Content-Type: multipart/mixed; boundary="-------------0212060413476" This is a multi-part message in MIME format. ---------------0212060413476 Content-Type: text/plain; name="02-503.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-503.comments" E-mail: S.B.Kuksin@ma.hw.ac.uk, Armen.Shirikyan@math.u-psud.fr ---------------0212060413476 Content-Type: text/plain; name="02-503.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-503.keywords" randomly forced Navier-Stokes equations, a priori estimates, stationary measure, ergodicity, exponential mixing ---------------0212060413476 Content-Type: application/postscript; name="limit.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="limit.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: limit.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o limit.ps limit.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2002.12.06:1108 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 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 dup 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 dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 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}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /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 TeXDict begin 39158280 55380996 1000 300 300 (limit.dvi) @start /Fa 8 122 df97 D99 D<133E130CA41318A4EA03B0EA0C701218EA3030136012 60A3EAC0C013C812401241EA62D0EA1C700F177C9612>I<1203120712061200A6121812 2C124CA2128C1218A312301232126212641224123808177D960B>105 D<123E120CA41218A41230A41260A412C012C8A312D0126007177D9609>108 D110 D114 D121 D E /Fb 3 118 df<3907FE1FF83900E00380A43901C0 0700A43803800EA2EBFFFEEB800E48485AA4000E5BA4485B38FF83FE1D177F961D>72 D<130E13131337133613301360A4EA03FCEA00C0A5EA0180A5EA0300A41202126612E65A 1278101D7E9611>102 D117 D E /Fc 2 51 df<1218127812981218AC12FF08107D8F0F> 49 D<121FEA6180EA40C0EA806012C01200A213C0EA0180EA030012065AEA10201220EA 7FC012FF0B107F8F0F>I E /Fd 1 81 df<3903F003E0380FFC0F39187C18F039301E20 7000601340EB0FC000E0EB807F39F007007E00F81438007C14101520003E1460001E14E0 000E14F0120C0018147812300000143C151CA2150CA201FF130848EBC01848EBF8303906 0FFFE0486C13C00008EB1F00000090C7FCA65B130E130820257F9B22>80 D E /Fe 2 49 df0 D<1218A31230A31260A312C0A2050B7E8B 09>48 D E /Ff 6 111 df34 D99 D<1208A21200A41270129812B01230A21260126412681270060F7D8E0B>105 D<12381218A35A13C0EA3360EA3440EA7800127E12631320EAC340EAC1800B0E7E8D10> 107 D109 DI E /Fg 48 124 df<903801F03C9038071C47010C13C7EC19C690381C0180140313181338 A2EC0700A20003B512F03900700700A3140EA213E0A35CA2EA01C0A35CA2EA0380A21430 EB0070A248136038C630E038E638C038CC3180D8781EC7FC2025819C19>11 D<14FE90380301801306EB0C03EB1C0191C7FC13181338A43803FFFE3800700EA35CA213 E0A25CA3EA01C01472A438038034141891C7FC90C8FCA25A12C612E65A12781925819C17 >I<1218123CA31204A21208A21210122012401280060C779C0D>39 D<12181238127812381208A21210A212201240A21280050C7D830D>44 DI<1230127812F0126005047C830D>I<1304130C131813381378 EA07B8EA0070A413E0A4EA01C0A4EA0380A4EA0700A45AEAFFF00E1C7B9B15>49 D<133EEB4180EB80C0EA0100000213E0EA0440A21208A3381081C0A238110380000E1300 EA00065B5B136013800003C7FC12044813404813805AEB0100EA7F07EA43FEEA81FCEA80 78131D7D9B15>I<131FEB60C013803801006012021340000413E0A3EB81C0EA03013800 0380EB070013FC131C1306A21307A41270EAE00E12805BEA40185BEA20E0EA1F80131D7D 9B15>I<1418A21438A21478A214B8EB0138A2EB023C141C1304130C13081310A21320A2 EB7FFCEBC01C1380EA0100141E0002130EA25A120C001C131EB4EBFFC01A1D7E9C1F>65 D<903803F02090381E0C6090383002E09038E003C03801C001EA038048C7FC000E148012 1E121C123C15005AA35AA41404A35C12705C6C5B00185B6C485AD80706C7FCEA01F81B1E 7A9C1E>67 D<48B5FC39003C03C090383800E0A21570A24913781538A215785BA4484813 F0A315E03803800115C0140315803907000700140E5C5C000E13E0B512801D1C7E9B1F> I<48B512F038003C00013813301520A35BA214081500495AA21430EBFFF03801C020A439 038040801400A2EC0100EA07005C14021406000E133CB512FC1C1C7E9B1C>I<903803F0 2090381E0C6090383002E09038E003C03801C001EA038048C7FC000E1480121E121C123C 15005AA35AA2903801FF809038001E00141CA400705BA27E001813786C139038070710D8 01F8C7FC1B1E7A9C20>71 D73 D<3801FFC038003C001338A45BA45BA4485AA43803 8002A31404EA0700140C14181438000E13F0B5FC171C7E9B1A>76 DII<3801FFFE39003C0380903838 01C0EC00E0A3EB7001A315C0EBE0031580EC0700141C3801FFF001C0C7FCA3485AA448C8 FCA45AEAFFE01B1C7E9B1C>80 D83 D<001FB512C0381C070138300E00002014 80126012405B1280A2000014005BA45BA45BA4485AA41203EA7FFE1A1C799B1E>I<397F F03FE0390F000700000E13061404A3485BA4485BA4485BA4485BA35CA249C7FCEA60025B 6C5AEA1830EA07C01B1D789B1F>I<3AFF83FF07F03A3C007001C00038158002F01300A2 90380170025D13025D13045D13085D131001305B1320D81C405BA2D98071C7FCA2381D00 72A2001E1374A2001C1338A20018133014201210241D779B29>87 D97 D<123F1207A2120EA45AA4EA39E0EA3A18EA3C0C12381270130EA3EAE0 1CA31318133813301360EA60C0EA3180EA1E000F1D7C9C13>I<13F8EA0304120EEA1C0E EA181CEA30001270A25AA51304EA60081310EA3060EA0F800F127C9113>II<13F8EA0704120CEA1802EA38041230EA7008EA7FF0 EAE000A5EA60041308EA30101360EA0F800F127C9113>IIIII107 DI<391C1E078039266318C0394683A0E0384703 C0008E1380A2120EA2391C0701C0A3EC0380D8380E1388A2EC0708151039701C03203930 0C01C01D127C9122>II<13F8EA030CEA0E06487E121812 3000701380A238E00700A3130EA25BEA60185BEA30E0EA0F8011127C9115>I<38038780 3804C860EBD03013E0EA09C014381201A238038070A31460380700E014C0EB0180EB8300 EA0E86137890C7FCA25AA45AB4FC151A809115>III< EA01F0EA0608120C131CEA1818EA1C00121F13C0EA0FF01207EA00781338EA603012E012 C0EA8060EA60C0EA1F000E127D9111>I<12035AA3120EA4EAFFE0EA1C00A35AA45AA4EA E080A2EAE100A2126612380B1A7C990E>I<381C0180EA2E03124EA2388E0700A2121CA2 EA380EA438301C80A3EA383C38184D00EA0F8611127C9116>II<381E 0183382703871247148338870701A2120EA2381C0E02A31404EA180C131C1408EA1C1E38 0C26303807C3C018127C911C>I<38038780380CC840380870E012103820E0C014001200 A2485AA4EA03811263EAE38212C5EA8584EA787813127E9113>I<381C0180EA2E03124E A2388E0700A2121CA2EA380EA4EA301CA3EA383CEA1878EA0FB8EA003813301370EAE060 5BEA81800043C7FC123C111A7C9114>I123 D E /Fh 25 122 df<1238127C12FEA3127C123807077C8610>46 D<13181378EA01F812FFA21201B3A7387FFFE0A213207C9F1C>49 DI<13FE3807FFC0380F07E0381E03F0123FEB81F8A3EA1F 0314F0120014E0EB07C0EB1F803801FE007F380007C0EB01F014F8EB00FCA2003C13FE12 7EB4FCA314FCEA7E01007813F8381E07F0380FFFC03801FE0017207E9F1C>I68 D80 D<3801FE023807FF86381F01FE383C007E007C131E0078130EA200F81306A27E 1400B4FC13E06CB4FC14C06C13F06C13F86C13FC000313FEEA003F1303EB007F143FA200 C0131FA36C131EA26C133C12FCB413F838C7FFE00080138018227DA11F>83 D<007FB61280A2397E03F80F00781407007014030060140100E015C0A200C01400A40000 1500B3A248B512F0A222227EA127>I97 DIII<13FE3807FF80380F87C0381E01E0003E13F0EA7C0014F812FCA2B5FCA200FCC7FCA3 127CA2127E003E13186C1330380FC0703803FFC0C6130015167E951A>I I104 D<121C123E127FA3123E121CC7FCA7B4FCA2121FB2EAFFE0A20B247EA310>I108 D<3AFF07F007F090391FFC1FFC3A1F303E30 3E01401340496C487EA201001300AE3BFFE0FFE0FFE0A22B167E9530>I<38FF07E0EB1F F8381F307CEB403CEB803EA21300AE39FFE1FFC0A21A167E951F>I<13FE3807FFC0380F 83E0381E00F0003E13F848137CA300FC137EA7007C137CA26C13F8381F01F0380F83E038 07FFC03800FE0017167E951C>I114 DI<487EA41203A21207A2120F123FB5FCA2EA0F80ABEB8180A5EB8300 EA07C3EA03FEEA00F811207F9F16>I<38FF01FEA2381F003EAF147E14FE380F81BE3907 FF3FC0EA01FC1A167E951F>I<39FFE01FE0A2391F800700000F1306EBC00E0007130C13 E000035BA26C6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA2130EA2130CA25B1278EA FC3813305BEA69C0EA7F80001FC8FC1B207F951E>121 D E /Fi 37 122 df<13201340EA0180120313001206120E5AA2123C1238A21278A312F85AA97E12 78A31238A2123C121CA27E12067E13801201EA004013200B297C9E13>40 D<7E12401230123812187E120E7EA213801203A213C0A313E01201A9120313C0A31380A2 12071300A2120E120C5A1238123012405A0B297D9E13>I<127812FCA4127806067D850D> 46 D48 D<1360EA01E0120F12FF12F31203 B3A2387FFF80A2111B7D9A18>IIII< 38380180383FFF005B5B5B13C00030C7FCA4EA31F8EA361E38380F80EA3007000013C014 E0A3127812F8A214C012F038600F8038381F00EA1FFEEA07F0131B7E9A18>I<1260387F FFE0A214C01480A238E00300EAC0065B5BC65AA25B13E0A212015B1203A41207A66C5A13 1C7D9B18>55 DI65 D<90381FE0209038FFF8E03803F80F3807C003380F 800148C7FC123E1560127E127C00FC1400A8007C1460127E123E15C07E390F8001803907 C003003803F80E3800FFFCEB1FE01B1C7D9B22>67 D<39FFFC3FFFA2390FC003F0AA90B5 FCA2EBC003AC39FFFC3FFFA2201C7E9B25>72 D76 D80 D<007FB512E0A238781F81007013800060146000E0147000C01430A400001400B03807FF FEA21C1C7E9B21>84 D97 D99 DII<3803F0F0380E1F38EA 3C0F3838073000781380A400381300EA3C0FEA1E1CEA33F00030C7FCA3EA3FFF14C06C13 E014F0387801F838F00078A300701370007813F0381E03C03807FF00151B7F9118>103 DI<12 1E123FA4121EC7FCA6127FA2121FAEEAFFC0A20A1E7F9D0E>I<137813FCA413781300A6 EA03FCA2EA007CB2127012F8137813F0EA70E0EA1F800E26839D0F>III<39FF0FC07E903831E18F3A 1F40F20780D980FC13C0A2EB00F8AB3AFFE7FF3FF8A225127F9128>I<38FF0FC0EB31E0 381F40F0EB80F8A21300AB38FFE7FFA218127F911B>II<38FF3F80EBE1E0381F80F0EB0078147C143C143EA6143C147C1478EB80F0EBC1 E0EB3F0090C7FCA6EAFFE0A2171A7F911B>I114 DI<1203A45AA2 5AA2EA3FFC12FFEA1F00A9130CA4EA0F08EA0798EA03F00E1A7F9913>I<38FF07F8A2EA 1F00AC1301120F380786FFEA01F818127F911B>I<39FF8FF8FEA2391F03E030A3390F87 F06013869038C6F8E03907CC78C0A23903FC7D80EBF83D143F3901F01F00A20000131EEB E00EA21F127F9122>119 D<38FFC1FCA2381F00601380000F13C0A23807C180A23803E3 00A213F7EA01F613FE6C5AA21378A21330A25B1270EAF8E05BEAF9800073C7FC123E161A 7F9119>121 D E /Fj 18 122 df<126012F0A212600404798312>46 D64 DII75 D83 D97 D99 D<12F8A21238A4EA39E0EA3FF0EA3E38123C1238A8EAFE3EA20F 147F9312>104 D<1206120FA21206C7FCA3127FA21207AAEAFFF0A20C157D9412>I<12F8 A21238A4EA3BFCA2EA38F0EA39E0EA3BC0EA3F80A213C0EA3DE0123813701338EAFC7EA2 0F147F9312>107 D 109 DI114 DI117 D119 D121 D E /Fk 56 122 df<13FCEA0782EA0E07121C130290C7FCA4B5FCEA1C07AC387F1FC012 17809614>12 D14 D<120112021204120C1218A21230A212701260A312E0 AA1260A312701230A21218A2120C12041202120108227D980E>40 D<12801240122012301218A2120CA2120E1206A31207AA1206A3120E120CA21218A21230 12201240128008227E980E>I<126012F0A212701210A21220A21240A2040A7D830A>44 DI<126012F0A2126004047D830A>I48 D<1206120E12FE120EB1EAFFE00B157D9412>III<1330A2137013F0120113 70120212041208121812101220124012C0EAFFFEEA0070A5EA03FE0F157F9412>I54 D<1240EA7FFE13FC13F8EAC008EA801013 20EA00401380A2EA01005AA212021206A2120EA512040F167E9512>III<126012F0A212601200A6126012F0A21260040E7D8D0A>I<126012F0A212601200A6 126012F0A212701210A21220A21240A204147D8D0A>I<13101338A3135CA3138EA3EA01 07A238020380A33807FFC0EA0401A2380800E0A2001813F0123838FE03FE17177F961A> 65 D68 DIII<38FF83FE381C0070AA381FFFF0381C0070AA 38FF83FE17177F961A>II<38FF80FE381C0078 146014401480EB0100130613085B13381378139CEA1D0E121EEA1C07EB0380EB01C0A2EB 00E014701478147C38FF80FF18177F961B>75 D<00FEEB03F8001E14C000171305A33813 8009A23811C011A33810E021A2EB7041A3EB3881A2EB1D01A2130EA2123839FE040FF81D 177F9620>77 D82 DI<38FF80FE381C00381410B06C132012066C1340380181 8038007E0017177F961A>85 D<3AFF07FC3F803A3C00E00E00001C1404A2EB0170000E5C A2EB023800075CA2EB041CD803845BA2EB880ED801C85BA2EBD007D800F05BA3EBE00301 6090C7FCA221177F9624>87 D<12FCA212C0B3AB12FCA206217D980A>91 D<12FCA2120CB3AB12FCA2062180980A>93 D97 D<12F81238A8EA39F0EA3E 0CEA380613077F1480A414005B1306EA361CEA21F011177F9614>II<133E130EA8EA07 CEEA1C3EEA300E1270126012E0A412601270EA301EEA182E3807CF8011177F9614>III< EA0F9EEA18E3EA3060EA7070A3EA3060EA18C0EA2F80EA20001260EA3FE013F813FCEA60 0EEAC006A3EA600CEA3838EA0FE010157F8D12>I<12F81238A813F8EA3B1CEA3C0E1238 AA38FE3F8011177F9614>I<12301278A212301200A512F81238AC12FE07177F960A>I<12 F81238A8133E13381330134013801239EA3FC0EA39E0123813F01378133CA2EAFE7F1017 7F9613>107 D<12F81238B3A312FE07177F960A>I<38F8F83E383B1CC7393C0F0380EA38 0EAA39FE3F8FE01B0E7F8D1E>II< EA07C0EA1830EA3018EA600CA2EAE00EA5EA701CEA3018EA1830EA07C00F0E7F8D12>I< EAF9F0EA3E1CEA380613077F1480A414005B130EEA3E1CEA39F00038C7FCA512FE11147F 8D14>IIII<1208A31218A21238EAFFC0EA3800A71340A4EA1C80EA0F000A147F930E>III<38FEFE7C383838381410133C 001C1320134C381E4E60380ECE401387000713801303A200031300EA0201160E7F8D19> I121 D E /Fl 2 122 df<120CA2EACCC012EDEA7F80EA0C00 EA7F80EAEDC012CCEA0C00A20A0B7D8B10>3 D<1218A512FFA21218AF08167D900E>121 D E /Fm 9 107 df0 D<1204A3EAC460EAF5E0EA3F80EA0E00EA 3F80EAF5E0EAC460EA0400A30B0D7E8D11>3 D<14C01303EB0700131C1378EA01E0EA07 80000EC7FC123812F0A21238120E6C7EEA01E0EA0078131C1307EB03C013001400A5387F FF80B512C0121B7D931A>20 D<14101418A280A28080B612E0A2C7EA030014065CA25CA2 14101B107E8E21>33 D<1204120EA2121CA31238A212301270A21260A212C0A2070F7F8F 0A>48 D<000F131E393BC06180396060804038403100D8801A1320130EA3130B39401180 40903820C0C03930C07B80390F001E001B0D7E8C21>II<1303A21306A2130C13 18A21330A2136013C0A2EA0180A2EA0300A212065AA25AA25A5AA25A1240101A7C9300> 54 D<12C0B3AB021D7D950A>106 D E /Fn 32 122 df11 D13 DI18 D 23 D26 DII 34 D<13201360A213C0A3EA0180A3EA0300A31206A25AA35AA35AA35AA35AA30B1D7E95 11>61 D<3807FFE03800E0383801C018141CA338038038147014E0EBFFC0380700E01470 14301438000E1370A214E0A2381C038038FFFE0016147F9319>66 DI<3907FC7FC03900E00E0048485AA448485AA3EB FFF838070070A4000E5BA448485A38FF8FF81A147F931B>72 D<3907FC1FC03900E00600 3801C0085C14601480D80381C7FC1386138E139EEA07671387EB0380A2380E01C0A26D7E A2487F38FF83FC1A147F931C>75 D<3907E01FC00000EB060038017004A21338A238021C 08A2130EA2486C5AA2EB0390A2380801E0A21300A20018134012FE1A147F931A>78 D<3807FFC03800E0703801C018141CA338038038147014C0EBFF00380701C0A2EB00E0A2 380E01C0A314C2001C13C438FF807817147F9319>82 D<381FFFF838381C18EA20381408 12401410EA807000001300A35BA4485AA4485AEA3FF81514809314>84 D<123C120C5AA45AEA3380EA3C60EA3020EA6030A4EAC060A2EA40C0EA6080EA2300121E 0C147F930F>98 DI<1206120712061200A41238124CA2128C12981218A212301232A212 64A2123808147F930C>105 D<1330133813301300A4EA01C0EA0260EA04301360120812 00A213C0A4EA0180A4EA630012E312C612780D1A81930E>I<121E12065AA45A1338135C 139CEA3118EA36001238EA3F80EA61C0EA60C8A3EAC0D013600E147F9312>I<123C120C 1218A41230A41260A412C012C8A312D0126006147F930A>I<3830F87C38590C86384E0D 06EA9C0EEA980C1218A248485A15801418A23960301900140E190D7F8C1D>II112 D114 D<1207EA1880EA19C0EA3180EA3800121E7EEA0380124112E1EAC1001282127C0A0D7E8C 10>I<1204120CA35AEAFF80EA1800A25AA45A1261A212621264123809127F910D>III121 