b(of)h(the)f(major)515 3392 y(dif)n(ferences)19 b(is)j(that)f(we)h(are)f(not)g(able)g(to)g (use)g(the)g(theory)f(of)h(Mark)o(o)o(v)f(semigroups.)26 b(Moreo)o(v)o(er)m(,)515 3491 y(we)21 b(were)f(not)g(able)h(to)f(get)h (an)o(y)f(uniform)e(bound)h(\(in)h Fu(t)p FB(\))h(on)f(the)h(distrib)n (ution)e(of)h(solutions)g(when)515 3591 y(the)i(dri)n(ving)f(force)h Fu(\030)28 b FB(is)23 b(a)g(space-time)f(white)h(noise.)32 b(W)-7 b(e)24 b(are)f(able)f(to)h(establish)g(such)f(a)h(bound)515 3691 y(only)18 b(for)h(stationary)f(solutions.)24 b(Therefore,)17 b(we)i(will)h(construct)e(the)h(stationary)f(process)h Fu(u)g FB(as)h(a)515 3790 y(limit)g(of)g(the)h(unique)d(in)m(v)n (ariant)h(stationary)g(solutions)h(to)g(the)g(Galerkin)f (approximations.)639 3890 y(The)h(paper)e(is)j(or)o(ganized)16 b(as)21 b(follo)n(ws.)j(In)19 b(Section)g(2,)h(we)g(introduce)e(the)h (spectral)h(Galerkin)515 3989 y(approximation)g(and)k(present)f(our)g (main)g(result.)36 b(The)24 b(ne)o(xt)f(Section)g(3)h(presents)g (compactness)515 4089 y(results)30 b(and)f(the)h(proof)f(of)h(the)g (main)f(result.)54 b(In)30 b(Sections)g(4)g(and)g(5,)i(we)e(will)h(gi)n (v)o(e)e(a-priori)515 4189 y(estimates)20 b(for)g(the)g(solutions.)25 b(The)19 b(\002nal)i(Section)f(6)g(contains)f(technical)h(results.)515 4363 y Fv(Ackno)o(wledgements)515 4493 y Fx(D.)26 b(B.)f(wishes)i(to)f (thank)i(Jean-Pierre)e(Eckmanns)h(mathematical)g(physics)g(group)h(in)e (Gene)n(v)n(a)i(for)e(their)515 4592 y(w)o(arm)f(hospitality)-5 b(.)43 b(M.)25 b(H.)g(thanks)h(Jean-Pierre)f(Eckmann,)j(Guillaume)d(v)n (an)h(Baalen,)h(and)f(Emmanuel)515 4692 y(Zabe)o(y)i(for)f(se)n(v)o (eral)h(discussions.)49 b(The)28 b(latter)e(authors)i(w)o(ork)g(w)o(as) g(partially)f(supported)i(by)f(the)f(F)o(onds)515 4791 y(National)19 b(Suisse.)p 515 4845 1146 4 v 605 4900 a Fl(1)634 4924 y Fo(The)e(symbol)g Fk(L)p Fo(\()p Fj(X)5 b Fo(\))17 b(will)h(al)o(w)o(ays)h(denote)f(the)g(la)o(w)g(of)f(the)g (random)h(v)n(ariable)i Fj(X)5 b Fo(.)p eop %%Page: 3 3 3 2 bop 517 232 a FB(N)t Fo(O)q(T)n(A)m(T)t(I)t(O)t(N)24 b(A)t(N)t(D)h(F)t(O)t(R)t(M)t(U)t(L)t(A)m(T)t(I)t(O)t(N)g(O)t(F)g(T)t (H)t(E)f(M)t(A)t(I)t(N)h(R)t(E)t(S)t(U)t(L)n(T)953 b Fv(3)515 523 y FA(2)99 b(Notation)25 b(and)g(f)n(ormulation)g(of)g(the) h(main)e(r)n(esult)515 697 y FB(De\002ne)19 b(the)h(space)f(L)1121 662 y Fq(2)1182 697 y Fs(:=)j Fr(f)p Fu(f)32 b Fr(2)23 b FB(L)1525 667 y Fq(2)1562 697 y FB(\([)p Fs(0)p Fu(;)14 b(L)p FB(]\))20 b Fs(:)1876 631 y Fn(R)1932 651 y Ft(L)1916 727 y Fq(0)1995 697 y Fu(f)9 b FB(\()p Fu(x)p FB(\))p Fu(dx)23 b Fs(=)g(0)p Fr(g)c FB(with)h(standard)e(L)2961 667 y Fq(2)2998 697 y FB(-norm)g Fr(k)d(\001)h(k)p FB(.)515 797 y(W)-7 b(e)21 b(de\002ne)f Fu(A)h FB(as)g(the)f(linear)g (self-adjoint)f(operator)f(in)i(L)2194 767 y Fq(2)2252 797 y FB(formally)e(gi)n(v)o(en)h(by)1635 961 y Fu(A)k Fs(:=)g Fr(\000)p Fu(@)1945 927 y Fq(4)1940 981 y Ft(x)2000 961 y Fs(+)18 b Fu(\027)5 b(@)2178 927 y Fq(2)2173 981 y Ft(x)2238 961 y FB(,)515 1125 y(where)15 b(the)g(domain)g(of)g (de\002nition)g Fu(D)r FB(\()p Fu(A)p FB(\))g(consists)i(of)e(all)h (functions)e Fu(f)32 b Fr(2)23 b Fu(H)2757 1095 y Fq(4)2794 1125 y FB(\([)p Fs(0)p Fu(;)14 b(L)p FB(]\))g(satisfying)515 1224 y Fu(M)9 b FB(\()p Fu(f)g FB(\))22 b Fs(=)i(0)d FB(and)f(boundary)e(conditions)h(gi)n(v)o(en)h(by)g(the)h(equation.)k (W)-7 b(e)22 b(will)g(write)f Fu(H)3041 1194 y Fq(4)3102 1224 y Fs(=)i Fu(D)r FB(\()p Fu(A)p FB(\))515 1324 y(for)16 b(short.)23 b(Moreo)o(v)o(er)m(,)14 b(it)k(is)f(well-kno)n(wn)e(that)i Fu(A)g FB(generates)f(an)g(analytic)g(semigroup)f Fr(f)p Fu(e)3124 1294 y Ft(tA)3203 1324 y Fr(g)3245 1336 y Ft(t)p Fi(\025)p Fq(0)3358 1324 y FB(,)515 1424 y(and)20 b(we)g(use)h(the)f (fractional)f(po)n(wers)h(of)g Fu(A)h FB(to)g(de\002ne)f(the)g (standard)f(fractional)g(Sobole)n(v)g(spaces)515 1523 y Fu(H)591 1493 y Ft(s)647 1523 y FB(for)g Fu(s)k Fr(\025)g Fs(0)d FB(with)h(dual)e(spaces)i Fu(H)1622 1493 y Fi(\000)p Ft(s)1709 1523 y FB(.)639 1623 y(In)e(the)h(sequel)f(we)h(will)g(need)f (spaces)h(of)f(functions)f(on)h(the)h(whole)f(real)g(line)h(with)f(v)n (alues)g(in)515 1722 y(Sobole)n(v)14 b(spaces.)24 b(W)-7 b(e)17 b(recall)f(the)f(de\002nition)g(of)h(the)f(space)h Fr(C)5 b FB(\()p Fv(R)o Fu(;)14 b(H)2494 1692 y Ft(s)2529 1722 y FB(\),)j(which)e(is)i(gi)n(v)o(en)d(as)j(the)f(set)515 1822 y(of)21 b(all)i(functions)d(such)i(that)g(for)f(an)o(y)g Fu(a)27 b(<)e(b)p FB(,)e(the)f(restriction)f(to)h([)p Fu(a;)14 b(b)p FB(])21 b(is)i(in)f Fr(C)5 b FB(\([)p Fu(a;)14 b(b)p FB(])p Fu(;)g(H)3158 1792 y Ft(s)3191 1822 y FB(\).)30 b(W)-7 b(e)515 1922 y(say)20 b(that)h Fu(f)833 1934 y Ft(n)898 1922 y FB(con)m(v)o(er)o(ges)d(to)i Fu(f)30 b FB(in)20 b Fr(C)5 b FB(\()p Fv(R)o Fu(;)14 b(H)1740 1892 y Ft(s)1775 1922 y FB(\))21 b(if)f(and)g(only)g(if)g Fu(f)2317 1934 y Ft(n)2385 1922 y Fr(!)j Fu(f)30 b FB(in)20 b(an)o(y)g Fr(C)5 b FB(\([)p Fu(a;)14 b(b)p FB(])p Fu(;)g(H)3150 1892 y Ft(s)3183 1922 y FB(\))20 b(with)515 2021 y Fu(a)27 b(<)h(b)23 b FB(equipped)e(with)h(the)h(standard)f(maximum)f(norm.)31 b(W)-7 b(e)24 b(de\002ne)f(the)f(spaces)h(L)3018 1991 y Fq(2)3040 2044 y Fm(loc)3116 2021 y FB(\()p Fv(R)o Fu(;)14 b(H)3316 1991 y Ft(s)3351 2021 y FB(\))515 2121 y(in)20 b(an)g(analogous)f(w)o(ay)-5 b(.)639 2221 y(W)e(e)26 b(write)f Fr(f)p Fu(e)1050 2233 y Ft(k)1090 2221 y Fr(g)1132 2233 y Ft(k)q Fi(2)p Fh(N)1284 2221 y FB(for)f(a)h(complete)f (orthonormal)e(set)j(in)g(L)2481 2190 y Fq(2)2543 2221 y FB(of)f(eigen)m(v)o(ectors)e(of)j Fu(A)g FB(and)515 2320 y(denote)17 b(by)g Fs(\005)918 2332 y Ft(N)1000 2320 y FB(the)h(orthogonal)d(projector)i(onto)g(the)h(subspace)f(of)h (L)2549 2290 y Fq(2)2604 2320 y FB(spanned)f(by)g Fu(e)3035 2332 y Fq(1)3072 2320 y Fu(;)d(:)g(:)g(:)g(;)g(e)3296 2332 y Ft(N)3358 2320 y FB(.)515 2420 y(Then)19 b(the)h Fu(N)9 b FB(th)20 b(spectral)g(Galerkin)f(approximations)f Fu(u)2165 2432 y Ft(N)2248 2420 y FB(of)i(\(1.1\))f(is)i(gi)n(v)o(en)e (by)g(the)i(solution)e(of)1232 2584 y Fu(@)1276 2596 y Ft(t)1306 2584 y Fu(u)1354 2596 y Ft(N)1439 2584 y Fs(=)k Fu(Au)1637 2596 y Ft(N)1718 2584 y Fs(+)18 b(\005)1863 2596 y Ft(N)1926 2584 y Fu(@)1975 2549 y Fq(2)1970 2604 y Ft(x)2012 2584 y FB(\()p Fu(@)2084 2596 y Ft(x)2126 2584 y Fu(u)2174 2596 y Ft(N)2236 2584 y FB(\))2264 2549 y Fq(2)2319 2584 y Fs(+)h(\005)2465 2596 y Ft(N)2561 2563 y Fs(_)2528 2584 y Fu(W)35 b FB(,)558 b(\(2.1\))515 2747 y(where)15 b Fu(A)h FB(can)f(be)h(interpreted)d(either)j(as)g(an)f Fu(N)10 b Fr(\002)q Fu(N)f FB(-matrix)14 b(or)h(as)i(a)f(dif)n (ferential)d(operator)h(acting)515 2847 y(on)21 b(the)g(range)g(of)g Fs(\005)1103 2859 y Ft(N)1167 2847 y FB(.)29 b(When)22 b(considering)d(\(2.1\),)h(we)i(will)h(al)o(w)o(ays)f(tak)o(e)f (initial)h(conditions)e(in)515 2947 y(the)h(range)e(of)i Fs(\005)996 2959 y Ft(N)1059 2947 y FB(.)27 b(In)21 b(this)g(equation,) 1701 2926 y Fs(_)1668 2947 y Fu(W)33 b FB(is)22 b(the)e(generalized)f (time-deri)n(v)n(ati)n(v)o(e)g(of)h(a)h(tw)o(o-sided)515 3046 y(c)o(ylindrical)c(W)m(iener)i(process)g Fu(W)32 b FB(on)19 b(L)1671 3016 y Fq(2)1727 3046 y FB(with)g(co)o(v)n(ariance) e(operator)h Fu(Q)p FB(.)25 b(\(See)19 b([DPZ92)n(])h(for)f(the)515 3146 y(de\002nition)g(of)h(a)g(c)o(ylindrical)f(W)m(iener)h(process.\)) k(W)-7 b(e)21 b(mak)o(e)f(the)g(follo)n(wing)f(assumption)g(on)g Fu(Q)p FB(:)515 3334 y Fv(Assumption)i(2.1)40 b Fp(Ther)m(e)21 b(e)n(xist)g(positive)f(number)o(s)g Fu(\013)2126 3346 y Ft(k)2187 3334 y Fp(and)g(a)g(constant)f Fu(C)27 b Fp(suc)o(h)20 b(that)1324 3498 y Fu(Qe)1429 3510 y Ft(k)1492 3498 y Fs(=)j Fu(\013)1633 3510 y Ft(k)1674 3498 y Fu(e)1713 3510 y Ft(k)1919 3498 y Fp(and)165 b Fr(j)p Fu(\013)2286 3510 y Ft(k)2327 3498 y Fr(j)23 b(\024)g Fu(C)29 b Fp(,)515 3662 y(for)20 b(all)h Fu(k)26 b(>)c Fs(0)p Fp(.)515 3822 y FB(Notice)i(that)g(this)g(assumption)f(co)o(v)o(ers)g(the)h(case)g (of)g(space-time)f(white)h(noise)g(\()p Fu(\013)2957 3834 y Ft(k)3028 3822 y Fs(=)30 b(1)p FB(\).)36 b(The)515 3922 y(assumption)18 b(that)i Fu(Q)g FB(and)f Fu(A)i FB(ha)n(v)o(e)e(the)h(same)g(eigen)m(v)o(ectors)d(implies)j(that)g(we)g (restrict)g(ourselv)o(es)515 4021 y(to)i(translationally)f(in)m(v)n (ariant)g(noise,)h(which)g(is)i(also)e(called)g(homogeneous)e(in)i(the) h(physics)e(lit-)515 4121 y(erature.)j(This)c(assumption)f(is)i (crucial)f(to)g(v)o(erify)f(technical)g(results)i(lik)o(e)f(Lemma)g (6.1.)639 4221 y(Since)c(\(2.1\))e(is)i(actually)f(a)g(stochastic)h (dif)n(ferential)d(equation)h(in)i Fv(R)2586 4185 y Ft(N)2665 4221 y FB(with)f(locally)g(Lipschitz)515 4320 y(coef)n(\002cients,)20 b(it)h(is)h(well-kno)n(wn)d(\(see)i Fp(e)o(.g)o(.)e FB([Has80)o(])i(or) f([Arn74)n(]\))h(that)g(it)g(possesses)h(\(locally\))d(a)515 4420 y(strong)d(solution.)24 b(Standard)16 b(ar)o(guments)f(allo)n(w)j (to)f(sho)n(w)h(the)f(follo)n(wing)f(proposition,)g(the)h(proof)515 4519 y(of)j(which)f(will)i(be)f(gi)n(v)o(en)f(in)i(Section)e(4)i(belo)n (w)-5 b(.)515 4707 y Fv(Pr)o(oposition)19 b(2.2)40 b Fp(F)-9 b(or)24 b(e)o(very)g Fu(N)39 b Fr(\025)29 b Fs(1)24 b Fp(and)f(any)h(initial)g(condition)e(in)i Fs(\005)2705 4719 y Ft(N)2768 4707 y FB(L)2808 4677 y Fq(2)2845 4707 y Fp(,)h(Equation)d(\(2.1\))515 4807 y(possesses)i(a)g(unique)e(global) g(str)l(ong)i(solution.)34 b(Furthermor)m(e)o(,)24 b(the)g(law)g(of)g (this)g(solution)e(con-)515 4907 y(ver)m(g)o(es)g(in)g(variation)e (norm)i(towar)m(ds)f(a)h(unique)e(in)m(variant)h(measur)m(e)g Fr(Q)2651 4919 y Ft(N)2736 4907 y Fp(whic)o(h)h(has)f(bounded)515 5006 y(moments)f(of)g(second)f(or)m(der)-9 b(.)p eop %%Page: 4 4 4 3 bop 517 232 a FB(P)t Fo(R)q(O)t(O)t(F)26 b(O)t(F)e(T)t(H)t(E)h(M)t (A)t(I)t(N)g(R)t(E)t(S)t(U)t(L)n(T)1802 b Fv(4)639 523 y FB(Consider)27 b(processes)g Fr(f)p Fu(u)1403 535 y Ft(N)1465 523 y FB(\()p Fu(t)p FB(\))p Fr(g)1593 535 y Ft(t)p Fi(2)p Fh(R)1736 523 y FB(gi)n(v)o(en)f(as)i(stationary)f (solutions)g(of)g(the)g Fu(N)9 b FB(th)28 b(spectral)515 623 y(Galerkin)19 b(approximation)e(corresponding)g(to)j(the)g(in)m(v)n (ariant)e(measure)i Fr(Q)2710 635 y Ft(N)2773 623 y FB(.)25 b(Hence,)20 b Fu(u)3121 635 y Ft(N)3204 623 y FB(satis-)515 722 y(\002es)h(the)f(follo)n(wing)f(stochastic)h(ODE.)1269 922 y Fu(@)1313 934 y Ft(t)1342 922 y Fu(u)1390 934 y Ft(N)1476 922 y Fs(=)j Fu(Au)1674 934 y Ft(N)1755 922 y Fs(+)18 b(\005)1900 934 y Ft(N)1963 922 y Fu(@)2012 887 y Fq(2)2007 942 y Ft(x)2049 922 y FB(\()p Fu(@)2121 934 y Ft(x)2163 922 y Fu(u)2211 934 y Ft(N)2273 922 y FB(\))2301 887 y Fq(2)2356 922 y Fs(+)2473 901 y(_)2439 922 y Fu(W)2517 934 y Ft(N)2604 922 y FB(,)595 b(\(2.2\))515 1069 y(where)22 b Fu(W)819 1081 y Ft(N)906 1069 y FB(is)h(gi)n(v)o(en)f (by)g Fu(W)1373 1081 y Ft(N)1437 1069 y FB(\()p Fu(t)p FB(\))27 b Fs(=)1642 1006 y Fn(P)1730 1027 y Ft(N)1730 1093 y(k)q Fq(=1)1869 1069 y Fu(\013)1922 1081 y Ft(k)1963 1069 y Fu(e)2002 1081 y Ft(k)2056 1069 y Fu(w)2115 1081 y Ft(k)2156 1069 y FB(\()p Fu(t)p FB(\))c(with)g(the)g Fr(f)p Fu(w)2662 1081 y Ft(k)2703 1069 y Fr(g)2745 1081 y Ft(k)q Fi(2)p Fh(N)2895 1069 y FB(being)f(a)h(f)o(amily)515 1168 y(of)d(independent)e(tw)o(o-sided)i(standard)f(Bro)n(wnian)g (motions)h(de\002ned)g(on)g(the)g(probability)f(space)515 1268 y(underlying)j Fu(W)12 b FB(.)39 b(Since)25 b Fu(u)1308 1280 y Ft(N)1395 1268 y FB(is)h(stationary)-5 b(,)24 b(we)h(ha)n(v)o(e)g Fr(L)p Fs(\()p Fu(u)2284 1280 y Ft(N)2347 1268 y FB(\()p Fu(t)p FB(\))p Fs(\))31 b Fr(\021)g(Q)2660 1280 y Ft(N)2748 1268 y FB(for)24 b(an)o(y)g Fu(t)31 b Fr(2)h Fv(R)p FB(.)39 b(By)515 1376 y Fr(P)580 1337 y Fm([)p Fq(0)p Ft(;T)9 b Fm(])573 1401 y Ft(N)747 1376 y FB(we)24 b(denote)e(the)h(path)g(measure)f(of)h Fr(f)p Fu(u)1888 1388 y Ft(N)1950 1376 y FB(\()p Fu(t)p FB(\))p Fr(g)2078 1388 y Ft(t)p Fi(2)p Fm([)p Fq(0)p Ft(;T)9 b Fm(])2291 1376 y FB(,)24 b(and)f(by)g Fr(P)2645 1388 y Ft(N)2731 1376 y FB(the)g(measure)g(for)g(the)515 1476 y(whole)c(process)h Fu(u)1056 1488 y Ft(N)1139 1476 y FB(in)h(path)e(space.)639 1576 y(It)d(is)g(well-kno)n(wn)d(\(see)j Fp(e)o(.g)o(.)e FB([DPZ92)n(]\))h(that,)h(for)f(an)o(y)f(pair)h Fu(t)23 b(>)g(t)2533 1588 y Fq(0)2570 1576 y FB(,)17 b(the)e(process)g Fu(u)3038 1588 y Ft(N)3116 1576 y FB(satis\002es)515 1675 y(\(with)20 b(probability)e Fs(1)p FB(\))i(the)g(follo)n(wing)f(v) n(ariation)f(of)i(constants)g(formula:)588 1891 y Fu(u)636 1903 y Ft(N)698 1891 y FB(\()p Fu(t)p FB(\))j Fs(=)f Fu(e)933 1857 y Fm(\()p Ft(t)p Fi(\000)p Ft(t)1054 1865 y Fg(0)1087 1857 y Fm(\))p Ft(A)1160 1891 y Fu(u)1208 1903 y Ft(N)1271 1891 y FB(\()p Fu(t)1329 1903 y Fq(0)1365 1891 y FB(\))d Fs(+)1495 1778 y Fn(Z)1578 1799 y Ft(t)1541 1967 y(t)1566 1975 y Fg(0)1621 1891 y Fu(@)1670 1857 y Fq(2)1665 1912 y Ft(x)1707 1891 y Fu(e)1746 1857 y Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))p Ft(A)1946 1891 y Fs(\005)2008 1903 y Ft(N)2072 1891 y FB(\()p Fu(@)2144 1903 y Ft(x)2185 1891 y Fu(u)2233 1903 y Ft(N)2296 1891 y FB(\()p Fu(s)p FB(\)\))2419 1857 y Fq(2)2455 1891 y Fu(ds)f Fs(+)2638 1778 y Fn(Z)2721 1799 y Ft(t)2684 1967 y(t)2709 1975 y Fg(0)2764 1891 y Fu(e)2803 1857 y Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))p Ft(A)3004 1891 y Fu(dW)3125 1903 y Ft(N)3189 1891 y FB(\()p Fu(s)p FB(\))p Fu(:)3220 2040 y FB(\(2.3\))515 2139 y(Again,)g(we)j(consider)d(the)i(dif)n (ferential)f(operators)f(either)h(as)i(operators)d(on)i(the)g(range)f (of)g Fs(\005)3226 2151 y Ft(N)3310 2139 y FB(or)515 2239 y(as)i Fu(N)27 b Fr(\002)18 b Fu(N)9 b FB(-matrices.)639 2339 y(As)32 b(our)e(solutions)g(of)h(\(1.1\))f(do)g(not)h(ha)n(v)o(e)f (enough)f(re)o(gularity)-5 b(,)31 b(we)g(will)h(focus)e(on)h(mild)515 2438 y(solutions,)19 b(which)h(are)g(solutions)g(of)f(such)h(inte)o (gral)g(equations.)j(Our)d(main)g(result)g(is)515 2625 y Fv(Theor)o(em)g(2.3)40 b Fp(Consider)18 b(equation)d(\(1.1\))h(with)i (periodic)f(or)h(Neumann)e(b)m(.c.)24 b(in)18 b(the)f(stable)g(and)515 2724 y(only)i(Neumann)g(b)m(.c.)25 b(in)20 b(the)g(unstable)f(case)o(.) 