D E /Fo 6 53 df<13C0A8B51280A23800C000A811127E8D15>43 D<121EEA61801240EAC0C0A7EA40801261EA1E000A0D7E8C0E>48 D<121812F81218AA12FF080D7D8C0E>I<123EEA4180EA80C012C01200A2EA0180EA0300 1204EA08401230EA7F8012FF0A0D7E8C0E>I<123E1241EA61801201EA0300121EEA0180 EA00C0A212C0A2EA4180EA3E000A0D7E8C0E>I<12035AA2120B12131223126312C3EAFF C0EA0300A3EA0FC00A0D7E8C0E>I E /Fp 1 85 df84 D E /Fq 14 122 df<132013401380EA01005A1206A25AA25AA212381230A21270A3126012E0AD1260 1270A31230A212381218A27EA27EA27E7EEA0080134013200B317A8113>0 D<7E12407E7E12187EA27EA27EA213801201A213C0A3120013E0AD13C01201A31380A212 031300A21206A25AA25A12105A5A5A0B317F8113>I<13181360EA01C0EA0380EA0700B1 120E5A123012C01230121C7E7EB1EA0380EA01C0EA006013180D317B8118>8 D<12C01230121C7E7EB1EA0380EA01C0EA006013181360EA01C0EA0380EA0700B1120E5A 123012C00D317B8118>I<12C0B3A9021B7A800E>12 DI<1306 130C131813301370136013C012011380120313005A1206120E120C121CA212181238A312 301270A65AB21270A612301238A31218121CA2120C120E120612077E1380120113C01200 1360137013301318130C13060F4A788119>16 D<12C012607E7E121C120C7E12077E1380 120113C0120013E013601370A213301338A31318131CA6130EB2131CA613181338A31330 1370A2136013E013C012011380120313005A12065A121C12185A5A5A0F4A7F8119>I<12 C0B3A9021B73801C>63 D80 D<14F8EB0184EB0306EB060E1404EB0E00A3 5BAA5BAA5BAA5BA3EA40C012E0EAC1800043C7FC123E172E7E7F14>82 D88 D90 D<13C0B000C013C0EAF0C33838C700EA0EDCEA07F86C5A6C5A 6C5A12187B7F1C>121 D E /Fr 14 121 df10 D<120212041208121812101230122012601240A212C0AA1240 A212601220123012101218120812041202071E7D950D>40 D<1280124012201230121012 181208120C1204A21206AA1204A2120C1208121812101230122012401280071E7E950D> I<1360AAB512F0A238006000AA14167E9119>43 D<120FEA30C0EA6060A2EA4020EAC030 A9EA4020EA6060A2EA30C0EA0F000C137E9211>48 D<120C121C12EC120CAFEAFFC00A13 7D9211>I<121FEA60C01360EAF07013301260EA0070A2136013C012011380EA02005AEA 08101210EA2020EA7FE012FF0C137E9211>II<136013 E0A2EA016012021206120C120812101220126012C0EAFFFCEA0060A5EA03FC0E137F9211 >II<387FFFE0B512F0C8FCA6B512F06C13E0140A7E8B19> 61 D<127FEAE1C0EAE040EA40601200EA07E0EA3860126012C01364A2EA61E4EA3E380E 0D7E8C11>97 D<38F3E1F03834321838381C0CEA3018A938FC7E3F180D7F8C1B>109 D120 D E /Fs 5 91 df69 D80 D82 D84 D<003FB512803921C08100EA260038580102386002041240495A1200495A495AA2495A49 5AA20181C7FCA2EA0102EA0204A248481340485A15C03910200140EC0280382040043840 80181460B6FC1A1C7F9B2C>90 D E /Ft 28 115 df0 D<126012F0A2126004047C8B0C>I15 D17 D20 D<12C012F0123C120FEA03C0EA00F0133C130FEB03C0EB00F0143C140FEC0380EC0F0014 3C14F0EB03C0010FC7FC133C13F0EA03C0000FC8FC123C127012C0C9FCA7007FB5FCB612 8019227D9920>I<90387FFF800003B5FCD80780C7FC000CC8FC5A5AA25AA25AA81260A2 7EA27E120E6C7E0001B512806C7E191A7D9620>26 D<153081A381A281811680ED00C0B7 12F8A2C912C0ED0380160015065DA25DA35D25167E942A>33 D49 DI<1460A214C0A2EB0180A3EB03 00A21306A25BA25BA35BA25BA25BA2485AA248C7FCA31206A25AA25AA25AA35AA25A1240 13287A9D00>54 D66 D68 D<0103B5FC131F903830E0 0E01C1130C00011400EA0381EB01C01206EA0003A25CA2130791C7FCECFFC0491300010E C7FC131E131CA25BA25BA2EA30E01270EAFD80B4C8FC123C201D7F9B1E>70 D<90380C1F809038107FE090382087F0EBC38138018700D8030613F8EA060ED80E1C1378 121C5BEA382090C7FC127800701470A200F014F015E0A2140115C015806C1303EC070000 781306007C5B007E13106C5B381FC1C06CB4C7FCEA03F81D1E7E9C21>79 DI<011F131C90383F807C01EF133CEA01 87390007C0381530010313601580ECC10014C614C814F014C0A21307130B13331343EA01 83EA02031204121800307F127039F001E180ECF70000F813FC38E000F81E1C7E9B1E>88 D<13FE3803FF80380F01E0381C00700030131848130CA2481306B200401302171A7E981C >92 D<13101338A2136CA313C6A2EA0183A238030180A2380600C0A3481360A2481330A2 481318A348130CA24813061402171A7E981C>94 D<00C0130214060060130CA26C1318A3 6C1330A26C1360A26C13C0A338030180A238018300A2EA00C6A2136CA31338A21310171A 7E981C>I<133C13E0EA01C013801203AD13005A121C12F0121C12077E1380AD120113C0 EA00E0133C0E297D9E15>102 D<12F0121C12077E1380AD120113C0EA00E0133C13E0EA 01C013801203AD13005A121C12F00E297D9E15>I<134013C0EA0180A3EA0300A21206A3 5AA25AA35AA25AA35AA21260A37EA27EA37EA27EA37EA2EA0180A3EA00C013400A2A7D9E 10>I<12C0A21260A37EA27EA37EA27EA37EA2EA0180A3EA00C0A2EA0180A3EA0300A212 06A35AA25AA35AA25AA35AA20A2A7E9E10>I<12C0B3B3A502297B9E0C>II<164016C0ED0180A2ED0300A21506A25DA25DA25DA25DA2 5DA24A5AA24AC7FC120C003C1306124E008E5B12075CEA03805CA26C6C5AA26C6C5AA2EB 7180A2013BC8FCA2131EA2130CA2222A7E8123>112 D114 D E /Fu 61 122 df<13F8EA030C380E0604EA1C07383803080030138800701390A200E0 13A0A214C01480A3EA6007EB0B8838307190380F80E016127E911B>11 DI<38078010EA1FC0383FE020EA7FF038603040EAC0183880 088012001304EB0500A21306A31304A3130CA35BA45BA21320141B7F9115>I<1338137F EB87803801030090C7FC7FA27F12007FA2137013F8EA03B8EA063CEA0C1C121812381270 A212E0A413181338EA6030EA70606C5AEA0F80111E7F9D12>I<1310A3EB1FC0EB3040EB 6F80EB800048C7FC120212065A120812185AA212201260A312E05A7EA41270127C123FEA 1FE0EA07F81200133C131C130C5BEA0310EA00E012257F9C12>16 DI<137813CCEA0186EA03061206120E120CEA1C07 1218EA3806A2EA300E1270A2EA7FFEEAE01CA31318133812C013701360A213C0EAC18012 61EA6200123C101D7E9C13>I21 D<3801803000031370A3380700E0A4380E01C0A4381C0388 A3EA1E07383E1990383BE0E00038C7FCA25AA45AA25A151B7F9119>I<007E1360000E13 E0A3381C01C0A2EB03801400485A130E130C5B485A5BEA71C00073C7FC12FC12F013127E 9115>I<1310A3EB1F8013F03801CF00EA038048C7FC120EA6EA06FCEA0384EA06FC0008 C7FC12185A12201260A212E0A31270127CEA3F80EA1FE0EA03F8C67E131E130E130CEA01 08EA00F01125809C12>I<380FFFF85A5A386084001241EA81041201EA030CA212021206 A2120E120CEA1C0EA21238EA180615127E9118>I<131EEB7180EBC0C0EA01801203EB00 E05AA2380E01C0A31480EA1C0314001306EA1E0CEA3A18EA39E00038C7FCA25AA45AA25A 131B7F9115>I<3801FFF85A120F381E1E00EA180EEA38061270A2EAE00EA3130C131C13 185BEA60606C5A001FC7FC15127E9118>I<380FFFE05A5A3860C0001240485A12001201 A348C7FCA35AA3120E120613127E9112>I<0008130648130EA248130614025A13081318 00801304A2EB300C140800C01318EBF030B512F0387F9FE0EB1FC0383E0F00171280911A >33 D<13FCEA03FEEA0E07EA180012105AA2EA10C0EA1720EA1FE0EA20005AA25AA21304 EA4008EA6030EA3FE0EA0F8010147F9213>I<153081A281A2811507ED0380ED01C0ED00 E0B712F8A2250C7E942A>42 D<126012F0A2126004047C830C>58 D<126012F0A212701210A41220A212401280040C7C830C>II<130113031306A3130CA31318A31330A3 1360A213C0A3EA0180A3EA0300A31206A25AA35AA35AA35AA35AA210297E9E15>I<12E0 1278121EEA0780EA01E0EA0078131EEB0780EB01E0EB0078141EEC0780A2EC1E001478EB 01E0EB0780011EC7FC1378EA01E0EA0780001EC8FC127812E019187D9520>I<133E13C3 380100C0120200031360EA0780147038030030C7FCA3EBFC70EA0382EA0601120C381800 F0003813E012301270EB01C012E0A21480130314001306A2EA600C6C5AEA1860EA0F8014 1F7E9D16>64 D<48B5FC39003C01C090383800E015F01570A25B15F0A2EC01E09038E003 C0EC0780EC1F00EBFFFC3801C00FEC0780EC03C0A2EA0380A439070007801500140E5C00 0E1378B512C01C1C7E9B1F>66 D<903801F80890380E0618903838013890386000F04848 1370485A48C71230481420120E5A123C15005AA35AA45CA300701302A200305B00385B6C 5B6C136038070180D800FEC7FC1D1E7E9C1E>I<48B5128039003C01E090383800701538 153C151C5B151EA35BA44848133CA3153848481378157015F015E039070001C0EC0380EC 0700141C000E1378B512C01F1C7E9B22>I<48B512F839003C0078013813181510A35BA2 14081500495AA21430EBFFF03801C020A4390380404014001580A23907000100A25C1406 000E133EB512FC1D1C7E9B1F>I<48B512F038003C00013813301520A35BA21408150049 5AA21430EBFFF03801C020A448485A91C7FCA348C8FCA45AEAFFF01C1C7E9B1B>I<3A01 FFC3FF803A003C00780001381370A4495BA449485AA390B5FC3901C00380A4484848C7FC A43807000EA448131E39FFE1FFC0211C7E9B23>72 D<3801FFC038003C001338A45BA45B A4485AA4485AA448C7FCA45AEAFFE0121C7E9B12>I<3A01FFC07F803A003C001E000138 131815205D5DD97002C7FC5C5C5CEBE04014C0EBE1E013E23801C47013D0EBE03813C0EA 038080A280EA0700A280A2488039FFE03FF0211C7E9B23>75 D<3801FFE038003C001338 A45BA45BA4485AA438038002A31404EA0700140C14181438000E13F0B5FC171C7E9B1C> III<48B5FC39003C03C090383800 E015F01570A24913F0A315E0EBE001EC03C0EC0700141E3801FFF001C0C7FCA3485AA448 C8FCA45AEAFFE01C1C7E9B1B>80 D<3801FFFE39003C03C090383800E015F01570A24913 F0A3EC01E001E013C0EC0780EC1E00EBFFF03801C038140C140EA2EA0380A43807001E15 08A2151048130FD8FFE01320C7EA03C01D1D7E9B20>82 DI<001FB512 F0391C03807039300700300020142012601240130E1280A2000014005BA45BA45BA45BA4 1201EA7FFF1C1C7F9B18>I<397FF03FE0390F000700000E13061404A3485BA4485BA448 5BA4485BA35CA249C7FCEA60025B6C5AEA1830EA07C01B1D7D9B1C>I<39FFC00FF0391C 00038015001402A25C5C121E000E5B143014205CA25C49C7FC120FEA07025BA25BA25B5B EA03A013C05BA290C8FCA21C1D7D9B18>I<123F1207A2120EA45AA4EA39E0EA3A30EA3C 1812381270131CA3EAE038A313301370136013C01261EA2300121E0E1D7E9C12>98 DIIIII105 D<1307130FA213061300A61378139CEA010C1202131C12041200A21338A41370A413E0A4 EA01C01261EAF180EAF30012E6127C1024809B11>III<39381F81F0394E20C618394640E81CEB80F0EA8F0000 8E13E0120EA2391C01C038A315703938038071A215E115E23970070064D8300313382012 7E9124>II<380787803809C8603808D03013E0 EA11C014381201A238038070A31460380700E014C0EB0180EB8300EA0E86137890C7FCA2 5AA4123CB4FC151A819115>112 D114 DI< 13C01201A3EA0380A4EAFFF0EA0700A3120EA45AA4EA3820A21340A2EA1880EA0F000C1A 80990F>I<001C13C0EA27011247A238870380A2120EA2381C0700A438180E20A3EA1C1E 380C26403807C38013127E9118>II<380787803808C8403810F0C038 20F1E0EBE3C03840E1803800E000A2485AA43863808012F3EB810012E5EA84C6EA787813 127E9118>120 D<001C13C0EA27011247A238870380A2120EA2381C0700A4EA180EA3EA 1C1EEA0C3CEA07DCEA001C1318EA6038EAF0305B485AEA4180003EC7FC121A7E9114>I E /Fv 87 128 df0 D<1303A2497EA2497E130BEB13E01311EB21F01320497E1478EB807C143C3801003E 141E0002131F8048148014074814C0140348EB01E0003014F000201300006014F8007FB5 FCB612FCA21E1D7E9C23>I5 D<137F3803C1E038070070001C131C003C131E0038130E0078130F00707F00F01480A500 70140000785BA20038130E6C5BA26C5B00061330A20083EB608000811340A2394180C100 007F13FFA3191D7E9C1E>10 D<137E3801C180EA0301380703C0120EEB018090C7FCA5B5 12C0EA0E01B0387F87F8151D809C17>12 DI<90383F07E03901C09C18380380F0D80701133C000E13 E00100131892C7FCA5B612FC390E00E01CB03A7FC7FCFF80211D809C23>I34 D<13E0EA0310EA0608A2120EA45BA25B6C5AEC3FE09038800F80EC060000031304120738 09C00800115BEA30E03820F020EA607038E03840EB3C80131C90380F00207F0070EB8040 383009C0391830E180390FC03F001B1F7E9D20>38 D<126012F012F812681208A31210A2 122012401280050C7C9C0C>I<1380EA0100120212065AA25AA25AA35AA412E0AC1260A4 7EA37EA27EA27E12027EEA0080092A7C9E10>I<7E12407E12307EA27EA27EA37EA41380 AC1300A41206A35AA25AA25A12205A5A092A7E9E10>I<1306ADB612E0A2D80006C7FCAD 1B1C7E9720>43 D<126012F0A212701210A41220A212401280040C7C830C>II<126012F0A2126004047C830C>I<130113031306A3130CA31318A31330A3 1360A213C0A3EA0180A3EA0300A31206A25AA35AA35AA35AA35AA210297E9E15>II<5A1207123F12C71207B3A5EAFFF80D1C7C9B15>III<130CA2131C133C A2135C13DC139CEA011C120312021204120C1208121012301220124012C0B512C038001C 00A73801FFC0121C7F9B15>II<13F0EA030CEA0404EA0C0EEA181E1230130CEA7000A21260EAE3E0EAE430EAE818 EAF00C130EEAE0061307A51260A2EA7006EA300E130CEA1818EA0C30EA03E0101D7E9B15 >I<1240387FFF801400A2EA4002485AA25B485AA25B1360134013C0A212015BA21203A4 1207A66CC7FC111D7E9B15>III<126012F0A212601200AA126012F0A2126004127C91 0C>I<126012F0A212601200AA126012F0A212701210A41220A212401280041A7C910C>I< 126012F0A212601200A51260AD12F0A71260041E7C940C>I<007FB512C0B612E0C9FCA8 B612E06C14C01B0C7E8F20>I<1306A3130FA3EB1780A2EB37C01323A2EB43E01341A2EB 80F0A338010078A2EBFFF83802003CA3487FA2000C131F80001E5BB4EBFFF01C1D7F9C1F >65 DI<90381F8080EBE0613801801938070007000E 13035A14015A00781300A2127000F01400A8007014801278A212386CEB0100A26C13026C 5B380180083800E030EB1FC0191E7E9C1E>IIII<90381F8080EBE06138018019380700 07000E13035A14015A00781300A2127000F01400A6ECFFF0EC0F80007013071278A21238 7EA27E6C130B380180113800E06090381F80001C1E7E9C21>I<39FFF0FFF0390F000F00 AC90B5FCEB000FAD39FFF0FFF01C1C7F9B1F>I I<3807FF8038007C00133CB3127012F8A21338EA7078EA4070EA30E0EA0F80111D7F9B15 >I<39FFF01FE0390F000780EC060014045C5C5C5C5C49C7FC13021306130FEB17801327 EB43C0EB81E013016D7E1478A280143E141E80158015C039FFF03FF01C1C7F9B20>IIII80 D82 D<3807E080EA1C19EA300513 03EA600112E01300A36C13007E127CEA7FC0EA3FF8EA1FFEEA07FFC61380130FEB07C013 0313011280A300C01380A238E00300EAD002EACC0CEA83F8121E7E9C17>I<007FB512C0 38700F010060130000401440A200C014201280A300001400B1497E3803FFFC1B1C7F9B1E >I<39FFF01FF0390F000380EC0100B3A26C1302138000035BEA01C03800E018EB7060EB 0F801C1D7F9B1F>I<39FFE00FF0391F0003C0EC01806C1400A238078002A213C000035B A2EBE00C00011308A26C6C5AA213F8EB7820A26D5AA36D5AA2131F6DC7FCA21306A31C1D 7F9B1F>I<3AFFE1FFC0FF3A1F003E003C001E013C13186C6D1310A32607801F1320A33A 03C0278040A33A01E043C080A33A00F081E100A39038F900F3017913F2A2017E137E013E 137CA2013C133C011C1338A20118131801081310281D7F9B2B>I<39FFF003FC390F8001 E00007EB00C06D13800003EB01006D5A000113026C6C5A13F8EB7808EB7C18EB3C10EB3E 20131F6D5A14C06D5AABEB7FF81E1C809B1F>89 D<12FEA212C0B3B312FEA207297C9E0C >91 DI<12FEA21206B3B312FEA20729809E0C>I<120C12121221EA4080EA80400A057B9B15 >I<12E0A403047C9C0C>I97 D<12FC121CAA137CEA1D87 381E0180381C00C014E014601470A6146014E014C0381E018038190700EA10FC141D7F9C 17>IIII< 13F8EA018CEA071E1206EA0E0C1300A6EAFFE0EA0E00B0EA7FE00F1D809C0D>II<12FC121CAA137C1387EA1D03001E1380121CAD38FF9FF0141D7F9C17>I<1218 123CA21218C7FCA712FC121CB0EAFF80091D7F9C0C>I<13C0EA01E0A2EA00C01300A7EA 07E01200B3A21260EAF0C012F1EA6180EA3E000B25839C0D>I<12FC121CAAEB0FE0EB07 80EB06005B13105B5B13E0121DEA1E70EA1C781338133C131C7F130F148038FF9FE0131D 7F9C16>I<12FC121CB3A9EAFF80091D7F9C0C>I<39FC7E07E0391C838838391D01901800 1EEBE01C001C13C0AD3AFF8FF8FF8021127F9124>IIII<3803E080EA0E19EA1805EA3807EA7003A212E0A61270A2EA38071218EA0E 1BEA03E3EA0003A7EB1FF0141A7F9116>III<1204A4120CA2121C123C EAFFE0EA1C00A91310A5120CEA0E20EA03C00C1A7F9910>I<38FC1F80EA1C03AD130712 0CEA0E1B3803E3F014127F9117>I<38FF07E0383C0380381C0100A2EA0E02A2EA0F06EA 0704A2EA0388A213C8EA01D0A2EA00E0A3134013127F9116>I<39FF3FC7E0393C0703C0 001CEB01801500130B000E1382A21311000713C4A213203803A0E8A2EBC06800011370A2 EB8030000013201B127F911E>I<38FF0FE0381E0700EA1C06EA0E046C5AEA039013B0EA 01E012007F12011338EA021C1204EA0C0E487E003C138038FE1FF014127F9116>I<38FF 07E0383C0380381C0100A2EA0E02A2EA0F06EA0704A2EA0388A213C8EA01D0A2EA00E0A3 1340A25BA212F000F1C7FC12F312661238131A7F9116>III126 DI E /Fw 33 122 df 45 D48 D<130E131E137EEA07FE12FFA212F81200B3ABB512FEA317277BA622>III65 D<91387FE003903907FFFC07011FEBFF0F90397FF00F9F9039FF00 01FFD801FC7F4848147F4848143F4848141F485A160F485A1607127FA290C9FC5AA97E7F 1607123FA26C7E160E6C7E6C6C141C6C6C143C6C6C14786CB4EB01F090397FF007C0011F B512800107EBFE009038007FF028297CA831>67 D73 D75 D82 D<9038FF80600003EBF0E0000F13F8381F80FD383F 001F003E1307481303A200FC1301A214007EA26C140013C0EA7FFCEBFFE06C13F86C13FE 80000714806C14C0C6FC010F13E0EB007FEC1FF0140F140700E01303A46C14E0A26C1307 6C14C0B4EB0F80EBE03F39E3FFFE0000E15B38C01FF01C297CA825>I<3803FF80000F13 F0381F01FC383F80FE147F801580EA1F00C7FCA4EB3FFF3801FC3FEA0FE0EA1F80EA3F00 127E5AA4145F007E13DF393F839FFC381FFE0F3803FC031E1B7E9A21>97 DIIIII<9038FF80F00003EBE3F839 0FC1FE1C391F007C7C48137E003EEB3E10007EEB3F00A6003E133E003F137E6C137C380F C1F8380BFFE00018138090C8FC1238A2123C383FFFF814FF6C14C06C14E06C14F0121F38 3C0007007CEB01F8481300A4007CEB01F0A2003FEB07E0390FC01F806CB5120038007FF0 1E287E9A22>II<1207EA0F80EA1FC0EA3FE0A3EA1FC0EA 0F80EA0700C7FCA7EAFFE0A3120FB3A3EAFFFEA30F2B7EAA12>I107 DI<26FFC07FEB1FC0903AC1FFC07FF0903A C307E0C1F8D80FC49038F101FC9039C803F20001D801FE7F01D05BA201E05BB03CFFFE3F FF8FFFE0A3331B7D9A38>I<38FFC07E9038C1FF809038C30FC0D80FC413E0EBC80701D8 13F013D0A213E0B039FFFE3FFFA3201B7D9A25>II<38FFE1FE9038EFFF80 9038FE0FE0390FF803F09038F001F801E013FC140015FEA2157FA8157E15FEA215FC1401 01F013F89038F807F09038FC0FE09038EFFF809038E1FC0001E0C7FCA9EAFFFEA320277E 9A25>I<90383F80703901FFE0F03803F079380FE01D381F800F123FEB00075AA2127E12 FEA8127FA27E1380001F130F380FC01F3807F0773801FFE738007F87EB0007A9EC7FFFA3 20277E9A23>I<38FFC1F0EBC7FCEBC63E380FCC7F13D813D0A2EBF03EEBE000B0B5FCA3 181B7F9A1B>I<3803FE30380FFFF0EA3E03EA7800127000F01370A27E00FE1300EAFFE0 6CB4FC14C06C13E06C13F0000713F8C6FCEB07FC130000E0137C143C7E14387E6C137038 FF01E038E7FFC000C11300161B7E9A1B>I<13E0A41201A31203A21207120F381FFFE0B5 FCA2380FE000AD1470A73807F0E0000313C03801FF8038007F0014267FA51A>I<39FFE0 7FF0A3000F1307B2140FA2000713173903F067FF3801FFC738007F87201B7D9A25>I<39 FFFC03FFA3390FF000F0000714E07F0003EB01C0A2EBFC0300011480EBFE070000140013 FFEB7F0EA2149EEB3F9C14FC6D5AA26D5AA36D5AA26D5AA2201B7F9A23>I<39FFFC03FF A3390FF000F0000714E07F0003EB01C0A2EBFC0300011480EBFE070000140013FFEB7F0E A2149EEB3F9C14FC6D5AA26D5AA36D5AA26D5AA25CA21307003890C7FCEA7C0FEAFE0E13 1E131C5BEA74F0EA3FE0EA0F8020277F9A23>121 D E /Fx 30 124 df<126012F0A212701210A31220A21240A2040B7D830B>44 D<126012F0A2126004047D 830B>46 D50 D68 D<00FEEB7FC0000FEB0E001404EA0B80EA09C0A2EA08E01370A21338131CA213 0E1307EB0384A2EB01C4EB00E4A21474143CA2141C140C121C38FF80041A1A7F991D>78 D83 D<39FFE07FC0390E000E001404B200065B12076C5B6C6C5A3800E0C0013FC7FC1A1A7F99 1D>85 D<3AFF81FF07F03A3C007801C0001CEC0080A36C90389C0100A33907010E02A339 03830F04EB8207A2150C3901C40388A33900E801D0A390387000E0A301305B0120134024 1A7F9927>87 D97 D<12FC121CA913FCEA1D07381E0380381C01C01300 14E0A6EB01C01480381E0300EA1906EA10F8131A809915>II<133F1307A9EA03E7EA0C17EA 180F487E127012E0A6126012706C5AEA1C373807C7E0131A7F9915>I II<121812 3CA212181200A612FC121CAE12FF081A80990A>105 D<12FC121CA9EB1FC0EB0F00130C 5B13205B13E0121DEA1E70EA1C7813387F131E7F148038FF9FE0131A809914>107 D<12FC121CB3A6EAFF80091A80990A>I<38FC7C1F391D8E6380391E0781C0A2001C1301 AB39FF9FE7F81D107F8F20>I IIIIII<1208A41218A21238EAFFC0 EA3800A81320A41218EA1C40EA07800B177F960F>I<38FC1F80EA1C03AB1307120CEA0E 0B3803F3F01410808F15>I<38FF0F80383C0700EA1C061304A26C5AA26C5AA3EA03A0A2 EA01C0A36C5A11107F8F14>I<39FE7F1F8039381C0700003C1306381C0C04130E380E16 081317A238072310149013A33803C1A014E0380180C0A319107F8F1C>I<38FF0F80383C 0700EA1C061304A26C5AA26C5AA3EA03A0A2EA01C0A36C5AA248C7FCA212E112E212E412 7811177F8F14>121 D123 D E /Fy 7 117 df<1303497EA2497EA3EB1BE0A2EB3BF01331A2EB60F8A2EBE0FCEBC07CA248487EEBFF FE487FEB001F4814800006130FA248EB07C039FF803FFCA21E1A7F9921>65 D97 D<12FCA2123CA713FE383F8780383E01C0003C13E0EB00F0A214F8 A514F0A2EB01E0003E13C0383B07803830FE00151A7E9919>II114 DI<1206A4120EA2121EEA3FF012FFEA1E00A81318A5 EA0F30EA03E00D187F9711>I E /Fz 2 122 df<1202A3EAC218EAF278EA3AE0EA0F80A2 EA3AE0EAF278EAC218EA0200A30D0E7E8E12>3 D<1206A8EAFFF0A2EA0600B30C1D7E96 11>121 D E /FA 15 122 df<497EA3497EA3EB05E0A2EB09F01308A2EB1078A3497EA3 497EA2EBC01F497EA248B51280EB0007A20002EB03C0A348EB01E0A348EB00F0121C003E EB01F839FF800FFF20237EA225>65 D75 D<3803F020380C0C60EA 1802383001E0EA70000060136012E0A21420A36C1300A21278127FEA3FF0EA1FFE6C7E00 03138038003FC0EB07E01301EB00F0A214707EA46C1360A26C13C07E38C8018038C60700 EA81FC14247DA21B>83 D97 D101 D<14703803F198380E1E 18EA1C0E38380700A200781380A400381300A2EA1C0EEA1E1CEA33F00020C7FCA2123012 38EA3FFE381FFFC06C13E0383000F0481330481318A400601330A2003813E0380E038038 03FE0015217F9518>103 D<120E12FE121E120EABEB1F80EB60C0EB80E0380F0070A212 0EAF38FFE7FF18237FA21B>I<121C123EA3121CC7FCA8120E127E121E120EB1EAFFC00A 227FA10E>I<120E12FE121E120EABEB03FCEB01F014C01480EB02005B5B5B133813F8EA 0F1CEA0E1E130E7F1480EB03C0130114E0EB00F014F838FFE3FE17237FA21A>107 D<390E1FC07F3AFE60E183803A1E807201C03A0F003C00E0A2000E1338AF3AFFE3FF8FFE 27157F942A>109 D<380E1F8038FE60C0381E80E0380F0070A2120EAF38FFE7FF18157F 941B>I114 DI<000E137038FE07F0EA1E0000 0E1370AD14F0A238060170380382783800FC7F18157F941B>117 D<38FF80FE381E00781430000E1320A26C1340A2EB80C000031380A23801C100A2EA00E2 A31374A21338A31310A25BA35B12F05B12F10043C7FC123C171F7F941A>121 D E /FB 25 124 df50 D68 D78 D<90387F80203801FFE03907C07860380F001C001EEB06E0481303003813010078130012 70156012F0A21520A37E1500127C127E7E13C0EA1FF86CB47E6C13F06C13FCC613FF010F 1380010013C0EC1FE01407EC03F01401140015F8A26C1478A57E15706C14F015E07E6CEB 01C000ECEB038000C7EB070038C1F01E38807FFCEB0FF01D337CB125>83 D<13FE380303C0380C00E00010137080003C133C003E131C141EA21208C7FCA3EB0FFEEB FC1EEA03E0EA0F80EA1F00123E123C127C481404A3143EA21278007C135E6CEB8F08390F 0307F03903FC03E01E1F7D9E21>97 D99 D<15F0141FA214011400AFEB0FC0EB70 303801C00C3803800238070001120E001E13005AA2127C1278A212F8A71278A2127C123C A27E000E13016C1302380380046C6C487E3A00F030FF80EB1FC021327EB125>III<15F090387F03083901C1C41C380380E83907007008 48EB7800001E7FA2003E133EA6001E133CA26C5B6C13706D5A3809C1C0D8087FC7FC0018 C8FCA5121C7E380FFFF86C13FF6C1480390E000FC00018EB01E048EB00F0007014704814 38A500701470A26C14E06CEB01C00007EB07003801C01C38003FE01E2F7E9F21>I<120F EA1F80A4EA0F00C7FCABEA0780127FA2120F1207B3A6EA0FC0EAFFF8A20D307EAF12> 105 D107 DI<260780FEEB1FC03BFF83078060F0903A8C03C180783B0F9001E2003C D807A013E4DA00F47F01C013F8A2495BB3A2486C486C133F3CFFFC1FFF83FFF0A2341F7E 9E38>I<380780FE39FF83078090388C03C0390F9001E0EA07A06E7E13C0A25BB3A2486C 487E3AFFFC1FFF80A2211F7E9E25>II<380781FC39FF 86078090388801C0390F9000E0D807A0137001C01378497F153E151E151FA2811680A716 005DA2151E153E153C6D5B01A013705D90389803C0D9860FC7FCEB81F80180C8FCAB487E EAFFFCA2212D7E9E25>I<90380FC01090387830303801E0083803800439070002704813 01001E14F0123E003C1300127CA2127812F8A71278127CA2123C123E001E13017E6C1302 380380043801C0083800F030EB1FC090C7FCAB4A7E91381FFF80A2212D7E9E23>I<3807 83E038FF8418EB887CEA0F90EA07A01438EBC000A35BB3487EEAFFFEA2161F7E9E19>I< 3801FC10380E0330381800F048137048133012E01410A37E6C1300127EEA3FF06CB4FC6C 13C0000313E038003FF0EB01F813006C133CA2141C7EA27E14186C1338143000CC136038 C301C03880FE00161F7E9E1A>I<1340A513C0A31201A212031207120F381FFFE0B5FC38 03C000B01410A80001132013E000001340EB78C0EB1F00142C7FAB19>III121 D123 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 339 436 a FB(Some)21 b(limiting)f(prop)r(erties)i(of)g (randomly)e(forced)572 527 y(2D)h(Na)n(vier{Stok)n(es)h(equations)571 648 y FA(Sergei)16 b(Kuksin)866 629 y Fz(\003)1013 648 y FA(Armen)e(Shiriky)o(an)1377 629 y Fz(y)889 839 y Fy(Abstract)419 904 y Fx(W)m(e)19 b(consider)j(random)e(p)q(erturbations)i(of)e(2D)f (Na)o(vier{Stok)o(es)j(equations.)361 949 y(Under)17 b(some)g(natural)h(conditions)i(on)c(random)i(forces,)f(w)o(e)g(study)g (asymptotic)361 995 y(prop)q(erties)e(of)e(solutions)i(and)f (stationary)g(measures.)257 1132 y Fw(0)67 b(In)n(tro)r(duction)257 1223 y Fv(W)m(e)14 b(consider)h(the)f(2D)f(Na)o(vier{Stok)o(es)h (\(NS\))h(system)e(p)q(erturb)q(ed)j(b)o(y)e(a)g(random)e(force:)557 1314 y(_)-20 b Fu(u)9 b Ft(\000)h Fu(\027)s Fv(\001)p Fu(u)e Fv(+)h(\()p Fu(u;)e Ft(r)p Fv(\))p Fu(u)h Fv(+)i Ft(r)p Fu(p)h Fv(=)h Fu(\021)q Fv(\()p Fu(t;)7 b(x)p Fv(\))p Fu(;)47 b Fv(div)7 b Fu(u)k Fv(=)h(0)p Fu(:)206 b Fv(\(0.1\))257 1406 y(The)11 b(space)h(v)n(ariable)d Fu(x)i Fv(b)q(elongs)f(either)i(to)e(a)g(b)q(ounded)i(domain,)c(and)j (then)g(the)g(Diric)o(hlet)257 1456 y(b)q(oundary)i(condition)e(is)h (imp)q(osed,)f(or)i(to)f(the)h(t)o(w)o(o-dimensional)c(torus)k Fs(T)1445 1441 y Fr(2)1473 1456 y Fv(=)e Fs(R)1549 1441 y Fr(2)1565 1456 y Fu(=)p Fv(2)p Fu(\031)q Fs(Z)1662 1441 y Fr(2)1678 1456 y Fv(,)257 1505 y(and)17 b(then)h(w)o(e)f(assume) g(that)743 1472 y Fq(R)762 1520 y Fp(T)785 1512 y Fo(2)807 1505 y Fu(u)7 b(dx)16 b Ft(\021)949 1472 y Fq(R)968 1520 y Fp(T)991 1512 y Fo(2)1013 1505 y Fu(\021)8 b(dx)16 b Ft(\021)h Fv(0.)27 b(Denoting)16 b(b)o(y)h Fu(H)j Fv(the)d(corre-)257 1555 y(sp)q(onding)12 b Fu(L)460 1540 y Fr(2)479 1555 y Fv(-space)h(of)e(div)o(ergence-free)i(v)o(ector)g(\014elds)f(with)g (the)g(natural)g(norm)e Ft(j)5 b(\001)g(j)11 b Fv(and)257 1605 y(b)o(y)i(\005)g(the)g(orthogonal)f(pro)r(jections)h(to)g Fu(H)s Fv(,)f(w)o(e)h(write)g(\(0.1\))f(as)h(a)f(random)f(system)i(in)f Fu(H)s Fv(:)652 1696 y(_)-19 b Fu(u)o Fv(\()p Fu(t)p Fv(\))10 b(+)f Fu(\027)s(Lu)p Fv(\()p Fu(t)p Fv(\))g(+)h Fu(B)r Fv(\()p Fu(u)p Fv(\()p Fu(t)p Fv(\))p Fu(;)d(u)p Fv(\()p Fu(t)p Fv(\)\))12 b(=)f Fu(\021)q Fv(\()p Fu(t)p Fv(\))p Fu(:)302 b Fv(\(0.2\))257 1788 y(Here)20 b Fu(L)g Fv(=)g Ft(\000)p Fv(\005\001,)f Fu(B)r Fv(\()p Fu(u;)7 b(u)p Fv(\))19 b(=)h(\005\()p Fu(u;)7 b Ft(r)p Fv(\))p Fu(u)p Fv(,)18 b(and)g Fu(\021)q Fv(\()p Fu(t)p Fv(\))i(=)g(\005)p Fu(\021)q Fv(\()p Fu(t;)7 b Ft(\001)p Fv(\))18 b(\(see)i([CF88)o(]\).) 32 b(W)m(e)257 1837 y(denote)22 b(b)o(y)e Ft(f)p Fu(e)502 1843 y Fn(j)520 1837 y Ft(g)g Fv(the)h(Hilb)q(ert)g(basis)g(in)f Fu(H)j Fv(formed)c(b)o(y)i(the)g(eigenfunctions)g(of)f(the)257 1887 y(op)q(erator)e Fu(L)f Fv(with)g(eigen)o(v)n(alues)f(0)h Fu(<)g(\013)902 1893 y Fr(1)936 1887 y Ft(\024)g Fu(\013)1012 1893 y Fr(2)1047 1887 y Ft(\024)g(\001)7 b(\001)g(\001)e Fv(.)28 b(Concerning)17 b(the)g(force)h Fu(\021)q Fv(,)f(w)o(e)257 1937 y(assume)h(that)g(it)g(is)g(smo)q(oth)f(in)h Fu(x)p Fv(,)g(whereas)i(as)e(a)g(function)g(of)f(time)g Fu(t)h Fv(it)g(is)g(either)h(a)257 1987 y(kic)o(k-pro)q(cess)d(or)d(a)h(white) g(noise.)k(Th)o(us,)c(w)o(e)g(consider)h(the)f(follo)o(wing)e(t)o(w)o (o)h(cases:)320 2070 y Ft(\017)20 b Fv(The)14 b(function)g Fu(\021)h Fv(mo)q(dels)e(random)f(kic)o(ks)i(with)f(some)g(p)q(erio)q (d)i Fu(")d(>)f Fv(0:)398 2192 y Fu(\021)i Fv(=)f Fu(\021)497 2198 y Fn(")514 2192 y Fv(\()p Fu(t;)7 b(x)p Fv(\))k(=)659 2160 y Ft(p)p 694 2160 20 2 v 32 x Fu(")755 2140 y Fm(1)741 2153 y Fq(X)720 2242 y Fn(k)q Fr(=)p Fm(\0001)830 2192 y Fu(\021)851 2198 y Fn(k)871 2192 y Fv(\()p Fu(x)p Fv(\))p Fu(\016)r Fv(\()p Fu(t)e Ft(\000)h Fu(k)q(")p Fv(\))p Fu(;)48 b(\021)1168 2198 y Fn(k)1188 2192 y Fv(\()p Fu(x)p Fv(\))12 b(=)1313 2140 y Fm(1)1299 2153 y Fq(X)1301 2241 y Fn(j)r Fr(=1)1366 2192 y Fu(b)1384 2198 y Fn(j)1402 2192 y Fu(\030)1420 2198 y Fn(j)r(k)1456 2192 y Fu(e)1475 2198 y Fn(j)1492 2192 y Fv(\()p Fu(x)p Fv(\))p Fu(;)44 b Fv(\(0.3\))p 257 2278 573 2 v 303 2304 a Fl(\003)321 2316 y Fk(Departmen)o(t)13 b(of)i(Mathematics,)f(Heriot-W)m(att)g(Univ) o(ersit)o(y)m(,)h(Edin)o(burgh)e(EH14)j(4AS,)f(Scotland,)257 2355 y(UK;)20 b(Steklo)o(v)d(Institute)g(of)i(Mathematics,)f(8)h (Gubkina)e(St.,)j(117966)e(Mosco)o(w,)i(Russia;)i(E-mail:)257 2395 y Fj(S.B.Kuksin@)o(ma.)o(hw.)o(ac.)o(uk)p Fk(.)305 2424 y Fl(y)321 2436 y Fk(Departmen)o(t)13 b(of)i(Mathematics,)f (Heriot-W)m(att)g(Univ)o(ersit)o(y)m(,)h(Edin)o(burgh)e(EH14)j(4AS,)f (Scotland,)257 2475 y(UK;)e(E-mail:)h Fj(A.Shiriky)o(an@)o(ma.)o(hw.)o (ac.)o(uk)o Fk(.)963 2628 y Fv(1)p eop %%Page: 2 2 2 1 bop 361 262 a Fv(where)17 b Ft(f)p Fu(b)522 268 y Fn(j)539 262 y Ft(g)e Fv(is)h(a)f(sequence)j(of)d(nonnegativ)o(e)g (constan)o(ts)h(suc)o(h)h(that)1482 230 y Fq(P)1526 274 y Fn(j)1551 262 y Fu(b)1569 246 y Fr(2)1569 272 y Fn(j)1601 262 y Fu(<)e Ft(1)361 311 y Fv(and)k Ft(f)p Fu(\030)486 317 y Fn(j)r(k)522 311 y Ft(g)g Fv(is)g(a)g(family)e(of)h(b)q(ounded)i (indep)q(enden)o(t)h(scalar)e(random)f(v)n(ariables)361 361 y(with)f Fu(k)q Fv(-indep)q(enden)o(t)h(distributions)g(that)f (satisfy)g(certain)h(regularit)o(y)e(assump-)361 411 y(tions)e(\(see)h(condition)e(\(H\))i(in)e(Section)h(3\).)320 493 y Ft(\017)20 b Fv(The)14 b(function)g Fu(\021)q Fv(\()p Fu(t)p Fv(\))g(is)g(the)h(time)d(deriv)n(ativ)o(e)h(of)h(an)f Fu(H)s Fv(-v)n(alued)g(Wiener)i(pro)q(cess:)559 603 y Fu(\021)e Fv(=)f Fu(\021)q Fv(\()p Fu(t;)7 b(x)p Fv(\))k(=)816 574 y Fu(@)p 809 593 40 2 v 809 631 a(@)r(t)853 603 y(\020)s Fv(\()p Fu(t;)c(x)p Fv(\))p Fu(;)48 b(\020)s Fv(\()p Fu(t;)7 b(x)p Fv(\))k(=)1190 563 y Fq(X)1212 651 y Fn(j)1256 603 y Fu(b)1274 609 y Fn(j)1292 603 y Fu(\014)1315 609 y Fn(j)1333 603 y Fv(\()p Fu(t)p Fv(\))p Fu(e)1399 609 y Fn(j)1417 603 y Fv(\()p Fu(x)p Fv(\))p Fu(;)119 b Fv(\(0.4\))361 735 y(where)19 b Ft(f)p Fu(b)524 741 y Fn(j)541 735 y Ft(g)f Fv(is)g(a)f(sequence)j(of)d(constan)o(ts)i(as)f(ab)q(o)o(v)o(e)g (and)g Ft(f)p Fu(\014)1385 741 y Fn(j)1402 735 y Ft(g)g Fv(is)g(a)f(family)e(of)361 785 y(indep)q(enden)o(t)g(standard)g(Bro)o (wnian)e(motions.)320 865 y(Under)h(the)h(assumption)d(that)749 953 y Fu(b)767 959 y Fn(j)796 953 y Ft(6)p Fv(=)g(0)41 b(for)g(1)11 b Ft(\024)h Fu(j)i Ft(\024)e Fu(N)r(;)413 b Fv(\(0.5\))257 1042 y(where)20 b Fu(N)25 b Fv(=)20 b Fu(N)525 1048 y Fn("\027)581 1042 y Ft(\025)g Fv(0)f(is)g(a)f (suitable)h(in)o(teger,)h(it)e(w)o(as)h(pro)o(v)o(ed)g(in)f([KS00])g (that)h(the)257 1091 y(NS)e(system)g(\(0.2\))o(,)g(\(0.3\))f(has)h(a)f (unique)h(stationary)f(measure)h Fu(\026)f Fv(=)g Fu(\026)1416 1097 y Fn("\027)1469 1091 y Fv(and)h(that)g(all)257 1141 y(solutions)d(con)o(v)o(erge)g(to)g Fu(\026)g Fv(in)f(distribution:)610 1248 y Fs(E)8 b Fu(f)d Fv(\()p Fu(u)p Fv(\()p Fu(t)p Fv(\)\))15 b Ft(!)837 1192 y Fq(Z)860 1286 y Fn(H)898 