25 b(Then)19 b(the)h(family)h(of)f(measur)m(es)g Fr(fQ)2925 2736 y Ft(N)2987 2724 y Fr(g)3029 2736 y Ft(N)6 b Fi(2)p Fh(N)3199 2724 y Fp(given)515 2824 y(by)20 b(Pr)l(oposition)f(2.2)h(is) h(tight)f(on)f FB(L)1540 2794 y Fq(2)1577 2824 y Fp(.)639 2923 y(Furthermor)m(e)o(,)g(for)i(any)e(of)h(its)h(accumulation)c (points)i Fr(Q)p Fp(,)h(ther)m(e)g(e)n(xists)h(a)f(pr)l(obability)f (space)515 3023 y FB(\()552 3002 y Fs(~)543 3023 y(\012)o Fu(;)662 3002 y Fs(~)639 3023 y Fr(F)8 b Fu(;)763 3002 y Fs(~)744 3023 y Fr(P)f FB(\))p Fp(,)38 b(a)d(two-sided)f Fu(Q)p Fp(-W)-5 b(iener)35 b(pr)l(ocess)2005 3002 y Fs(~)1981 3023 y Fu(W)12 b Fp(,)39 b(and)34 b(a)h Fv(stationary)e Fp(stoc)o(hastic)h(pr)l(ocess)515 3123 y Fr(f)p Fu(u)p FB(\()p Fu(t)p FB(\))p Fr(g)733 3135 y Ft(t)p Fi(2)p Fh(R)867 3123 y Fp(with)21 b Fu(u)i Fr(2)h(C)5 b FB(\()p Fv(R)o Fu(;)14 b(H)1430 3093 y Fi(\000)p Fq(3)1519 3123 y FB(\))k Fr(\\)h FB(L)1679 3093 y Fq(2)1701 3145 y Fm(loc)1776 3123 y FB(\()p Fv(R)o Fu(;)14 b(H)1976 3093 y Fq(1)2013 3123 y FB(\))p Fp(,)21 b(suc)o(h)e(that)h Fr(L)p FB(\()p Fu(u)p FB(\()p Fu(t)p FB(\)\))i Fr(\021)h(Q)d Fp(for)h(e)o(very)f Fu(t)k Fr(2)f Fv(R)p Fp(,)515 3222 y(and)c(suc)o(h)h(that)703 3421 y Fu(u)p FB(\()p Fu(t)p FB(\))h Fs(=)i Fu(e)985 3387 y Fm(\()p Ft(t)p Fi(\000)p Ft(t)1106 3395 y Fg(0)1138 3387 y Fm(\))p Ft(A)1212 3421 y Fu(u)p FB(\()p Fu(t)1318 3433 y Fq(0)1354 3421 y FB(\))18 b Fs(+)1483 3308 y Fn(Z)1566 3329 y Ft(t)1530 3497 y(t)1555 3505 y Fg(0)1610 3421 y Fu(@)1659 3387 y Fq(2)1654 3442 y Ft(x)1695 3421 y Fu(e)1734 3387 y Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))p Ft(A)1935 3421 y FB(\()p Fu(@)2007 3433 y Ft(x)2049 3421 y Fu(u)p FB(\()p Fu(s)p FB(\)\))2220 3387 y Fq(2)2255 3421 y Fu(ds)h Fs(+)2439 3308 y Fn(Z)2522 3329 y Ft(t)2485 3497 y(t)2510 3505 y Fg(0)2565 3421 y Fu(e)2604 3387 y Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))p Ft(A)2805 3421 y Fu(d)2872 3400 y Fs(~)2848 3421 y Fu(W)12 b FB(\()p Fu(s)p FB(\))187 b(\(2.4\))515 3651 y Fp(holds)19 b(for)i(all)f Fu(t)k Fr(\025)e Fu(t)1112 3663 y Fq(0)1149 3651 y Fp(,)1209 3630 y Fs(~)1191 3651 y Fr(P)6 b Fp(-almost)20 b(sur)m(ely)-5 b(.)639 3810 y FB(W)e(e)17 b(will)f(not)f(focus)g(on)h(optimal)e(re)o (gularity)-5 b(,)14 b(b)n(ut)i(we)g(could)e(slightly)i(impro)o(v)o(e)d (the)i(re)o(gularity)515 3910 y(of)27 b Fu(u)i FB(analogous)d(to)i (Corollary)f(3.2)g(and)h(3.3)f(of)h([BG02)o(].)48 b(Moreo)o(v)o(er)m(,) 27 b(we)h(could)f(pro)o(v)o(e)f(that)515 4010 y(support)g(of)i(the)g (measure)f Fr(Q)h FB(is)h(concentrated)c(in)j(a)g(smaller)g(space)g (than)f(L)2812 3980 y Fq(2)2849 4010 y FB(,)j(b)n(ut)e(we)g(are)g(f)o (ar)515 4109 y(from)19 b(getting)g(enough)f(re)o(gularity)h(to)h(pro)o (v)o(e)e(pathwise)i(uniqueness.)639 4209 y(In)31 b(the)g(stable)g (case,)i(it)f(is)g(easily)f(possible)f(to)h(pro)o(v)o(e)e(an)i(analog)f (of)g(Theorem)f(2.3)h(with)515 4309 y(Dirichlet)20 b(boundary)d (conditions,)h(b)n(ut)j(we)f(do)g(not)g(enter)g(into)f(details)i(here.) 515 4533 y FA(3)99 b(Pr)n(oof)25 b(of)g(the)h(main)f(r)n(esult)515 4707 y FB(The)d(main)g(step)h(of)g(the)f(proof)g(of)g(Theorem)f(2.3)h (is)i(a)f(bound)e(on)h(the)h(logarithmic)d(moments)i(of)515 4807 y Fr(Q)583 4819 y Ft(N)667 4807 y FB(that)f Fp(does)g(not)f FB(depend)f(on)i Fu(N)9 b FB(.)27 b(The)21 b(main)f(technical)h(dif)n (\002culty)e(is)j(that)f(It)7 b(\210)-35 b(o')-5 b(s)21 b(formula)f(can)515 4907 y(not)i(be)h(applied)g(to)g(\(1.1\))e(since)j (the)f(co)o(v)n(ariance)e(of)h(our)h(noise)g(is)h(not)e(necessarily)h (trace)g(class.)515 5006 y(W)-7 b(e)21 b(postpone)e(the)h(proof)e(of)i (Theorem)f(3.1)g(belo)n(w)h(to)g(sections)h(4)f(and)f(5.)p eop %%Page: 5 5 5 4 bop 517 232 a FB(P)t Fo(R)q(O)t(O)t(F)26 b(O)t(F)e(T)t(H)t(E)h(M)t (A)t(I)t(N)g(R)t(E)t(S)t(U)t(L)n(T)1802 b Fv(5)515 523 y(Theor)o(em)20 b(3.1)40 b Fp(Let)26 b Fr(Q)1202 535 y Ft(N)1290 523 y Fp(be)f(the)g(measur)m(e)f(on)h FB(L)1973 493 y Fq(2)2035 523 y Fp(in)m(variant)e(for)j(the)e Fu(N)9 b Fp(th)25 b(Galerkin)g(appr)l(oxi-)515 623 y(mation.)f(Then,)19 b(ther)m(e)i(e)n(xists)g(a)g(constant)e Fu(C)27 b Fp(independent)17 b(of)k Fu(N)29 b Fp(suc)o(h)20 b(that)1358 716 y Fn(Z)1404 905 y Fm(L)1430 888 y Fg(2)1480 829 y FB(log)1600 762 y Fn(\000)1638 829 y Fs(1)e(+)g Fr(k)p Fu(u)p Fr(k)1913 795 y Fq(2)1913 852 y Fi(C)1952 835 y Fg(1)1987 762 y Fn(\001)2039 829 y Fr(Q)2107 841 y Ft(N)2170 829 y FB(\()p Fu(du)p FB(\))k Fr(\024)g Fu(C)30 b Fp(,)515 1038 y(uniformly)19 b(in)i Fu(N)9 b Fp(.)639 1200 y FB(Using)20 b(this)h(result,)f(we)h (turn)e(to)h(the)515 1354 y Fp(Pr)l(oof)g(of)g(Theor)m(em)g(2.3.)40 b FB(The)30 b(tightness)g(of)g(the)g(f)o(amily)g Fr(fQ)2360 1366 y Ft(N)2422 1354 y Fr(g)g FB(follo)n(ws)g(immediately)f(from)515 1454 y(Theorem)20 b(3.1)h(and)h(the)g(compact)f(embedding)e(of)j Fr(C)2066 1424 y Fq(1)2126 1454 y FB(into)f(L)2317 1424 y Fq(2)2354 1454 y FB(.)31 b(W)-7 b(e)23 b(choose)e(an)o(y)h (accumulation)515 1553 y(point)e Fr(Q)i FB(of)f Fr(fQ)998 1565 y Ft(N)1060 1553 y Fr(g)g FB(and)g(assume)g(without)f(loss)i(of)f (generality)f(that)h Fr(Q)2616 1565 y Ft(N)2701 1553 y FB(con)m(v)o(er)o(ges)d(weakly)i(to)515 1653 y Fr(Q)i FB(in)f(the)h(space)f(of)g(Borel)h(measures)f(on)g(L)1795 1623 y Fq(2)1832 1653 y FB(.)29 b(Denote)21 b(by)g Fr(P)2307 1665 y Ft(N)2392 1653 y FB(the)g(la)o(w)h(of)f(the)g(\(unique)f(in)i (la)o(w\))515 1753 y(stationary)d(process)h(associated)g(to)g(the)g(in) m(v)n(ariant)f(measure)g Fr(Q)2378 1765 y Ft(N)2462 1753 y FB(by)g(Proposition)g(2.2.)639 1852 y(In)26 b(order)f(to)h(construct) f(the)g(process)h Fu(u)g FB(appearing)e(in)i(the)g(statement,)h(we)f (\002rst)h(sho)n(w)e(that)515 1952 y(the)18 b(f)o(amily)f(of)g (measures)h Fr(fP)1384 1964 y Ft(N)1446 1952 y Fr(g)g FB(is)h(tight)e(\(it)h(turns)g(out)f(that)h(it)h(is)f(so)h(on)e(the)h (space)f Fr(C)5 b FB(\()p Fv(R)p Fu(;)14 b(H)3198 1922 y Fi(\000)p Fq(3)3286 1952 y FB(\))c Fr(\\)515 2052 y FB(L)555 2021 y Fq(2)577 2074 y Fm(loc)652 2052 y FB(\()p Fv(R)p Fu(;)k(H)853 2021 y Fq(1)889 2052 y FB(\)\),)i(and)f(then)g(v)o (erify)f(that)i(the)f(limiting)g(process)g(obtained)f(by)h(the)h(usual) f(Prokhoro)o(v-)515 2151 y(Sk)o(ohorod)j(ar)o(gument)f(really)j (satis\002es)i(the)e(inte)o(gral)f(equation)g(\(2.4\).)639 2251 y(T)-7 b(o)24 b(pro)o(v)o(e)f(the)h(tightness)f(of)h(the)g(f)o (amily)g Fr(fP)1967 2263 y Ft(N)2029 2251 y Fr(g)p FB(,)h(we)f (consider)f Fu(u)2592 2263 y Ft(N)2679 2251 y FB(as)i(a)f(solution)f (of)h(\(2.3\))515 2350 y(with)j(initial)h(condition)d Fu(u)1301 2362 y Ft(N)1364 2350 y FB(\()p Fs(0)p FB(\))h(distrib)n (uted)g(according)g(to)h Fr(Q)2385 2362 y Ft(N)2448 2350 y FB(.)46 b(W)-7 b(e)29 b(denote)d(by)h Fu(W)3102 2320 y Ft(N)3090 2373 y(A)3164 2350 y FB(\()p Fu(t)p FB(\))h(the)515 2450 y(stochastic)20 b(con)m(v)n(olution)d(gi)n(v)o(en)i(by)1421 2673 y Fu(W)1511 2638 y Ft(N)1499 2693 y(A)1574 2673 y FB(\()p Fu(t)p FB(\))k Fs(=)1770 2560 y Fn(Z)1853 2580 y Ft(t)1816 2748 y Fq(0)1896 2673 y Fu(e)1935 2638 y Ft(A)p Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))2150 2673 y Fu(dW)2271 2685 y Ft(N)2334 2673 y FB(\()p Fu(s)p FB(\))g(,)515 2895 y(and)e(we)h(de\002ne)f Fu(v)1041 2907 y Ft(N)1104 2895 y FB(\()p Fu(t)p FB(\))k Fs(:=)g Fu(u)1376 2907 y Ft(N)1439 2895 y FB(\()p Fu(t)p FB(\))19 b Fr(\000)g Fu(W)1718 2865 y Ft(N)1706 2918 y(A)1781 2895 y FB(\()p Fu(t)p FB(\).)29 b(The)21 b(reason)g(is)i(that)e(the)h(stochastic)g (process)f Fu(v)3316 2907 y Ft(N)515 2994 y FB(e)o(xhibits)g (trajectories)h(with)g(much)f(more)g(time-re)o(gularity)f(than)h Fu(u)2494 3006 y Ft(N)2557 2994 y FB(.)31 b(The)22 b(process)f Fu(v)3072 3006 y Ft(N)3158 2994 y FB(is)i(then)515 3094 y(pathwise)d(a)g(strong)g(solution)f(of)h(the)g(random)e(PDE)j(gi)n(v)o (en)e(by)866 3262 y Fu(@)910 3274 y Ft(t)940 3262 y Fu(v)980 3274 y Ft(N)1066 3262 y Fs(=)j Fu(Av)1255 3274 y Ft(N)1337 3262 y Fr(\000)c Fu(@)1469 3228 y Fq(2)1464 3282 y Ft(x)1506 3262 y Fs(\005)1568 3274 y Ft(N)1632 3262 y FB(\()p Fu(@)1704 3274 y Ft(x)1745 3262 y Fu(v)1785 3274 y Ft(N)1867 3262 y Fs(+)g Fu(@)1994 3274 y Ft(x)2036 3262 y Fu(W)2126 3228 y Ft(N)2114 3282 y(A)2189 3262 y FB(\))2217 3228 y Fq(2)2277 3262 y FB(,)165 b Fu(v)2503 3274 y Ft(N)2567 3262 y FB(\()p Fs(0)p FB(\))21 b Fs(=)i Fu(u)2822 3274 y Ft(N)2885 3262 y FB(\()p Fs(0)p FB(\))e Fu(:)193 b FB(\(3.1\))639 3430 y(W)-7 b(e)29 b(will)e(need)g(the)g(follo)n(wing)f (technical)g(lemma,)i(the)f(proof)f(of)h(which)f(is)i(postponed)d(to) 515 3529 y(sections)20 b(4)g(and)g(5.)515 3720 y Fv(Lemma)h(3.2)40 b Fp(F)l(ix)21 b Fu(";)14 b(T)33 b(>)23 b Fs(0)d Fp(and)f(assume)i (that)f(ther)m(e)g(e)n(xists)h Fu(R)j(>)f Fs(0)d Fp(suc)o(h)g(that)1257 3888 y Fv(P)p Fs(\()p Fr(k)p Fu(u)1430 3900 y Ft(N)1492 3888 y FB(\()p Fs(0)p FB(\))p Fr(k)i Fu(>)g(R)q Fs(\))h Fu(<)g(")166 b Fp(for)20 b(all)h Fu(N)31 b Fr(2)24 b Fv(N)p Fp(.)515 4055 y(Assume)32 b(furthermor)m(e)g(that)g Fu(u)p FB(\()p Fs(0)p FB(\))f Fp(is)i(independent)d(of)i(the)g(W)-5 b(iener)33 b(incr)m(ements)f(for)h(positive)515 4155 y(times.)25 b(Then)20 b(ther)m(e)h(e)n(xists)1333 4134 y Fs(~)1315 4155 y Fu(R)j(>)f Fs(0)d Fp(independent)d(of)k Fu(N)29 b Fp(suc)o(h)20 b(that)876 4323 y Fv(P)927 4256 y Fn(\000)964 4323 y Fr(k)p Fu(v)1046 4335 y Ft(N)1109 4323 y Fr(k)1151 4338 y Fi(C)s Fm(\()p Fq(0)p Ft(;T)5 b(;)p Fm(L)1352 4321 y Fg(2)1384 4338 y Fm(\))1426 4323 y Fs(+)18 b Fr(k)p Fu(v)1591 4335 y Ft(N)1653 4323 y Fr(k)1695 4338 y Fm(L)1721 4321 y Fg(2)1753 4338 y Fm(\()p Fq(0)p Ft(;T)5 b(;H)1947 4321 y Fg(2)1980 4338 y Fm(\))2027 4323 y Fu(>)2133 4302 y Fs(~)2115 4323 y Fu(R)2178 4256 y Fn(\001)2239 4323 y Fu(<)23 b(")166 b Fp(for)20 b(all)h Fu(N)32 b Fr(2)23 b Fv(N)p Fu(:)639 4508 y FB(Using)g(this)g(result,)h (we)f(v)o(erify)e(the)i(tightness)g(of)f Fr(fP)2217 4468 y Fm([)p Fq(0)p Ft(;T)9 b Fm(])2210 4532 y Ft(N)2360 4508 y Fr(g)23 b FB(on)f(the)h(space)g Fr(C)5 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)3189 4478 y Fi(\000)p Fq(3)3276 4508 y FB(\))20 b Fr(\\)515 4608 y FB(L)555 4578 y Fq(2)591 4608 y FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)867 4578 y Fq(1)903 4608 y FB(\))27 b(in)f(a)i(similar)e(w)o(ay)h (as)h(in)e([BG02)o(,)j(Section)d(5],)i(so)f(we)g(only)f(brie\003y)g(sk) o(etch)h(the)515 4707 y(main)20 b(ideas)g(here.)639 4807 y(Gi)n(v)o(en)32 b Fu(")46 b(>)g