1248 y Fu(f)t Fv(\()p Fu(v)q Fv(\))p Fu(\026)p Fv(\()p Fu(dv)q Fv(\))44 b(as)e Fu(t)11 b Ft(!)g(1)p Fu(;)273 b Fv(\(0.6\))257 1357 y(where)12 b Fu(f)i Fv(is)c(an)o(y)g(b)q(ounded)g (con)o(tin)o(uous)g(functional)f(on)h Fu(H)s Fv(.)17 b(In)10 b(subsequen)o(t)i(publications,)257 1407 y(w)o(e)20 b(ha)o(v)o(e)g(sho)o(wn)f(that)h(the)g(con)o(v)o(ergence)h(is)f(exp)q (onen)o(tially)e(fast)i(pro)o(vided)f(that)h(the)257 1457 y(functional)13 b Fu(f)18 b Fv(is)13 b(Lipsc)o(hitz)g(con)o(tin)o (uous;)g(see)i([Kuk02)o(])e(for)g(discussions)h(and)f(references.)320 1507 y(The)21 b(NS)g(system)f(\(0.2\),)i(\(0.4\))o(,)g(\(0.5\))e(w)o (as)h(considered)h(later)f(b)o(y)f(E,)h(Mattingly)m(,)257 1557 y(Sinai)16 b([EMS01])h(and)g(Bricmon)o(t,)f(Kupiainen,)i(Lefev)o (ere)g([BKL02].)27 b(Under)18 b(the)g(addi-)257 1606 y(tional)e(restriction)j(that)e(the)h(sum)e(in)h(\(0.4\))g(is)g (\014nite,)h(they)g(pro)o(v)o(ed)f(that)h(the)g(system)257 1656 y(has)g(a)e(unique)h(stationary)g(measure)g(and)g(studied)h(con)o (v)o(ergence)g(of)f(solutions)f(to)h(this)257 1706 y(measure;)k(see)f ([Kuk02])e(for)h(discussions)h(and)e(references.)36 b(In)19 b(the)h(w)o(ork)e([KS02])g(w)o(e)257 1756 y(applied)d(to)h(the)g (white-forced)h(NS)f(system)f(\(0.2\))o(,)h(\(0.4\))o(,)g(\(0.5\))f (the)h(tec)o(hniques)h(dev)o(el-)257 1806 y(op)q(ed)d(for)f(studying)g (the)h(kic)o(k)o(ed)f(equations.)k(It)d(w)o(as)f(pro)o(v)o(ed)g(that)g (con)o(v)o(ergence)i(\(0.6\))e(is)257 1856 y(exp)q(onen)o(tially)f (fast)h(for)f(an)o(y)g(b)q(ounded)i(Lipsc)o(hitz)f(functional)e Fu(f)18 b Fv(and)12 b(an)o(y)h(solution)e Fu(u)p Fv(\()p Fu(t)p Fv(\))257 1905 y(of)j(the)g(NS)g(system)g(suc)o(h)g(that)g Fs(E)9 b Ft(j)p Fu(u)p Fv(\(0\))p Ft(j)898 1890 y Fr(2)930 1905 y Fu(<)j Ft(1)p Fv(.)320 1955 y(In)17 b(this)h(w)o(ork,)g(w)o(e)g (sp)q(ecify)g(con)o(v)o(ergence)i(\(0.6\))d(in)g(the)i(case)f(of)f(the) i(white-forced)257 2005 y(equation)13 b(and)g(study)g(relations)g(b)q (et)o(w)o(een)h(the)g(stationary)e(measures)h(for)g(the)g(kic)o(k-)g (and)257 2055 y(white-forced)i(equations.)j(The)c(pap)q(er)h(is)f (organised)f(as)h(follo)o(ws.)320 2105 y(In)g(Section)i(1,)e(w)o(e)h (establish)g(a)f(priori)g(estimates)h(for)f(momen)o(ts)f(of)h(Sob)q (olev)g(norms)257 2154 y(of)f(solutions)f(for)g(the)i(white-forced)f (NS)g(system)g(under)g(the)h(p)q(erio)q(dic)f(b)q(oundary)g(condi-)257 2204 y(tions.)18 b(In)c(Section)h(2,)e(w)o(e)h(use)h(these)g(estimates) f(to)g(pro)o(v)o(e)g(Theorem)f(2.1,)f(whic)o(h)i(claims)257 2254 y(that)g(con)o(v)o(ergence)h(\(0.6\))e(holds)h(for)f(functionals)g Fu(f)19 b Fv(that)13 b(are)h(de\014ned)h(and)f(H\177)-21 b(older)13 b(con-)257 2304 y(tin)o(uous)f(on)f(a)g(Sob)q(olev)h(space)g Fu(H)788 2289 y Fn(k)820 2304 y Fv(\(with)f(an)h(arbitrary)f Fu(k)i Ft(\025)e Fv(0\))h(and)f(ha)o(v)o(e)h(a)f(p)q(olynomial)257 2354 y(gro)o(wth)17 b(at)g(in\014nit)o(y)m(.)26 b(This)17 b(generalisation)f(is)h(imp)q(ortan)o(t)e(since)j(functionals)e(relev)n (an)o(t)257 2403 y(for)j(statistical)g(h)o(ydro)q(dynamics)f(are)h(un)o (b)q(ounded)h(and)f(usually)f(are)i(not)f(con)o(tin)o(uous)257 2453 y(on)e(the)h(space)g Fu(H)s Fv(.)27 b(F)m(or)17 b(example,)f(the)i(correlation)f(tensor)h(giv)o(es)f(rise)g(to)g (functionals)257 2503 y Fu(f)t Fv(\()p Fu(u)p Fv(\))c(=)f Fu(u)418 2509 y Fn(i)431 2503 y Fv(\()p Fu(x)p Fv(\))p Fu(u)511 2509 y Fn(j)528 2503 y Fv(\()p Fu(y)q Fv(\),)h(where)g Fu(x)f Fv(and)f Fu(y)j Fv(are)e(p)q(oin)o(ts)g(in)f(the)h(space)h (domain)d(and)h Fu(i;)c(j)14 b Fv(=)e(1)p Fu(;)7 b Fv(2.)963 2628 y(2)p eop %%Page: 3 3 3 2 bop 320 262 a Fv(Section)16 b(3)g(is)g(dev)o(oted)h(to)f(a)g (comparison)f(of)h(solutions)f(and)h(stationary)g(measures)257 311 y(for)e(kic)o(k-)f(and)g(white-forced)h(equations.)k(Namely)m(,)11 b(let)j Fu(u)1183 317 y Fn(")1201 311 y Fv(\()p Fu(t)p Fv(\))g(and)f Fu(u)p Fv(\()p Fu(t)p Fv(\))h(b)q(e)g(solutions)f(of)257 361 y(the)i(NS)g(system)f(\(0.2\))f(that)i(corresp)q(ond)h(to)e(a)g (deterministic)g(initial)e(function)i Fu(u)1581 367 y Fr(0)1612 361 y Ft(2)e Fu(H)257 411 y Fv(and)i(righ)o(t-hand)f(sides)i (\(0.3\))e(and)h(\(0.4\))o(,)f(resp)q(ectiv)o(ely)m(.)20 b(Assuming)12 b(that)770 502 y Fs(E)c Fu(\030)819 508 y Fn(j)r(k)870 502 y Ft(\021)j Fv(0)p Fu(;)48 b Fs(E)9 b Fu(\030)1046 485 y Fr(2)1044 513 y Fn(j)r(k)1094 502 y Ft(\021)j Fv(1)p Fu(;)433 b Fv(\(0.7\))257 594 y(w)o(e)14 b(\014rst)g(sho)o(w)g(that)f Fu(u)619 600 y Fn(")637 594 y Fv(\()p Fu(t)p Fv(\))g(con)o(v)o(erges)i(to)e Fu(u)p Fv(\()p Fu(t)p Fv(\))g(in)g(distribution.)18 b(F)m(urthermore,)12 b(if)h(all)f Fu(b)1672 600 y Fn(j)257 643 y Fv(are)k(nonzero,)g(then)f (the)h(kic)o(k-forced)f(equation)g(\(0.2\))o(,)g(\(0.3\))g(has)g(a)g (unique)g(stationary)257 693 y(measure)20 b Fu(\026)450 699 y Fn(")487 693 y Fv(for)e(an)o(y)h Fu(")i(>)g Fv(0.)34 b(The)19 b(main)f(results)i(of)f(this)g(pap)q(er)h(is)f(Theorem)g(3.1,) 257 743 y(according)14 b(to)g(whic)o(h)g(w)o(e)g(ha)o(v)o(e)786 834 y Fu(\026)811 840 y Fn(")840 834 y Fu(*)d(\026)41 b Fv(as)h Fu(")12 b Ft(!)f Fv(0)p Fu(;)449 b Fv(\(0.8\))257 926 y(where)16 b Fu(\026)f Fv(is)g(the)h(unique)f(stationary)f(measure) h(for)g(the)g(white-forced)h(equation.)21 b(Since)257 976 y Ft(D)q Fv(\()p Fu(u)330 982 y Fn(")348 976 y Fv(\()p Fu(t)p Fv(\)\))14 b Fu(*)f(\026)505 982 y Fn(")538 976 y Fv(as)i Fu(t)e Ft(!)g(1)i Fv(and)g Ft(D)q Fv(\()p Fu(u)885 982 y Fn(")903 976 y Fv(\()p Fu(t)p Fv(\)\))e Fu(*)g Ft(D)q Fv(\()p Fu(u)p Fv(\()p Fu(t)p Fv(\)\))j(as)f Fu(")f Ft(!)f Fv(0,)h(con)o(v)o(ergence)j(\(0.8\))257 1025 y(implies)12 b(that)i(w)o(e)g(ha)o(v)o(e)g(the)h(follo)o(wing)c(comm)o(utativ)n(e)h (diagram:)789 1125 y Ft(D)822 1091 y Fq(\000)841 1125 y Fu(u)865 1131 y Fn(")883 1125 y Fv(\()p Fu(t)p Fv(\))930 1091 y Fq(\001)993 1101 y Fn(t)p Fm(!1)970 1125 y Ft(\000)-14 b(\000)-9 b(\000)g(\000)c(!)20 b Fu(\026)1140 1131 y Fn(")790 1207 y(")p Fm(!)p Fr(0)855 1160 y Fq(?)855 1185 y(?)855 1210 y(y)1123 1160 y(?)1123 1185 y(?)1123 1210 y(y)1150 1207 y Fn(")p Fm(!)p Fr(0)798 1293 y Ft(D)831 1259 y Fq(\000)850 1293 y Fu(u)p Fv(\()p Fu(t)p Fv(\))921 1259 y Fq(\001)993 1269 y Fn(t)p Fm(!1)970 1293 y Ft(\000)-14 b(\000)-9 b(\000)g(\000)c(!)29 b Fu(\026)320 1384 y Fv(Con)o(v)o (ergence)20 b(\(0.8\))f(manifests)f(the)i(fact)f(that)h(the)g(white)f (forced)h(equations)f(o)q(c-)257 1434 y(cup)o(y)14 b(a)f(sp)q(ecial)g (place)h(among)d(randomly)g(forced)j(equations.)j(The)d(results)g(pro)o (v)o(ed)g(and)257 1484 y(discussed)i(in)d(this)h(pap)q(er)h(suggests)g (the)g(follo)o(wing)c(conjecture:)320 1558 y Fi(Conjecture.)17 b Fv(Let)d(us)g(set)687 1681 y Fu(\021)708 1687 y Fn(")725 1681 y Fv(\()p Fu(t;)7 b(x)p Fv(\))k(=)870 1649 y Ft(p)p 904 1649 20 2 v 904 1681 a Fu(")944 1629 y Fm(1)931 1641 y Fq(X)932 1730 y Fn(j)r Fr(=1)998 1681 y Fu(b)1016 1687 y Fn(j)1033 1681 y Fu(\030)1051 1687 y Fn(j)1068 1681 y Fv(\()p Fu(t=")p Fv(\))p Fu(e)1174 1687 y Fn(j)1193 1681 y Fv(\()p Fu(x)p Fv(\))p Fu(;)257 1816 y Fv(where)i(0)e Fu(<)h(")g Ft(\024)f Fv(1,)g Ft(f)p Fu(\030)608 1822 y Fn(j)626 1816 y Ft(g)g Fv(is)g(a)g(sequence)i(of)d(i.i.d.)16 b(stationary)11 b(pro)q(cesses)i(that)f(are)f(mixing)257 1866 y(\(in)i(an)f(appropriate)g(sense\))j(and)d(satisfy)g(the)i (conditions)e Fs(E)c Fu(\030)1246 1872 y Fn(j)1266 1866 y Fv(\()p Fu(t)p Fv(\))k Ft(\021)g Fv(0)g(and)g Fs(E)d Fu(\030)1533 1851 y Fr(2)1531 1877 y Fn(j)1555 1866 y Fv(\()p Fu(t)p Fv(\))i Ft(\021)h Fv(1.)257 1916 y(Then)j(the)f (assertions)h(b)q(elo)o(w)f(tak)o(e)g(place)g(on)f(condition)h(that)g Fu(b)1289 1922 y Fn(j)1317 1916 y Ft(6)p Fv(=)e(0)i(for)f(all)g Fu(j)h Ft(\025)e Fv(1:)297 1999 y(\(i\))20 b(Equation)13 b(\(0.2\))h(with)f Fu(\021)g Fv(=)f Fu(\021)833 2005 y Fn(")850 1999 y Fv(\()p Fu(t;)7 b(x)p Fv(\))13 b(has)h(a)g(unique)g (stationary)f(measure)h Fu(\026)1581 2005 y Fn(")1599 1999 y Fv(.)285 2082 y(\(ii\))20 b(All)e(solutions)g(for)h(\(0.2\))f (with)g(deterministic)h(initial)e(data)h(con)o(v)o(erge)i(to)e Fu(\026)1618 2088 y Fn(")1655 2082 y Fv(in)361 2132 y(distribution)13 b(as)h Fu(t)e Ft(!)f(1)p Fv(.)274 2215 y(\(iii\))19 b(W)m(e)c(ha)o(v)o (e)f Fu(\026)555 2221 y Fn(")586 2215 y Fu(*)f(\026)i Fv(as)f Fu(")g Ft(!)f Fv(0,)h(where)i Fu(\026)f Fv(is)f(the)i(unique)f (stationary)f(measure)h(for)361 2264 y(the)g(white-forced)f(equation.) 320 2372 y Fi(Ac)o(kno)o(wledgemen)o(t)o(s.)22 b Fv(W)m(e)15 b(thank)h(I.)f(Gy\177)-21 b(ongy)15 b(for)h(discussions.)25 b(This)16 b(researc)o(h)257 2422 y(w)o(as)e(supp)q(orted)h(b)o(y)f (gran)o(t)g(GR/N63055/01)d(from)h(EPSR)o(C.)963 2628 y(3)p eop %%Page: 4 4 4 3 bop 257 262 a Fw(1)67 b(A)22 b(priori)j(estimates)d(for)g(higher)h (Sob)r(olev)g(norms)257 361 y Fh(1.1)56 b(P)n(erio)r(dic)18 b(b)r(oundary)g(conditions)257 437 y Fv(Let)i Fu(H)375 422 y Fn(s)393 437 y Fv(\()p Fs(T)440 422 y Fr(2)456 437 y Fv(\))f(b)q(e)h(the)h(Sob)q(olev)e(space)i(of)e(order)h Fu(s)g Fv(on)f Fs(T)1208 422 y Fr(2)1243 437 y Fv(endo)o(w)o(ed)h(with) g(the)g(norm)257 487 y Ft(k)p Fu(u)p Ft(k)323 493 y Fn(s)352 487 y Fv(=)12 b Ft(j)p Fu(L)436 472 y Fn(s=)p Fr(2)487 487 y Fu(u)p Ft(j)g Fv(and)g(let)h Fu(H)711 472 y Fn(s)740 487 y Fv(=)f Fu(H)822 472 y Fn(s)839 487 y Fv(\()p Fs(T)886 472 y Fr(2)902 487 y Fv(\))7 b Ft(\\)g Fu(H)s Fv(.)16 b(W)m(e)c(shall)g(also)g(use)i(the)f(notation)e Ft(k)c(\001)g(k)j Fv(=)257 537 y Ft(k)f(\001)f(k)328 543 y Fr(1)346 537 y Fv(.)18 b(In)13 b(this)h(subsection,)g(w)o(e)g(deriv)o(e)g(some)f (estimates)g(for)g(the)h(exp)q(ectation)h Fs(E)8 b Ft(k)p Fu(u)p Ft(k)1644 522 y Fn(m)1644 547 y(s)1678 537 y Fv(,)257 587 y(where)17 b Fu(m)c Ft(\025)h Fv(1)h(and)g Fu(s)f Ft(\025)f Fv(0)i(are)h(arbitrary)f(in)o(tegers)g(and)g Fu(u)p Fv(\()p Fu(t)p Fv(\))g(is)g(a)g(solution)f(of)h(the)g(NS)257 637 y(system)f(\(0.2\))o(,)g(\(0.4\))f(under)i(the)f(p)q(erio)q(dic)g (b)q(oundary)g(conditions.)320 686 y(Let)19 b(us)g(set)g Fu(b)544 692 y Fr(max)626 686 y Fv(=)h(sup)740 697 y Fn(j)765 686 y Fu(b)783 692 y Fn(j)818 686 y Fv(and)f Fu(B)935 692 y Fn(k)975 686 y Fv(=)1026 655 y Fq(P)1070 699 y Fn(j)1095 686 y Fu(\013)1122 671 y Fn(k)1122 697 y(j)1142 686 y Fu(b)1160 671 y Fr(2)1160 697 y Fn(j)1178 686 y Fv(.)32 b(In)19 b(what)f(follo)o(ws,)g(w)o(e)g(shall)257 736 y(alw)o(a)o(ys)13 b(assume)h(that)729 832 y Fu(B)760 838 y Fr(0)791 832 y Fu(<)d Ft(1)p Fu(;)48 b Fs(E)9 b Fu(e)987 815 y Fn(\033)q(\027)r Fm(j)p Fn(u)1056 819 y Fo(0)1074 815 y Fm(j)1084 802 y Fo(2)1114 832 y Fu(<)j Ft(1)p Fu(;)392 b Fv(\(1.1\))257 922 y(where)17 b Fu(u)403 928 y Fr(0)436 922 y Fv(is)e(the)h(initial)d(function)i(and)g Fu(\033)g(>)f Fv(0)h(is)g(so)g(small)e(that)i(2)p Fu(\033)q(b)1400 907 y Fr(2)1400 932 y(max)1477 922 y Ft(\024)f Fu(\013)1550 928 y Fr(1)1568 922 y Fv(.)22 b(Con-)257 972 y(dition)15 b(\(1.1\))f(ensures)j(that)f(the)g(Cauc)o(h)o(y)f(problem)f(has)h(a)g (unique)g(solution)g Fu(u)p Fv(\()p Fu(t)p Fv(\))g(in)g(an)257 1022 y(appropriate)k(functional)e(space,)j(and)e(the)g(follo)o(wing)e (inequalit)o(y)h(holds)h(for)f(a)h(second)257 1071 y(exp)q(onen)o(tial) c(momen)o(t)d(of)i Fu(u)p Fv(\()p Fu(t)p Fv(\))h(\(see)i([BKL00)o(,)e (Lemma)d(1])i(and)h([Shi02)n(,)g(Remark)e(3.2]\):)593 1169 y Fs(E)d Fu(e)644 1152 y Fn(\033)r(\027)r Fm(j)p Fn(u)p Fr(\()p Fn(t)p Fr(\))p Fm(j)763 1140 y Fo(2)794 1169 y Ft(\024)j Fu(e)857 1152 y Fm(\000)p Fn(\033)q(\027)r(t)936 1169 y Fs(E)d Fu(e)987 1152 y Fn(\033)r(\027)r Fm(j)p Fn(u)1057 1156 y Fo(0)1075 1152 y Fm(j)1085 1140 y Fo(2)1112 1169 y Fv(+)h Fu(C)q(;)47 b(t)11 b Ft(\025)h Fv(0)p Fu(;)257 b Fv(\(1.2\))257 1259 y(where)15 b Fu(C)g(>)c Fv(0)j(dep)q(ends)h(only) f(on)f Fu(\033)i Fv(and)f Fu(B)960 1265 y Fr(0)979 1259 y Fv(.)320 1309 y(F)m(or)f(an)o(y)h Fu(T)j(>)12 b Fv(0)h(and)h(an)o(y)g (in)o(teger)g Fu(k)e Ft(\025)g Fv(0,)h(w)o(e)h(set)523 1426 y Fu(V)547 1432 y Fn(k)568 1426 y Fv(\()p Fu(T)6 b Fv(\))11 b(=)33 b(sup)685 1461 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)790 1380 y Fq(\020)815 1426 y Fu(t)830 1409 y Fn(k)850 1426 y Ft(k)p Fu(u)p Fv(\()p Fu(t)p Fv(\))p Ft(k)963 1409 y Fr(2)963 1436 y Fn(k)993 1426 y Fv(+)9 b Fu(\027)1064 1369 y Fq(Z)1106 1380 y Fn(t)1087 1464 y Fr(0)1127 1426 y Fu(s)1146 1409 y Fn(k)1167 1426 y Ft(k)p Fu(u)p Fv(\()p Fu(s)p Fv(\))p Ft(k)1284 1409 y Fr(2)1284 1436 y Fn(k)q Fr(+1)1347 1426 y Fu(ds)1388 1380 y Fq(\021)1412 1426 y Fu(:)180 b Fv(\(1.3\))257 1544 y(In)15 b(the)g(theorem)f(b)q(elo)o(w,)f(w)o(e)i(con\014ne)g (ourselv)o(es)g(to)f(the)h(case)g Fu(\027)g Fv(=)d(1.)19 b(The)c(dep)q(endence)257 1593 y(of)f(the)g(constan)o(ts)h(on)f Fu(\027)i Fv(will)c(b)q(e)i(sp)q(eci\014ed)i(later.)257 1676 y Fi(Theorem)f(1.1.)21 b Fg(Supp)n(ose)13 b(that)f Fu(B)823 1682 y Fn(k)856 1676 y Fu(<)g Ft(1)f Fg(for)h(an)g(inte)n(ger) g Fu(k)g Ft(\025)g Fv(0)p Fg(.)18 b(Then)12 b(for)g(any)g Fu(T)18 b(>)12 b Fv(0)257 1725 y Fg(and)k(any)g(inte)n(ger)e Fu(p)d Ft(\025)h Fv(1)j Fg(ther)n(e)f(is)h(a)g(c)n(onstant)g Fu(C)1046 1731 y Fn(k)q(p)1095 1725 y Fv(=)d Fu(C)1169 1731 y Fn(k)q(p)1206 1725 y Fv(\()p Fu(\033)o(;)7 b(T)f Fv(\))11 b Fu(>)h Fv(0)j Fg(such)g(that)752 1825 y Fs(E)9 b Fu(V)808 1831 y Fn(k)831 1825 y Fv(\()p Fu(T)d Fv(\))893 1808 y Fn(p)924 1825 y Ft(\024)12 b Fu(C)998 1831 y Fn(k)q(p)1035 1825 y Fs(E)d Fu(e)1086 1808 y Fn(\033)q Fm(j)p Fn(u)1136 1812 y Fo(0)1155 1808 y Fm(j)1165 1795 y Fo(2)1183 1825 y Fu(:)409 b Fv(\(1.4\))320 1915 y(Inequalit)o(y)15 b(\(1.4\))h(imm)o (ediately)d(implies)h(a)i(priori)f(estimates)h(for)g(momen)o(ts)d(of)j (the)257 1965 y(random)d(v)n(ariables)499 2073 y Fu(U)527 2079 y Fn(k)548 2073 y Fv(\()p Fu(t)579 2079 y Fr(0)598 2073 y Fu(;)7 b(T)f Fv(\))11 b(=)66 b(sup)718 2108 y Fn(t)731 2112 y Fo(0)746 2108 y Fm(\024)p Fn(t)p Fm(\024)p Fn(t)824 2112 y Fo(0)840 2108 y Fr(+)p Fn(T)889 2027 y Fq(\020)914 2073 y Ft(k)p Fu(u)p Fv(\()p Fu(t)p Fv(\))p Ft(k)1027 2056 y Fr(2)1027 2083 y Fn(k)1056 2073 y Fv(+)10 b Fu(\027)1128 2017 y Fq(Z)1169 2027 y Fn(t)1151 2111 y(t)1164 2115 y Fo(0)1191 2073 y Ft(k)p Fu(u)p Fv(\()p Fu(s)p Fv(\))p Ft(k)1308 2056 y Fr(2)1308 2083 y Fn(k)q Fr(+1)1370 2073 y Fu(ds)1411 2027 y Fq(\021)1436 2073 y Fu(:)156 b Fv(\(1.5\))257 2191 y(Namely)m(,)12 b(w)o(e)i(ha)o(v)o(e)f (the)i(follo)o(wing)c(corollary:)257 