Fs(0)p FB(,)36 b(we)d(look)f(for)g(a)h (compact)f(set)i Fu(K)2190 4819 y Ft(")2258 4807 y FB(such)e(that)h Fr(P)2666 4767 y Fm([)p Fq(0)p Ft(;T)9 b Fm(])2659 4831 y Ft(N)2810 4807 y FB(\()p Fu(K)2909 4819 y Ft(")2943 4807 y FB(\))33 b(is)h(bounded)515 4907 y(from)24 b(belo)n(w)h(by)g Fs(1)d Fr(\000)f Fu(")26 b FB(for)f(all)h Fu(N)9 b FB(.)41 b(Combining)23 b(Theorem)h(3.1)g(with)i(Lemma)f(3.2,)g(there)g(e)o(x-) 515 5006 y(ists)k Fu(R)f FB(such)g(that,)h(with)f(probability)d(lar)o (ger)i(than)g Fs(1)c Fr(\000)h Fu(")p FB(,)29 b Fu(v)2344 5018 y Ft(N)2436 5006 y FB(lies)f(in)g(a)g(ball)g(of)f(radius)g Fu(R)i FB(of)p eop %%Page: 6 6 6 5 bop 517 232 a FB(T)t Fo(H)t(E)24 b(S)t(T)n(A)t(B)t(L)t(E)g(C)t(A)t (S)t(E)2176 b Fv(6)515 523 y Fr(C)5 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b FB(L)804 493 y Fq(2)838 523 y FB(\))19 b Fr(\\)f FB(L)998 493 y Fq(2)1035 523 y FB(\()p Fs(0)p Fu(;)c(T)7 b(;)14 b(H)1311 493 y Fq(2)1346 523 y FB(\))21 b(and)f Fr(k)p Fu(v)1618 535 y Ft(N)1680 523 y FB(\()p Fs(0)p FB(\))p Fr(k)i(\024)h Fu(R)q FB(.)j(Furthermore,)17 b Fu(W)2587 493 y Ft(N)2575 546 y(A)2673 523 y Fr(!)24 b Fu(W)2858 535 y Ft(A)2933 523 y FB(in)d Fr(C)5 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b Fr(C)3317 493 y Fq(1)3351 523 y FB(\))515 623 y Fr(P)7 b FB(-a.s.)49 b(Using)28 b(standard)f(compactness)g(results)h(\(e.g.)49 b([Gat93)n(,)31 b(Proposition)c(1])h(or)g([DPZ92)n(,)515 722 y(Proposition)g(8.4]\))g (for)g(the)i(inte)o(gral)e(operator)g(appearing)f(in)j(\(2.3\),)g(we)f (can)h(check)e(that)i(the)515 822 y(abo)o(v)o(e)22 b(bounds)g(imply)h (the)h(e)o(xistence)e(of)i(a)g(compact)e(subset)i Fu(K)2443 792 y Fq(1)2437 842 y Ft(")2504 822 y FB(of)g Fu(C)6 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)2939 792 y Fi(\000)p Fq(3)3026 822 y FB(\))24 b(such)f(that)515 922 y Fu(v)555 934 y Ft(N)641 922 y Fr(2)g Fu(K)796 891 y Fq(1)790 942 y Ft(")854 922 y FB(with)d(probability)e(lar)o(ger)h (than)h Fs(1)e Fr(\000)f Fu(")p FB(.)26 b(Since)20 b Fu(v)2257 934 y Ft(N)2341 922 y FB(is)h(also)f(bounded)e(in)i(L)3003 891 y Fq(2)3040 922 y FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)3316 891 y Fq(2)3351 922 y FB(\))515 1021 y(with)28 b(high)g(probability)-5 b(,)27 b(we)i(obtain)e(by)h(an)g(interpolation) e(theorem)h(\(e.g.)49 b([VF88)o(,)30 b(Theorem)515 1121 y(IV)-11 b(.4.1]\))24 b(the)i(e)o(xistence)f(of)h(a)h(compact)e(set)h Fu(K)1923 1091 y Fq(2)1917 1141 y Ft(")1994 1121 y Fr(\032)33 b FB(L)2132 1091 y Fq(2)2169 1121 y FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)2445 1091 y Fq(1)2481 1121 y FB(\))26 b(such)g(that)g Fu(v)2905 1133 y Ft(N)3002 1121 y Fr(2)34 b Fu(K)3168 1091 y Fq(2)3162 1141 y Ft(")3231 1121 y FB(with)515 1220 y(probability)18 b(lar)o(ger)h(than)g Fs(1)f Fr(\000)g Fu(")p FB(.)639 1320 y(Hence,)f Fr(fP)997 1290 y Ft(v)1030 1298 y Ff(N)1087 1320 y Fr(g)1129 1332 y Ft(N)6 b Fi(2)p Fh(N)1296 1320 y FB(is)18 b(tight)e(on)h(the)g(space) g Fr(C)5 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)2285 1290 y Fi(\000)p Fq(3)2372 1320 y FB(\))7 b Fr(\\)g FB(L)2509 1290 y Fq(2)2544 1320 y FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)2820 1290 y Fq(1)2856 1320 y FB(\).)24 b(By)17 b(the)g(de\002ni-)515 1420 y(tion)e(of)h(the)f(projection)f Fs(\005)1278 1390 y Ft(N)1342 1420 y FB(,)i(we)h(readily)d(obtain)h (the)h(con)m(v)o(er)o(gence)c(of)j Fu(W)2686 1390 y Ft(N)2674 1443 y(A)2772 1420 y Fs(=)23 b(\005)2922 1390 y Ft(N)2985 1420 y Fu(W)3063 1432 y Ft(A)3140 1420 y Fr(!)h Fu(W)3325 1432 y Ft(A)515 1519 y FB(in)18 b Fr(C)5 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)923 1489 y Fq(1)958 1519 y FB(\),)k(as)g Fu(W)1190 1531 y Ft(A)1263 1519 y FB(is)h(already)d(in)i(that)g(space.) 24 b(Combining)16 b(both)h(ar)o(guments,)f(we)i(thus)f(ob-)515 1619 y(tain)h(the)h(tightness)f(of)g(the)h(f)o(amily)f Fr(fP)1644 1579 y Fm([)p Fq(0)p Ft(;T)9 b Fm(])1637 1643 y Ft(N)1787 1619 y Fr(g)1829 1631 y Ft(N)d Fi(2)p Fh(N)1997 1619 y FB(on)18 b(the)h(space)f Fr(C)5 b FB(\()p Fs(0)p Fu(;)14 b(T)7 b(;)14 b(H)2748 1589 y Fi(\000)p Fq(3)2835 1619 y FB(\))e Fr(\\)g FB(L)2982 1589 y Fq(2)3019 1619 y FB(\()p Fs(0)p Fu(;)i(T)7 b(;)14 b(H)3295 1589 y Fq(1)3331 1619 y FB(\).)515 1719 y(Since)25 b(this)h(holds)e(for)h(arbitrary)f (time)h(interv)n(als,)g(it)h(is)h(straightforw)o(ard)22 b(to)k(e)o(xtend)d(this)j(to)g(the)515 1818 y(whole)19 b(line,)g(so)h Fr(fP)1097 1830 y Ft(N)1160 1818 y Fr(g)f FB(is)i(tight)e(on)g Fr(C)5 b FB(\()p Fv(R)p Fu(;)14 b(H)1822 1788 y Fi(\000)p Fq(3)1910 1818 y FB(\))i Fr(\\)h FB(L)2066 1788 y Fq(2)2088 1841 y Fm(loc)2163 1818 y FB(\()p Fv(R)p Fu(;)d(H)2364 1788 y Fq(1)2401 1818 y FB(\).)25 b(W)-7 b(e)20 b(call)h Fr(P)2809 1788 y Fi(\003)2866 1818 y FB(one)e(of)h(its)g(limit-)515 1918 y(ing)h(measures)g(and)f(we) i(obtain)e(a)i(subsequence)e Fr(fP)2064 1930 y Ft(N)2117 1939 y Ff(k)2156 1918 y Fr(g)h FB(that)h(con)m(v)o(er)o(ges)c(weakly)j (to)g Fr(P)3131 1888 y Fi(\003)3191 1918 y FB(in)h(the)515 2017 y(abo)o(v)o(ementioned)16 b(space.)639 2117 y(No)n(w)k(we)g(can)g (use)g(Sk)o(ohorod')-5 b(s)17 b(Theorem)i(to)g(obtain)g(a)i(ne)n(w)e (probability)f(space)i(\()3074 2096 y Fs(~)3065 2117 y(\012)o Fu(;)3184 2096 y Fs(~)3161 2117 y Fr(F)8 b Fu(;)3284 2096 y Fs(~)3266 2117 y Fr(P)f FB(\),)515 2217 y(a)27 b Fu(Q)p FB(-W)m(iener)e(process)1240 2196 y Fs(~)1216 2217 y Fu(W)39 b FB(on)26 b(that)g(space,)i(stochastic)f(processes)k Fs(~)-47 b Fu(u)2576 2229 y Ft(k)2643 2217 y FB(with)27 b(la)o(ws)3014 2196 y Fs(~)2996 2217 y Fr(P)3054 2229 y Ft(k)3129 2217 y Fs(=)34 b Fr(P)3286 2229 y Ft(N)3339 2238 y Ff(k)515 2316 y FB(solving)20 b(\(2.3\))h(with)g Fs(\005)1194 2328 y Ft(N)1247 2337 y Ff(k)1312 2295 y Fs(~)1288 2316 y Fu(W)34 b FB(instead)21 b(of)g Fu(W)1826 2328 y Ft(N)1889 2316 y FB(,)i(as)f(well)g(as)g(a)g(stochastic)g (process)k Fs(~)-47 b Fu(u)21 b FB(with)h(prob-)515 2416 y(ability)g(distrib)n(ution)f Fr(L)p FB(\()5 b Fs(~)-47 b Fu(u)o FB(\))26 b Fs(=)g Fr(P)1496 2386 y Fi(\003)1557 2416 y FB(such)c(that)27 b Fs(~)-47 b Fu(u)1927 2428 y Ft(k)1994 2416 y Fr(!)32 b Fs(~)-48 b Fu(u)2192 2395 y Fs(~)2174 2416 y Fr(P)6 b FB(-a.s.)31 b(in)22 b(L)2535 2386 y Fq(2)2557 2439 y Fm(loc)2632 2416 y FB(\()p Fv(R)p Fu(;)14 b(H)2833 2386 y Fq(1)2870 2416 y FB(\))19 b Fr(\\)i(C)5 b FB(\()p Fv(R)o Fu(;)14 b(H)3242 2386 y Fi(\000)p Fq(3)3331 2416 y FB(\).)515 2524 y(Hence,)28 b Fs(~)-47 b Fu(u)820 2536 y Ft(k)860 2524 y FB(\()p Fu(t)p FB(\))28 b Fr(!)34 b Fs(~)-48 b Fu(u)p FB(\()p Fu(t)p FB(\))23 b(in)g Fu(H)1406 2494 y Fi(\000)p Fq(3)1495 2524 y FB(,)h(and)f(additionally)e(we)j(ha)n (v)o(e)2420 2503 y Fs(~)2402 2524 y Fr(P)2467 2494 y Ft(t)2524 2524 y Fs(=)k Fr(Q)c FB(for)e(all)i Fu(t)k Fr(2)h Fv(R)24 b FB(by)e(our)515 2624 y(initial)e(choice)g(of)g(a)g (subsequence.)639 2723 y(T)-7 b(o)25 b(sho)n(w)g(that)30 b Fs(~)-48 b Fu(u)25 b FB(is)h(actually)e(stationary)-5 b(,)24 b(we)h(\002rst)h(remark)d(that)30 b Fs(~)-47 b Fu(u)2644 2735 y Ft(k)2716 2723 y Fr(!)36 b Fs(~)-47 b Fu(u)25 b FB(in)g Fr(C)5 b FB(\()p Fv(R)o Fu(;)14 b(H)3242 2693 y Fi(\000)p Fq(3)3331 2723 y FB(\).)515 2823 y(Hence,)j(for)g(an)o (y)f(choice)h(of)g(\()p Fu(t)1397 2835 y Fq(1)1434 2823 y Fu(;)d(:)g(:)g(:)g(;)g(t)1649 2835 y Ft(m)1712 2823 y FB(\))22 b Fr(2)i Fv(R)1901 2788 y Ft(m)1982 2823 y FB(we)17 b(readily)g(obtain)f(in)i(the)f(weak)h(con)m(v)o(er)o(gence) 515 2923 y(of)26 b(measures)f(on)h(\()p Fu(H)1160 2893 y Fi(\000)p Fq(3)1248 2923 y FB(\))1276 2893 y Ft(m)1366 2923 y FB(that)g Fr(L)p FB(\(\()5 b Fs(~)-47 b Fu(u)1678 2935 y Ft(k)1718 2923 y FB(\()p Fu(t)1776 2935 y Fq(1)1812 2923 y FB(\))p Fu(;)14 b(:)g(:)g(:)g(;)19 b Fs(~)-47 b Fu(u)2073 2935 y Ft(k)2113 2923 y FB(\()p Fu(t)2171 2935 y Ft(m)2234 2923 y FB(\)\))33 b Fr(!)h(L)p FB(\(\()5 b Fs(~)-47 b Fu(u)o FB(\()p Fu(t)2658 2935 y Fq(1)2695 2923 y FB(\))p Fu(;)14 b(:)g(:)g(:)f(;)19 b Fs(~)-47 b Fu(u)o FB(\()p Fu(t)3012 2935 y Ft(m)3075 2923 y FB(\)\).)43 b(Since)520 3022 y Fs(~)-47 b Fu(u)563 3034 y Ft(k)624 3022 y FB(is)21 b(stationary)-5 b(,)19 b(this)h(immediately)f(implies)h (the)h(stationarity)e(of)25 b Fs(~)-47 b Fu(u)o FB(.)639 3122 y(Using)15 b(the)989 3101 y Fs(~)971 3122 y Fr(P)6 b FB(-a.s.)24 b(con)m(v)o(er)o(gence)11 b(as)16 b(in)f([BG02)o(,)i (Theorem)c(3.1],)i(it)h(is)g(technical)f(b)n(ut)g(straight-)515 3222 y(forw)o(ard)f(to)j(v)o(erify)d(that)22 b Fs(~)-47 b Fu(u)16 b FB(is)h(actually)f(a)g(solution)g(of)g(\(2.4\))e(with)j (respect)f(to)2767 3201 y Fs(~)2743 3222 y Fu(W)c FB(.)24 b(This)16 b(completes)515 3321 y(the)k(proof)f(of)g(Theorem)g(2.3.)3379 3321 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 3321 a 515 3548 a FA(4)99 b(The)26 b(stable)f(case)515 3723 y FB(This)33 b(section)h(pro)o(vides)d(the)j(postponed)d(proofs)h (of)h(the)h(pre)n(vious)d(sections)j(in)g(the)f(case)h(of)515 3822 y(strictly)27 b(ne)o(gati)n(v)o(e)e Fu(A)p FB(.)47 b(W)-7 b(e)28 b(will)g(discuss)f(the)h(necessary)e(changes)g(in)h (order)f(to)i(co)o(v)o(er)d(the)i(un-)515 3922 y(stable)20 b(case)h(in)f(Section)g(5)g(belo)n(w)-5 b(.)24 b(W)-7 b(e)22 b(start)e(with)h(the)515 4098 y Fp(Pr)l(oof)f(of)g(Pr)l (oposition)f(2.2.)40 b FB(The)18 b(claim)g(follo)n(ws)g(from)f([DPZ96)o (,)h(Has80)o(])h(if)f(we)h(can)f(sho)n(w)f(that)515 4197 y(there)23 b(e)o(xists)g(a)h(constant)e Fu(C)30 b FB(such)23 b(that)h Fv(E)p Fr(k)p Fu(u)1836 4209 y Ft(N)1898 4197 y FB(\()p Fu(t)p FB(\))p Fr(k)2026 4167 y Fq(2)2091 4197 y Fr(\024)k Fu(C)i FB(uniformly)21 b(in)j Fu(t)p FB(.)34 b(By)24 b(It)7 b(\210)-35 b(o')-5 b(s)23 b(formula,)515 4297 y(we)d(ha)n(v)o(e)545 4534 y Fu(d)p Fr(k)p Fu(u)678 4546 y Ft(N)741 4534 y Fr(k)783 4500 y Fq(2)843 4534 y Fs(=)i(2)p Fr(h)p Fu(u)1052 4546 y Ft(N)1114 4534 y Fu(;)14 b(Au)1261 4546 y Ft(N)1324 4534 y Fr(i)g Fu(dt)19 b Fs(+)f(2)p Fr(h)p Fu(u)1667 4546 y Ft(N)1729 4534 y Fu(;)c(@)1815 4500 y Fq(2)1810 4555 y Ft(x)1852 4534 y FB(\()p Fu(@)1924 4546 y Ft(x)1966 4534 y Fu(u)2014 4546 y Ft(N)2076 4534 y FB(\))2104 4500 y Fq(2)2141 4534 y Fr(i)g Fu(dt)19 b Fs(+)f(2)p Fr(h)p Fu(u)2484 4546 y Ft(N)2546 4534 y Fu(;)c(dW)2704 4546 y Ft(N)2768 4534 y FB(\()p Fu(t)p FB(\))p Fr(i)k Fs(+)3018 4430 y Ft(N)2987 4455 y Fn(X)2987 4634 y Ft(k)q Fq(=1)3121 4534 y Fu(\013)3174 4500 y Fq(2)3174 4555 y Ft(k)3229 4534 y Fu(dt)23 b(:)3220 4707 y FB(\(4.1\))515 4807 y(Since)17 b Fr(h)p Fu(u)797 4819 y Ft(N)859 4807 y Fu(;)d(@)945 4777 y Fq(2)940 4827 y Ft(x)982 4807 y FB(\()p Fu(@)1054 4819 y Ft(x)1096 4807 y Fu(u)1144 4819 y Ft(N)1206 4807 y FB(\))1234 4777 y Fq(2)1271 4807 y Fr(i)24 b Fs(=)e(0)17 b FB(and)g Fu(A)g FB(is)h(a)g(strictly)f(ne)o(gati)n(v)o(e)d(de\002nite)j(operator)m(,)e (the)i(claim)g(fol-)515 4907 y(lo)n(ws)h(after)f(inte)o(grating)e(\()p Fs(4)p Fu(:)p Fs(1)p FB(\))i(on)g(both)f(sides,)j(taking)d(e)o (xpectations,)g(and)h(applying)f(Gronw)o(all')-5 b(s)515 5006 y(formula.)36 b(Notice)25 b(that)g(the)f(bound)f(on)h(the)h (second)f(momenta)f(obtained)g(with)i(this)g(procedure)p eop %%Page: 7 7 7 6 bop 517 232 a FB(T)t Fo(H)t(E)24 b(S)t(T)n(A)t(B)t(L)t(E)g(C)t(A)t (S)t(E)2176 b Fv(7)515 523 y FB(di)n(v)o(er)o(ges)22 b(with)i Fu(N)33 b FB(and)24 b(it)h(remains)e(an)h(open)f(problem)f(to) j(establish)f(a)g(bound)e(independent)g(of)515 623 y Fu(N)29 b FB(for)20 b(arbitrary)e(solutions.)3379 623 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 623 a 639 804 a FB(T)-7 b(o)21 b(pro)o(v)o(e)d(Theorem)g(3.1)i (for)f(the)i(stable)f(case,)h(we)f(\002rst)h(v)o(erify)e(an)h(L)2661 774 y Fq(2)2698 804 y FB(-bound.)515 1002 y Fv(Theor)o(em)g(4.1)40 b Fp(Let)26 b Fr(Q)1202 1014 y Ft(N)1290 1002 y Fp(be)f(the)g(in)m (variant)e(measur)m(e)i(on)f FB(L)2299 972 y Fq(2)2361 1002 y Fp(for)i(the)e Fu(N)9 b Fp(th)25 b(Galerkin)g(appr)l(oxi-)515 1102 y(mation.)f(Ther)m(e)c(e)n(xists)i(a)e(constant)f Fu(C)27 b Fp(suc)o(h)20 b(that)1377 1193 y Fn(Z)1423 1381 y Fm(L)1449 1365 y Fg(2)1499 1306 y FB(log)1619 1239 y Fn(\000)1657 1306 y Fs(1)e(+)g Fr(k)p Fu(u)p Fr(k)1932 1272 y Fq(2)1968 1239 y Fn(\001)2020 1306 y Fr(Q)2088 1318 y Ft(N)2151 1306 y FB(\()p Fu(du)p FB(\))k Fr(\024)g Fu(C)30 b Fp(,)515 1528 y(for)20 b(all)h Fu(N)9 b Fp(.)515 