2273 y Fi(Corollary)k(1.2.)21 b Fg(Under)13 b(the)f(c)n(onditions)i(of)e(The)n(or)n(em)h(1.1,)g(for)f (any)h Fu(k)q Fg(,)g Fu(p)p Fg(,)g Fu(t)1503 2279 y Fr(0)1533 2273 y Ft(\025)f Fv(1)p Fg(,)h(and)257 2323 y Fu(T)18 b(>)12 b Fv(0)i Fg(we)h(have)562 2413 y Fs(E)9 b Fu(U)622 2419 y Fn(k)646 2413 y Fv(\()p Fu(t)677 2419 y Fr(0)695 2413 y Fu(;)e(T)f Fv(\))760 2396 y Fn(p)791 2413 y Ft(\024)11 b Fu(C)867 2396 y Fm(0)864 2423 y Fn(k)q(p)902 2413 y Fv(\()p Fu(\033)o(;)c(T)f Fv(\))1013 2379 y Fq(\000)1031 2413 y Fu(e)1050 2396 y Fm(\000)p Fn(\033)q(t)1109 2400 y Fo(0)1128 2413 y Fs(E)i Fu(e)1179 2396 y Fn(\033)q Fm(j)p Fn(u)1229 2400 y Fo(0)1248 2396 y Fm(j)1258 2383 y Fo(2)1285 2413 y Fv(+)h(1)1347 2379 y Fq(\001)1366 2413 y Fu(;)226 b Fv(\(1.6\))257 2503 y Fg(wher)n(e)15 b(the)g(p)n(ositive)f(c)n(onstant)i Fu(C)792 2488 y Fm(0)789 2515 y Fn(k)q(p)826 2503 y Fv(\()p Fu(T)6 b Fv(\))15 b Fg(do)n(es)g(not)g(dep)n(end)h(on)g Fu(t)1280 2509 y Fr(0)1313 2503 y Fg(and)g Fu(u)1418 2509 y Fr(0)1436 2503 y Fg(.)963 2628 y Fv(4)p eop %%Page: 5 5 5 4 bop 320 262 a Fv(T)m(o)11 b(establish)h(inequalit)o(y)g(\(1.6\))o (,)g(it)g(su\016ces)h(to)g(apply)e(an)h(analogue)f(of)h(Theorem)f(1.1) 257 317 y(for)j(the)g(in)o(terv)n(al)f([)p Fu(t)569 323 y Fr(0)597 317 y Ft(\000)c Fv(1)p Fu(;)e(t)693 323 y Fr(0)720 317 y Fv(+)j Fu(T)c Fv(])13 b(and)h(use)h(\(1.2\))e(to)h (estimate)f Fs(E)c Fu(e)1337 302 y Fn(\033)q Fm(j)p Fn(u)p Fr(\()p Fn(t)1413 306 y Fo(0)1431 302 y Fm(\000)p Fr(1\))p Fm(j)1497 289 y Fo(2)1515 317 y Fv(.)257 399 y Fg(Pr)n(o)n(of)15 b(of)g(The)n(or)n(em)f(1.1.)21 b Fv(F)m(or)g Fu(k)26 b Fv(=)f(0,)e(inequalit)o(y)e(\(1.4\))g(follo)o(ws)f(from)g(Lemma)f (1.3)257 449 y(in)d([KS02].)24 b(W)m(e)16 b(no)o(w)g(assume)g(that)h Fu(k)f Fv(=)g Fu(m)g Ft(\025)g Fv(1)g(and)g(that)h(for)e Fu(k)i Ft(\024)f Fu(m)11 b Ft(\000)g Fv(1)16 b(inequal-)257 499 y(it)o(y)h(\(1.4\))e(is)h(already)g(established.)27 b(Application)15 b(of)h(It^)-21 b(o's)16 b(form)o(ula)d(to)k(the)f (functional)257 549 y Fu(f)277 555 y Fn(m)309 549 y Fv(\()p Fu(t)p Fv(\))c(=)g Fu(t)427 533 y Fn(m)458 549 y Ft(k)p Fu(u)p Fv(\()p Fu(t)p Fv(\))p Ft(k)571 533 y Fr(2)571 559 y Fn(m)616 549 y Fv(results)j(in)299 656 y Fu(d)-7 b(f)334 662 y Fn(m)365 656 y Fv(\()p Fu(t)p Fv(\))12 b(=)g Fu(mt)519 639 y Fn(m)p Fm(\000)p Fr(1)594 656 y Ft(k)p Fu(u)p Ft(k)660 639 y Fr(2)660 666 y Fn(m)690 656 y Fu(dt)d Ft(\000)h Fv(2)p Fu(t)814 639 y Fn(m)852 621 y Fq(\015)852 646 y(\015)875 656 y Fu(L)908 626 y Ff(m)p Fo(+1)p 908 632 61 2 v 932 649 a(2)976 656 y Fu(u)1000 621 y Fq(\015)1000 646 y(\015)1023 631 y Fr(2)1042 656 y Fu(dt)e Ft(\000)414 731 y(\000)h Fv(2)p Fu(t)491 714 y Fn(m)529 698 y Fq(\000)548 731 y Fu(L)581 701 y Ff(m)p Fo(+1)p 582 707 V 605 724 a(2)650 731 y Fu(u;)e(L)726 701 y Ff(m)p Fe(\000)p Fo(1)p 725 707 63 2 v 749 724 a(2)794 731 y Fu(B)r Fv(\()p Fu(u;)g(u)p Fv(\))926 698 y Fq(\001)945 731 y Fu(dt)i Fv(+)g Fu(B)1063 737 y Fn(m)1095 731 y Fu(t)1110 714 y Fn(m)1142 731 y Fu(dt)g Fv(+)g(2)p Fu(t)1265 714 y Fn(m)1296 698 y Fq(\000)1316 731 y Fu(L)1349 703 y Ff(m)p 1349 708 26 2 v 1354 724 a Fo(2)1381 731 y Fu(u;)e(L)1457 703 y Ff(m)p 1457 708 V 1462 724 a Fo(2)1489 731 y Fu(d\020)1532 698 y Fq(\001)1551 731 y Fu(:)41 b Fv(\(1.7\))257 818 y(In)o(tegrating)19 b(\(1.7\))f(with)g(resp)q(ect) j(to)e Fu(t)f Fv(and)g(using)h(the)g(inequalit)o(y)e(\(see)j([KS01)o(,) g(Sec-)257 868 y(tion)14 b(6.3]\))447 935 y Fq(\014)447 960 y(\014)461 937 y(\000)480 971 y Fu(L)513 941 y Ff(m)p Fo(+1)p 513 947 61 2 v 537 964 a(2)581 971 y Fu(u;)7 b(L)657 941 y Ff(m)p Fe(\000)p Fo(1)p 657 947 63 2 v 681 964 a(2)726 971 y Fu(B)r Fv(\()p Fu(u;)g(u)p Fv(\))858 937 y Fq(\001)877 935 y(\014)877 960 y(\014)902 971 y Ft(\024)12 b Fu(c)964 954 y Fm(0)964 981 y Fn(m)996 935 y Fq(\015)996 960 y(\015)1019 971 y Fu(u)1043 935 y Fq(\015)1043 960 y(\015)1070 933 y Fo(4)p Ff(m)p Fe(\000)p Fo(1)p 1070 939 77 2 v 1089 956 a(2)p Ff(m)1066 987 y Fn(m)p Fr(+1)1154 935 y Fq(\015)1154 960 y(\015)1177 971 y Fu(u)1201 935 y Fq(\015)1201 960 y(\015)1229 933 y Ff(m)p Fo(+1)p 1229 939 61 2 v 1239 956 a(2)p Ff(m)1297 935 y Fq(\014)1297 960 y(\014)1310 971 y Fu(u)1334 935 y Fq(\014)1334 960 y(\014)1353 935 y Fo(1)p 1353 940 15 2 v 1353 956 a(2)902 1046 y Ft(\024)951 1030 y Fr(1)p 951 1037 17 2 v 951 1061 a(4)973 1046 y Ft(k)p Fu(u)p Ft(k)1039 1029 y Fr(2)1039 1057 y Fn(m)p Fr(+1)1121 1046 y Fv(+)d Fu(c)1180 1052 y Fn(m)1212 1046 y Ft(k)p Fu(u)p Ft(k)1278 1029 y Fr(2\()p Fn(m)p Fr(+1\))1393 1046 y Ft(j)p Fu(u)p Ft(j)1441 1029 y Fr(2)p Fn(m)1488 1046 y Fu(;)257 1133 y Fv(w)o(e)14 b(deriv)o(e)299 1250 y Fu(t)314 1233 y Fn(m)345 1250 y Ft(k)p Fu(u)p Fv(\()p Fu(t)p Fv(\))p Ft(k)458 1233 y Fr(2)458 1261 y Fn(m)499 1250 y Fv(+)9 b Fu(\027)570 1194 y Fq(Z)612 1204 y Fn(t)593 1288 y Fr(0)633 1250 y Fu(s)652 1233 y Fn(m)684 1250 y Ft(k)p Fu(u)p Fv(\()p Fu(s)p Fv(\))p Ft(k)801 1233 y Fr(2)801 1261 y Fn(m)p Fr(+1)875 1250 y Fu(ds)i Ft(\024)h Fu(m)1014 1194 y Fq(Z)1056 1204 y Fn(t)1038 1288 y Fr(0)1078 1250 y Fu(s)1097 1233 y Fn(m)p Fm(\000)p Fr(1)1171 1250 y Ft(k)p Fu(u)p Ft(k)1237 1233 y Fr(2)1237 1261 y Fn(m)1268 1250 y Fu(ds)d Fv(+)743 1369 y(+)789 1352 y Fn(B)814 1356 y Ff(m)842 1352 y Fn(t)855 1339 y Ff(m)p Fo(+1)p 789 1359 129 2 v 818 1383 a Fn(m)p Fr(+1)932 1369 y Fv(+)g(2)p Fu(c)1012 1375 y Fn(m)1051 1312 y Fq(Z)1092 1323 y Fn(t)1074 1407 y Fr(0)1114 1369 y Fu(s)1133 1352 y Fn(m)1165 1369 y Ft(k)p Fu(u)p Ft(k)1231 1352 y Fr(2\()p Fn(m)p Fr(+1\))1346 1369 y Ft(j)p Fu(u)p Ft(j)1394 1352 y Fr(2)p Fn(m)1441 1369 y Fu(ds)g Fv(+)h Fu(\030)1551 1375 y Fn(m)1582 1369 y Fv(\()p Fu(t)p Fv(\))p Fu(;)257 1476 y Fv(where)15 b Fu(c)395 1482 y Fn(m)438 1476 y Fv(=)d(0)i(for)f Fu(m)f Fv(=)g(1)i(and)502 1588 y Fu(\030)520 1594 y Fn(m)552 1588 y Fv(\()p Fu(t)p Fv(\))e(=)f(2)682 1531 y Fq(Z)724 1541 y Fn(t)705 1625 y Fr(0)745 1588 y Fu(s)764 1570 y Fn(m)796 1554 y Fq(\000)815 1588 y Fu(L)848 1559 y Ff(m)p 848 1564 26 2 v 854 1580 a Fo(2)881 1588 y Fu(u;)c(L)957 1559 y Ff(m)p 956 1564 V 962 1580 a Fo(2)989 1588 y Fu(d\020)1032 1554 y Fq(\001)1060 1588 y Ft(\000)1107 1560 y Fv(1)p 1107 1578 21 2 v 1107 1616 a(2)1139 1531 y Fq(Z)1181 1541 y Fn(t)1162 1625 y Fr(0)1202 1588 y Fu(s)1221 1570 y Fn(m)1253 1588 y Ft(k)p Fu(u)p Ft(k)1319 1570 y Fr(2)1319 1598 y Fn(m)p Fr(+1)1392 1588 y Fu(ds:)257 1695 y Fv(It)14 b(follo)o(ws)f(that)292 1782 y Fu(V)316 1788 y Fn(m)347 1782 y Fv(\()p Fu(T)6 b Fv(\))12 b Ft(\024)g Fu(mV)525 1788 y Fn(m)p Fm(\000)p Fr(1)600 1782 y Fv(\()p Fu(T)6 b Fv(\))j(+)718 1765 y Fn(B)743 1769 y Ff(m)770 1765 y Fn(T)794 1752 y Ff(m)p Fo(+1)p 718 1772 140 2 v 752 1796 a Fn(m)p Fr(+1)872 1782 y Fv(+)g(2)p Fu(c)952 1788 y Fn(m)983 1782 y Fu(V)1007 1788 y Fr(0)1026 1782 y Fv(\()p Fu(T)d Fv(\))1088 1765 y Fn(m)p Fr(+1)1162 1782 y Fu(V)1186 1788 y Fr(1)1205 1782 y Fv(\()p Fu(T)g Fv(\))1267 1765 y Fn(m)1308 1782 y Fv(+)31 b(sup)1350 1817 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)1461 1782 y Fu(\030)1479 1788 y Fn(m)1511 1782 y Fv(\()p Fu(t)p Fv(\))p Fu(:)j Fv(\(1.8\))257 1898 y(W)m(e)13 b(no)o(w)f(rep)q(eat)i(the)f(argumen)o(t)e(used)j(in)e(the)h(pro)q(of)g (of)f(Lemma)e(1.3)h(in)h([KS02])g(to)g(sho)o(w)257 1948 y(that,)i(if)f Fu(\015)h(>)e Fv(0)i(is)f(su\016cien)o(tly)h(small,)d (then)636 2045 y Fs(P)664 1999 y Fq(\020)714 2045 y Fv(sup)693 2080 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)805 2045 y Fu(\030)823 2051 y Fn(m)854 2045 y Fv(\()p Fu(t)p Fv(\))h Ft(\025)g Fu(\032)978 1999 y Fq(\021)1015 2045 y Ft(\024)f Fu(e)1077 2027 y Fm(\000)p Fn(\015)r(\032)1142 2045 y Fu(;)48 b(\032)12 b Ft(\025)g Fv(0)p Fu(:)292 b Fv(\(1.9\))257 2171 y(Indeed,)19 b(the)f(quadratic)f(v)n(ariation)f(of)g(the)i(in)o (tegral)f Fu(M)1170 2177 y Fn(t)1202 2171 y Fv(:=)1263 2138 y Fq(R)1291 2148 y Fn(t)1283 2186 y Fr(0)1312 2171 y Fu(s)1331 2156 y Fn(m)1363 2171 y Fv(\()p Fu(L)1412 2145 y Ff(m)p 1413 2150 26 2 v 1418 2166 a Fo(2)1445 2171 y Fu(u;)7 b(L)1521 2145 y Ff(m)p 1521 2150 V 1526 2166 a Fo(2)1553 2171 y Fu(d\020)s Fv(\))17 b(has)257 2221 y(the)e(form)406 2326 y Ft(h)p Fu(M)5 b Ft(i)483 2332 y Fn(t)509 2326 y Fv(=)553 2286 y Fq(X)575 2375 y Fn(j)620 2326 y Fu(\013)647 2309 y Fr(2)p Fn(m)647 2336 y(j)695 2326 y Fu(b)713 2309 y Fr(2)713 2336 y Fn(j)738 2269 y Fq(Z)780 2280 y Fn(t)761 2364 y Fr(0)801 2326 y Fu(s)820 2309 y Fr(2)p Fn(m)869 2326 y Fu(u)893 2309 y Fr(2)893 2336 y Fn(j)911 2326 y Fv(\()p Fu(s)p Fv(\))i Fu(ds)12 b Ft(\024)g Fu(B)1097 2332 y Fn(m)p Fm(\000)p Fr(1)1178 2326 y Fu(T)1208 2309 y Fn(m)1247 2269 y Fq(Z)1288 2280 y Fn(t)1270 2364 y Fr(0)1310 2326 y Fu(s)1329 2309 y Fn(m)1361 2326 y Ft(k)p Fu(u)p Ft(k)1427 2309 y Fr(2)1427 2336 y Fn(m)p Fr(+1)1500 2326 y Fu(ds)509 2465 y Ft(\024)570 2437 y Fv(1)p 558 2456 45 2 v 558 2494 a(4)p Fu(\015)614 2409 y Fq(Z)656 2419 y Fn(t)638 2503 y Fr(0)677 2465 y Fu(s)696 2448 y Fn(m)728 2465 y Ft(k)p Fu(u)p Ft(k)794 2448 y Fr(2)794 2476 y Fn(m)p Fr(+1)867 2465 y Fu(ds;)963 2628 y Fv(5)p eop %%Page: 6 6 6 5 bop 257 262 a Fv(where)15 b(w)o(e)g(set)f Fu(\015)h Fv(=)c(\(4)p Fu(T)649 246 y Fn(m)681 262 y Fu(B)712 268 y Fn(m)p Fm(\000)p Fr(1)786 262 y Fv(\))802 246 y Fm(\000)p Fr(1)847 262 y Fv(.)18 b(It)c(follo)o(ws)e(that)760 348 y Fu(\030)778 354 y Fn(m)810 348 y Fv(\()p Fu(t)p Fv(\))g Ft(\024)g Fv(2)p Fu(M)974 354 y Fn(t)997 348 y Ft(\000)e Fv(2)p Fu(\015)r Ft(h)p Fu(M)5 b Ft(i)1160 354 y Fn(t)1175 348 y Fu(:)257 435 y Fv(Since)18 b(exp\(2)p Fu(\015)r(M)532 441 y Fn(t)559 435 y Ft(\000)12 b Fv(2)p Fu(\015)647 420 y Fr(2)666 435 y Ft(h)p Fu(M)5 b Ft(i)743 441 y Fn(t)758 435 y Fv(\))17 b(is)g(a)g(sup)q(ermartingale)f(whose)h(mean)f(v)n(alue) h(do)q(es)h(not)257 485 y(exceed)e(1,)c(the)i(required)h(estimate)e (\(1.9\))f(follo)o(ws)g(from)g(the)i(classical)f(sup)q(ermartingale)257 535 y(inequalit)o(y)g(\(see)i(Theorem)f(VI.T1)f(in)g([Mey66])g(or)h (Theorem)f(I)q(I)q(I.6.11)g(in)g([Kry95]\).)320 584 y(Com)o(bining)d (inequalities)j(\(1.8\))f(and)h(\(1.9\))g(and)g(the)h(induction)f(h)o (yp)q(othesis,)h(w)o(e)f(see)257 634 y(that)21 b(all)e(the)h(momen)o (ts)e(of)i Fu(V)756 640 y Fn(m)788 634 y Fv(\()p Fu(T)6 b Fv(\))20 b(are)h(\014nite.)37 b(This)20 b(completes)g(the)g(pro)q(of) g(of)g(the)257 684 y(theorem.)p 1659 684 2 29 v 1661 658 25 2 v 1661 684 V 1686 684 2 29 v 320 766 a(W)m(e)11 b(no)o(w)h(study)g(the)h(dep)q(endence)h(of)d(the)i(constan)o(t)f Fu(C)1189 751 y Fm(0)1186 778 y Fn(k)q(p)1235 766 y Fv(in)g(\(1.6\))f (on)h Fu(\027)s Fv(.)k(T)m(o)11 b(simplify)257 816 y(the)k(presen)o (tation,)f(w)o(e)g(shall)f(only)g(consider)i(the)g(case)g(of)e(a)g (stationary)h(solution.)320 866 y(Let)f(us)g(b)q(egin)f(with)g(the)i (situation)d(in)i(whic)o(h)f(the)h(righ)o(t-hand)f(side)h(is)f (analytic)g(in)g Fu(x)p Fv(.)257 916 y(More)j(precisely)m(,)f(w)o(e)g (assume)f(that)766 981 y Fm(1)752 994 y Fq(X)754 1082 y Fn(j)r Fr(=1)819 1033 y Fv(exp)q(\(2)p Fu(\032)941 1007 y Ft(p)p 976 1007 45 2 v 26 x Fu(\013)1003 1039 y Fn(j)1027 1033 y Fv(\))p Fu(b)1061 1016 y Fr(2)1061 1044 y Fn(j)1091 1033 y Fu(<)f Ft(1)p Fu(;)394 b Fv(\(1.10\))257 1162 y(where)15 b Fu(\032)d(>)g Fv(0)i(is)g(a)f(constan)o(t)i(not)e (dep)q(ending)i(on)f Fu(\027)s Fv(.)257 1241 y Fi(Prop)q(osition)e (1.3.)21 b Fg(L)n(et)14 b Fu(u)p Fv(\()p Fu(t;)7 b(x)p Fv(\))13 b Fg(b)n(e)h(a)g(stationary)g(solution)g(of)23 b Fv(\(0.2\))o Fg(,)14 b Fv(\(0.3\))o Fg(.)19 b(Then)14 b(for)257 1291 y(any)19 b(inte)n(gers)f Fu(k)q(;)7 b(m)18 b Ft(\025)g Fv(1)g Fg(and)h(any)g Fu(")f(>)g Fv(0)g Fg(and)h Fu(T)24 b(>)18 b Fv(0)g Fg(ther)n(e)g(is)g(a)g(c)n(onstant)h Fu(C)1590 1297 y Fn(k)q(m)1657 1291 y Fv(=)257 1341 y Fu(C)287 1347 y Fn(k)q(m)337 1341 y Fv(\()p Fu(";)7 b(T)f Fv(\))12 b Fu(>)f Fv(0)k Fg(such)g(that)493 1440 y Fs(E)525 1394 y Fq(\020)614 1440 y Fv(sup)559 1475 y Fn(t)572 1479 y Fo(0)588 1475 y Fm(\024)p Fn(t)p Fm(\024)p Fn(t)666 1479 y Fo(0)681 1475 y Fr(+)p Fn(T)731 1405 y Fq(\015)731 1430 y(\015)754 1440 y Fu(u)p Fv(\()p Fu(t)p Fv(\))825 1405 y Fq(\015)825 1430 y(\015)848 1415 y Fn(m)848 1457 y(k)880 1394 y Fq(\021)916 1440 y Ft(\024)d Fu(C)990 1446 y Fn(k)q(m)1039 1440 y Fu(\027)1063 1423 y Fm(\000)p Fr(2)p Fn(k)q(m)p Fr(+)1183 1412 y Fo(3)p Ff(m)p 1183 1417 40 2 v 1196 1433 a Fo(2)1228 1423 y Fm(\000)p Fn(")1272 1440 y Fu(;)48 b(t)1347 1446 y Fr(0)1378 1440 y Ft(\025)11 b Fv(0)p Fu(:)129 b Fv(\(1.11\))320 1555 y(W)m(e)16 b(note)h(that,)f (for)h Fu(k)f Fv(=)h(1,)f(a)g(stronger)i(assertion)f(holds)f(if)g(w)o (e)h(are)g(in)o(terested)h(in)257 1604 y(estimating)c(the)i(mean)e(v)n (alue)g(at)h(a)g(\014xed)h(p)q(oin)o(t)f Fu(t)g Fv(\(see)h([Shi02)o(,)f (Theorem)g(2.2]\):)k(there)257 1654 y(is)14 b(a)g(constan)o(t)g Fu(C)g(>)e Fv(0)i(suc)o(h)g(that)673 1741 y Fs(E)19 b Fv(exp)778 1707 y Fq(\000)797 1741 y Fu(\033)q(\027)s Ft(k)p Fu(u)p Fv(\()p Fu(t;)7 b Ft(\001)p Fv(\))p Ft(k)990 1724 y Fr(2)1007 1707 y Fq(\001)1037 1741 y Ft(\024)12 b Fu(C)q(;)47 b(t)12 b Ft(\025)g Fv(0)p Fu(:)308 b Fv(\(1.12\))257 1838 y(In)14 b(particular,)f(since)i Fu(y)635 1823 y Fn(m)679 1838 y Ft(\024)d Fu(C)753 1844 y Fn(m)784 1838 y Fu(\027)808 1823 y Fm(\000)838 1812 y Ff(m)p 838 1817 26 2 v 844 1833 a Fo(2)871 1838 y Fu(e)890 1823 y Fn(\033)q(\027)r(y) 947 1810 y Fo(2)965 1838 y Fv(,)h(for)h(an)o(y)f Fu(m)f Ft(\025)g Fv(1)i(w)o(e)g(ha)o(v)o(e)683 1925 y Fs(E)8 b Ft(k)p Fu(u)p Fv(\()p Fu(t;)f Ft(\001)p Fv(\))p Ft(k)858 1908 y Fn(m)903 1925 y Ft(\024)12 b Fu(C)977 1931 y Fn(m)1015 1925 y Fu(\027)1039 1908 y Fm(\000)1069 1896 y Ff(m)p 1069 1901 V 1075 1918 a Fo(2)1102 1925 y Fu(;)47 b(t)12 b Ft(\025)g Fv(0)p Fu(:)257 2011 y Fg(Pr)n(o)n(of)j(of)g(Pr)n(op)n (osition)f(1.3.)21 b Fv(Since)12 b Fu(u)p Fv(\()p Fu(t;)7 b(x)p Fv(\))j(is)h(a)f(stationary)h(solution,)f(it)h(su\016ces)h(to)f (con-)257 2061 y(sider)18 b(the)f(case)h Fu(t)540 2067 y Fr(0)575 2061 y Fv(=)f(0.)27 b(According)17 b(to)g(Theorems)f(2.1)g (and)h(2.2)f(in)g([Shi02)o(],)g(for)h(an)o(y)257 2111 y Fu(\016)d(>)e Fv(0)g(and)f Fu(T)18 b(>)12 b Fv(0)g(there)h(is)f(a)g (p)q(ositiv)o(e)g(random)e(v)n(ariable)h Fu(r)1215 2117 y Fn(\027)1248 2111 y Fv(suc)o(h)h(that)h(for)e(an)o(y)h(in)o(teger)257 2161 