1709 y(Pr)l(oof)o(.)40 b FB(By)21 b(\(3.1\),)d(the)j(L)1242 1679 y Fq(2)1278 1709 y FB(-norm)e Fr(k)p Fu(v)1584 1721 y Ft(N)1647 1709 y FB(\()p Fu(t)p FB(\))p Fr(k)1775 1679 y Fq(2)1831 1709 y FB(satis\002es)623 1890 y Fu(@)667 1902 y Ft(t)697 1890 y Fr(k)p Fu(v)779 1902 y Ft(N)841 1890 y Fr(k)883 1856 y Fq(2)943 1890 y Fs(=)k(2)p Fr(h)p Fu(v)1145 1902 y Ft(N)1208 1890 y Fu(;)14 b(Av)1347 1902 y Ft(N)1410 1890 y Fr(i)19 b(\000)f Fs(2)p Fr(h)p Fu(v)1658 1902 y Ft(N)1721 1890 y Fu(;)c(@)1807 1856 y Fq(2)1802 1911 y Ft(x)1844 1890 y FB(\()p Fu(@)1916 1902 y Ft(x)1957 1890 y Fu(v)1997 1902 y Ft(N)2079 1890 y Fs(+)k Fu(@)2206 1902 y Ft(x)2248 1890 y Fu(W)2338 1856 y Ft(N)2326 1911 y(A)2401 1890 y FB(\))2429 1856 y Fq(2)2466 1890 y Fr(i)943 2077 y Fs(=)23 b(2)p Fr(h)p Fu(v)1145 2089 y Ft(N)1208 2077 y Fu(;)14 b(Av)1347 2089 y Ft(N)1410 2077 y Fr(i)19 b(\000)f Fs(2)1600 1964 y Fn(Z)1682 1984 y Ft(L)1645 2152 y Fq(0)1746 2077 y Fu(@)1790 2089 y Ft(x)1832 2077 y FB(\()p Fu(@)1904 2089 y Ft(x)1945 2077 y Fu(v)1985 2089 y Ft(N)2048 2077 y FB(\))2076 2042 y Fq(2)2127 2077 y Fu(dx)h Fr(\000)f Fs(4)2375 1964 y Fn(Z)2458 1984 y Ft(L)2421 2152 y Fq(0)2521 2077 y Fu(@)2570 2042 y Fq(2)2565 2097 y Ft(x)2607 2077 y Fu(v)2647 2089 y Ft(N)2724 2077 y Fu(@)2768 2089 y Ft(x)2810 2077 y Fu(v)2850 2089 y Ft(N)2927 2077 y Fu(@)2971 2089 y Ft(x)3013 2077 y Fu(W)3103 2042 y Ft(N)3091 2097 y(A)3180 2077 y Fu(dx)1105 2318 y Fr(\000)g Fs(2)1244 2205 y Fn(Z)1326 2225 y Ft(L)1289 2393 y Fq(0)1389 2318 y Fu(@)1438 2283 y Fq(2)1433 2338 y Ft(x)1475 2318 y Fu(v)1515 2330 y Ft(N)1592 2318 y FB(\()p Fu(@)1664 2330 y Ft(x)1706 2318 y Fu(W)1796 2283 y Ft(N)1784 2338 y(A)1859 2318 y FB(\))1887 2283 y Fq(2)1938 2318 y Fu(dx)1192 b FB(\(4.2\))943 2501 y Fr(\024)23 b(\000k)p Fu(@)1187 2466 y Fq(2)1182 2521 y Ft(x)1223 2501 y Fu(v)1263 2513 y Ft(N)1326 2501 y Fr(k)1368 2466 y Fq(2)1423 2501 y Fr(\000)18 b Fu(\027)5 b Fr(k)p Fu(@)1638 2513 y Ft(x)1680 2501 y Fu(v)1720 2513 y Ft(N)1783 2501 y Fr(k)1825 2466 y Fq(2)1880 2501 y Fs(+)18 b(4)p Fr(k)p Fu(@)2096 2466 y Fq(2)2091 2521 y Ft(x)2132 2501 y Fu(v)2172 2513 y Ft(N)2236 2501 y Fr(k)c(k)p Fu(@)2378 2513 y Ft(x)2418 2501 y Fu(v)2458 2513 y Ft(N)2521 2501 y Fr(k)g(k)p Fu(@)2663 2513 y Ft(x)2704 2501 y Fu(W)2794 2466 y Ft(N)2782 2521 y(A)2857 2501 y Fr(k)2899 2513 y Fi(1)1105 2625 y Fs(+)k(2)p Fr(k)p Fu(@)1321 2591 y Fq(2)1316 2646 y Ft(x)1357 2625 y Fu(v)1397 2637 y Ft(N)1460 2625 y Fr(k)c(k)p Fu(@)1602 2637 y Ft(x)1642 2625 y Fu(W)1732 2591 y Ft(N)1720 2646 y(A)1795 2625 y Fr(k)1837 2591 y Fq(2)1837 2646 y(4)943 2789 y Fr(\024)23 b(\000)1106 2733 y Fs(1)p 1106 2770 42 4 v 1106 2846 a(4)1157 2789 y Fr(k)p Fu(@)1248 2754 y Fq(2)1243 2809 y Ft(x)1284 2789 y Fu(v)1324 2801 y Ft(N)1387 2789 y Fr(k)1429 2754 y Fq(2)1485 2789 y Fr(\000)18 b Fu(\027)5 b Fr(k)p Fu(@)1700 2801 y Ft(x)1741 2789 y Fu(v)1781 2801 y Ft(N)1845 2789 y Fr(k)1887 2754 y Fq(2)1942 2789 y Fs(+)18 b(8)p Fr(k)p Fu(v)2149 2801 y Ft(N)2211 2789 y Fr(k)2253 2754 y Fq(2)2290 2789 y Fr(k)p Fu(@)2376 2801 y Ft(x)2417 2789 y Fu(W)2507 2754 y Ft(N)2495 2809 y(A)2570 2789 y Fr(k)2612 2754 y Fq(4)2612 2809 y Fi(1)2700 2789 y Fs(+)g(4)p Fr(k)p Fu(@)2911 2801 y Ft(x)2952 2789 y Fu(W)3042 2754 y Ft(N)3030 2809 y(A)3105 2789 y Fr(k)3147 2754 y Fq(4)3147 2809 y(4)3207 2789 y Fu(:)515 2992 y FB(Using)h(the)g(Poincar)5 b(\264)-33 b(e)18 b(inequality)g(and)h(the)g(f)o(act)g(that)h(we)f(consider)f (only)h(solutions)f(with)i(v)n(anish-)515 3092 y(ing)i(mean,)f(we)i (see)g(that)f(there)f(e)o(xists)i(a)g(positi)n(v)o(e)e(constant)g Fu(\013)i FB(independent)d(of)i Fu(N)31 b FB(\(b)n(ut)22 b(depen-)515 3192 y(ding)d(on)h Fu(L)p FB(\))g(such)g(that)919 3373 y Fu(@)963 3385 y Ft(t)992 3373 y Fr(k)p Fu(v)1074 3385 y Ft(N)1137 3373 y Fr(k)1179 3339 y Fq(2)1239 3373 y Fr(\024)i(\000)p Fu(\013)p Fr(k)p Fu(v)1526 3385 y Ft(N)1589 3373 y Fr(k)1631 3339 y Fq(2)1686 3373 y Fs(+)c(8)p Fr(k)p Fu(v)1893 3385 y Ft(N)1956 3373 y Fr(k)1998 3339 y Fq(2)2034 3373 y Fr(k)p Fu(@)2120 3385 y Ft(x)2162 3373 y Fu(W)2252 3339 y Ft(N)2240 3393 y(A)2315 3373 y Fr(k)2357 3339 y Fq(4)2357 3393 y Fi(1)2445 3373 y Fs(+)g(4)p Fr(k)p Fu(@)2656 3385 y Ft(x)2697 3373 y Fu(W)2787 3339 y Ft(N)2775 3393 y(A)2850 3373 y Fr(k)2892 3339 y Fq(4)2892 3393 y(4)2952 3373 y Fu(:)515 3554 y FB(W)-7 b(e)21 b(de\002ne)f(no)n(w)f(for)h(an)o(y)f(interv)n(al)h([)p Fu(s;)14 b(t)p FB(])19 b(the)i(quantity)e Fu(W)2246 3524 y Ft(N)2234 3576 y Fm([)p Ft(s;t)p Fm(])2374 3554 y FB(by)1394 3805 y Fu(W)1484 3771 y Ft(N)1472 3826 y Fm([)p Ft(s;t)p Fm(])1614 3805 y Fs(=)1702 3692 y Fn(Z)1785 3713 y Ft(t)1748 3881 y(s)1828 3805 y Fs(8)p Fr(k)p Fu(@)1956 3817 y Ft(x)1997 3805 y Fu(W)2087 3771 y Ft(N)2075 3826 y(A)2150 3805 y FB(\()p Fu(r)r FB(\))p Fr(k)2287 3771 y Fq(4)2287 3826 y Fi(1)2371 3805 y Fu(dr)26 b(:)515 4027 y FB(As)21 b(a)f(consequence,) e(we)j(ha)n(v)o(e)e(the)h(follo)n(wing)f Fp(a-priori)h FB(estimate)g(on)g(the)g(norm)f(of)h Fu(v)3040 4039 y Ft(N)3103 4027 y FB(:)600 4264 y Fr(k)p Fu(v)682 4276 y Ft(N)745 4264 y FB(\()p Fu(t)p FB(\))p Fr(k)873 4229 y Fq(2)932 4264 y Fr(\024)i Fu(e)1058 4229 y Fi(\000)p Ft(\013t)p Fq(+)p Ft(W)1300 4204 y Ff(N)1291 4246 y Fe([)p Fg(0)p Ff(;t)p Fe(])1398 4264 y Fr(k)p Fu(v)1480 4276 y Ft(N)1543 4264 y FB(\()p Fs(0)p FB(\))p Fr(k)1683 4229 y Fq(2)1737 4264 y Fs(+)c Fu(C)1899 4151 y Fn(Z)1982 4171 y Ft(t)1945 4339 y Fq(0)2025 4264 y Fu(e)2064 4229 y Fi(\000)p Ft(\013)p Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))p Fq(+)p Ft(W)2427 4204 y Ff(N)2418 4246 y Fe([)p Ff(s;t)p Fe(])2525 4196 y Fn(\000)2563 4264 y Fs(1)g(+)g Fr(k)p Fu(@)2792 4276 y Ft(x)2834 4264 y Fu(W)2924 4229 y Ft(N)2912 4284 y(A)2987 4264 y FB(\()p Fu(s)p FB(\))p Fr(k)3124 4229 y Fq(4)3124 4284 y(4)3160 4196 y Fn(\001)3212 4264 y Fu(ds)932 4460 y Fr(\024)k Fu(e)1058 4426 y Fi(\000)p Ft(\013t)p Fq(+)p Ft(W)1300 4401 y Ff(N)1291 4442 y Fe([)p Fg(0)p Ff(;t)p Fe(])1398 4460 y Fr(k)p Fu(v)1480 4472 y Ft(N)1543 4460 y FB(\()p Fs(0)p FB(\))p Fr(k)1683 4426 y Fq(2)1737 4460 y Fs(+)c Fu(C)6 b(e)1924 4426 y Ft(W)1995 4401 y Ff(N)1986 4442 y Fe([)p Fg(0)p Ff(;t)p Fe(])2093 4460 y FB(\()p Fu(W)2211 4426 y Ft(N)2199 4481 y Fm([)p Fq(0)p Ft(;t)p Fm(])2338 4460 y Fs(+)18 b Fu(t)p FB(\))23 b Fu(:)695 b FB(\(4.3\))515 4652 y(Since)20 b Fu(u)768 4664 y Ft(N)854 4652 y Fs(=)i Fu(v)981 4664 y Ft(N)1063 4652 y Fs(+)c Fu(W)1236 4621 y Ft(N)1224 4674 y(A)1320 4652 y FB(and)h Fu(W)1550 4621 y Ft(N)1538 4674 y(A)1613 4652 y FB(\()p Fs(0)p FB(\))j Fs(=)h(0)d FB(we)g(obtain)g(for)f(some)h Fu(")j(>)g Fs(0)d FB(\002x)o(ed)f(later)i(on:)607 4833 y Fr(k)p Fu(u)697 4845 y Ft(N)759 4833 y FB(\()p Fu(t)p FB(\))p Fr(k)887 4798 y Fq(2)947 4833 y Fr(\024)h FB(\()p Fs(1)c(+)g Fu(")p FB(\))p Fr(k)p Fu(v)1354 4845 y Ft(N)1416 4833 y FB(\()p Fu(t)p FB(\))p Fr(k)1544 4798 y Fq(2)1598 4833 y Fs(+)g Fu(C)6 b Fr(k)p Fu(W)1878 4798 y Ft(N)1866 4853 y(A)1941 4833 y FB(\()p Fu(t)p FB(\))p Fr(k)2069 4798 y Fq(2)947 4974 y Fr(\024)22 b FB(\()p Fs(1)c(+)g Fu(")p FB(\))p Fu(e)1311 4940 y Fi(\000)p Ft(\013t)p Fq(+)p Ft(W)1553 4915 y Ff(N)1544 4957 y Fe([)p Fg(0)p Ff(;t)p Fe(])1650 4974 y Fr(k)p Fu(u)1740 4986 y Ft(N)1802 4974 y FB(\()p Fs(0)p FB(\))p Fr(k)1942 4940 y Fq(2)1996 4974 y Fs(+)g Fu(C)6 b(e)2183 4940 y Ft(W)2254 4915 y Ff(N)2245 4957 y Fe([)p Fg(0)p Ff(;t)p Fe(])2352 4974 y FB(\()p Fu(W)2470 4940 y Ft(N)2458 4995 y Fm([)p Fq(0)p Ft(;t)p Fm(])2597 4974 y Fs(+)18 b Fu(t)p FB(\))g Fs(+)g Fu(C)6 b Fr(k)p Fu(W)3036 4940 y Ft(N)3024 4995 y(A)3099 4974 y FB(\()p Fu(t)p FB(\))p Fr(k)3227 4940 y Fq(2)3263 4974 y Fu(:)p eop %%Page: 8 8 8 7 bop 517 232 a FB(T)t Fo(H)t(E)24 b(S)t(T)n(A)t(B)t(L)t(E)g(C)t(A)t (S)t(E)2176 b Fv(8)515 523 y FB(Note)20 b(that)g(the)g(constants)g(may) g(depend)e(on)i Fu(")p FB(.)639 623 y(The)27 b(problem)e(at)i(this)g (point)f(is)h(that)g(the)g(e)o(xponential)d(moment)i(of)g(the)h(random) e(v)n(ariable)515 722 y Fu(W)605 692 y Ft(N)593 744 y Fm([)p Fq(0)p Ft(;t)p Fm(])735 722 y FB(is)c(in\002nite.)k(W)-7 b(e)21 b(therefore)d(tak)o(e)j(logarithms)d(on)i(both)f(sides,)i (yielding)681 927 y(log)801 860 y Fn(\000)839 927 y Fs(1)d(+)g Fr(k)p Fu(u)1072 939 y Ft(N)1134 927 y FB(\()p Fu(t)p FB(\))p Fr(k)1262 893 y Fq(2)1298 860 y Fn(\001)1359 927 y Fr(\024)23 b FB(log)1566 835 y Fn(\020)1616 927 y FB(\()p Fs(1)18 b(+)g Fu(")p FB(\))c Fu(e)1907 893 y Fi(\000)p Ft(\013t)p Fq(+)p Ft(W)2149 868 y Ff(N)2140 909 y Fe([)p Fg(0)p Ff(;t)p Fe(])2246 860 y Fn(\000)2284 927 y Fs(1)k(+)g Fr(k)p Fu(u)2517 939 y Ft(N)2579 927 y FB(\()p Fs(0)p FB(\))p Fr(k)2719 893 y Fq(2)2754 860 y Fn(\001)1686 1110 y Fs(+)g Fu(C)6 b(e)1873 1075 y Ft(W)1944 1050 y Ff(N)1935 1092 y Fe([)p Fg(0)p Ff(;t)p Fe(])2042 1042 y Fn(\000)2080 1110 y Fu(W)2170 1075 y Ft(N)2158 1130 y Fm([)p Fq(0)p Ft(;t)p Fm(])2298 1110 y Fs(+)18 b Fu(t)2411 1042 y Fn(\001)2468 1110 y Fs(+)g Fu(C)6 b Fr(k)p Fu(W)2748 1075 y Ft(N)2736 1130 y(A)2810 1110 y FB(\()p Fu(t)p FB(\))p Fr(k)2938 1075 y Fq(2)2993 1110 y Fs(+)18 b(1)3118 1017 y Fn(\021)3190 1110 y Fu(:)3220 1018 y FB(\(4.4\))639 1296 y(Using)32 b(Lemma)e(6.1)h(it)h(is)g(no)n(w) f(easy)g(to)h(v)o(erify)e(that)h(we)h(can)f(apply)g(Lemma)f(6.4)h (\(with)515 1396 y(a)f(constant)g Fu(K)36 b FB(independent)27 b(of)j Fu(N)9 b FB(\))30 b(to)g(the)h(right-hand)c(side)k(of)e (\(4.4\),)i(where)f(we)g(tak)o(e)h(the)515 1495 y(conditional)18 b(e)o(xpectation)g(w)-5 b(.r)g(.t.)20 b Fu(u)1543 1507 y Ft(N)1606 1495 y FB(\()p Fs(0)p FB(\).)k(Hence,)725 1702 y Fv(E)780 1610 y Fn(\020)830 1634 y(\000)882 1702 y FB(log)988 1634 y Fn(\000)1026 1702 y Fs(1)18 b(+)g Fr(k)p Fu(u)1259 1714 y Ft(N)1321 1702 y FB(\()p Fu(t)p FB(\))p Fr(k)1449 1667 y Fq(2)1485 1634 y Fn(\001\001)1561 1652 y Fq(2)1612 1606 y Fn(\014)1612 1656 y(\014)1612 1706 y(\014)1654 1702 y Fu(u)1702 1714 y Ft(N)1764 1702 y FB(\()p Fs(0)p FB(\))1862 1610 y Fn(\021)1934 1702 y Fr(\024)2021 1610 y Fn(\020)2071 1702 y FB(log)2191 1634 y Fn(\000)2229 1702 y Fs(1)g(+)g Fr(k)p Fu(u)2462 1714 y Ft(N)2524 1702 y FB(\()p Fs(0)p FB(\))p Fr(k)2664 1667 y Fq(2)2700 1634 y Fn(\001)2738 1610 y(\021)2787 1627 y Fq(2)3220 1702 y FB(\(4.5\))1007 1863 y Fs(+)g(2)1132 1796 y Fn(\000)1169 1863 y Fu(")g Fs(+)g FB(log)o(\()p Fs(1)g(+)g Fu(")p FB(\))g Fr(\000)g Fu(\013t)h Fs(+)f Fv(E)p Fu(W)2084 1829 y Ft(N)2072 1883 y Fm([)p Fq(0)p Ft(;t)p Fm(])2193 1796 y Fn(\001)2250 1863 y Fr(\001)g FB(log)2411 1796 y Fn(\000)2449 1863 y Fs(1)g(+)g Fr(k)p Fu(u)2682 1829 y Ft(N)2744 1863 y FB(\()p Fs(0)p FB(\))p Fr(k)2884 1829 y Fq(2)2920 1796 y Fn(\001)2976 1863 y Fs(+)g Fu(C)30 b FB(,)515 2030 y(for)21 b(some)g(constant)g Fu(C)28 b FB(depending)19 b(on)i Fu(")g FB(and)g(on)g(the)h(parameters) e(of)h(the)h(problem,)d(b)n(ut)j(not)f(on)515 2130 y Fu(N)9 b FB(.)639 2229 y(At)21 b(this)g(point,)e(we)i(choose)e(\002rst) i Fu(t)g FB(suf)n(\002ciently)e(small)h(such)g(that)1690 2431 y Fv(E)p Fu(W)1835 2396 y Ft(N)1823 2451 y Fm([)p Fq(0)p Ft(;t)p Fm(])1967 2431 y Fr(\024)2065 2374 y Fu(\013t)p 2065 2411 84 4 v 2086 2488 a Fs(2)2181 2431 y Fu(:)515 2620 y FB(This)g(can)g(be)g(done)f(uniformly)f(in)j Fu(N)29 b FB(by)20 b(Lemma)f(6.1.)25 b(Then)19 b(\002x)h Fu(")h FB(so)f(small)h(such)f(that)1587 2821 y Fu(")e Fs(+)g FB(log)o(\()p Fs(1)g(+)g Fu(")p FB(\))k Fr(\024)2191 2765 y Fu(\013t)p 2191 2802 V 2212 2878 a Fs(4)2284 2821 y Fu(:)515 3011 y FB(T)-7 b(aking)19 b(e)o(xpectations)f(on)h(both)g (sides)i(of)e(\(4.5\))g(and)g(using)g(the)h(stationarity)f(of)h Fu(u)2919 3023 y Ft(N)2981 3011 y FB(\()p Fu(t)p FB(\),)g(we)g(ha)n(v)o (e)1460 3218 y Fv(E)14 b FB(log)1650 3150 y Fn(\000)1688 3218 y Fs(1)k(+)g Fr(k)p Fu(u)1921 3230 y Ft(N)1983 3218 y FB(\()p Fs(0)p FB(\))p Fr(k)2123 3183 y Fq(2)2158 3150 y Fn(\001)2219 3218 y Fr(\024)2326 3162 y Fu(C)p 2317 3199 V 2317 3275 a(\013t)515 3407 y FB(for)h(\002x)o(ed)h Fu(t)h FB(suf)n(\002ciently)e(small,)h(therefore)f(concluding)e(the)k (proof)d(of)i(Theorem)f(4.1.)3379 3407 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 3407 a 515 3597 a Fv(Remark)h(4.2)40 b FB(Theorem)26 b(4.1)h(establishes)h (only)f(a)h(uniform)d(bound)h(\(in)h Fu(t)p FB(\))h(for)f(stationary)g (so-)515 3697 y(lutions)e(of)g(our)g(problem.)38 b(Estimates)26 b(lik)o(e)g(\(4.4\))e(or)h(\(4.5\))f(are)i(not)f(suf)n(\002cient)g(to)g (get)h(uniform)515 3796 y(bounds)18 b(for)i(arbitrary)e(solutions,)i (and)g(this)g(question)f(remains)h(open.)639 3958 y(Let)h(us)f(no)n(w)g (turn)f(to)i(the)515 4125 y Fp(Pr)l(oof)f(of)g(Theor)m(em)g(3.1.)40 b FB(Using)20 b(\(4.2\))f(we)i(obtain)e(after)h(inte)o(gration)915 4236 y Fn(Z)998 4257 y Fq(1)961 4425 y(0)1049 4349 y Fr(k)p Fu(@)1140 4315 y Fq(2)1135 4370 y Ft(x)1176 4349 y Fu(v)1216 4361 y Ft(N)1279 4349 y FB(\()p Fu(t)p FB(\))p Fr(k)1407 4315 y Fq(2)1444 4349 y Fu(dt)j Fr(\024)f Fu(C)1707 4236 y Fn(Z)1790 4257 y Fq(1)1753 4425 y(0)1841 4349 y Fr(k)p Fu(v)1923 4361 y Ft(N)1985 4349 y FB(\()p Fu(t)p FB(\))p Fr(k)2113 4315 y Fq(2)2149 4349 y FB(\()p Fs(1)c(+)g Fr(k)p Fu(@)2406 4361 y Ft(x)2447 4349 y Fu(W)2537 4315 y Ft(N)2525 4370 y(A)2600 4349 y FB(\()p Fu(t)p FB(\))p Fr(k)2728 4315 y Fq(4)2728 4370 y Fi(1)2797 4349 y FB(\))c Fu(dt)308 b FB(\(4.6\))1618 4588 y Fs(+)18 b Fu(C)1780 4475 y Fn(Z)1863 4495 y Fq(1)1826 4663 y(0)1914 4588 y Fr(k)p Fu(@)2000 4600 y Ft(x)2042 4588 y Fu(W)2132 4554 y Ft(N)2120 4608 y(A)2195 4588 y FB(\()p Fu(t)p FB(\))p Fr(k)2323 4554 y Fq(4)2323 4608 y(4)2373 4588 y Fu(dt)g Fs(+)g Fu(C)6 b Fr(k)p Fu(v)2694 4600 y Ft(N)2757 4588 y FB(\()p Fs(0)p FB(\))p Fr(k)2897 4554 y Fq(2)2956 4588 y Fu(:)515 4807 y FB(Using)30 b(Y)-9 b(oung')k(s)30 b(inequality)f(and)h(the)g(Sobole)n(v)g(embedding)e(of)i Fu(H)2572 4777 y Fq(1)2640 4807 y FB(into)g(L)2840 4777 y Fi(1)2910 4807 y FB(,)k(we)d(ha)n(v)o(e)f(the)515 4907 y(bound)1184 5006 y Fr(k)p Fu(u)1274 5018 y Ft(N)1336 5006 y Fr(k)1378 4972 y Fq(2)1378 5029 y Fi(C)1417 5012 y Fg(1)1476 5006 y Fr(\024)22 b Fu(C)6 b Fr(k)p Fu(@)1719 4972 y Fq(2)1714 5027 y Ft(x)1756 5006 y Fu(v)1796 5018 y Ft(N)1859 5006 y Fr(k)1901 4972 y Fq(2)1957 5006 y Fs(+)18 b Fu(C)6 b Fr(k)p Fu(@)2191 5018 y Ft(x)2232 5006 y Fu(W)2322 4972 y Ft(N)2310 5027 y(A)2385 5006 y Fr(k)2427 4972 y Fq(4)2427 5027 y Fi(1)2516 5006 y Fs(+)18 b Fu(C)29 b(:)p eop %%Page: 9 9 9 8 bop 517 232 a FB(T)t Fo(H)t(E)24 b(S)t(T)n(A)t(B)t(L)t(E)g(C)t(A)t (S)t(E)2176 b Fv(9)515 523 y FB(This)20 b(yields)649 643 y Fn(Z)732 664 y Fq(1)695 832 y(0)769 756 y Fr(k)p Fu(u)859 768 y Ft(N)921 756 y FB(\()p Fu(t)p FB(\))p Fr(k)1049 722 y Fq(2)1049 778 y Fi(C)1088 762 y Fg(1)1138 756 y Fu(dt)j Fr(\024)f Fu(C)1401 643 y Fn(Z)1484 664 y Fq(1)1447 832 y(0)1535 756 y Fr(k)p Fu(v)1617 768 y Ft(N)1679 756 y FB(\()p Fu(t)p FB(\))p Fr(k)1807 722 y Fq(2)1843 689 y Fn(\000)1881 756 y Fs(1)c(+)g Fr(k)p Fu(@)2110 768 y Ft(x)2152 756 y Fu(W)2242 722 y Ft(N)2230 777 y(A)2305 756 y FB(\()p Fu(t)p FB(\))p Fr(k)2433 722 y Fq(4)2433 777 y Fi(1)2502 689 y Fn(\001)2554 756 y Fu(dt)871 995 y Fs(+)g Fu(C)1033 882 y Fn(Z)1116 902 y Fq(1)1079 1070 y(0)1167 995 y Fr(k)p Fu(@)1253 1007 y Ft(x)1294 995 y Fu(W)1384 961 y Ft(N)1372 1015 y(A)1447 995 y FB(\()p Fu(t)p FB(\))p Fr(k)1575 961 y Fq(4)1575 1015 y(4)1625 995 y Fu(dt)h Fs(+)f Fu(C)25 b Fs(+)18 b Fu(C)6 b Fr(k)p Fu(u)2122 1007 y Ft(N)2184 995 y FB(\()p Fs(0)p FB(\))p Fr(k)2324 961 y Fq(2)792 1234 y Fr(\024)23 b Fu(C)959 1121 y Fn(Z)1042 1141 y Fq(1)1005 1309 y(0)1093 1141 y Fn(\020)1143 1234 y Fu(e)1182 1199 y Fi(\000)p Ft(\013t)p Fq(+)p Ft(W)1424 1174 y Ff(N)1415 1216 y Fe([)p Fg(0)p Ff(;t)p Fe(])1522 1234 y Fr(k)p Fu(u)1612 1246 y Ft(N)1674 1234 y FB(\()p Fs(0)p FB(\))p Fr(k)1814 1199 y Fq(2)1868 1234 y Fs(+)18 b Fu(e)1990 1199 y Ft(W)2061 1174 y Ff(N)2052 1216 y Fe([)p Fg(0)p Ff(;t)p Fe(])2158 1166 y Fn(\000)2197 1234 y Fu(W)2287 1199 y Ft(N)2275 1254 y Fm([)p Fq(0)p Ft(;t)p Fm(])2414 1234 y Fs(+)g Fu(t)2527 1166 y Fn(\001)2565 1141 y(\021)871 1472 y Fr(\002)954 1405 y Fn(\000)992 1472 y Fs(1)g(+)g Fr(k)p Fu(@)1221 1484 y Ft(x)1262 1472 y Fu(W)1352 1438 y Ft(N)1340 1493 y(A)1415 1472 y FB(\()p Fu(t)p FB(\))p Fr(k)1543 1438 y Fq(4)1543 1493 y Fi(1)1612 1405 y Fn(\001)1664 1472 y Fu(dt)h Fs(+)f Fu(C)1918 1359 y Fn(Z)2001 1380 y Fq(1)1964 1548 y(0)2052 1472 y Fr(k)p Fu(@)2138 1484 y Ft(x)2179 1472 y Fu(W)2269 1438 y Ft(N)2257 1493 y(A)2332 1472 y FB(\()p Fu(t)p FB(\))p Fr(k)2460 1438 y Fq(4)2460 1493 y(4)2510 1472 y Fu(dt)h Fs(+)f Fu(C)25 b Fs(+)18 b Fu(C)6 b Fr(k)p Fu(u)3007 1484 y Ft(N)3069 1472 y FB(\()p Fs(0)p FB(\))p Fr(k)3209 1438 y Fq(2)792 1677 y Fr(\024)23 b Fu(C)945 1585 y Fn(\020)995 1609 y(\000)1033 1677 y Fu(e)1072 1642 y Ft(W)1143 1617 y Ff(N)1134 1659 y Fe([)p Fg(0)p Ff(;)p Fg(1)p Fe(])1264 1677 y Fs(+)18 b(1)1389 1609 y Fn(\001)1427 1677 y Fr(k)p Fu(u)1517 1689 y Ft(N)1579 1677 y FB(\()p Fs(0)p FB(\))p Fr(k)1719 1642 y Fq(2)1773 1677 y Fs(+)g Fu(e)1895 1642 y Ft(W)1966 1617 y Ff(N)1957 1659 y Fe([)p Fg(0)p Ff(;)p Fg(1)p Fe(])2068 1609 y Fn(\000)2106 1677 y Fu(W)2196 1642 y Ft(N)2184 1697 y Fm([)p Fq(0)p Ft(;)p Fq(1)p Fm(])2332 1677 y Fs(+)g(1)2457 1609 y Fn(\001)2513 1677 y Fs(+)g(1)2638 1585 y Fn(\021)2687 1609 y(\000)2725 1677 y Fs(1)g(+)g Fu(W)2958 1642 y Ft(N)2946 1697 y Fm([)p Fq(0)p Ft(;)p Fq(1)p Fm(])3075 1609 y Fn(\001)3136 1677 y FB(,)515 1872 y(where)h(we)i(used)f(\(4.3\).)j(Using)d(Theorem)f(4.1) g(and)h(Lemma)f(6.1,)h(we)g(immediately)f(obtain)1342 2105 y Fv(E)14 b FB(log)1531 1988 y Fn(\022)1592 1992 y(Z)1675 2013 y Fq(1)1638 2181 y(0)1726 2105 y Fr(k)p Fu(u)1816 2117 y Ft(N)1878 2105 y FB(\()p Fu(t)p FB(\))p Fr(k)2006 2071 y Fq(2)2006 2128 y Fi(C)2045 2111 y Fg(1)2080 2105 y Fu(dt)19 b Fs(+)f(1)2297 1988 y Fn(\023)2381 2105 y Fr(\024)k Fu(C)q(:)515 2326 y FB(Finally)-5 b(,)19 b(Jensen')-5 b(s)21 b(inequality)d(and)i(the)g(stationarity)g(of)g Fu(u)2232 2338 y Ft(N)2315 2326 y FB(yield)887 2559 y Fv(E)14 b FB(log)1076 2492 y Fn(\000)1114 2559 y Fr(k)p Fu(u)1204 2571 y Ft(N)1266 2559 y FB(\()p Fs(0)p FB(\))p Fr(k)1406 2525 y Fq(2)1406 2581 y Fi(C)1445 2565 y Fg(1)1498 2559 y Fs(+)k(1)1623 2492 y Fn(\001)1684 2559 y Fs(=)k Fv(E)1840 2446 y Fn(Z)1924 2467 y Fq(1)1887 2635 y(0)1975 2559 y FB(log)2095 2492 y Fn(\000)2133 2559 y Fr(k)p Fu(u)2223 2571 y Ft(N)2285 2559 y FB(\()p Fu(t)p FB(\))p Fr(k)2413 2525 y Fq(2)2413 2581 y Fi(C)2452 2565 y Fg(1)2505 2559 y Fs(+)c(1)2630 2492 y Fn(\001)2682 2559 y Fu(dt)1684 2798 y Fr(\024)k Fv(E)14 b FB(log)1960 2681 y Fn(\022)2022 2685 y(Z)2105 2705 y Fq(1)2068 2873 y(0)2156 2798 y Fr(k)p Fu(u)2246 2810 y Ft(N)2308 2798 y FB(\()p Fu(t)p FB(\))p Fr(k)2436 2764 y Fq(2)2436 2820 y Fi(C)2475 2804 y Fg(1)2510 2798 y Fu(dt)19 b Fs(+)f(1)2727 2681 y Fn(\023)2810 2798 y Fr(\024)23 b Fu(C)29 b FB(,)515 3018 y(concluding)17 b(the)k(proof)d(of)i(Theorem)f(3.1.)3379 3018 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 3018 a 639 3195 a FB(W)-7 b(e)19 b(no)n(w)e(turn)f(to)i(the)f(proof)f (of)h(Lemma)g(3.2.)23 b(This)18 b(proof)d(will)k(not)e(use)g(the)h (strict)g(ne)o(gati)n(vity)515 3294 y(of)25 b Fu(A)h FB(and)e(is)i(thus)f(still)i(v)n(alid)d(in)i(the)f(unstable)f(case.)40 b(Since)25 b(we)h(need)e(this)i(bound)d(only)i(for)f(a)515 3394 y(\002x)o(ed)19 b(time)i(interv)n(al)e([)p Fs(0)p Fu(;)14 b(T)e FB(],)18 b(we)j(can)f(bound)e(the)i(terms)h(in)f(a)g (rather)g(crude)f(w)o(ay)-5 b(.)515 3555 y Fp(Pr)l(oof)20 b(of)g(Lemma)g(3.2.)40 b FB(De\002ne)1430 3732 y Fu(W)1520 3697 y Ft(N)1508 3752 y(T)1606 3732 y Fs(:=)70 b FB(sup)1717 3803 y Ft(t)p Fi(2)p Fm([)p Fq(0)p Ft(;T)9 b Fm(])1940 3732 y Fr(k)p Fu(@)2026 3744 y Ft(x)2067 3732 y Fu(W)2157 3697 y Ft(N)2145 3752 y(A)2220 3732 y FB(\()p Fu(t)p FB(\))p Fr(k)2348 3697 y Fq(4)2348 3752 y Fi(1)2441 3732 y Fu(:)515 3964 y FB(Using)23 b(the)h(f)o(actorization)e(method)g(and)i (Sobole)n(v)e(embedding)f(it)k(is)f(straightforw)o(ard)e(to)h(check)515 4064 y(that)j Fv(E)p Fu(W)811 4034 y Ft(N)799 4087 y(T)909 4064 y Fu(<)34 b(C)f FB(uniformly)24 b(in)i Fu(N)9 b FB(.)44 b(This)27 b(result)f(is)h(established)f(completely)f(analogous) f(to)515 4164 y([BMPS01)o(,)33 b(Lemma)c(5.1].)53 b(Note)30 b(that)g(the)g(uniformity)e(in)i Fu(N)40 b FB(is)31 b(not)e(tri)n (vial,)j(as)f(the)f(f)o(amily)515 4263 y Fr(f)p Fs(\005)619 4233 y Ft(N)681 4263 y Fr(g)723 4275 y Ft(N)6 b Fi(2)p Fh(N)893 4263 y FB(is)22 b(not)d(uniformly)f(bounded)g(as)j(operators)e (on)g Fu(L)2336 4233 y Fi(1)2427 4263 y FB(or)h Fr(C)2566 4233 y Fq(0)2603 4263 y FB(.)639 4363 y(Using)h(this)h(and)f(the)g (assumption)f(on)h Fr(k)p Fu(u)1861 4375 y Ft(N)1923 4363 y FB(\()p Fs(0)p FB(\))p Fr(k)p FB(,)f(we)i(see)g(that)f(for)g(e)n (v)o(ery)f Fu(")25 b(>)f Fs(0)e FB(there)e(is)j(an)515 4463 y Fu(R)g(>)g Fs(0)d FB(such)g(that)1175 4639 y Fv(P)1226 4572 y Fn(\000)1264 4639 y Fu(W)1354 4605 y Ft(N)1342 4660 y(T)1440 4639 y Fu(<)j(R)83 b FB(and)f Fr(k)p Fu(u)1967 4651 y Ft(N)2029 4639 y FB(\()p Fs(0)p FB(\))p Fr(k)22 b Fu(<)g(R)2342 4572 y Fn(\001)2403 4639 y Fu(>)h Fs(1)18 b Fr(\000)g Fu(")k(:)502 b FB(\(4.7\))515 4815 y(Combining)18 b(\(4.7\))h(and)h(\(4.3\),)e(we)j(see)g(that)f(with)g(probability)e (lar)o(ger)h(than)h Fs(1)e Fr(\000)g Fu(")i FB(one)g(has)1141 5006 y Fr(k)p Fu(v)1223 5018 y Ft(N)1286 5006 y FB(\()p Fu(t)p FB(\))p Fr(k)1414 4972 y Fq(2)1473 5006 y Fr(\024)j Fu(e)1600 4972 y Ft(W)1671 4947 y Ff(N)1662 4989 y(T)1728 5006 y Fr(k)p Fu(v)1810 5018 y Ft(N)1873 5006 y FB(\()p Fs(0)p FB(\))p Fr(k)2013 4972 y Fq(2)2067 5006 y Fs(+)18 b Fu(C)6 b(e)2254 4972 y Ft(W)2325 4947 y Ff(N)2316 4989 y(T)2383 5006 y FB(\()p Fu(W)2501 4972 y Ft(N)2489 5027 y(T)2582 5006 y Fs(+)18 b Fu(T)12 b FB(\))p eop %%Page: 10 10 10 9 bop 517 232 a FB(T)t Fo(H)t(E)24 b(U)t(N)t(S)t(T)n(A)t(B)t(L)t(E)g (C)t(A)t(S)t(E)2031 b Fv(10)1473 523 y Fr(\024)23 b Fu(e)1600 489 y Ft(R)1654 523 y Fu(R)1718 489 y Fq(2)1773 523 y Fs(+)18 b Fu(C)6 b(e)1960 489 y Ft(R)2015 523 y FB(\()p Fu(R)18 b Fs(+)g Fu(T)12 b FB(\))22 b Fs(=:)h Fu(R)2493 489 y Fi(0)2539 523 y FB(,)515 697 y(for)c(an)o(y)h Fu(t)j Fr(2)g FB([)p Fs(0)p Fu(;)14 b(T)e FB(].)23 b(Using)d(\(4.6\))f(in)i (the)f(same)g(w)o(ay)-5 b(,)20 b(we)g(get)657 816 y Fn(Z)740 837 y Ft(T)703 1005 y Fq(0)807 929 y Fr(k)p Fu(@)898 895 y Fq(2)893 950 y Ft(x)934 929 y Fu(v)974 941 y Ft(N)1037 929 y FB(\()p Fu(t)p FB(\))p Fr(k)1165 895 y Fq(2)1224 929 y Fu(dt)k Fr(\024)e Fu(C)6 b Fr(k)p Fu(v)1555 941 y Ft(N)1618 929 y FB(\()p Fs(0)p FB(\))p Fr(k)1758 895 y Fq(2)1812 929 y Fs(+)18 b Fu(C)1974 816 y Fn(Z)2057 837 y Ft(T)2020 1005 y Fq(0)2123 929 y Fr(k)p Fu(v)2205 941 y Ft(N)2268 929 y FB(\()p Fu(t)p FB(\))p Fr(k)2396 895 y Fq(2)2432 929 y FB(\()p Fu(W)2550 895 y Ft(N)2538 950 y(T)2631 929 y Fs(+)g(1)p FB(\))p Fu(dt)g Fs(+)g Fu(C)6 b(T)12 b(W)3174 895 y Ft(N)3162 950 y(T)1321 1105 y Fr(\024)22 b Fu(C)6 b FB(\()p Fu(R)19 b Fs(+)f Fu(T)12 b(R)1791 1070 y Fi(0)1813 1105 y FB(\()p Fu(R)19 b Fs(+)f(1)p FB(\))g Fs(+)g Fu(T)12 b(R)q FB(\))21 b(,)515 1278 y(with)f (probability)e(lar)o(ger)h(than)h Fs(1)e Fr(\000)g Fu(")p FB(,)i(thus)g(concluding)e(the)i(proof)f(of)h(Lemma)f(3.2.)3379 1278 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 1278 a 515 1505 a FA(5)99 b(The)26 b(unstable)g(case)515 1679 y FB(This)j(section)g(deals)h(with)f(the)h(case)f(where)g(the)g (operator)f Fu(A)i FB(is)g(no)f(longer)f(strictly)i(ne)o(gati)n(v)o(e) 515 1779 y(de\002nite.)39 b(In)25 b(order)f(to)h(treat)g(this)h(case,)g (we)g(mak)o(e)e(use)h(of)g(a)h(trick)f(that)g(w)o(as)h(used)f(in)g ([NST85)n(,)515 1879 y(CEES93)o(])e(to)g(get)g(bounds)e(on)h(the)h (deterministic)f(K)o(uramoto-Si)n(v)n(ashinsk)o(y)c(equation.)32 b(It)23 b(turns)515 1978 y(out)j(that)h(the)g(present)f(model)f(is)j (suf)n(\002ciently)e(close)h(to)g(that)f(equation)g(to)g(mak)o(e)h (that)g(trick)f(go)515 2078 y(through.)j(Ne)n(v)o(ertheless,)22 b(we)g(can)h(only)e(treat)i(Neumann)d(boundary)g(conditions)h(\(which)g (is)j(the)515 2178 y(same)k(as)h(considering)e(the)h(restriction)g(on)g ([)p Fs(0)p Fu(;)14 b(L)p FB(])27 b(of)h(functions)f(that)h(are)h(e)n (v)o(en)e(and)h(periodic)515 2277 y(with)21 b(period)f Fs(2)p Fu(L)p FB(\).)27 b(In)21 b(a)g(similar)h(w)o(ay)f(we)g(can)g (treat)g(also)h(Dirichlet)f(boundary)d(conditions,)i(b)n(ut)515 2377 y(periodic)f(boundary)e(conditions)i(are)h(still)h(open.)639 2476 y(Most)d(of)e(the)h(proofs)f(are)h(similar)g(to)g(the)g(pre)n (vious)f(section,)h(so)g(we)g(will)h(only)e(state)i(the)f(main)515 2576 y(dif)n(ferences.)23 b(Instead)c(of)h(de\002ning)f Fu(v)1611 2588 y Ft(N)1695 2576 y FB(as)i(pre)n(viously)-5 b(,)18 b(we)i(de\002ne)g Fu(v)2551 2588 y Ft(N)2635 2576 y FB(by)1395 2750 y Fu(v)1435 2762 y Ft(N)1499 2750 y FB(\()p Fu(t)p FB(\))i Fs(=)h Fu(u)1743 2762 y Ft(N)1805 2750 y FB(\()p Fu(t)p FB(\))18 b Fr(\000)g Fu(W)2082 2715 y Ft(N)2070 2770 y(A)2145 2750 y FB(\()p Fu(t)p FB(\))g Fr(\000)g Fs(\010)2392 2762 y Ft(N)2478 2750 y FB(,)721 b(\(5.1\))515 2923 y(where)23 b Fs(\010)802 2935 y Ft(N)895 2923 y Fs(=)29 b(\005)1051 2935 y Ft(N)1115 2923 y Fs(\010)24 b FB(for)f(some)h(function)e Fs(\010)i FB(to)g(be)g(chosen)f(later)h(and)g Fu(W)2759 2893 y Ft(N)2747 2946 y(A)2846 2923 y FB(is)h(the)f(stochastic)515 3023 y(con)m(v)n(olution)f(de\002ned)i(in)h(the)g(pre)n(vious)e (section.)42 b(The)26 b(stochastic)f(process)h Fu(v)2857 3035 y Ft(N)2947 3023 y FB(then)f(satis\002es)515 3123 y(the)20 b(follo)n(wing)e(random)h(PDE:)901 3296 y Fu(@)945 3308 y Ft(t)974 3296 y Fu(v)1014 3308 y Ft(N)1100 3296 y Fs(=)k Fu(Av)1290 3308 y Ft(N)1372 3296 y Fs(+)18 b Fu(A)p Fs(\010)1577 3308 y Ft(N)1658 3296 y Fr(\000)g Fs(\005)1803 3308 y Ft(N)1867 3296 y Fu(@)1916 3262 y Fq(2)1911 3317 y Ft(x)1953 3296 y FB(\()p Fu(@)2025 3308 y Ft(x)2066 3296 y Fu(v)2106 3308 y Ft(N)2188 3296 y Fs(+)g Fu(@)2315 3308 y Ft(x)2357 3296 y Fu(W)2447 3262 y Ft(N)2435 3317 y(A)2528 3296 y Fs(+)g Fu(@)2655 3308 y Ft(x)2697 3296 y Fs(\005)2759 3308 y Ft(N)2822 3296 y Fs(\010)p FB(\))2910 3262 y Fq(2)2970 3296 y Fu(:)227 b FB(\(5.2\))515 3470 y(W)-7 b(e)21 b(can)f(re)n(write)g(this)h(as)575 3644 y Fu(@)619 3656 y Ft(t)649 3644 y Fu(v)689 3656 y Ft(N)775 3644 y Fs(=)885 3623 y(~)863 3644 y Fu(Av)965 3656 y Ft(N)1047 3644 y Fr(\000)d Fu(@)1179 3609 y Fq(2)1174 3664 y Ft(x)1216 3644 y FB(\()p Fu(@)1288 3656 y Ft(x)1329 3644 y Fu(v)1369 3656 y Ft(N)1451 3644 y Fs(+)g Fu(@)1578 3656 y Ft(x)1620 3644 y Fu(W)1710 3609 y Ft(N)1698 3664 y(A)1773 3644 y FB(\))1801 3609 y Fq(2)1856 3644 y Fr(\000)g Fu(@)1988 3609 y Fq(2)1983 3664 y Ft(x)2025 3644 y FB(\()p Fu(@)2097 3656 y Ft(x)2138 3644 y Fs(\005)2200 3656 y Ft(N)2264 3644 y Fs(\010)p FB(\))2352 3609 y Fq(2)2407 3644 y Fs(+)g Fu(A)p Fs(\010)h Fr(\000)f Fs(2)p Fu(@)2805 3609 y Fq(2)2800 3664 y Ft(x)2841 3644 y FB(\()p Fu(@)2913 3656 y Ft(x)2955 3644 y Fs(\010)c Fu(@)3073 3656 y Ft(x)3114 3644 y Fu(W)3192 3656 y Ft(A)3247 3644 y FB(\))22 b(,)515 3817 y(where)d(the)i(operator)1180 3796 y Fs(~)1158 3817 y Fu(A)g FB(is)g(de\002ned)e(as)1498 3970 y Fs(~)1476 3991 y Fu(A)q(v)26 b Fs(=)d Fu(Av)f Fr(\000)c Fs(2)p Fu(@)1991 3957 y Fq(2)1986 4012 y Ft(x)2027 3991 y FB(\()p Fu(@)2099 4003 y Ft(x)2141 3991 y Fs(\010)c Fu(@)2259 4003 y Ft(x)2300 3991 y Fu(v)s FB(\))23 b Fu(:)515 4165 y FB(Using)k(e)o(xactly)g(the)h(same)g(technique)e(as)i(in)g(the)g(pre) n(vious)e(section,)j(we)f(see)g(that)g(in)g(order)e(to)515 4264 y(get)f(uniform)e(bounds)g(on)h(the)h(Galerkin)f(approximations)e (of)j Fu(v)s FB(,)i(it)e(suf)n(\002ces)g(to)g(\002nd)g(a)g(smooth)515 4364 y(function)18 b Fs(\010)j FB(such)f(that)1591 4464 y Fr(h)p Fu(v)s(;)1726 4443 y Fs(~)1703 4464 y Fu(A)q(v)s Fr(i)k(\024)e(\000)p Fu(c)p Fr(k)p Fu(@)2144 4429 y Fq(2)2139 4484 y Ft(x)2180 4464 y Fu(v)s Fr(k)2265 4429 y Fq(2)3220 4464 y FB(\(5.3\))515 4608 y(for)f(some)h(constant)g Fu(c)k(>)h Fs(0)p FB(.)k(Using)22 b(this)g(function,)f(it)i(is)g(easy)f (to)h(v)o(erify)d(that)j(the)f(assertions)g(of)515 4707 y(Proposition)13 b(2.2)i(and)g(Theorems)f(3.1)g(and)h(4.1)g(hold)f(in)i (the)f(unstable)f(case,)j(too.)23 b(The)15 b(only)f(major)515 4807 y(changes)22 b(appear)h(in)g(the)h(v)n(alues)f(of)g(the)h (constants,)f(which)g(do)h(no)n(w)f(depend)e(on)j(the)f(choice)g(of)515 4907 y Fs(\010)p FB(.)h(W)-7 b(e)19 b(will)f(therefore)e(not)i(go)f (through)e(the)j(proofs)e(of)h(these)h(assertions)g(for)f(the)g (unstable)g(case,)515 5006 y(b)n(ut)j(we)g(will)h(sk)o(etch)f(ho)n(w)g (to)g(\002nd)g(a)h(function)d Fs(\010)j FB(such)f(that)2312 4985 y Fs(~)2290 5006 y Fu(A)h FB(is)h(strictly)e(ne)o(gati)n(v)o(e)e (de\002nite.)p eop %%Page: 11 11 11 10 bop 517 232 a FB(T)t Fo(H)t(E)24 b(U)t(N)t(S)t(T)n(A)t(B)t(L)t(E) g(C)t(A)t(S)t(E)2031 b Fv(11)639 523 y FB(Inte)o(grating)18 b(by)i(parts,)g(we)g(see)h(that)f(the)h(bilinear)e(form)g(\(5.3\))g (can)h(be)g(written)g(as)1051 691 y Fr(h)p Fu(v)s(;)1185 670 y Fs(~)1163 691 y Fu(A)q(v)s Fr(i)j Fs(=)g Fr(\000k)p Fu(@)1568 657 y Fq(2)1563 712 y Ft(x)1604 691 y Fu(v)s Fr(k)1689 657 y Fq(2)1744 691 y Fr(\000)18 b Fu(\027)5 b Fr(k)p Fu(@)1959 703 y Ft(x)2001 691 y Fu(v)s Fr(k)2086 657 y Fq(2)2141 691 y Fr(\000)18 b(h)p Fu(@)2300 703 y Ft(x)2343 691 y Fu(v)s(;)c FB(\()p Fu(@)2500 657 y Fq(2)2495 712 y Ft(x)2536 691 y Fs(\010)p FB(\))g Fu(@)2682 703 y Ft(x)2724 691 y Fu(v)s Fr(i)23 b FB(,)377 b(\(5.4\))515 859 y(where)27 b Fu(\027)33 b FB(is)c(ne)o(gati)n(v)o(e.)45 b(The)27 b(problem)f(is)i(therefore)e(reduced)g(to)i(\002nding)e(a)i (smooth)f(periodic)515 959 y(function)18 b Fs(\010)j FB(such)f(that)g(the)g(Schr)7 b(\250)-35 b(odinger)18 b(operator)1532 1127 y Fr(H)1602 1139 y Fq(\010)1677 1127 y Fs(=)k Fr(\000)1839 1094 y Fq(1)p 1839 1108 34 4 v 1839 1156 a(2)1882 1127 y Fu(@)1931 1097 y Fq(2)1926 1148 y Ft(x)1986 1127 y Fs(+)c Fu(@)2118 1097 y Fq(2)2113 1148 y Ft(x)2155 1127 y Fs(\010)p FB(\()p Fu(x)p FB(\))23 b(,)515 1295 y(with)d Fp(Diric)o(hlet)g FB(boundary)d(conditions)i (satis\002es)i Fr(h)p Fu(u;)14 b Fr(H)2173 1307 y Fq(\010)2225 1295 y Fu(u)p Fr(i)22 b(\025)h(j)p Fu(\027)5 b Fr(jk)p Fu(u)p Fr(k)2639 1265 y Fq(2)2696 1295 y FB(for)19 b(all)i(functions)d Fu(u)i FB(in)515 1395 y(its)27 b(domain.)41 b(The)26 b(idea)g(appearing)e(in)i([NST85)n(])h(is)g(to)f(choose)f Fs(\010)i FB(such)e(that,)j(a)o(w)o(ay)e(from)f(the)515 1494 y(boundary)-5 b(,)24 b Fu(@)921 1464 y Fq(2)916 1515 y Ft(x)958 1494 y Fs(\010)j FB(is)g(for)e(all)i(practical)f (purposes)f(constant)g(and)h(suf)n(\002ciently)f(lar)o(ge)g(\(say)h (equal)515 1594 y(to)31 b(about)f Fs(2)p Fr(j)p Fu(\027)5 b Fr(j)p FB(\).)58 b(The)31 b(problem)f(is)i(that,)h(in)f(order)e(for)g (\(5.2\))g(to)h(hold,)i Fs(\010)f FB(has)f(to)h(belong)d(to)515 1694 y Fu(D)r FB(\()p Fu(A)p FB(\))19 b(and)f(must)g(therefore)f (satisfy)i(the)g(same)g(boundary)c(conditions)i(as)j Fu(u)p FB(.)k(As)c(a)f(consequence)515 1806 y Fu(@)564 1776 y Fq(2)559 1827 y Ft(x)601 1806 y Fs(\010)31 b FB(must)g(satisfy) 1133 1739 y Fn(R)1189 1760 y Ft(L)1173 1836 y Fq(0)1252 1806 y Fu(@)1301 1776 y Fq(2)1296 1827 y Ft(x)1338 1806 y Fs(\010)p FB(\()p Fu(s)p FB(\))14 b Fu(dx)43 b Fs(=)g(0)p FB(,)33 b(which)d(is)i(of)f(course)f(impossible)h(for)f(a)h(constant) 515 1906 y(\(non-zero\))19 b(function.)29 b(Looking)20 b(at)i(\(5.4\),)f(we)i(notice)f(that)g Fu(@)2348 1876 y Fq(2)2343 1926 y Ft(x)2385 1906 y Fs(\010)p FB(\()p Fu(x)p FB(\))k Fs(=)g(2)p Fr(j)p Fu(\027)5 b Fr(j)p Fs(\()q(1)17 b Fr(\000)i Fu(\016)s FB(\()p Fu(x)p FB(\))p Fs(\))j FB(w)o(ould)515 2005 y(formally)i(\002t)i(our)e(needs,)i(since)g(the)f (delta-peak)f(is)i(inte)o(grated)e(against)g Fu(@)2733 2017 y Ft(x)2775 2005 y Fu(v)s FB(,)j(which)e(v)n(anishes)515 2105 y(at)f(the)g(boundaries,)f(due)g(to)i(the)f(Neumann)e(conditions.) 35 b(The)24 b(function)e Fs(\010)i FB(obtained)f(this)i(w)o(ay)515 2205 y(does)16 b(of)h(course)f(not)h(belong)e(to)j Fu(D)r FB(\()p Fu(A)p FB(\),)f(so)g(we)h(look)e(for)g(an)h(approximation)d(of) i(it)i(which)f(is)g(more)515 2304 y(re)o(gular)-5 b(.)639 2404 y(Since)20 b Fs(\010)h FB(satis\002es)h(Neumann)c(boundary)g (conditions,)g(it)j(is)g(natural)f(to)g(write)g(it)h(as)1394 2636 y Fs(\010)p FB(\()p Fu(x)p FB(\))i Fs(=)g(2)p Fr(j)p Fu(\027)5 b Fr(j)1845 2532 y Fi(1)1818 2557 y Fn(X)1816 2733 y Ft(n)p Fq(=1)1955 2636 y Fu(')2009 2648 y Ft(n)2082 2636 y FB(cos)2192 2544 y Fn(\020)2252 2580 y Fs(2)p Fu(\031)s(n)p 2252 2617 142 4 v 2295 2693 a(L)2404 2544 y Fn(\021)2476 2636 y Fu(:)721 b FB(\(5.5\))515 2874 y(\(The)19 b(sum)h(starts)h(at)g Fs(1)f FB(because)g(we)h(are)f (interested)f(only)h(in)g(functions)f(with)h(v)n(anishing)f(mean.\))515 2974 y(If)d(we)i(choose)e Fu(')1004 2986 y Ft(n)1072 2974 y Fs(=)23 b(2)p Fu(n)1252 2944 y Fi(\000)p Fq(2)1340 2974 y FB(,)18 b(we)f(see)h(that)f Fu(@)1808 2944 y Fq(2)1803 2994 y Ft(x)1845 2974 y Fs(\010)g FB(is)h(gi)n(v)o(en)d(by)i Fu(@)2343 2944 y Fq(2)2338 2994 y Ft(x)2380 2974 y Fs(\010)p FB(\()p Fu(x)p FB(\))22 b Fs(=)h(2)p Fr(j)p Fu(\027)5 b Fr(j)p Fs(\(1)18 b Fr(\000)g Fu(\016)s FB(\()p Fu(x)p FB(\))p Fs(\))p FB(,)g(which)515 3074 y(is)j(what)f(we)h(w)o(ould)e (lik)o(e)i(to)f(approximate.)i(In)e(order)f(to)i(get)f(a)g(re)o(gular)f (function,)f(we)j(de\002ne)1362 3289 y Fu( )1416 3301 y Ft(n)1485 3289 y Fs(:=)h Fu(n)1645 3254 y Fq(2)1682 3289 y Fu(')1736 3301 y Ft(n)1805 3289 y Fs(=)1893 3172 y Fn(\032)1969 3240 y Fs(2)82 b FB(for)20 b Fu(n)j Fr(\024)f Fs(2)p Fu(n)2463 3252 y Fi(\003)2501 3240 y FB(,)1969 3340 y Fs(0)82 b FB(for)20 b Fu(n)j(>)f Fs(2)p Fu(n)2463 3352 y Fi(\003)2501 3340 y FB(,)3220 3289 y(\(5.6\))515 3501 y(where)d Fu(n)788 3513 y Fi(\003)847 3501 y FB(is)i(some)f(\(suf) n(\002ciently)f(lar)o(ge\))f(constant)i(to)g(be)g(\002x)o(ed)f(later)h (on.)25 b(W)m(ith)20 b(this)h(de\002nition,)515 3601 y(we)f(ha)n(v)o(e:)515 3791 y Fv(Pr)o(oposition)f(5.1)40 b Fp(F)-9 b(or)18 b(e)o(very)g Fu(L;)c(C)29 b(>)22 b Fs(0)p Fp(,)d(ther)m(e)f(e)n(xists)h(a)f(value)f Fu(n)2467 3803 y Fi(\003)2528 3791 y Fu(>)23 b Fs(0)18 b Fp(suc)o(h)f(that)h(the) g(quadr)o(a-)515 3891 y(tic)j(form)f Fr(H)866 3903 y Fq(\010)939 3891 y Fp(with)g Fs(\010)h Fp(de\002ned)e(as)h(in)g (\(5.5\))f(and)h(\(5.6\),)e(satis\002es)1384 4004 y Fn(Z)1467 4025 y Ft(L)1430 4193 y Fq(0)1531 4117 y Fu(u)p FB(\()p Fu(x)p FB(\))p Fs(\()o Fr(H)1783 4129 y Fq(\010)1835 4117 y Fu(u)p Fs(\))p FB(\()p Fu(x)p FB(\))c Fu(dx)23 b Fr(\025)g Fu(C)6 b Fr(k)p Fu(u)p Fr(k)2430 4083 y Fq(2)2489 4117 y Fp(,)515 4326 y(for)20 b(e)o(very)h Fu(u)f Fp(in)g(the)g(domain) f(of)i Fr(H)1533 4338 y Fq(\010)1584 4326 y Fp(.)515 4517 y Fv(Remark)f(5.2)40 b FB(Notice)21 b(that)g(the)g(function)e Fs(\010)i FB(de\002ned)f(by)g(\(5.5\))g(and)g(\(5.6\))f(is)j(actually)e (analytic,)515 4616 y(so)25 b(the)f(e)o(xpressions)g(appearing)e(in)j (\(5.2\))e(and)h(containing)f Fs(\010)i FB(can)f(all)h(be)g(bounded)d (uniformly)515 4716 y(in)e Fu(N)30 b FB(\(not)19 b(in)h Fu(n)986 4728 y Fi(\003)1045 4716 y FB(of)g(course,)f(b)n(ut)i Fu(n)1569 4728 y Fi(\003)1627 4716 y FB(is)g(chosen)f(independently)d (of)j Fu(N)9 b FB(\).)515 4907 y Fv(Remark)20 b(5.3)40 b FB(As)29 b(in)e([CEES93)o(])h(we)g(could)e(choose)h(a)h(slo)n(w)g (decay)e(of)i Fu( )2771 4919 y Ft(n)2844 4907 y FB(for)f Fu(n)37 b(>)f Fs(2)p Fu(n)3249 4919 y Fi(\003)3314 4907 y FB(to)515 5006 y(optimize)19 b(the)h Fu(L)p FB(-dependence)d(of)j (our)f(bound,)f(b)n(ut)j(we)f(ne)o(glected)f(this)i(for)e(simplicity)-5 b(.)p eop %%Page: 12 12 12 11 bop 517 232 a FB(T)t Fo(E)t(C)t(H)t(N)t(I)t(C)t(A)t(L)25 b(E)t(S)t(T)t(I)t(M)t(A)m(T)t(E)t(S)1943 b Fv(12)515 523 y Fp(Pr)l(oof)o(.)40 b FB(Applying)18 b(the)i(ar)o(guments)d(of)i ([CEES93)o(,)h(Prop.)f(2.1],)f(we)i(see)g(that)g(it)g(suf)n(\002ces)g (to)g(sho)n(w)515 623 y(that)g(the)g(quantity)842 856 y Fs(\000)j(:=)1084 777 y Fn(X)1028 956 y Ft(k)q(>m>)p Fq(0)1284 800 y Fr(j)p Fu( )1361 812 y Ft(k)q Fq(+)p Ft(m)1531 800 y Fr(\000)18 b Fu( )1668 812 y Ft(k)q Fi(\000)p Ft(m)1819 800 y Fr(j)1842 769 y Fq(2)p 1284 837 559 4 v 1450 913 a Fu(E)1511 925 y Ft(k)1552 913 y Fu(E)1613 925 y Ft(m)1875 856 y FB(,)83 b(with)g Fu(E)2271 868 y Ft(n)2339 856 y Fs(=)23 b Fu(\013n)2530 821 y Fq(2)2591 856 y FB(,)82 b Fu(\013)24 b Fs(=)2868 800 y(2)p Fu(\031)2960 769 y Fq(2)p 2868 837 130 4 v 2886 913 a Fu(L)2943 889 y Fq(2)3031 856 y FB(,)515 1116 y(can)15 b(be)h(made)f(arbitrarily)f (small)i(by)f(choosing)f Fu(n)1946 1128 y Fi(\003)2000 1116 y FB(suf)n(\002ciently)h(lar)o(ge.)22 b(The)15 b(only)g(non-v)n (anishing)515 1216 y(terms)h(of)h(this)g(sum)f(are)h(those)f(where)g Fs(0)22 b Fr(\024)h Fu(k)8 b Fr(\000)d Fu(m)22 b Fr(\024)h Fs(2)p Fu(n)2174 1228 y Fi(\003)2229 1216 y FB(and)16 b Fu(k)8 b Fs(+)d Fu(m)22 b Fr(\025)g