y Fu(m)g Ft(\025)g Fv(1)i(and)f(an)h(appropriate)g(constan)o(t)g Fu(K)948 2167 y Fn(m)992 2161 y Fu(>)e Fv(0)h(w)o(e)h(ha)o(v)o(e)339 2258 y Fs(E)371 2212 y Fq(\020)427 2258 y Fv(sup)406 2293 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)511 2223 y Fq(\015)511 2247 y(\015)534 2258 y Fv(exp\()p Fu(r)632 2264 y Fn(\027)653 2258 y Fu(\027)677 2241 y Fr(2+)p Fn(\016)736 2258 y Fu(L)769 2230 y Fo(1)p 769 2235 15 2 v 769 2251 a(2)791 2258 y Fv(\))p Fu(u)p Fv(\()p Fu(t;)7 b Ft(\001)p Fv(\))909 2223 y Fq(\015)909 2247 y(\015)931 2233 y Fn(m)962 2212 y Fq(\021)999 2258 y Ft(\024)k Fu(K)1077 2264 y Fn(m)1109 2258 y Fu(\027)1133 2241 y Fm(\000)1163 2230 y Ff(m)p 1163 2235 26 2 v 1169 2251 a Fo(2)1196 2258 y Fu(;)48 b Fs(E)8 b Fu(r)1307 2241 y Fm(\000)p Fn(m)1306 2268 y(\027)1379 2258 y Ft(\024)k Fu(K)1458 2264 y Fn(m)1490 2258 y Fu(:)81 b Fv(\(1.13\))257 2372 y(Let)15 b(us)f(\014x)g(an)f(arbitrary)h(in)o(teger)h Fu(k)d Ft(\025)g Fv(1.)18 b(Direct)c(v)o(eri\014cation)g(sho)o(ws)g (that)367 2465 y Ft(k)p Fu(u)p Ft(k)433 2448 y Fr(2)433 2475 y Fn(k)464 2465 y Fv(=)508 2426 y Fq(X)530 2514 y Fn(j)575 2465 y Fu(\013)602 2448 y Fn(k)602 2475 y(j)622 2465 y Ft(j)p Fu(u)658 2471 y Fn(j)675 2465 y Ft(j)687 2448 y Fr(2)716 2465 y Ft(\024)e Fu(C)790 2471 y Fn(k)810 2465 y Fu(\027)834 2448 y Fm(\000)p Fr(2\()p Fn(k)q Fm(\000)p Fr(1\)\(2+)p Fn(\016)q Fr(\))1056 2465 y Fu(r)1076 2448 y Fm(\000)p Fr(2\()p Fn(k)q Fm(\000)p Fr(1\))1075 2475 y Fn(\027)1207 2430 y Fq(\015)1207 2455 y(\015)1230 2465 y Fv(exp)q(\()p Fu(r)1329 2471 y Fn(\027)1349 2465 y Fu(\027)1373 2448 y Fr(2+)p Fn(\016)1432 2465 y Fu(L)1465 2437 y Fo(1)p 1466 2442 15 2 v 1466 2458 a(2)1487 2465 y Fv(\))p Fu(u)1527 2430 y Fq(\015)1527 2455 y(\015)1550 2440 y Fr(2)1568 2465 y Fu(;)963 2628 y Fv(6)p eop %%Page: 7 7 7 6 bop 257 262 a Fv(where)15 b Fu(C)407 268 y Fn(k)439 262 y Fu(>)d Fv(0)h(dep)q(ends)j(only)d(on)h Fu(k)q Fv(.)k(It)c(follo)o (ws)e(that)311 360 y(sup)290 395 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)402 360 y Ft(k)p Fu(u)p Ft(k)468 343 y Fn(m)468 370 y(k)510 360 y Ft(\024)g Fu(C)591 324 y Ff(m)p 591 329 26 2 v 597 345 a Fo(2)584 372 y Fn(k)631 360 y Fu(\027)655 343 y Fm(\000)p Fn(m)p Fr(\()p Fn(k)q Fm(\000)p Fr(1\)\(2+)p Fn(\016)q Fr(\))889 360 y Fu(r)909 343 y Fm(\000)p Fn(m)p Fr(\()p Fn(k)q Fm(\000)p Fr(1\))908 370 y Fn(\027)1082 360 y Fv(sup)1060 395 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)1165 324 y Fq(\015)1165 349 y(\015)1188 360 y Fv(exp)q(\()p Fu(r)1287 366 y Fn(\027)1307 360 y Fu(\027)1331 343 y Fr(2+)p Fn(\016)1391 360 y Fu(L)1424 331 y Fo(1)p 1424 336 15 2 v 1424 353 a(2)1445 360 y Fv(\))p Fu(u)1485 324 y Fq(\015)1485 349 y(\015)1508 335 y Fn(m)1540 360 y Fu(:)31 b Fv(\(1.14\))257 481 y(Com)o(bining)12 b(\(1.13\))i(and)g(\(1.14\))f(and)h(c)o(ho)q(osing)g(a)g(su\016cien)o (tly)g(small)e Fu(\016)i Fv(=)f Fu(\016)1488 487 y Fn(k)q(m)1538 481 y Fv(\()p Fu(T)6 b Fv(\))13 b Fu(>)f Fv(0,)257 531 y(w)o(e)i(obtain)g(\(1.11\))o(.)p 1659 531 2 29 v 1661 504 25 2 v 1661 531 V 1686 531 2 29 v 320 614 a(W)m(e)j(conclude)h (this)f(subsection)i(with)e(a)g(short)h(discussion)g(of)f(the)h(case)h (when)f(the)257 664 y(righ)o(t-hand)c(side)g Fu(\021)h Fv(is)f(smo)q(oth,)e(but)i(not)g(analytic<)e(with)i(resp)q(ect)i(to)e Fu(x)p Fv(.)257 747 y Fi(Prop)q(osition)e(1.4.)21 b Fg(L)n(et)d Fu(B)716 753 y Fn(k)753 747 y Fu(<)e Ft(1)h Fg(for)g(an)h(inte)n(ger)f Fu(k)g Ft(\025)g Fv(0)g Fg(and)h(and)g(let)f Fu(u)p Fv(\()p Fu(t;)7 b(x)p Fv(\))17 b Fg(b)n(e)g(a)257 796 y(stationary)i(solution.) 32 b(Then)20 b(for)f(any)h(inte)n(ger)e Fu(m)i Ft(\025)g Fv(1)f Fg(and)h(any)g Fu(T)25 b(>)20 b Fv(0)f Fg(ther)n(e)f(is)h(a)257 846 y(c)n(onstant)d Fu(C)453 852 y Fn(k)q(m)502 846 y Fv(\()p Fu(T)6 b Fv(\))12 b Fu(>)g Fv(0)j Fg(not)g(dep)n(ending)h(on)f Fu(\027)i Fg(and)f Fu(u)p Fv(\()p Fu(t;)7 b(x)p Fv(\))14 b Fg(such)h(that)486 950 y Fs(E)518 904 y Fq(\020)607 950 y Fv(sup)553 985 y Fn(t)566 989 y Fo(0)581 985 y Fm(\024)p Fn(t)p Fm(\024)p Fn(t)659 989 y Fo(0)674 985 y Fr(+)p Fn(T)724 915 y Fq(\015)724 940 y(\015)747 950 y Fu(u)p Fv(\()p Fu(t)p Fv(\))818 915 y Fq(\015)818 940 y(\015)841 925 y Fn(m)841 967 y(k)873 904 y Fq(\021)909 950 y Ft(\024)d Fu(C)983 956 y Fn(k)q(m)1032 950 y Fv(\()p Fu(T)6 b Fv(\))h Fu(\027)1125 933 y Fm(\000)p Fr(3)p Fn(k)q(m)p Fm(\000)1246 922 y Ff(m)p 1246 927 26 2 v 1252 943 a Fo(2)1279 950 y Fu(;)48 b(t)1354 956 y Fr(0)1385 950 y Ft(\025)11 b Fv(0)p Fu(:)122 b Fv(\(1.15\))320 1071 y(T)m(o)11 b(establish)i(inequalit)o(y)f(\(1.15\))o(,)g(it)h (su\016ces)g(to)g(rep)q(eat)h(the)f(pro)q(of)f(of)g(Theorem)g(1.1,)257 1121 y(follo)o(wing)g(the)i(dep)q(endence)j(of)c(the)h(constan)o(ts)h (on)f Fu(\027)s Fv(.)j(W)m(e)d(shall)f(not)g(dw)o(ell)h(on)f(it.)257 1237 y Fh(1.2)56 b(Diric)n(hlet)17 b(b)r(oundary)i(condition)257 1314 y Fv(In)12 b(this)g(subsection,)h(w)o(e)f(brie\015y)h(discuss)g (the)f(case)h(in)e(whic)o(h)h(Eq.)g(\(0.2\))o(,)g(\(0.3\))f(is)h (studied)257 1364 y(in)g(a)h(b)q(ounded)g(domain)d Fu(D)j Ft(\032)f Fs(R)776 1349 y Fr(2)804 1364 y Fv(with)g(smo)q(oth)g(b)q (oundary)g(and)h(is)f(supplemen)o(ted)h(with)257 1413 y(the)i(Diric)o(hlet)e(b)q(oundary)h(condition.)320 1463 y(As)k(in)f(the)h(case)h(of)e(the)h(p)q(erio)q(dic)g(b)q(oundary)g (conditions,)f(w)o(e)h(assume)f(that)h(\(1.1\))257 1513 y(holds.)h(This)14 b(implies)e(that)i(the)h(Cauc)o(h)o(y)f(problem)e (has)j(a)e(unique)i(solution.)i(Moreo)o(v)o(er,)257 1563 y(inequalit)o(y)g(\(1.2\))g(remains)f(v)n(alid)g(for)h(2)p Fu(\033)q(b)950 1548 y Fr(2)950 1573 y(max)1031 1563 y Ft(\024)g Fu(\013)1107 1569 y Fr(1)1143 1563 y Fv(\(see)i([Shi02)n(,) f(Remark)e(3.2]\).)27 b(In)257 1613 y(particular,)14 b(for)f(an)o(y)g(stationary)h(solution)f Fu(u)p Fv(\()p Fu(t;)7 b(x)p Fv(\))13 b(w)o(e)h(ha)o(v)o(e)751 1712 y Fs(E)9 b Fu(e)802 1695 y Fn(\033)r(\027)r Fm(j)p Fn(u)p Fr(\()p Fn(t)p Fr(\))p Fm(j)921 1682 y Fo(2)952 1712 y Ft(\024)j Fu(C)q(;)47 b(t)11 b Ft(\025)h Fv(0)p Fu(;)394 b Fv(\(1.16\))257 1803 y(whence)16 b(it)d(follo)o(ws)f(that)622 1894 y Fs(E)654 1859 y Fq(\014)654 1884 y(\014)670 1894 y Fu(u)p Fv(\()p Fu(t)p Fv(\))741 1859 y Fq(\014)741 1884 y(\014)755 1869 y Fn(m)798 1894 y Ft(\024)g Fu(C)872 1900 y Fn(m)903 1894 y Fu(\027)927 1877 y Fm(\000)957 1866 y Ff(m)p 957 1871 V 963 1887 a Fo(2)990 1894 y Fu(;)48 b(t)11 b Ft(\025)h Fv(0)p Fu(;)48 b(m)12 b Ft(\025)g Fv(1)p Fu(:)257 b Fv(\(1.17\))257 2032 y Fw(2)67 b(Con)n(v)n(ergence)22 b(to)h(the)f(stationary)h(measure)257 2123 y Fv(In)18 b(this)g(section,)h(w)o(e)f(assume)g(that)g Fu(\027)i Fv(=)e(1)g(and)g(that)f Fu(B)1199 2129 y Fn(k)q Fr(+1)1281 2123 y Fu(<)h Ft(1)f Fv(for)h(some)f Fu(k)i Ft(\025)f Fv(0.)257 2172 y(W)m(e)g(denote)g(b)o(y)f Fu(C)561 2178 y Fn(b)578 2172 y Fv(\()p Fu(H)s Fv(\))g(the)i(space)f(of)f(b)q(ounded) h(con)o(tin)o(uous)g(functionals)f(on)g Fu(H)k Fv(and)257 2222 y(b)o(y)15 b Ft(P)s Fv(\()p Fu(H)s Fv(\))h(the)g(set)g(of)f(all)f (probabilit)o(y)f(Borel)j(measures)g(on)f Fu(H)s Fv(.)22 b(Let)15 b Fu(P)1419 2228 y Fn(t)1434 2222 y Fv(\()p Fu(u;)7 b Fv(\000\),)14 b Fu(u)g Ft(2)f Fu(H)s Fv(,)257 2272 y(\000)h Ft(2)f(B)q Fv(\()p Fu(H)s Fv(\),)i(b)q(e)h(the)f (transition)g(function)g(asso)q(ciated)g(with)g(the)g(NS)h(system)e (\(0.2\))h(and)257 2322 y(let)f Fd(P)351 2328 y Fn(t)371 2322 y Fv(:)d Fu(C)424 2328 y Fn(b)440 2322 y Fv(\()p Fu(H)s Fv(\))h Ft(!)f Fu(C)605 2328 y Fn(b)621 2322 y Fv(\()p Fu(H)s Fv(\))j(and)g Fd(P)820 2307 y Fm(\003)820 2332 y Fn(t)851 2322 y Fv(:)d Ft(P)s Fv(\()p Fu(H)s Fv(\))h Ft(!)f(P)s Fv(\()p Fu(H)s Fv(\))j(b)q(e)h(the)f(corresp)q(onding)h (Mark)o(o)o(v)257 2372 y(semigroups:)448 2482 y Fd(P)482 2488 y Fn(t)497 2482 y Fu(f)t Fv(\()p Fu(u)p Fv(\))d(=)633 2425 y Fq(Z)656 2520 y Fn(H)694 2482 y Fu(P)721 2488 y Fn(t)736 2482 y Fv(\()p Fu(u;)7 b(dv)q Fv(\))p Fu(f)t Fv(\()p Fu(v)q Fv(\))p Fu(;)49 b Fd(P)1026 2465 y Fm(\003)1026 2492 y Fn(t)1045 2482 y Fu(\026)p Fv(\(\000\))12 b(=)1184 2425 y Fq(Z)1207 2520 y Fn(H)1245 2482 y Fu(P)1272 2488 y Fn(t)1287 2482 y Fv(\()p Fu(u;)7 b Fv(\000\))p Fu(\026)p Fv(\()p Fu(dv)q Fv(\))p Fu(;)963 2628 y Fv(7)p eop %%Page: 8 8 8 7 bop 257 262 a Fv(where)15 b Fu(f)i Ft(2)11 b Fu(C)483 268 y Fn(b)500 262 y Fv(\()p Fu(H)s Fv(\))j(and)g Fu(\026)d Ft(2)h(P)s Fv(\()p Fu(H)s Fv(\).)18 b(A)c(measure)g Fu(\026)e Ft(2)f(P)s Fv(\()p Fu(H)s Fv(\))k(is)e(said)h(to)g(b)q(e)h Fg(stationary)257 311 y Fv(if)e Fd(P)329 296 y Fm(\003)329 322 y Fn(t)349 311 y Fu(\026)e Fv(=)h Fu(\026)i Fv(for)g(all)e Fu(t)g Ft(\025)g Fv(0.)320 361 y(W)m(e)18 b(denote)h(b)o(y)g Fu(u)p Fv(\()p Fu(t)p Fv(;)7 b Fu(u)695 367 y Fr(0)712 361 y Fv(\))19 b(a)g(solution)e(for)i(\(0.2\))e(equal)i(to)f(a)h (random)e(v)n(ariable)g Fu(u)1671 367 y Fr(0)257 411 y Fv(at)h Fu(t)f Fv(=)g(0.)28 b(In)18 b([KS02)o(],)f(w)o(e)h(pro)o(v)o (e)f(that)h(the)g(NS)f(equation)g(has)g(a)g(unique)h(stationary)257 461 y(measure)c Fu(\026)g Fv(and)356 513 y Fq(\014)356 538 y(\014)356 563 y(\014)370 561 y Fs(E)9 b Fu(f)427 528 y Fq(\000)449 561 y Fu(u)p Fv(\()p Fu(t)p Fv(;)e Fu(u)547 567 y Fr(0)564 561 y Fv(\))580 528 y Fq(\001)609 561 y Ft(\000)650 505 y Fq(Z)673 599 y Fn(H)712 561 y Fu(f)t Fv(\()p Fu(v)q Fv(\))p Fu(\026)p Fv(\()p Fu(dv)q Fv(\))889 513 y Fq(\014)889 538 y(\014)889 563 y(\014)916 561 y Ft(\024)12 b Fu(C)1000 528 y Fq(\000)1019 561 y Fv(1)d(+)g Fs(E)g Ft(j)p Fu(u)1158 567 y Fr(0)1179 561 y Ft(j)1191 544 y Fr(2)1209 528 y Fq(\001)1235 561 y Fu(e)1254 544 y Fm(\000)p Fn(\013\015)r(t)1336 561 y Fu(;)48 b(t)11 b Ft(\025)h Fv(0)p Fu(;)105 b Fv(\(2.1\))257 666 y(for)14 b(an)o(y)f(function)h Fu(f)i Ft(2)11 b Fu(C)667 672 y Fn(b)684 666 y Fv(\()p Fu(H)s Fv(\))j(satisfying)f(the)h (conditions)620 772 y(sup)615 806 y Fn(u)p Fm(2)p Fn(H)687 736 y Fq(\014)687 761 y(\014)700 772 y Fu(f)t Fv(\()p Fu(u)p Fv(\))780 736 y Fq(\014)780 761 y(\014)807 772 y Ft(\024)d Fv(1)p Fu(;)48 b Fv(sup)931 807 y Fn(u)p Fm(6)p Fr(=)p Fn(v)1006 743 y Ft(j)p Fu(f)t Fv(\()p Fu(u)p Fv(\))9 b Ft(\000)h Fu(f)t Fv(\()p Fu(v)q Fv(\))p Ft(j)p 1006 762 234 2 v 1051 800 a(j)p Fu(u)f Ft(\000)g Fu(v)q Ft(j)1170 788 y Fn(\013)1256 772 y Ft(\024)i Fv(1)p Fu(:)257 881 y Fv(Here)k Fu(\013)e Fv(is)h(an)o(y)f(n)o(um)o(b)q(er)f(suc)o(h)j (that)e(0)f Fu(<)f(\013)h Ft(\024)f Fv(1,)i(and)h(the)g(constan)o(ts)g Fu(C)g Ft(\025)e Fv(1)h(and)h Fu(\015)g(>)e Fv(0)257 931 y(are)j(indep)q(enden)o(t)g(of)e Fu(\013)p Fv(.)18 b(Inequalit)o(y)13 b(\(2.1\))h(implies)e(that)532 1013 y Ft(D)565 979 y Fq(\000)584 1013 y Fu(u)p Fv(\()p Fu(t)p Fv(;)7 b Fu(u)682 1019 y Fr(0)700 1013 y Fv(\))716 979 y Fq(\001)747 1013 y Fu(*)k(\026)41 b Fv(as)h Fu(t)11 b Ft(!)g(1)p Fu(;)48 b Fv(for)13 b(all)41 b Fu(u)1299 1019 y Fr(0)1329 1013 y Ft(2)11 b Fu(H)q(:)188 b Fv(\(2.2\))257 1094 y(Belo)o(w)10 b(w)o(e)h(use)g(the)f(results)i(from)c(the)j (previous)f(section)h(to)f(sho)o(w)g(that)g(estimates)g(similar)257 1144 y(to)k(\(2.1\))f(hold)h(for)f(a)h(m)o(uc)o(h)e(larger)i(class)h (of)e(functionals)g Fu(f)t Fv(.)320 1194 y(F)m(or)h Fu(\013)f Ft(2)h Fv(\(0)p Fu(;)7 b Fv(1])13 b(and)i(in)o(tegers)h Fu(p)d Ft(\025)h Fv(1)h(and)g Fu(l)g Ft(\025)e Fv(0,)i(w)o(e)g(denote)h (b)o(y)f Ft(O)1424 1174 y Fn(p)1423 1206 y(l)1443 1194 y Fv(\()p Fu(\013)p Fv(\))g(the)h(set)g(of)257 1244 y(con)o(tin)o(uous) e(functionals)f Fu(g)j Fv(on)d Fu(H)807 1228 y Fn(l)834 1244 y Fv(suc)o(h)h(that)664 1290 y Fq(\014)664 1315 y(\014)678 1325 y Fu(g)q Fv(\()p Fu(u)p Fv(\))755 1290 y Fq(\014)755 1315 y(\014)780 1325 y Ft(\024)e Fv(1)d(+)h Ft(k)p Fu(u)p Ft(k)962 1305 y Fn(p)962 1338 y(l)980 1325 y Fu(;)619 b Fv(\(2.3\))538 1360 y Fq(\014)538 1385 y(\014)552 1395 y Fu(g)q Fv(\()p Fu(u)p Fv(\))9 b Ft(\000)h Fu(g)q Fv(\()p Fu(v)q Fv(\))754 1360 y Fq(\014)754 1385 y(\014)780 1395 y Ft(\024)i(k)p Fu(u)d Ft(\000)g Fu(v)q Ft(k)961 1378 y Fn(\013)961 1406 y(l)985 1362 y Fq(\000)1004 1395 y Fv(1)g(+)h Ft(k)p Fu(u)p Ft(k)1142 1375 y Fn(p)p Fm(\000)p Fr(1)1142 1408 y Fn(l)1212 1395 y Fv(+)g Ft(k)p Fu(v)q Ft(k)1317 1375 y Fn(p)p Fm(\000)p Fr(1)1317 1408 y Fn(l)1378 1362 y Fq(\001)1397 1395 y Fu(;)202 b Fv(\(2.4\))257 1481 y(where)22 b Fu(u;)7 b(v)23 b Ft(2)f Fu(H)558 1466 y Fn(l)570 1481 y Fv(.)37 b(Let)21 b(us)g(tak)o(e)f(an)o(y)g(constan)o (t)h Fu(R)h Ft(\025)g Fv(1)e(and)g(consider)h(t)o(w)o(o)f(solu-)257 1531 y(tions)c(~)-23 b Fu(u)383 1537 y Fr(1)401 1531 y Fv(\()p Fu(t)p Fv(\))14 b(and)i(~)-24 b Fu(u)566 1537 y Fr(2)584 1531 y Fv(\()p Fu(t)p Fv(\))14 b(that)f(corresp)q(ond)i(to)f (initial)d(functions)j Fu(u)1318 1537 y Fr(10)1366 1531 y Fv(and)f Fu(u)1470 1537 y Fr(20)1518 1531 y Fv(satisfying)257 1587 y(the)k(inequalit)o(y)d Fs(E)8 b Fu(e)576 1572 y Fn(\033)q Fm(j)p Fn(u)626 1576 y Ff(i)p Fo(0)656 1572 y Fm(j)666 1559 y Fo(2)698 1587 y Ft(\024)15 b Fu(R)f(<)h Ft(1)g Fv(for)g(some)g Fu(\033)g(>)g Fv(0.)23 b(It)15 b(is)h(pro)o(v)o(ed)g(in)f([KS02)o(])g(\(see)257 1637 y(there)g(relation)e(\(2.39\)\))g(that)g(there)i(are)f(w)o(eak)f (solutions)g Fu(u)1215 1643 y Fr(1)1247 1637 y Fv(and)g Fu(u)1351 1643 y Fr(2)1383 1637 y Fv(for)g(\(0.2\))g(that)g(are)257 1686 y(distributed)i(as)i(~)-24 b Fu(u)545 1692 y Fr(1)577 1686 y Fv(and)16 b(~)-23 b Fu(u)682 1692 y Fr(2)700 1686 y Fv(,)14 b(resp)q(ectiv)o(ely)m(,)g(and)g(satisfy)f(the)i(inequalit)o (y)373 1768 y Fs(P)402 1734 y Fq(\010)423 1768 y Ft(j)p Fu(u)459 1774 y Fr(1)477 1768 y Fv(\()p Fu(t)p Fv(\))9 b Ft(\000)h Fu(u)599 1774 y Fr(2)617 1768 y Fv(\()p Fu(t)p Fv(\))p Ft(j)i(\024)f Fu(C)761 1774 y Fr(1)780 1768 y Fu(e)799 1751 y Fm(\000)p Fn(\015)r(t)859 1734 y Fq(\011)895 1768 y Ft(\025)g Fv(1)e Ft(\000)h Fu(C)1040 1774 y Fr(1)1058 1734 y Fq(\000)1077 1768 y Fv(1)f(+)h Fs(E)e Ft(j)p Fu(u)1216 1774 y Fr(10)1254 1768 y Ft(j)1266 1751 y Fr(2)1293 1768 y Fv(+)i Fu(E)f Ft(j)p Fu(u)1411 1774 y Fr(20)1445 1768 y Ft(j)1457 1751 y Fr(2)1475 1734 y Fq(\001)1494 1768 y Fu(e)1513 1751 y Fm(\000)p Fn(\015)r(t)895 1835 y Ft(\025)i Fv(1)e Ft(\000)h Fu(C)1040 1841 y Fr(2)1058 1835 y Fu(R)d(e)1116 1817 y Fm(\000)p Fn(\015)r(t)1176 1835 y Fu(:)416 b Fv(\(2.5\))257 1916 y(Denoting)14 b(b)o(y)f(\012)524 1922 y Fr(1)543 1916 y Fv(\()p Fu(t)p Fv(\))h(the)h(ev)o(en)o(t)f(in)g(the)g(left-hand) f(side)i(of)h(\(2.5\),)d(w)o(e)h(ha)o(v)o(e)710 1998 y Fs(P)738 1964 y Fq(\000)754 1998 y Fv(\012)784 2004 y Fr(1)803 1998 y Fv(\()p Fu(t)p Fv(\))850 1980 y Fn(c)867 1964 y Fq(\001)897 1998 y Ft(\024)e Fu(C)971 2004 y Fr(2)990 1998 y Fu(R)6 b(e)1047 1980 y Fm(\000)p Fn(\015)r(t)1107 1998 y Fu(;)485 b Fv(\(2.6\))633 2064 y Ft(j)p Fu(u)669 2070 y Fr(1)687 2064 y Fv(\()p Fu(t)p Fv(\))9 b Ft(\000)h Fu(u)809 2070 y Fr(2)827 2064 y Fv(\()p Fu(t)p Fv(\))p Ft(j)h(\024)h Fu(C)971 2070 y Fr(1)990 2064 y Fu(e)1009 2047 y Fm(\000)p Fn(\015)r(t)1069 2064 y Fu(;)48 b(!)12 b Ft(2)g Fv(\012)1237 2070 y Fr(1)1255 2064 y Fv(\()p Fu(t)p Fv(\))p Fu(:)290 b Fv(\(2.7\))257 2146 y(Let)17 b(us)g(\014x)f(an)o(y)g Fu(p)g Ft(\025)g Fv(1,)g Fu(k)h Ft(\025)f Fv(0,)g(and)g Fu(\013)g Ft(2)f Fv(\(0)p Fu(;)7 b Fv(1])15 b(and)h(tak)o(e)h(a)f(functional)f Fu(g)i Ft(2)f(O)1600 2126 y Fn(p)1599 2158 y(k)1619 2146 y Fv(\()p Fu(\013)p Fv(\).)257 2196 y(Assuming)d(that)h Fu(B)569 2202 y Fn(k)q Fr(+1)644 2196 y Fu(<)d Ft(1)p Fv(,)i(w)o(e)h(conclude)h (from)d(Corollary)h(1.2)g(that)653 2277 y Fs(E)685 2242 y Fq(\015)685 2267 y(\015)711 2277 y Fu(u)735 2283 y Fn(i)748 2277 y Fv(\()p Fu(t)p Fv(\))795 2242 y Fq(\015)795 2267 y(\015)819 2252 y Fn(m)819 2294 y(r)862 2277 y Ft(\024)f Fu(K)s Fv(\()p Fu(r)o(;)7 b(m;)g(\033)q Fv(\))p Fu(R;)47 b(t)12 b Ft(\025)f Fv(1)p Fu(;)317 b Fv(\(2.8\))257 2359 y(where)19 b Fu(i)e Fv(=)g(1)p Fu(;)7 b Fv(2,)16 b(0)g Ft(\024)h Fu(r)h Ft(\024)f Fu(k)12 b Fv(+)g(1,)17 b Fu(m)g Ft(\025)g Fv(1,)g(and)g Fu(K)s Fv(\()p Fu(r)o(;)7 b(m;)g(\033)q Fv(\))17 b Fu(>)g Fv(0)f(is)h(a)g(constan)o(t)h(not)257 2409 y(dep)q(ending)d(on)f(solutions.)j(Using)d(the)g(in)o(terp)q (olation)f(inequalit)o(y)771 2468 y Fq(\015)771 2493 y(\015)794 2503 y Fu(u)818 2468 y Fq(\015)818 2493 y(\015)840 2520 y Fn(k)872 2503 y Ft(\024)916 2468 y Fq(\014)916 2493 y(\014)930 2503 y Fu(u)954 2468 y Fq(\014)954 2493 y(\014)991 2467 y Fo(1)p 973 2472 52 2 v 973 2488 a Ff(k)p Fo(+1)1031 2468 y Fq(\015)1031 2493 y(\015)1054 2503 y Fu(u)1078 2468 y Fq(\015)1078 2493 y(\015)1124 2467 y Ff(k)p 1106 2472 V 1106 2488 a(k)p Fo(+1)1101 2520 y Fn(k)q Fr(+1)1165 2503 y Fu(;)963 2628 y Fv(8)p eop %%Page: 9 9 9 8 bop 257 262 a Fv(w)o(e)14 b(deriv)o(e)h(from)d(\(2.4\))h(and)h (\(2.7\))f(that)340 304 y Fq(\014)340 329 y(\014)354 339 y Fu(g)q Fv(\()p Fu(u)415 345 y Fr(1)434 339 y Fv(\()p Fu(t)p Fv(\)\))d Ft(\000)f Fu(g)q Fv(\()p Fu(u)609 345 y Fr(2)628 339 y Fv(\()p Fu(t)p Fv(\)\))691 304 y Fq(\014)691 329 y(\014)717 339 y Ft(\024)761 304 y Fq(\015)761 329 y(\015)784 339 y Fu(u)808 345 y Fr(1)826 339 y Fv(\()p Fu(t)p Fv(\))h Ft(\000)f Fu(u)948 345 y Fr(2)966 339 y Fv(\()p Fu(t)p Fv(\))1013 304 y Fq(\015)1013 329 y(\015)1037 314 y Fn(\013)1037 356 y(k)1060 306 y Fq(\000)1080 339 y Fv(1)f(+)i Ft(k)p Fu(u)1196 345 y Fr(1)1214 339 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1282 320 y Fn(p)p Fm(\000)p Fr(1)1282 352 y Fn(k)1353 339 y Fv(+)g Ft(k)p Fu(u)1440 345 y Fr(2)1458 339 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1526 320 y Fn(p)p Fm(\000)p Fr(1)1526 352 y Fn(k)1587 306 y Fq(\001)717 408 y Ft(\024)i Fu(C)791 414 y Fr(3)816 408 y Fu(e)835 391 y Fm(\000)p Fn(\015)878 395 y Ff(k)897 391 y Fn(\013t)933 374 y Fq(\000)952 408 y Fv(1)d(+)g Ft(k)p Fu(u)1068 414 y Fr(1)1087 408 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1155 388 y Fn(p)1155 421 y(k)q Fr(+1)1226 408 y Fv(+)h Ft(k)p Fu(u)1313 414 y Fr(2)1331 408 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1399 388 y Fn(p)1399 421 y(k)q Fr(+1)1461 374 y Fq(\001)1480 408 y Fu(;)112 b Fv(\(2.9\))257 486 y(where)15 b Fu(!)e Ft(2)e Fv(\012)485 492 y Fr(1)504 486 y Fv(\()p Fu(t)p Fv(\))j(and)g Fu(\015)667 492 y Fn(k)699 486 y Fv(=)769 467 y Fn(\015)p 748 476 61 2 v 748 500 a(k)q Fr(+1)814 486 y Fv(.)j(It)d(follo)o(ws)f (from)f(\(2.9\))h(and)h(\(2.8\))f(that)312 584 y Fs(E)343 538 y Fq(\020)371 584 y Fu(I)389 591 y Fr(\012)413 595 y Fo(1)429 591 y Fr(\()p Fn(t)p Fr(\))470 549 y Fq(\014)470 573 y(\014)483 584 y Fu(g)q Fv(\()p Fu(u)544 590 y Fr(1)563 584 y Fv(\()p Fu(t)p Fv(\)\))d Ft(\000)f Fu(g)q Fv(\()p Fu(u)738 590 y Fr(2)757 584 y Fv(\()p Fu(t)p Fv(\)\))820 549 y Fq(\014)820 573 y(\014)835 538 y(\021)871 584 y Ft(\024)j Fu(C)945 590 y Fr(4)963 584 y Fu(K)s Fv(\()p Fu(k)e Fv(+)g(1)p Fu(;)d(p;)g(\033)q Fv(\))p Fu(R)f(e)1269 567 y Fm(\000)p Fn(\015)1312 571 y Ff(k)1330 567 y Fn(\013t)1367 584 y Fu(;)48 b(t)11 b Ft(\025)h Fv(0)p Fu(:)53 b Fv(\(2.10\))257 674 y(F)m(urthermore,)14 b(in)f(view)h(of)i(\(2.3\))d(and)h(\(2.6\),)f (w)o(e)h(ha)o(v)o(e)333 765 y Fs(E)365 719 y Fq(\020)393 765 y Fu(I)411 772 y Fr(\012)435 776 y Fo(1)451 772 y Fr(\()p Fn(t)p Fr(\))490 764 y Ff(c)507 729 y Fq(\014)507 754 y(\014)521 765 y Fu(g)q Fv(\()p Fu(u)582 771 y Fr(1)600 765 y Fv(\()p Fu(t)p Fv(\)\))c Ft(\000)g Fu(g)q Fv(\()p Fu(u)776 771 y Fr(2)794 765 y Fv(\()p Fu(t)p Fv(\)\))857 729 y Fq(\014)857 754 y(\014)872 719 y(\021)908 765 y Ft(\024)i Fs(E)983 719 y Fq(\020)1011 765 y Fu(I)1029 772 y Fr(\012)1053 776 y Fo(1)1069 772 y Fr(\()p Fn(t)p Fr(\))1108 764 y Ff(c)1125 731 y Fq(\000)1144 765 y Fv(2)d(+)h Ft(k)p Fu(u)1261 771 y Fr(1)1279 765 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1347 745 y Fn(p)1347 777 y(k)1377 765 y Fv(+)f Ft(k)p Fu(u)1463 771 y Fr(2)1481 765 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1549 745 y Fn(p)1549 777 y(k)1570 731 y Fq(\001)1589 719 y(\021)428 858 y Ft(\024)j Fs(P)500 824 y Fq(\000)516 858 y Fu(I)534 865 y Fr(\012)558 869 y Fo(1)574 865 y Fr(\()p Fn(t)p Fr(\))613 857 y Ff(c)630 824 y Fq(\001)654 822 y Fo(1)p 654 827 15 2 v 654 843 a(2)682 812 y Fq(\020)707 858 y Fv(2)d(+)779 824 y Fq(\000)798 858 y Fs(E)f Ft(k)p Fu(u)874 864 y Fr(1)895 858 y Fv(\()p Fu(t)p Fv(\))p Ft(k)963 838 y Fr(2)p Fn(p)963 870 y(k)999 824 y Fq(\001)1023 822 y Fo(1)p 1023 827 V 1023 843 a(2)1053 858 y Fv(+)1095 824 y Fq(\000)1114 858 y Fs(E)h Ft(k)p Fu(u)1191 864 y Fr(1)1212 858 y Fv(\()p Fu(t)p Fv(\))p Ft(k)1280 838 y Fr(2)p Fn(p)1280 870 y(k)1316 824 y Fq(\001)1340 822 y Fo(1)p 1340 827 V 1340 843 a(2)1361 812 y Fq(\021)428 943 y Ft(\024)j Fu(C)502 949 y Fr(5)520 943 y Fu(K)s Fv(\()p Fu(k)q(;)7 b Fv(2)p Fu(p;)g(\033)q Fv(\))723 915 y Fo(1)p 722 920 V 722 936 a(2)743 943 y Fu(R)g(e)801 926 y Fm(\000)p Fn(\015)r(t=)p Fr(2)895 943 y Fu(:)676 b Fv(\(2.11\))257 1021 y(Com)o(bining)12 b(\(2.10\))h(and)h(\(2.11\))o (,)f(w)o(e)h(obtain)477 1070 y Fq(\014)477 1095 y(\014)491 1105 y Fs(E)9 b Fu(g)q Fv(\()p Fu(u)584 1111 y Fr(1)605 1105 y Fv(\()p Fu(t)p Fv(\)\))h Ft(\000)g Fs(E)e Fu(g)q Fv(\()q Fu(u)813 1111 y Fr(2)834 1105 y Fv(\()p Fu(t)p Fv(\)\))897 1070 y Fq(\014)897 1095 y(\014)923 1105 y Ft(\024)k Fu(C)s Fv(\()p Fu(k)q(;)7 b(p;)g(\033)q Fv(\))g Fu(R)e(e)1202 1088 y Fm(\000)p Fn(\015)1247 1075 y Fe(0)1245 1096 y Ff(k)1264 1088 y Fn(\013t)1300 1105 y Fu(;)48 b(t)12 b Ft(\025)f Fv(1)p Fu(;)120 b Fv(\(2.12\))257 1183 y(where)18 b Fu(g)f Ft(2)f(O)495 1163 y Fn(p)494 1195 y(k)514 1183 y Fv(\()p Fu(\013)p Fv(\).)26 b(In)16 b(particular,)h(assuming)e(that)h Fu(u)1171 1189 y Fr(2)1189 1183 y Fv(\()p Fu(t)p Fv(\))h(is)f(a)h(stationary)f(solution,)257 1233 y(w)o(e)e(arriv)o(e)g(at)g(the)h(follo)o(wing)c(result:)257 1305 y Fi(Theorem)k(2.1.)21 b Fg(L)n(et)16 b Fu(B)659 1311 y Fn(k)q Fr(+1)736 1305 y Fu(<)e Ft(1)h Fg(for)h(some)g(inte)n (ger)f Fu(k)g Ft(\025)f Fv(0)p Fg(,)i(let)f Fu(\026)h Fg(b)n(e)g(the)h(stationary)257 1355 y(me)n(asur)n(e)h(for)g(the)h(NS)f (system)j Fv(\(0.2\))p Fg(,)e Fv(\(0.3\))e Fg(with)h(the)g(p)n(erio)n (dic)g(b)n(oundary)h(c)n(onditions)257 1405 y(and)d(let)f Fu(u)p Fv(\()p Fu(t)p Fv(\))e(=)f Fu(u)p Fv(\()p Fu(t)p Fv(;)7 b Fu(u)623 1411 y Fr(0)641 1405 y Fv(\))16 b Fg(b)n(e)f(a)h (solution)f(for)k Fv(\(0.2\))c Fg(with)g(initial)f(function)i Fu(u)1491 1411 y Fr(0)1525 1405 y Fg(such)f(that)257 1460 y Fu(R)d Fv(:=)g Fs(E)d Fu(e)408 1445 y Fn(\033)q Fm(j)p Fn(u)458 1449 y Fo(0)477 1445 y Fm(j)487 1433 y Fo(2)517 1460 y Fu(<)j Ft(1)p Fg(,)j(wher)n(e)f Fu(\033)f(>)g Fv(0)i Fg(is)f(a)i(c)n(onstant.)k(Then)c(for)e(any)i Fu(p)c Ft(\025)g Fv(1)p Fg(,)j Fu(\013)c Ft(2)h Fv(\(0)p Fu(;)7 b Fv(1])p Fg(,)257 1510 y(and)16 b Fu(g)d Ft(2)e(O)444 1490 y Fn(p)443 1523 y(k)463 1510 y Fv(\()p Fu(\013)p Fv(\))k Fg(we)g(have)537 1561 y Fq(\014)537 1586 y(\014)551 1596 y Fs(E)8 b Fu(g)604 1562 y Fq(\000)626 1596 y Fu(u)p Fv(\()p Fu(t)p Fv(\))697 1562 y Fq(\001)725 1596 y Ft(\000)h Fv(\()p Fu(\026;)e(g)q Fv(\))863 1561 y Fq(\014)863 1586 y(\014)889 1596 y Ft(\024)12 b Fu(C)s Fv(\()p Fu(k)q(;)7 b(p;)g(\033)q Fv(\))g Fu(R)e(e)1168 1579 y Fm(\000)p Fn(\015)1213 1566 y Fe(0)1225 1579 y Fn(t)1240 1596 y Fu(;)48 b(t)12 b Ft(\025)g Fv(0)p Fu(;)179 b Fv(\(2.13\))257 1674 y Fg(wher)n(e)15 b Fu(\015)398 1659 y Fm(0)422 1674 y Fu(>)c Fv(0)k Fg(dep)n(ends)h(on)f Fu(\013)g Fg(and)g Fu(k)q Fg(.)320 1746 y Fv(The)i(results)h(from)e([KS02)o(])h(giv)o(en)f (in)h(\(2.1\))f(and)h(\(2.5\))f(apply)h(to)g(solutions)f(of)h(the)257 1796 y(NS)c(equations)f(b)q(oth)g(under)h(the)g(Diric)o(hlet)e(and)h(p) q(erio)q(dic)h(b)q(oundary)f(conditions.)17 b(Since)257 1846 y(our)f(pro)q(of)e(of)h(Theorem)g(2.1)f(uses)i(estimates)f (\(2.8\))f(for)h Fu(r)g Fv(=)f Fu(k)d Fv(+)f(1)j Ft(\025)h Fv(1,)h(it)g(applies)g(for)257 1896 y(the)h(p)q(erio)q(dic)f(case)h (only)m(.)j(Still,)13 b(if)h(w)o(e)h(tak)o(e)g Fu(k)f Fv(=)f(0,)i(then)g(inequalit)o(y)f(\(2.8\))g(with)h Fu(r)e Fv(=)h(0)257 1946 y(remains)i(v)n(alid)f(\(see)i(\(1.2\))f(and)g (Subsection)h(1.2\),)f(while)g(\(2.8\))g(with)g Fu(r)g Fv(=)g(1)g(w)o(as)h(used)257 1995 y(only)e(in)g(the)g(deriv)n(ation)g (of)f(inequalit)o(y)h(\(2.10\))o(.)22 b(The)16 b(latter)f(is)g(no)o(w)g (implied)e(b)o(y)j(\(2.6\))o(,)257 2045 y(\(2.7\),)d(and)h(\(2.8\))g (with)g Fu(r)f Fv(=)f(0.)19 b(After)c(\(2.8\))e(and)h(\(2.10\))f(are)i (established,)f(the)h(pro)q(of)f(of)257 2095 y(inequalit)o(y)g (\(2.12\))e(go)q(es)j(as)f(b)q(efore.)19 b(Therefore,)14 b(w)o(e)g(ha)o(v)o(e)g(the)h(follo)o(wing)c(result:)257 2167 y Fi(Prop)q(osition)h(2.2.)21 b Fg(If)c Fu(k)g Fv(=)g(0)p Fg(,)g(then)h(the)f(assertion)g(of)g(The)n(or)n(em)g(2.1)h(holds)f (true)g(for)257 2217 y(solutions)e(of)g(the)g(NS)g(system)g(under)g (Dirichlet's)f(b)n(oundary)i(c)n(ondition.)320 2290 y Fv(In)e(fact,)f(the)i(argumen)o(ts)f(used)h(to)f(pro)o(v)o(e)g (\(2.13\))f(with)h Fu(k)f Fv(=)f(0)i(apply)f(to)h(functionals)257 2339 y Fu(g)f Ft(2)e Fu(C)359 2345 y Fn(b)376 2339 y Fv(\()p Fu(H)s Fv(\))j(suc)o(h)g(that)624 2390 y Fq(\014)624 2415 y(\014)638 2425 y Fu(g)q Fv(\()p Fu(u)p Fv(\))c Ft(\000)f Fu(g)q Fv(\()p Fu(v)q Fv(\))840 2390 y Fq(\014)840 2415 y(\014)867 2425 y Ft(\024)j(j)p Fu(u)c Ft(\000)i Fu(v)q Ft(j)1037 2392 y Fq(\000)1056 2425 y Fu(e)1075 2408 y Fn(")p Fm(j)p Fn(u)p Fm(j)1131 2395 y Fo(2)1158 2425 y Fv(+)f Fu(e)1218 2408 y Fn(")p Fm(j)p Fn(u)p Fm(j)1274 2395 y Fo(2)1292 2392 y Fq(\001)1311 2425 y Fu(;)257 2503 y Fv(where)15 b Fu(u;)7 b(v)13 b Ft(2)e Fu(H)s Fv(,)i(and)h Fu(")e(>)g Fv(0)h(is)h(a)g(su\016cien)o(tly)f(small)f(constan)o(t.)963 2628 y(9)p eop %%Page: 10 10 10 9 bop 257 262 a Fw(3)67 b(Kic)n(k-forced)23 b(equations)257 361 y Fh(3.1)56 b(Statemen)n(t)17 b(of)i(the)f(results)257 437 y Fv(In)i(this)g(section,)i(w)o(e)e(consider)h(the)f(NS)g(system)g (\(0.2\))f(under)i(the)f(p)q(erio)q(dic)h(b)q(ound-)257 487 y(ary)d(conditions,)g(p)q(erturb)q(ed)h(b)o(y)f(short)g(short-p)q (erio)q(d)h(random)d(kic)o(k-force)h Fu(\021)1525 493 y Fn(")1560 487 y Fv(de\014ned)257 537 y(in)d(\(0.3\))o(:)682 626 y(_)-20 b Fu(u)698 632 y Fn(")725 626 y Fv(+)10 b Fu(\027)s(Lu)843 632 y Fn(")869 626 y Fv(+)f Fu(B)r Fv(\()p Fu(u)983 632 y Fn(")1002 626 y Fu(;)e(u)1045 632 y Fn(")1062 626 y Fv(\))k(=)h Fu(\021)1154 632 y Fn(")1171 626 y Fv(\()p Fu(t;)7 b(x)p Fv(\))p Fu(:)331 b Fv(\(3.1\))257 714 y(W)m(e)15 b(assume)g(that)g(0)e Fu(<)h(")g Ft(\024)g Fv(1,)g(the)i(constan)o(ts)g Fu(b)1048 720 y Fn(j)1079 714 y Ft(\025)d Fv(0)i(are)h(the)f(same)g(as)g(in)f(\(0.4\),)h(and)257 764 y(the)i(distributions)f Ft(D)q Fv(\()p Fu(\030)643 770 y Fn(j)r(k)679 764 y Fv(\))h(of)e(the)i(indep)q(enden)o(t)h(random) c(v)n(ariables)i Fu(\030)1420 770 y Fn(j)r(k)1472 764 y Fv(satisfy)g(\(0.7\))257 814 y(and)e(the)h(follo)o(wing)c(condition:) 266 895 y Fi(\(H\))20 b Ft(D)q Fv(\()p Fu(\030)428 901 y Fn(j)r(k)464 895 y Fv(\))12 b(=)g Fu(p)557 901 y Fn(j)574 895 y Fv(\()p Fu(s)p Fv(\))7 b Fu(ds)p Fv(,)12 b(where)g Fu(p)835 901 y Fn(j)852 895 y Fv('s)f(are)g(functions)f(of)g(b)q (ounded)i(total)e(v)n(ariation)f(suc)o(h)361 954 y(that)14 b(supp)8 b Fu(p)565 960 y Fn(j)593 954 y Ft(\032)k Fv([)p Ft(\000)p Fv(1)p Fu(;)7 b Fv(1])12 b(and)847 921 y Fq(R)875 931 y Fn(\016)867 969 y Fm(\000)p Fn(\016)918 954 y Fu(p)939 960 y Fn(j)956 954 y Fv(\()p Fu(s)p Fv(\))7 b Fu(ds)12 b(>)g Fv(0)i(for)f(all)g Fu(j)h Ft(\025)e Fv(1)h(and)h Fu(\016)g(>)e Fv(0;)320 1035 y(T)m(o)d(simplify)e(notations,)j(in)f (what)h(follo)o(ws)e(it)i(is)f(assumed)h(that)g Fu(\027)j Fv(=)f(1.)17 b(W)m(e)9 b(normalise)257 1085 y(solutions)i(of)j(\(3.1\)) c(to)h(b)q(e)g(con)o(tin)o(uous)g(from)e(the)j(righ)o(t)f(and)f(set)i Fu(u)1303 1070 y Fn(k)1303 1096 y(")1335 1085 y Fv(=)g Fu(u)1403 1091 y Fn(")1420 1085 y Fv(\()p Fu(k)q(")p Fv(\).)18 b(Denoting)257 1135 y(b)o(y)d Fu(S)341 1141 y Fn(")374 1135 y Fv(the)h(op)q(erator)g(of)e(time)g Fu(")h Fv(shift)g(along)f(tra)r(jectories)i(of)f(the)h(free)f(NS)h (system,)e(w)o(e)257 1185 y(see)h(that)788 1274 y Fu(u)812 1256 y Fn(k)812 1284 y(")844 1274 y Fv(=)d Fu(S)913 1280 y Fn(")931 1240 y Fq(\000)950 1274 y Fu(u)974 1256 y Fn(k)q Fm(\000)p Fr(1)974 1284 y Fn(")1036 1240 y Fq(\001)1065 1274 y Fv(+)d Fu(\021)1127 1280 y Fn(k)1147 1274 y Fu(:)445 b Fv(\(3.2\))257 1362 y(Since)17 b Ft(j)p Fu(S)405 1368 y Fn(")422 1362 y Fv(\()p Fu(u)p Fv(\))p Ft(j)e(\024)f Fu(e)570 1347 y Fm(\000)p Fn(")614 1362 y Ft(j)p Fu(u)p Ft(j)h Fv(and)h Fs(E)8 b Fu(\021)812 1368 y Fn(k)850 1362 y Fv(=)15 b(0)g(due)h(to)g(\(0.7\))o(,)g(it)f(follo)o(ws)g(from)e (the)k(indep)q(en-)257 1412 y(dence)f(of)d Fu(\021)441 1418 y Fn(k)475 1412 y Fv(and)h Fu(u)580 1397 y Fn(k)580 1422 y(")614 1412 y Fv(that)461 1509 y Fs(E)492 1474 y