Fs(2)p Fu(n)2761 1228 y Fi(\003)2799 1216 y FB(.)i(W)-7 b(e)18 b(can)e(estimate)515 1315 y(these)k(terms)g(by)1013 1575 y Fs(\000)j Fr(\024)1210 1519 y Fs(4)p 1186 1556 91 4 v 1186 1632 a Fu(\013)1239 1608 y Fq(2)1300 1471 y Ft(n)1341 1479 y Fd(\003)1376 1471 y Fi(\000)p Fq(1)1321 1496 y Fn(X)1309 1672 y Ft(m)p Fq(=1)1519 1471 y(2)p Ft(n)1593 1479 y Fd(\003)1628 1471 y Fq(+)p Ft(m)1569 1496 y Fn(X)1475 1675 y Ft(k)q Fq(=2)p Ft(n)1636 1683 y Fd(\003)1671 1675 y Fi(\000)p Ft(m)1882 1519 y Fs(1)p 1806 1556 194 4 v 1806 1632 a Fu(m)1879 1608 y Fq(2)1916 1632 y Fu(k)1962 1608 y Fq(2)2028 1575 y Fs(+)2145 1519 y(4)p 2121 1556 91 4 v 2121 1632 a Fu(\013)2174 1608 y Fq(2)2295 1471 y Fi(1)2268 1496 y Fn(X)2235 1671 y Ft(m)p Fq(=)p Ft(n)2386 1679 y Fd(\003)2435 1471 y Fq(2)p Ft(n)2509 1479 y Fd(\003)2543 1471 y Fq(+)p Ft(m)2484 1496 y Fn(X)2471 1675 y Ft(k)q Fq(=)p Ft(m)2753 1519 y Fs(1)p 2677 1556 194 4 v 2677 1632 a Fu(m)2750 1608 y Fq(2)2787 1632 y Fu(k)2833 1608 y Fq(2)1088 1875 y Fr(\024)1254 1819 y Fs(8)p 1186 1856 179 4 v 1186 1932 a Fu(\013)1239 1908 y Fq(2)1277 1932 y Fu(n)1327 1944 y Fi(\003)1388 1771 y Ft(n)1429 1779 y Fd(\003)1464 1771 y Fi(\000)p Fq(1)1409 1796 y Fn(X)1397 1972 y Ft(m)p Fq(=1)1607 1819 y Fs(1)p 1573 1856 111 4 v 1573 1932 a Fu(m)1646 1908 y Fq(2)1711 1875 y Fs(+)1873 1819 y(8)p 1804 1856 179 4 v 1804 1932 a Fu(\013)1857 1908 y Fq(2)1895 1932 y Fu(n)1945 1944 y Fi(\003)2067 1772 y(1)2040 1796 y Fn(X)2007 1971 y Ft(m)p Fq(=)p Ft(n)2158 1979 y Fd(\003)2251 1819 y Fs(1)p 2216 1856 111 4 v 2216 1932 a Fu(m)2289 1908 y Fq(2)2359 1875 y Fr(\024)2503 1819 y Fs(4)p Fu(\031)2595 1789 y Fq(2)p 2457 1856 221 4 v 2457 1932 a Fs(3)p Fu(\013)2552 1908 y Fq(2)2589 1932 y Fu(n)2639 1944 y Fi(\003)2710 1875 y Fu(:)515 2136 y FB(In)e(both)g(sums,)h(we)g(used)g(the)g(f)o (act)g(that)f Fu(k)26 b FB(is)c(lar)o(ger)f(than)g Fu(n)2276 2148 y Fi(\003)2336 2136 y FB(and)h(that)f(there)h(are)f(less)i(than)e Fs(2)p Fu(n)3341 2148 y Fi(\003)515 2236 y FB(terms)30 b(in)f(the)h(inner)f(sum.)54 b(Thus,)31 b Fs(\000)g FB(can)e(clearly)g (be)h(made)f(arbitrarily)g(small)h(by)f(choosing)515 2336 y Fu(n)565 2348 y Fi(\003)626 2336 y FB(suf)n(\002ciently)22 b(lar)o(ge.)31 b(This)23 b(pro)o(v)o(es)e(Proposition)g(5.1)h(and)g (concludes)f(our)h(e)o(xposition)f(of)i(the)515 2435 y(unstable)c(case.)3379 2435 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 2435 a 515 2664 a FA(6)99 b(T)-9 b(echnical)26 b(estimates)515 2838 y FB(In)g(this)h (section,)g(we)g(pro)o(v)o(e)e(the)h(tw)o(o)h(technical)f(estimates)h (required)d(for)i(the)h(proof)e(of)h(Theo-)515 2938 y(rem)20 b(4.1)f(abo)o(v)o(e.)515 3137 y Fv(Lemma)i(6.1)40 b Fp(Ther)m(e)20 b(e)n(xists)i(a)e(constant)f Fu(C)27 b Fp(independent)18 b(of)i Fu(N)30 b Fp(suc)o(h)19 b(that)1525 3319 y Fv(E)p Fr(k)p Fu(@)1666 3331 y Ft(x)1708 3319 y Fu(W)1798 3285 y Ft(N)1786 3340 y(A)1861 3319 y FB(\()p Fu(t)p FB(\))p Fr(k)1989 3285 y Fq(4)1989 3340 y Fi(1)2081 3319 y Fr(\024)k Fu(C)6 b(t)2264 3285 y Fq(1)p Ft(=)p Fq(8)515 3502 y Fp(for)20 b(all)h Fu(t)i Fr(\024)f Fs(1)p Fp(.)515 3701 y Fv(Remark)e(6.2)40 b FB(The)23 b(po)n(wer)g Fs(1)p Fu(=)p Fs(8)f FB(in)i(the)f(abo)o(v)o(e)f(lemma)h(is)h(not)g(optimal)e (b)n(ut)i(it)g(is)g(suf)n(\002cient)f(for)515 3801 y(our)c(needs.)25 b(All)c(we)f(need)g(is)h Fv(E)p Fr(k)p Fu(@)1521 3813 y Ft(x)1562 3801 y Fu(W)1652 3771 y Ft(N)1640 3824 y(A)1715 3801 y FB(\()p Fu(t)p FB(\))p Fr(k)1843 3771 y Fq(4)1843 3822 y Fi(1)1936 3801 y Fs(=)h Fu(o)p FB(\()p Fs(1)p FB(\))e(uniformly)e(in)i Fu(N)9 b FB(.)515 4000 y Fv(Remark)20 b(6.3)40 b FB(The)21 b(constant)f(in)h(the)g(abo)o(v)o(e)f(lemma)g (depends)g(only)g(on)h(the)g(coef)n(\002cients)f(of)h(the)515 4100 y(problem)e(and)h(the)h(bound)e(on)h(the)h Fu(\013)1586 4112 y Ft(k)1627 4100 y FB(.)28 b(It)21 b(is)h(possible)e(to)h(allo)n (w)g(for)f(slo)n(wly)h(gro)n(wing)e Fu(\013)3116 4112 y Ft(k)3157 4100 y FB(,)i(using)515 4199 y(the)f(Sobole)n(v)f (embedding)e(of)j(L)1447 4169 y Fi(1)1538 4199 y FB(into)g(the)g (fractional)f(Sobole)n(v)g(space)h Fu(W)2735 4169 y Ft(s;p)2845 4199 y FB(for)g Fu(sp)i(>)h Fs(1)p FB(.)515 4382 y Fp(Pr)l(oof)o(.)40 b FB(F)o(or)22 b Fu(f)36 b Fr(2)28 b FB(L)1098 4352 y Fi(1)1168 4382 y FB(\([)p Fs(0)p Fu(;)14 b(L)p FB(]\))20 b(with)j(v)n(anishing)e(mean,)h(denote)f(by)i Fr(f)p Fu(f)2609 4394 y Ft(k)2649 4382 y Fr(g)2691 4394 y Ft(k)q Fi(2)p Fh(N)2841 4382 y FB(its)g(F)o(ourier)e(coef-)515 4482 y(\002cients.)35 b(Since)24 b(the)g(eigenfunctions)d Fu(e)1698 4494 y Ft(k)1762 4482 y FB(of)j Fu(A)g FB(are)g(uniformly)d (bounded)g(in)j(L)2863 4452 y Fi(1)2933 4482 y FB(,)h(we)f(ha)n(v)o(e)f (the)515 4581 y(follo)n(wing)18 b(estimate)j(on)f Fr(k)p Fu(f)9 b Fr(k)1392 4593 y Fi(1)1460 4581 y FB(:)622 4831 y Fr(k)p Fu(f)g Fr(k)756 4843 y Fi(1)848 4831 y Fr(\024)23 b Fu(C)1042 4727 y Fi(1)1015 4752 y Fn(X)1015 4930 y Ft(k)q Fq(=1)1149 4831 y Fr(j)p Fu(f)1213 4843 y Ft(k)1254 4831 y Fr(j)g(\024)g Fu(C)1453 4738 y Fn(\020)1530 4727 y Fi(1)1503 4752 y Fn(X)1503 4930 y Ft(k)q Fq(=1)1637 4831 y Fr(j)p Fu(k)s Fr(j)1729 4796 y Fq(1)p Ft(=)p Fq(2)1833 4831 y Fr(j)p Fu(f)1897 4843 y Ft(k)1938 4831 y Fr(j)1961 4796 y Fq(4)p Ft(=)p Fq(3)2065 4738 y Fn(\021)2115 4756 y Fq(3)p Ft(=)p Fq(4)2219 4738 y Fn(\020)2296 4727 y Fi(1)2269 4752 y Fn(X)2269 4930 y Ft(k)q Fq(=1)2404 4831 y Fr(j)p Fu(k)s Fr(j)2496 4796 y Fi(\000)p Fq(3)p Ft(=)p Fq(2)2652 4738 y Fn(\021)2701 4756 y Fq(1)p Ft(=)p Fq(4)2829 4831 y Fr(\024)f Fu(C)6 b Fr(k)p Fu(K)g(f)j Fr(k)3192 4843 y Fq(4)3251 4831 y FB(,)p eop %%Page: 13 13 13 12 bop 517 232 a FB(T)t Fo(E)t(C)t(H)t(N)t(I)t(C)t(A)t(L)25 b(E)t(S)t(T)t(I)t(M)t(A)m(T)t(E)t(S)1943 b Fv(13)515 523 y FB(where)24 b Fu(K)30 b FB(is)25 b(the)g(operator)d(that)j(acts)g (on)f(F)o(ourier)f(coef)n(\002cients)h(as)h(\()p Fu(K)6 b(f)j FB(\))2715 535 y Ft(k)2785 523 y Fs(=)31 b Fr(j)p Fu(k)s Fr(j)2973 493 y Fq(3)p Ft(=)p Fq(8)3077 523 y Fu(f)3118 535 y Ft(k)3158 523 y FB(.)39 b(Here)515 623 y(we)20 b(used)g(the)g(usual)g(isometry)g(between)f(L)1772 593 y Ft(p)1831 623 y FB(and)g Fu(`)2006 593 y Ft(q)2063 623 y FB(for)h Fu(p)2223 593 y Fi(\000)p Fq(1)2330 623 y Fs(+)e Fu(q)2453 593 y Fi(\000)p Fq(1)2565 623 y Fs(=)23 b(1)p FB(.)639 722 y(Denote)d(by)g Fu(\025)1052 734 y Ft(k)1115 722 y FB(the)g(eigen)m(v)n(alues)f(corresponding)e(to)k(the)f (eigenfunctions)e Fu(e)2898 734 y Ft(k)2939 722 y FB(.)26 b(By)21 b(the)g(de\002-)515 822 y(nition)e(of)h Fu(A)p FB(,)h(there)f(e)o(xist)g(constants)g Fu(c)1652 834 y Ft(i)1700 822 y FB(such)g(that)1045 1002 y Fu(c)1081 1014 y Fq(1)1118 1002 y Fu(k)1164 968 y Fq(4)1224 1002 y Fr(\024)j(\000)p Fu(\025)1425 1014 y Ft(k)1489 1002 y Fr(\024)g Fu(c)1613 1014 y Fq(2)1650 1002 y Fu(k)1696 968 y Fq(4)1756 1002 y FB(,)166 b Fr(j)p Fs(\()p Fu(K)6 b(@)2119 1014 y Ft(x)2160 1002 y Fu(e)2199 1014 y Ft(k)2240 1002 y Fs(\))p FB(\()p Fu(x)p FB(\))p Fr(j)2398 968 y Fq(2)2458 1002 y Fr(\024)23 b Fu(c)2582 1014 y Fq(3)2619 1002 y Fu(k)2665 968 y Fq(11)p Ft(=)p Fq(4)2826 1002 y Fu(:)515 1182 y FB(W)m(ith)16 b(these)h(notations,)1225 1114 y Fn(\000)1263 1182 y Fu(K)6 b(@)1384 1194 y Ft(x)1425 1182 y Fu(W)1515 1152 y Ft(N)1503 1205 y(A)1578 1114 y Fn(\001)1616 1182 y FB(\()p Fu(t;)14 b(x)p FB(\))i(\(with)g(\002x)o (ed)g(v)n(alues)g(of)g Fu(x)h FB(and)f Fu(t)p FB(\))g(are)h(centered)e (Gaus-)515 1281 y(sian)20 b(random)f(v)n(ariables)g(gi)n(v)o(en)g(by) 909 1472 y Fn(\000)947 1540 y Fu(K)6 b(@)1068 1552 y Ft(x)1109 1540 y Fu(W)1199 1505 y Ft(N)1187 1560 y(A)1262 1472 y Fn(\001)1300 1540 y FB(\()p Fu(t;)14 b(x)p FB(\))23 b Fs(=)1612 1436 y Ft(N)1581 1461 y Fn(X)1581 1640 y Ft(k)q Fq(=1)1715 1540 y Fu(\013)1768 1552 y Ft(k)1809 1540 y Fs(\()p Fu(K)6 b(@)1962 1552 y Ft(x)2004 1540 y Fu(e)2043 1552 y Ft(k)2083 1540 y Fs(\))q FB(\()p Fu(x)p FB(\))2246 1427 y Fn(Z)2329 1447 y Ft(t)2292 1615 y Fq(0)2372 1540 y Fu(e)2411 1505 y Fi(\000)p Ft(\025)2502 1514 y Ff(k)2538 1505 y Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))2703 1540 y Fu(dw)2805 1552 y Ft(k)2847 1540 y FB(\()p Fu(s)p FB(\))22 b(,)515 1807 y(with)28 b(independent)e(W)m(iener)j(processes)f Fu(w)1803 1819 y Ft(k)1844 1807 y FB(\()p Fu(t)p FB(\).)50 b(The)28 b(v)n(ariance)g(of)2566 1740 y Fn(\000)2604 1807 y Fu(K)6 b(@)2725 1819 y Ft(x)2766 1807 y Fu(W)2856 1777 y Ft(N)2844 1830 y(A)2919 1740 y Fn(\001)2957 1807 y FB(\()p Fu(t;)14 b(x)p FB(\))29 b(is)h(thus)515 1907 y(bounded)17 b(by)575 2165 y Fv(E)630 2094 y Fn(\014)630 2144 y(\014)659 2098 y(\000)697 2165 y Fu(K)6 b(@)818 2177 y Ft(x)859 2165 y Fu(W)949 2131 y Ft(N)937 2185 y(A)1012 2098 y Fn(\001)1050 2165 y FB(\()p Fu(t;)14 b(x)p FB(\))1220 2094 y Fn(\014)1220 2144 y(\014)1247 2115 y Fq(2)1307 2165 y Fs(=)1426 2061 y Ft(N)1395 2086 y Fn(X)1395 2265 y Ft(k)q Fq(=1)1530 2165 y Fu(\013)1583 2131 y Fq(2)1583 2185 y Ft(k)1624 2165 y Fr(j)p Fs(\()p Fu(K)6 b(@)1800 2177 y Ft(x)1841 2165 y Fu(e)1880 2177 y Ft(k)1921 2165 y Fs(\))p FB(\()p Fu(x)p FB(\))p Fr(j)2079 2131 y Fq(2)2130 2052 y Fn(Z)2213 2072 y Ft(t)2176 2241 y Fq(0)2256 2165 y Fu(e)2295 2131 y Fq(2)p Ft(\025)2367 2140 y Ff(k)2403 2131 y Fm(\()p Ft(t)p Fi(\000)p Ft(s)p Fm(\))2568 2165 y Fu(ds)1307 2443 y Fr(\024)23 b Fu(C)i Fr(\001)1548 2340 y Fi(1)1521 2365 y Fn(X)1520 2543 y Ft(k)q Fq(=1)1655 2443 y Fr(j)p Fu(k)s Fr(j)1747 2409 y Fq(11)p Ft(=)p Fq(4)1898 2330 y Fn(Z)1981 2351 y Ft(t)1944 2519 y Fq(0)2024 2443 y Fu(e)2063 2409 y Fi(\000)p Ft(c)2145 2417 y Fg(1)2177 2409 y Ft(k)2213 2384 y Fg(4)2246 2409 y Ft(s)2281 2443 y Fu(ds)f Fr(\024)2502 2340 y Fi(1)2475 2365 y Fn(X)2474 2543 y Ft(k)q Fq(=1)2661 2387 y Fu(C)p 2619 2424 151 4 v 2619 2502 a(k)2665 2478 y Fq(5)p Ft(=)p Fq(4)2779 2351 y Fn(\020)2829 2443 y Fs(1)18 b Fr(\000)g Fu(e)3011 2409 y Fi(\000)p Ft(c)3093 2417 y Fg(2)3124 2409 y Ft(k)3160 2384 y Fg(4)3193 2409 y Ft(t)3222 2351 y Fn(\021)3295 2443 y Fu(:)515 2698 y FB(W)-7 b(e)21 b(no)n(w)f(tak)o(e)g(some)g Fu(k)1203 2710 y Fi(\003)1262 2698 y FB(to)g(\002x)o(ed)g(later)g(and)g(split)h(the)f(sum)g(into)g (tw)o(o)g(parts:)712 2958 y Fv(E)p Fr(j)p Fs(\()q Fu(K)6 b(@)944 2970 y Ft(x)986 2958 y Fu(W)1064 2970 y Ft(A)1118 2958 y Fs(\))p FB(\()p Fu(x;)14 b(t)p FB(\))p Fr(j)1343 2923 y Fq(2)1403 2958 y Fr(\024)23 b Fu(C)1570 2853 y Ft(k)1605 2861 y Fd(\003)1640 2853 y Fi(\000)p Fq(1)1588 2879 y Fn(X)1587 3058 y Ft(k)q Fq(=1)1739 2958 y Fu(k)1785 2923 y Fq(11)p Ft(=)p Fq(4)1922 2958 y Fu(t)c Fs(+)f Fu(C)2178 2854 y Fi(1)2152 2879 y Fn(X)2133 3058 y Ft(k)q Fq(=)p Ft(k)2255 3066 y Fd(\003)2369 2902 y Fs(1)p 2314 2939 V 2314 3016 a Fu(k)2360 2992 y Fq(5)p Ft(=)p Fq(4)2497 2958 y Fr(\024)23 b Fu(C)6 b(k)2696 2915 y Fq(15)p Ft(=)p Fq(4)2693 2969 y Fi(\003)2834 2958 y Fu(t)18 b Fs(+)3017 2902 y Fu(C)p 2975 2939 V 2975 3035 a(k)3021 2992 y Fq(1)p Ft(=)p Fq(4)3018 3047 y Fi(\003)3158 2958 y Fu(:)515 3236 y FB(Choosing)g Fu(k)896 3248 y Fi(\003)958 3236 y Fr(\031)k Fu(t)1075 3206 y Fi(\000)p Fq(1)p Ft(=)p Fq(4)1232 3236 y FB(,)e(we)g(ha)n(v)o(e)f(the)h(estimate)g Fv(E)p Fr(j)2061 3169 y Fn(\000)2100 3236 y Fu(K)6 b(@)2221 3248 y Ft(x)2262 3236 y Fu(W)2352 3206 y Ft(N)2340 3259 y(A)2415 3169 y Fn(\001)2453 3236 y FB(\()p Fu(x;)14 b(t)p FB(\))p Fr(j)2646 3206 y Fq(2)2706 3236 y Fr(\024)23 b Fu(C)6 b(t)2889 3206 y Fq(1)p Ft(=)p Fq(16)3026 3236 y FB(.)26 b(Since)20 b(the)515 3344 y(random)e(v)n(ariables)1108 3277 y Fn(\000)1146 3344 y Fu(K)6 b(@)1267 3356 y Ft(x)1308 3344 y Fu(W)1398 3314 y Ft(N)1386 3367 y(A)1461 3277 y Fn(\001)1499 3344 y FB(\()p Fu(x;)14 b(t)p FB(\))20 b(are)g(Gaussian)g(for)g(all)h(\002x)o(ed)e(v)n(alues)h(\()p Fu(x;)14 b(t)p FB(\),)20 b(we)h(ha)n(v)o(e)720 3594 y Fv(E)p Fr(k)p Fu(@)861 3606 y Ft(x)903 3594 y Fu(W)993 3560 y Ft(N)981 3615 y(A)1055 3594 y Fr(k)1097 3560 y Fq(4)1097 3615 y Fi(1)1190 3594 y Fr(\024)i Fu(C)6 b Fv(E)p Fr(k)p Fu(K)g(@)1561 3606 y Ft(x)1602 3594 y Fu(W)1692 3560 y Ft(N)1680 3615 y(A)1755 3594 y Fr(k)1797 3560 y Fq(4)1797 3615 y(4)1857 3594 y Fs(=)1945 3481 y Fn(Z)2028 3502 y Ft(L)1991 3670 y Fq(0)2091 3594 y Fv(E)p Fs(\()q Fu(K)g(@)2300 3606 y Ft(x)2341 3594 y Fu(W)2419 3606 y Ft(A)2474 3594 y Fs(\))p FB(\()p Fu(x;)14 b(t)p FB(\))2676 3560 y Fq(4)2727 3594 y Fu(dx)24 b Fr(\024)e Fu(C)6 b(t)3023 3560 y Fq(1)p Ft(=)p Fq(8)3151 3594 y Fu(:)3379 3796 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 3796 a 515 3994 a Fv(Lemma)21 b(6.4)40 b Fp(F)-9 b(or)18 b(e)o(very)g Fu(")23 b(>)f Fs(0)c Fp(and)f Fu(K)29 b(>)22 b Fs(0)c Fp(ther)m(e)g(e)n(xists)h(a)f(constant)f Fu(C)24 b Fp(depending)16 b(only)h(on)g Fu(")515 4093 y Fp(and)g Fu(K)6 b Fp(,)18 b(suc)o(h)g(that)f(for)i(any)e(two)i(r)o (andom)e(variables)g Fu(W)2165 4105 y Fq(1)2221 4093 y Fp(and)g Fu(W)2442 4105 y Fq(2)2499 4093 y Fp(with)h Fv(E)2715 4026 y Fn(\000)2754 4093 y Fr(j)p Fu(W)2855 4105 y Fq(1)2893 4093 y Fr(j)2916 4063 y Fq(2)2971 4093 y Fs(+)g Fr(j)p Fu(W)3155 4105 y Fq(2)3193 4093 y Fr(j)3216 4063 y Fq(2)3253 4026 y Fn(\001)3314 4093 y Fr(\024)515 4193 y Fu(K)26 b Fp(we)21 b(obtain)880 4396 y Fv(E)935 4303 y Fn(\020)984 4396 y FB(log)1104 4328 y Fn(\000)1142 4396 y Fu(xe)1228 4361 y Ft(W)1290 4369 y Fg(1)1346 4396 y Fs(+)d Fu(e)1468 4361 y Ft(W)1530 4369 y Fg(2)1567 4328 y Fn(\001)1605 4303 y(\021)1654 4321 y Fq(2)1715 4396 y Fr(\024)k Fs(\()q FB(log)o(\()p Fu(x)p FB(\))p Fs(\))2076 4354 y Fq(2)2131 4396 y Fs(+)c(2)p FB(\()p Fu(")g Fs(+)g Fv(E)p Fu(W)2557 4408 y Fq(1)2595 4396 y FB(\))c(log)e Fu(x)19 b Fs(+)f Fu(C)29 b Fp(,)515 4595 y(for)20 b(e)o(very)h Fu(x)i Fr(\025)g Fs(1)p Fp(.)515 4775 y(Pr)l(oof)o(.)40 b FB(Expanding)18 b(the)i(square,)f(we)i(get)610 4994 y Fv(E)665 4902 y Fn(\020)715 4994 y FB(log)835 4927 y Fn(\000)873 4994 y Fu(xe)959 4959 y Ft(W)1021 4967 y Fg(1)1076 4994 y Fs(+)d Fu(e)1198 4959 y Ft(W)1260 4967 y Fg(2)1297 4927 y Fn(\001)1335 4902 y(\021)1385 4919 y Fq(2)1445 4994 y Fs(=)23 b FB(\(log)12 b Fu(x)p FB(\))1755 4959 y Fq(2)1811 4994 y Fs(+)18 b(2)c FB(log)e Fu(x)2130 4902 y Fn(\020)2180 4994 y Fv(E)p Fu(W)2313 5006 y Fq(1)2370 4994 y Fs(+)18 b Fv(E)c FB(log)2642 4927 y Fn(\000)2680 4994 y Fs(1)k(+)g Fu(e)2862 4959 y Ft(W)2924 4967 y Fg(2)2956 4959 y Fi(\000)p Ft(W)3070 4967 y Fg(1)3107 4994 y Fu(=x)3196 4927 y Fn(\001)3234 4902 y(\021)p eop %%Page: 14 14 14 13 bop 517 232 a FB(R)t Fo(E)t(F)t(E)t(R)t(E)t(N)t(C)t(E)t(S)2314 b Fv(14)1523 552 y Fs(+)18 b Fv(E)1661 460 y Fn(\020)1711 552 y Fu(W)1789 564 y Fq(1)1845 552 y Fs(+)g FB(log)2048 485 y Fn(\000)2086 552 y Fs(1)g(+)g Fu(e)2268 518 y Ft(W)2330 526 y Fg(2)2363 518 y Fi(\000)p Ft(W)2477 526 y Fg(1)2514 552 y Fu(=x)2603 485 y Fn(\001)2641 460 y(\021)2690 477 y Fq(2)2751 552 y Fu(:)515 748 y FB(Since)i(we)h(assumed)e Fu(x)24 b Fr(\025)e Fs(1)p FB(,)e(we)h(ha)n(v)o(e)818 958 y Fv(E)873 866 y Fn(\020)923 958 y Fu(W)1001 970 y Fq(1)1057 958 y Fs(+)d FB(log)1260 891 y Fn(\000)1298 958 y Fs(1)g(+)g Fu(e)1480 924 y Ft(W)1542 932 y Fg(2)1575 924 y Fi(\000)p Ft(W)1689 932 y Fg(1)1726 958 y Fu(=x)1815 891 y Fn(\001)1853 866 y(\021)1902 883 y Fq(2)1963 958 y Fr(\024)k Fs(2)2092 891 y Fn(\000)2130 958 y Fv(E)p Fu(W)2275 924 y Fq(2)2263 978 y(1)2331 958 y Fs(+)c Fv(E)p FB(\()p Fu(W)2575 970 y Fq(2)2631 958 y Fr(\000)g Fu(W)2792 970 y Fq(1)2830 958 y FB(\))2858 924 y Fq(2)2913 958 y Fs(+)g(1)3038 891 y Fn(\001)1963 1102 y Fr(\024)k Fs(6)p Fu(K)i Fs(+)18 b(2)k Fu(:)515 1279 y FB(It)j(thus)g(suf)n(\002ces)g(to) g(sho)n(w)f(that)h(there)g(e)o(xists)g Fu(x)1922 1291 y Fq(0)1991 1279 y Fu(>)31 b Fs(0)25 b FB(depending)e(only)h(on)g Fu(K)31 b FB(and)24 b(on)h Fu(")g FB(such)515 1378 y(that)1393 1513 y Fu(E)1454 1525 y Ft(x)1519 1513 y Fs(:=)e Fv(E)14 b FB(log)1805 1421 y Fn(\020)1855 1513 y Fs(1)k(+)2008 1457 y Fu(e)2047 1427 y Ft(W)2109 1435 y Fg(2)2141 1427 y Fi(\000)p Ft(W)2255 1435 y Fg(1)p 2008 1494 285 4 v 2126 1570 a Fu(x)2302 1421 y Fn(\021)2375 1513 y Fr(\024)k Fu(")719 b FB(\(6.1\))515 1682 y(for)19 b Fu(x)i FB(lar)o(ger)e(than)h Fu(x)1124 1694 y Fq(0)1162 1682 y FB(.)639 1781 y(T)-7 b(o)21 b(v)o(erify)d(\(6.1\))h(consider)g(arbitrary)f Fu(")23 b(>)g Fs(0)p FB(.)i(W)-7 b(e)21 b(de\002ne)f Fu(y)25 b Fs(=)e Fr(j)p Fu(W)2597 1793 y Fq(2)2653 1781 y Fr(\000)18 b Fu(W)2814 1793 y Fq(1)2851 1781 y Fr(j)j FB(and)f(denote)f(the)515 1881 y(probability)f(distrib)n(ution)h(on)h Fv(R)1466 1893 y Fq(+)1541 1881 y FB(of)g Fu(y)k FB(by)c Fv(P)p FB(.)25 b(No)n(w)20 b(choose)f Fu(y)2369 1893 y Fq(0)2429 1881 y Fu(>)k Fs(1)d FB(lar)o(ge)f(enough)f(such)i(that) 1274 1990 y Fn(Z)1357 2011 y Fi(1)1320 2179 y Ft(y)1354 2187 y Fg(0)1441 2103 y Fu(y)c Fv(P)p FB(\()p Fu(dy)s FB(\))22 b Fr(\024)h Fu(")g FB(,)2064 1990 y Fn(Z)2147 2011 y Fi(1)2111 2179 y Ft(y)2145 2187 y Fg(0)2232 2103 y Fv(P)p FB(\()p Fu(dy)s FB(\))f Fr(\024)g Fu(")h(:)515 2343 y FB(W)-7 b(e)21 b(choose)e Fu(y)936 2355 y Fq(0)996 2343 y Fs(=)k(1)18 b(+)g FB(\()p Fv(E)p Fu(y)1354 2313 y Fq(2)1391 2343 y FB(\))p Fu(=")p FB(.)24 b(No)n(w)c(de\002ne)1556 2579 y Fu(x)1603 2591 y Fq(0)1663 2579 y Fs(=)1761 2523 y Fu(e)1800 2493 y Fq(1+2)p Ft(K=")p 1761 2560 282 4 v 1882 2636 a Fu(")2076 2579 y Fr(\025)2173 2523 y Fu(e)2212 2493 y Ft(y)2246 2501 y Fg(0)p 2173 2560 109 4 v 2208 2636 a Fu(")2315 2579 y(:)515 2778 y FB(W)-7 b(e)21 b(thus)f(ha)n(v)o (e)835 2982 y Fu(E)896 2994 y Ft(x)961 2982 y Fs(=)1049 2869 y Fn(Z)1132 2889 y Ft(y)1166 2897 y Fg(0)1095 3058 y Fq(0)1216 2982 y FB(log)1322 2890 y Fn(\020)1372 2982 y Fs(1)d(+)1524 2926 y Fu(e)1563 2896 y Ft(y)p 1524 2963 79 4 v 1540 3039 a Fu(x)1613 2890 y Fn(\021)1676 2982 y Fv(P)p FB(\()p Fu(dy)s FB(\))h Fs(+)1971 2869 y Fn(Z)2054 2889 y Fi(1)2017 3058 y Ft(y)2051 3066 y Fg(0)2138 2982 y FB(log)2244 2890 y Fn(\020)2294 2982 y Fs(1)g(+)2447 2926 y Fu(e)2486 2896 y Ft(y)p 2447 2963 V 2462 3039 a Fu(x)2535 2890 y Fn(\021)2599 2982 y Fv(P)p FB(\()p Fu(dy)s FB(\))961 3220 y Fr(\024)23 b FB(log)1155 3127 y Fn(\020)1204 3220 y Fs(1)18 b(+)1357 3163 y Fu(e)1396 3133 y Ft(y)1430 3141 y Fg(0)p 1357 3201 109 4 v 1388 3277 a Fu(x)1476 3127 y Fn(\021)1544 3220 y Fs(+)1627 3107 y Fn(Z)1710 3127 y Fi(1)1673 3295 y Ft(y)1707 3303 y Fg(0)1794 3220 y FB(log)1914 3152 y Fn(\000)1952 3220 y Fs(1)g(+)g Fu("e)2173 3185 y Ft(y)r Fi(\000)p Ft(y)2295 3193 y Fg(0)2331 3152 y Fn(\001)2383 3220 y Fv(P)p FB(\()p Fu(dy)s FB(\))961 3457 y Fr(\024)23 b Fu(")18 b Fs(+)1189 3344 y Fn(Z)1272 3365 y Fi(1)1235 3533 y Ft(y)1269 3541 y Fg(0)1356 3457 y FB(log)1476 3390 y Fn(\000)1514 3457 y FB(\()p Fs(1)g(+)g Fu(")p FB(\))p Fu(e)1791 3423 y Ft(y)r Fi(\000)p Ft(y)1913 3431 y Fg(0)1948 3390 y Fn(\001)2000 3457 y Fv(P)p FB(\()p Fu(dy)s FB(\))961 3695 y Fr(\024)23 b Fu(")18 b Fs(+)g Fu(")c FB(log)o(\()p Fs(1)j(+)h Fu(")p FB(\))g Fs(+)1686 3582 y Fn(Z)1769 3602 y Fi(1)1732 3771 y Ft(y)1766 3779 y Fg(0)1839 3695 y FB(\()p Fu(y)j Fr(\000)d Fu(y)2053 3707 y Fq(0)2090 3695 y FB(\))c Fv(P)p FB(\()p Fu(dy)s FB(\))22 b Fr(\024)g Fs(2)p Fu(")c Fs(+)g Fu(")c FB(log)o(\()p Fs(1)j(+)h Fu(")p FB(\))23 b Fu(:)515 3924 y FB(The)d(claim)g(follo)n(ws)g(immediately)-5 b(.)3379 3924 y @beginspecial 0.2 setlinewidth newpath -0.2 2 div 0 moveto -5.5 0 rlineto 0 5.5 rlineto 5.5 0 rlineto 0 -5.5 rlineto closepath 0 setgray stroke @endspecial 3379 3924 a 515 4228 a FA(Refer)n(ences)515 4394 y Fx([Arn74])138 b(L.)33 b(Arnold,)j Fc(Stoc)o(hastic)e(Dif)o(fer)m (ential)f(Equations:)52 b(Theory)34 b(and)g(Applications)p Fx(,)j(W)m(ile)o(y-)893 4486 y(Interscience)20 b(Publication,)f(John)h (W)m(ile)o(y)e(&)h(Sons,)f(Ne)n(w)h(Y)-8 b(ork,)19 b(1974.)515 4609 y([BG02])150 b(D.)20 b(Bl)6 b(\250)-31 b(omk)o(er)21 b(and)g(C.)f(Gugg,)h Fc(On)g(the)f(Existence)h(of)f(Solutions)i(for)e (Amorphous)i(Molecular)893 4700 y(Beam)f(Epitaxy)p Fx(,)g(Journal)g(of) g(Nonlinear)g(Analysis:)27 b(Series)20 b(B)h(Real)f(W)-6 b(orld)20 b(Applications)i Fb(3)893 4792 y Fx(\(2002\),)e(no.)f(1,)g (61\22673.)515 4915 y([BGR02])100 b(D.)23 b(Bl)6 b(\250)-31 b(omk)o(er)m(,)25 b(C.)e(Gugg,)i(and)g(M.)e(Raible,)h Fc(Thin-F)m(ilm-Gr)m(owth)e(Models:)34 b(Roughness)25 b(and)893 5006 y(Corr)m(elation)20 b(Functions)p Fx(,)f(T)-6 b(o)19 b(appear)h(in)f Fc(Eur)m(opean)h(J)n(ournal)g(of)f(Appl.)f (Math.)p Fx(,)h(2002.)p eop %%Page: 15 15 15 14 bop 517 232 a FB(R)t Fo(E)t(F)t(E)t(R)t(E)t(N)t(C)t(E)t(S)2314 b Fv(15)515 523 y Fx([BMPS01])54 b(D.)18 b(Bl)6 b(\250)-31 b(omk)o(er)m(,)18 b(S.)f(Maier)o(-P)o(aape,)h(and)h(G.)e(Schneider)m(,) i Fc(The)f(Stoc)o(hastic)h(Landau)g(Equation)g(as)893 614 y(an)25 b(Amplitude)g(Equation)p Fx(,)h(Discrete)e(and)h (Continuous)h(Dynamical)f(Systems,)g(Series)f(B)g Fb(1)893 706 y Fx(\(2001\),)c(no.)f(4,)g(527\226541.)515 830 y([BS95])162 b(A.)20 b(L.)g(Barabasi)g(and)h(H.)f(E.)f(Stanle)o(y)-5 b(,)21 b Fc(Fr)o(actal)f(Concepts)h(in)f(Surface)i(Gr)m(owth)p Fx(,)e(Cambridge)893 922 y(Uni)n(v)o(ersity)g(Press,)e(1995.)515 1046 y([CEES93])70 b(P)-8 b(.)20 b(Collet,)f(J.-P)-8 b(.)19 b(Eckmann,)j(H.)e(Epstein,)g(and)h(J.)f(Stubbe,)h Fc(A)f(Global)h(Attr)o(acting)f(Set)g(for)h(the)893 1137 y(Kur)o(amoto-Sivashinsk)o(y)30 b(Equation)p Fx(,)h(Commun.)d(Math.)g (Phys.)e Fb(152)j Fx(\(1993\),)h(no.)e(1,)i(203\226)893 1229 y(214.)515 1353 y([CG94])150 b(M.)23 b(Capinski)h(and)f(D.)g (Gatarek,)h Fc(Stoc)o(hastic)g(Equations)f(in)g(Hilbert)g(Space)h(with) f(Applica-)893 1445 y(tions)g(to)f(Navier)o(-Stok)o(es)g(Equation)h(in) f(any)h(Dimension)p Fx(,)g(Journal)g(of)f(Functional)g(Analysis)893 1536 y Fb(126)e Fx(\(1994\),)g(no.)f(1,)g(26\22635.)515 1660 y([DPZ92])112 b(G.)16 b(Da)f(Prato)h(and)g(J.)g(Zabczyk,)h Fc(Stoc)o(hastic)g(Equations)g(in)e(In\002nite)i(Dimensions)p Fx(,)g(Uni)n(v)o(ersity)893 1752 y(Press,)h(Cambridge,)i(1992.)515 1876 y([DPZ96])112 b(G.)23 b(Da)g(Prato)g(and)h(J.)f(Zabczyk,)i Fc(Er)m(godicity)e(for)h(In\002nite)f(Dimensional)h(Systems)p Fx(,)h(London)893 1968 y(Mathematical)d(Society)g(Lecture)f(Note)h (Series,)f(v)o(ol.)g(229,)h(Uni)n(v)o(ersity)g(Press,)f(Cambridge,)893 2059 y(1996.)515 2183 y([FG95])158 b(F)-6 b(.)31 b(Flandoli)g(and)h(D.) f(Gatarek,)k Fc(Martingale)d(and)h(Stationary)f(Solutions)h(for)e(Stoc) o(hastic)893 2275 y(Navier)o(-Stok)o(es)20 b(Solutions)p Fx(,)g(Probab)m(.)f(Theory)g(Relat.)f(Fields)h Fb(102)g Fx(\(1995\),)h(367\226391.)515 2399 y([Gat93])146 b(D.)20 b(Gatarek,)h Fc(A)f(Note)g(on)h(Stoc)o(hastic)g(Equations)h(in)e (Hilbert)g(Spaces)p Fx(,)i(Statistics)d(and)i(Prob-)893 2491 y(ability)e(Letters)f Fb(17)i Fx(\(1993\),)f(no.)g(5,)g (387\226394.)515 2615 y([Has80])138 b(R.)19 b(Z.)f(Has'minski)n(\020) -23 b(\021,)19 b Fc(Stoc)o(hastic)h(Stability)f(of)g(Dif)o(fer)m (ential)f(Equations)p Fx(,)i(Sijthof)n(f)e(&)h(Noord-)893 2707 y(hof)n(f,)g(1980.)515 2831 y([MMS99])80 b(S.)36 b(G.)g(Mayr)m(,)41 b(M.)36 b(Mosk)o(e,)42 b(and)c(K.)d(Samwer)m(,)41 b Fc(Identi\002cation)c(of)g(K)m(e)n(y)f(P)-6 b(ar)o(ameter)o(s)37 b(by)893 2922 y(Comparing)30 b(Experimental)e(and)h(Simulated)g(Gr)m (owth)e(of)h(V)-8 b(apor)29 b(Deposited)f(Amorphous)893 3014 y Fx(Zr)964 3022 y Fy(65)1029 3014 y Fx(Al)1104 3022 y Fy(7)p Fa(:)p Fy(5)1188 3014 y Fx(Cu)1275 3022 y Fy(27)p Fa(:)p Fy(5)1408 3014 y Fc(F)m(ilms)p Fx(,)17 b(Phys.)h(Re)n(v)-5 b(.)19 b(B)g Fb(60)g Fx(\(1999\),)h (16950\22616955.)515 3138 y([NST85])112 b(B.)20 b(Nicolaenk)o(o,)h(B.)f (Scheurer)m(,)g(and)h(R.)e(T)-5 b(emam,)20 b Fc(Some)h(Global)f (Dynamical)h(Pr)m(operties)f(of)893 3230 y(the)j(Kur)o (amoto-Sivashinsk)o(y)j(Equations:)31 b(Nonlinear)24 b(Stability)f(and)g(Attr)o(actor)o(s)p Fx(,)h(Phys.)e(D)893 3321 y Fb(16)e Fx(\(1985\),)f(no.)g(2,)g(155\226183.)515 3445 y([RML)702 3414 y Fy(+)752 3445 y Fx(00])42 b(M.)26 b(Raible,)g(S.)e(G.)h(Mayr)m(,)i(S.)d(J.)h(Linz,)h(M.)g(Mosk)o(e,)i(P) -8 b(.)23 b(H)t(\250)-29 b(anggi,)28 b(and)e(K.)e(Samwer)m(,)j Fc(Amor)o(-)893 3537 y(phous)22 b(Thin)f(F)m(ilm)e(Gr)m(owth:)26 b(Theory)21 b(Compar)m(ed)h(with)e(Experiment)p Fx(,)h(Europhysics)h (Letters)893 3628 y Fb(50)e Fx(\(2000\),)f(61\22667.)515 3753 y([SP94])170 b(M.)30 b(Sie)o(gert)e(and)i(M.)f(Plischk)o(e,)j Fc(Solid-on-Solid)f(Models)f(of)f(Molecular)o(-Beam)h(Epitaxy)p Fx(,)893 3844 y(Phys.)19 b(Re)n(v)-5 b(.)19 b(E)f Fb(50)h Fx(\(1994\),)h(917\226931.)515 3968 y([VF88])158 b(M.)30 b(V)l(isik)f(and)i(A.)e(Fursik)o(o)o(v)-5 b(,)34 b Fc(Mathematical)c (Pr)m(oblems)g(of)g(Statistical)g(Hydr)m(odynamic)p Fx(,)893 4060 y(Kluwer)m(,)19 b(1988.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0204230537129--