Fq(\014)492 1499 y(\014)509 1509 y Fu(u)533 1492 y Fn(k)533 1520 y(")553 1474 y Fq(\014)553 1499 y(\014)567 1484 y Fr(2)597 1509 y Fv(=)e Fs(E)673 1474 y Fq(\014)673 1499 y(\014)689 1509 y Fu(S)714 1515 y Fn(")733 1509 y Fv(\()p Fu(u)773 1492 y Fn(k)q Fm(\000)p Fr(1)773 1520 y Fn(")835 1509 y Fv(\))851 1474 y Fq(\014)851 1499 y(\014)865 1484 y Fr(2)893 1509 y Fv(+)e Fs(E)e Ft(j)p Fu(\021)999 1515 y Fn(k)1022 1509 y Ft(j)1034 1492 y Fr(2)1064 1509 y Ft(\024)j Fu(e)1126 1492 y Fm(\000)p Fr(2)p Fn(")1187 1509 y Fs(E)1219 1474 y Fq(\014)1219 1499 y(\014)1235 1509 y Fu(u)1259 1492 y Fn(k)q Fm(\000)p Fr(1)1259 1520 y Fn(")1322 1474 y Fq(\014)1322 1499 y(\014)1336 1484 y Fr(2)1364 1509 y Fv(+)e Fu("B)1455 1515 y Fr(0)1475 1509 y Fu(;)257 1598 y Fv(where)15 b(w)o(e)g(used)f(the)h(relation)614 1687 y Fs(E)9 b Ft(j)p Fu(\021)679 1693 y Fn(k)701 1687 y Ft(j)713 1670 y Fr(2)743 1687 y Fv(=)j Fu(")813 1647 y Fq(X)835 1736 y Fn(j)880 1687 y Fu(b)898 1670 y Fr(2)898 1697 y Fn(j)923 1687 y Fs(E)d Fu(\030)975 1670 y Fr(2)973 1697 y Fn(j)r(k)1023 1687 y Fv(=)j Fu(")1093 1647 y Fq(X)1116 1736 y Fn(j)1160 1687 y Fu(b)1178 1670 y Fr(2)1178 1697 y Fn(j)1208 1687 y Fv(=)g Fu("B)1302 1693 y Fr(0)1321 1687 y Fu(:)257 1820 y Fv(In)i(particular,)f(if)g Fu(u)574 1826 y Fn(")592 1820 y Fv(\(0\))f(=)f(0,)j(then)577 1930 y Fs(E)608 1894 y Fq(\014)608 1919 y(\014)625 1930 y Fu(u)649 1913 y Fn(k)649 1940 y(")669 1894 y Fq(\014)669 1919 y(\014)683 1905 y Fr(2)713 1930 y Ft(\024)803 1902 y Fu("B)853 1908 y Fr(0)p 762 1920 152 2 v 762 1958 a Fv(1)9 b Ft(\000)h Fu(e)853 1946 y Fm(\000)p Fr(2)p Fn(")930 1930 y Ft(\024)978 1902 y Fu(B)1009 1908 y Fr(0)p 978 1920 51 2 v 993 1958 a Fv(2)1075 1930 y(for)k(eac)o(h)42 b Fu(k)12 b Ft(\025)g Fv(0)p Fu(:)257 2040 y Fv(F)m(or)i(0)f Ft(\024)f Fu(\034)18 b(<)12 b(")p Fv(,)j(w)o(e)g(ha)o(v)o(e)f Ft(j)p Fu(u)730 2046 y Fn(")747 2040 y Fv(\()p Fu(k)q(")c Fv(+)g Fu(\034)5 b Fv(\))p Ft(j)12 b Fv(=)h Ft(j)p Fu(S)1002 2046 y Fn(\034)1022 2040 y Fv(\()p Fu(u)1062 2025 y Fn(k)1062 2050 y(")1083 2040 y Fv(\))p Ft(j)f(\024)g(j)p Fu(u)1203 2025 y Fn(k)1203 2050 y(")1223 2040 y Ft(j)p Fv(.)19 b(Therefore,)d(if)d Fu(u)1529 2046 y Fn(")1547 2040 y Fv(\(0\))f(=)h(0,)257 2090 y(then)676 2179 y Fs(E)708 2143 y Fq(\014)708 2168 y(\014)725 2179 y Fu(u)749 2185 y Fn(")766 2179 y Fv(\()p Fu(t)p Fv(\))813 2143 y Fq(\014)813 2168 y(\014)827 2154 y Fr(2)857 2179 y Ft(\024)906 2162 y Fr(1)p 906 2169 17 2 v 906 2193 a(2)928 2179 y Fu(B)959 2185 y Fr(0)1019 2179 y Fv(for)h(all)40 b Fu(t)12 b Ft(\025)f Fv(0)p Fu(:)333 b Fv(\(3.3\))257 2267 y(Since)17 b(\(3.1\))f(is)g(the)h (NS)f(system)g(under)i(the)e(p)q(erio)q(dic)h(b)q(oundary)f (conditions,)h(w)o(e)f(also)257 2317 y(ha)o(v)o(e)e Ft(k)p Fu(S)399 2323 y Fn(")417 2317 y Fv(\()p Fu(u)p Fv(\))p Ft(k)d(\024)h Fu(e)568 2302 y Fm(\000)p Fn(")612 2317 y Ft(k)p Fu(u)p Ft(k)p Fv(.)17 b(Rep)q(eating)d(the)h(ab)q(o)o(v)o(e)e (argumen)o(ts,)g(w)o(e)h(get)663 2414 y Fs(E)695 2379 y Fq(\015)695 2404 y(\015)721 2414 y Fu(u)745 2420 y Fn(")763 2414 y Fv(\()p Fu(t)p Fv(\))810 2379 y Fq(\015)810 2404 y(\015)833 2389 y Fr(2)863 2414 y Ft(\024)912 2398 y Fr(1)p 912 2405 V 912 2429 a(2)933 2414 y Fu(B)964 2420 y Fr(1)1025 2414 y Fv(for)g(all)40 b Fu(t)11 b Ft(\025)h Fv(0)p Fu(;)327 b Fv(\(3.4\))257 2503 y(if)13 b Fu(u)319 2509 y Fn(")337 2503 y Fv(\(0\))f(=)f(0.)953 2628 y(10)p eop %%Page: 11 11 11 10 bop 320 262 a Fv(F)m(or)13 b Fu(")f Ft(2)f Fv(\(0)p Fu(;)c Fv(1],)12 b(let)i(us)g(denote)h(b)o(y)f Fu(\020)900 268 y Fn(")918 262 y Fv(\()p Fu(t)p Fv(\))e Ft(2)f Fu(H)17 b Fv(the)d(piecewise)h(constan)o(t)g(pro)q(cess)698 380 y Fu(\020)716 386 y Fn(")734 380 y Fv(\()p Fu(t)p Fv(\))d(=)836 323 y Fq(Z)878 333 y Fn(t)p Fr(+0)859 418 y(0+)941 380 y Fu(\021)962 386 y Fn(")980 380 y Fv(\()p Fu(s)p Fv(\))7 b Fu(ds;)49 b(t)11 b Ft(\025)h Fv(0)p Fu(;)361 b Fv(\(3.5\))257 495 y(\(note)16 b(that)e Fu(\020)475 501 y Fn(")493 495 y Fv(\()p Fu(t)p Fv(\))h(v)n(anishes)g(at)f(zero)i(and)e(is)h(con)o (tin)o(uous)f(from)f(the)i(righ)o(t\).)20 b(W)m(e)15 b(denote)257 545 y(b)o(y)317 534 y(~)313 545 y Fu(\020)331 551 y Fn(")349 545 y Fv(\()p Fu(t)p Fv(\))e(the)g(pro)q(cess)h(equal)e (to)g Fu(\020)797 551 y Fn(")815 545 y Fv(\()p Fu(t)p Fv(\))g(for)g Fu(t)g Ft(2)f Fu(")p Fs(Z)1052 551 y Fr(+)1089 545 y Fv(and)h(extended)i(to)e Fs(R)1425 551 y Fr(+)1462 545 y Fv(b)o(y)g(the)h(linear)257 595 y(in)o(terp)q(olation.)30 b(Let)18 b(us)h(\014x)f(an)o(y)f Fu(T)24 b(>)19 b Fv(0)f(and)g (restrict)h(these)h(pro)q(cesses)g(to)e(the)h(time)257 644 y(in)o(terv)n(al)13 b([0)p Fu(;)7 b(T)f Fv(].)17 b(Then,)d(b)o(y)f(the)i(Donsk)o(er)f(theorem)f(\(see)j([Bil99)n(]\),) 735 733 y Ft(D)q Fv(\()788 722 y(~)784 733 y Fu(\020)802 739 y Fn(")820 733 y Fv(\))c Fu(*)f Ft(D)q Fv(\()p Fu(\020)s Fv(\))42 b(as)f Fu(")12 b Ft(!)f Fv(0)392 b(\(3.6\))257 821 y(in)15 b(the)g(space)i(of)d(measures)h(on)g Fu(C)s Fv(\([0)p Fu(;)7 b(T)f Fv(];)h Fu(H)978 806 y Fr(1)994 821 y Fv(\).)21 b(As)16 b(w)o(e)f(sho)o(w)g(b)q(elo)o(w)g(in)f (Corollary)g(3.3,)257 871 y(this)g(con)o(v)o(ergence)i(implies)c(that) 617 960 y Ft(D)650 926 y Fq(\000)669 960 y Fu(u)693 966 y Fn(")710 960 y Fv(\()p Fu(t;)7 b(u)784 966 y Fr(0)802 960 y Fv(\))818 926 y Fq(\001)849 960 y Fu(*)k Ft(D)935 926 y Fq(\000)954 960 y Fu(u)p Fv(\()p Fu(t;)c(u)1052 966 y Fr(0)1070 960 y Fv(\))1086 926 y Fq(\001)1147 960 y Fv(as)41 b Fu(")12 b Ft(!)f Fv(0)274 b(\(3.7\))257 1048 y(for)10 b(eac)o(h)h Fu(t)h Ft(2)f Fv([0)p Fu(;)c(T)f Fv(])j(and)h(eac)o(h)h Fu(u)767 1054 y Fr(0)797 1048 y Ft(2)g Fu(H)s Fv(,)g(where)g Fu(u)1037 1054 y Fn(")1055 1048 y Fv(\()p Fu(t)p Fv(;)c Fu(u)1129 1054 y Fr(0)1147 1048 y Fv(\))j(\(accordingly)m(,)g Fu(u)p Fv(\()p Fu(t)p Fv(;)d Fu(u)1513 1054 y Fr(0)1531 1048 y Fv(\)\))k(stands)257 1098 y(for)j(a)f(solution)h(for)f(\(3.2\))g(\(accordingly)m(,)g(for)g (\(0.2\)\))h(equal)f(to)h Fu(u)1283 1104 y Fr(0)1315 1098 y Fv(at)g Fu(t)d Fv(=)h(0.)320 1148 y(The)h(random)e(dynamical)f (system)i(\(3.2\))g(de\014nes)i(a)e(Mark)o(o)o(v)g(c)o(hain)g(in)g(the) h(space)h Fu(H)s Fv(.)257 1198 y(Let)k Fu(P)362 1204 y Fn("t)392 1198 y Fv(\()p Fu(u;)7 b Ft(\001)p Fv(\),)16 b Fu(t)h Ft(2)f Fu(")p Fs(Z)633 1204 y Fr(+)658 1198 y Fv(,)h(b)q(e)h(its)f(transition)f(function.)28 b(In)17 b([KS00)o(],)g(w)o(e)g(pro)o(v)o(e)g(that)g(if)257 1247 y(condition)d(\(0.5\))f(is)i(satis\014ed)g(for)e(a)h(su\016cien)o(tly)h (large)e(in)o(teger)i Fu(N)i Fv(=)c Fu(N)1416 1253 y Fn(")1446 1247 y Ft(\025)f Fv(1,)i(then)h(the)257 1297 y(Mark)o(o)o(v)f(c)o(hain)f(has)h(a)g(unique)g(stationary)f(measure)h Fu(\026)1146 1303 y Fn(")1178 1297 y Fv(and)1245 1282 y Fr(1)722 1386 y Fu(P)749 1392 y Fn("t)779 1386 y Fv(\()p Fu(u;)7 b Ft(\001)p Fv(\))j Fu(*)h(\026)954 1392 y Fn(")1014 1386 y Fv(as)41 b Fu(t)12 b Ft(!)f(1)p Fu(:)378 b Fv(\(3.8\))257 1474 y(In)15 b(particular,)f(due)h(to)f(this)h(con)o(v)o(ergence)h (with)f Fu(u)d Fv(=)h(0)h(and)h(\(3.3\))o(,)f(for)g(an)o(y)h Fu(M)i(>)c Fv(0)h(w)o(e)257 1524 y(ha)o(v)o(e)353 1491 y Fq(R)373 1539 y Fn(H)404 1490 y Fq(\000)423 1524 y Ft(j)p Fu(u)p Ft(j)471 1509 y Fr(2)498 1524 y Ft(_)9 b Fu(M)580 1490 y Fq(\001)605 1524 y Fu(\026)630 1530 y Fn(")648 1524 y Fv(\()p Fu(du)p Fv(\))i Ft(\024)786 1508 y Fr(1)p 786 1515 17 2 v 786 1538 a(2)808 1524 y Fu(B)839 1530 y Fr(0)858 1524 y Fv(.)18 b(Applying)13 b(the)h(Levi)g(theorem,)f(w)o(e)h(\014nd)g(that)622 1582 y Fq(Z)645 1676 y Fn(H)684 1638 y Ft(j)p Fu(u)p Ft(j)732 1621 y Fr(2)756 1638 y Fu(\026)781 1644 y Fn(")799 1638 y Fv(\()p Fu(du)p Fv(\))d Ft(\024)937 1622 y Fr(1)p 937 1629 V 937 1652 a(2)959 1638 y Fu(B)990 1644 y Fr(0)1050 1638 y Fv(for)41 b(0)12 b Fu(<)f(")h Ft(\024)g Fv(1)p Fu(:)279 b Fv(\(3.9\))257 1748 y(If)14 b Fu(B)330 1754 y Fr(1)361 1748 y Fu(<)d Ft(1)p Fv(,)i(then)i(due)f(to)g(similar)d (argumen)o(ts)i(w)o(e)i(ha)o(v)o(e)613 1798 y Fq(Z)636 1893 y Fn(H)675 1855 y Ft(k)p Fu(u)p Ft(k)741 1838 y Fr(2)765 1855 y Fu(\026)790 1861 y Fn(")808 1855 y Fv(\()p Fu(du)p Fv(\))c Ft(\024)946 1839 y Fr(1)p 946 1846 V 946 1869 a(2)968 1855 y Fu(B)999 1861 y Fr(1)1059 1855 y Fv(for)42 b(0)11 b Fu(<)h(")g Ft(\024)f Fv(1)p Fu(:)249 b Fv(\(3.10\))257 1967 y(By)18 b(\(3.10\))o(,)g(the)g(family)c Ft(f)p Fu(\026)713 1973 y Fn(")730 1967 y Fu(;)7 b(")17 b Ft(2)f Fv(\(0)p Fu(;)7 b Fv(1])p Ft(g)16 b Fv(regarded)i(as)f (measures)h(on)e Fu(H)k Fv(is)d(tigh)o(t.)27 b(In)257 2017 y(fact,)14 b(it)f(con)o(v)o(erges)i(to)f(a)g(limit)d(as)j Fu(")e Ft(!)f Fv(0:)257 2097 y Fi(Theorem)k(3.1.)21 b Fg(L)n(et)e Fu(B)662 2103 y Fr(1)700 2097 y Fu(<)h Ft(1)p Fg(,)f Fu(b)844 2103 y Fn(j)881 2097 y Ft(6)p Fv(=)g(0)g Fg(for)g(al)r(l)f Fu(j)k Ft(\025)e Fv(1)p Fg(,)f(and)h Fu(\026)f Fg(b)n(e)g(the)h(stationary)257 2147 y(me)n(asur)n(e)15 b(for)k Fv(\(0.2\))o Fg(.)g(Then)c Fu(\026)739 2153 y Fn(")769 2147 y Fu(*)c(\026)j Fg(as)h Fu(")d Ft(!)f Fv(0)p Fg(.)257 2263 y Fh(3.2)56 b(Pro)r(of)19 b(of)f(Theorem)f(3.1)257 2339 y Fv(Since)f(the)g(family)c(of)j(measures)g Ft(f)p Fu(\026)842 2345 y Fn(")860 2339 y Ft(g)g Fv(is)g(compact,)f(to)h(pro)o (v)o(e)h(the)f(theorem)g(it)g(su\016ces)257 2389 y(to)i(v)o(erify)g (the)h(follo)o(wing)d(prop)q(ert)o(y:)25 b(if)17 b(a)g(sequence)i Ft(f)p Fu(")1171 2395 y Fn(n)1194 2389 y Ft(g)e Fv(is)g(suc)o(h)h(that) f Fu(")1486 2395 y Fn(n)1526 2389 y Ft(!)g Fv(0)g(and)p 257 2425 573 2 v 304 2452 a Fc(1)321 2464 y Fk(In)12 b([KS00)o(],)g(w)o(e)h(pro)o(v)o(e)e(the)g(con)o(v)o(ergence)e(\(3.8\)) i(under)g(some)g(unessen)o(tial)e(additional)h(restrictions)257 2503 y(whic)o(h)h(w)o(ere)h(remo)o(v)o(ed)d(in)i(our)g(subsequence)d (publications,)g(see)j(in)g([Kuk02)o(].)953 2628 y Fv(11)p eop %%Page: 12 12 12 11 bop 257 262 a Fu(\026)282 268 y Fn(")298 272 y Ff(n)332 262 y Fu(*)11 b(\025)h Ft(2)f(P)s Fv(\()p Fu(H)s Fv(\),)h(then)h Fu(\025)e Fv(=)h Fu(\026)p Fv(.)17 b(Belo)o(w)12 b(w)o(e)g(\014x)g(a)f(sequence)j Ft(f)p Fu(")1291 268 y Fn(n)1314 262 y Ft(g)d Fv(as)h(ab)q(o)o(v)o(e)f(and)h(study)257 311 y(the)j(limiting)10 b(measure)k Fu(\025)p Fv(.)320 361 y(Let)j(us)f(consider)i(the)f(pro)q(cesses)i Fu(\020)891 367 y Fn(")925 361 y Fv(and)1012 350 y(~)1009 361 y Fu(\020)1027 367 y Fn(")1061 361 y Fv(\(see)f(\(3.5\))o(\))f(and)f(c)o(ho)q(ose)h (an)o(y)f Fu(T)22 b(>)16 b Fv(0.)257 411 y(Due)j(to)g(Sk)o(orokho)q (d's)e(represen)o(tation)k(theorem)d(\(see)i([Bil99)n(]\),)f(there)h (exist)f(random)257 465 y(pro)q(cesses)j Ft(f)467 454 y Fv(^)463 465 y Fu(\020)481 471 y Fn(")497 475 y Ff(n)519 465 y Fv(\()p Fu(t)p Fv(\))p Fu(;)7 b Fv(0)21 b Ft(\024)g Fu(t)f Ft(\024)h Fu(T)6 b Ft(g)19 b Fv(and)h Ft(f)950 454 y Fv(^)946 465 y Fu(\020)s Fv(\()p Fu(t)p Fv(\))p Fu(;)7 b Fv(0)20 b Ft(\024)h Fu(t)f Ft(\024)h Fu(T)6 b Ft(g)20 b Fv(de\014ned)g(on)f(a)g(common)257 520 y(probabilit)o(y)13 b(space)i(\(\012)p Fu(;)7 b Ft(F)t Fu(;)g Fs(P)-7 b Fv(\))11 b(suc)o(h)k(that)e Ft(D)q Fv(\()989 509 y(^)985 520 y Fu(\020)1003 526 y Fn(")1019 530 y Ff(n)1042 520 y Fv(\))f(=)g Ft(D)q Fv(\()1167 509 y(~)1163 520 y Fu(\020)1181 526 y Fn(")1197 530 y Ff(n)1219 520 y Fv(\),)i Ft(D)q Fv(\()1314 509 y(^)1310 520 y Fu(\020)s Fv(\))e(=)g Ft(D)q Fv(\()p Fu(\020)s Fv(\),)i(and)614 601 y(^)610 612 y Fu(\020)628 618 y Fn(")644 622 y Ff(n)667 612 y Fv(\()p Ft(\001)p Fv(\))21 b Ft(\000)-7 b(!)722 637 y Fn(n)p Fm(!1)824 601 y Fv(^)820 612 y Fu(\020)s Fv(\()p Ft(\001)p Fv(\))42 b(in)f Fu(C)s Fv(\([0)p Fu(;)7 b(T)f Fv(];)h Fu(H)1203 595 y Fr(1)1219 612 y Fv(\))41 b(a.s.)246 b(\(3.11\))257 717 y(W)m(e)14 b(de\014ne)h(pro)q(cesses)h Fu(\020)649 702 y Fm(0)646 728 y Fn(")662 732 y Ff(n)685 717 y Fv(\()p Fu(t)p Fv(\))11 b Ft(2)h Fu(H)821 702 y Fr(1)853 717 y Fv(using)i(the)g(follo)o(wing)d(relation:)603 814 y Fu(\020)624 797 y Fm(0)621 824 y Fn(")637 828 y Ff(n)659 814 y Fv(\()p Fu(t)p Fv(\))h(=)766 803 y(^)762 814 y Fu(\020)780 820 y Fn(")796 824 y Ff(n)818 814 y Fv(\()p Fu(t)p Fv(\))42 b(if)f Fu(k)q(")1015 820 y Fn(n)1049 814 y Ft(\024)12 b Fu(t)g(<)f Fv(\()p Fu(k)g Fv(+)e(1\))p Fu(")1309 820 y Fn(n)1332 814 y Fu(:)239 b Fv(\(3.12\))257 902 y(Clearly)m(,)13 b Ft(D)q Fv(\()p Fu(\020)480 887 y Fm(0)477 913 y Fn(")493 917 y Ff(n)515 902 y Fv(\))f(=)g Ft(D)q Fv(\()p Fu(\020)654 908 y Fn(")670 912 y Ff(n)692 902 y Fv(\).)320 961 y(Setting)i Fu(d)485 967 y Fn(n)518 961 y Fv(=)e(max)639 967 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fn(T)753 961 y Ft(k)778 950 y Fv(^)774 961 y Fu(\020)792 967 y Fn(")808 971 y Ff(n)831 961 y Fv(\()p Fu(t)p Fv(\))d Ft(\000)932 950 y Fv(^)929 961 y Fu(\020)s Fv(\()p Fu(t)p Fv(\))p Ft(k)p Fv(,)k(w)o(e)h(ha)o(v)o(e)732 1049 y Fu(d)754 1055 y Fn(n)788 1049 y Ft(!)d Fv(0)41 b(as)h Fu(n)11 b Ft(!)g(1)41 b Fv(a.s.)257 1137 y(Since)12 b Fu(B)394 1143 y Fr(1)425 1137 y Fu(<)g Ft(1)p Fv(,)f(then)630 1126 y(^)626 1137 y Fu(\020)k Fv(is)c(a)h(Wiener)g(pro)q(cess)h(in)e Fu(H)1099 1122 y Fr(1)1118 1137 y Fv(.)17 b(F)m(or)11 b(an)o(y)g Fu(K)k(>)d Fv(0,)f(let)h(us)g(consider)257 1187 y(the)j(ev)o(en)o(t)322 1275 y(\012)352 1281 y Fn(K)395 1275 y Fv(=)439 1241 y Fq(\010)463 1275 y Fu(!)e Ft(2)e Fv(\012)h(:)f Ft(k)631 1264 y Fv(^)627 1275 y Fu(\020)s Fv(\()p Fu(t)679 1281 y Fr(1)697 1275 y Fv(\))f Ft(\000)768 1264 y Fv(^)764 1275 y Fu(\020)t Fv(\()p Fu(t)817 1281 y Fr(2)835 1275 y Fv(\))p Ft(k)i(\024)g Fu(K)s Ft(j)p Fu(t)993 1281 y Fr(1)1020 1275 y Ft(\000)e Fu(t)1077 1281 y Fr(2)1095 1275 y Ft(j)1107 1258 y Fr(1)p Fn(=)p Fr(3)1173 1275 y Fv(for)j(0)f Ft(\024)f Fu(t)1327 1281 y Fr(1)1346 1275 y Fu(;)c(t)1380 1281 y Fr(2)1410 1275 y Ft(\024)k Fu(T)1483 1241 y Fq(\011)1508 1275 y Fu(:)63 b Fv(\(3.13\))257 1363 y(Then)15 b Fs(P)-6 b Fv(\(\012)440 1369 y Fn(K)469 1363 y Fv(\))12 b Ft(!)f Fv(1)j(as)g Fu(K)g Ft(!)d(1)p Fv(.)18 b(Due)c(to)g(\(3.12\))o(,)f(w)o(e)h(ha)o(v)o (e)539 1455 y Ft(k)p Fu(\020)581 1438 y Fm(0)578 1466 y Fn(")594 1470 y Ff(n)616 1455 y Fv(\()p Fu(t)p Fv(\))c Ft(\000)718 1445 y Fv(^)714 1455 y Fu(\020)t Fv(\()p Fu(t)p Fv(\))p Ft(k)h(\024)h Fu(d)881 1461 y Fn(n)903 1455 y Fv(\()p Fu(!)q Fv(\))e(+)g Fu(K)s(")1071 1438 y Fr(1)p Fn(=)p Fr(3)1071 1466 y Fn(n)1165 1455 y Fv(for)41 b Fu(!)13 b Ft(2)e Fv(\012)1364 1461 y Fn(K)1396 1455 y Fu(:)175 b Fv(\(3.14\))257 1544 y(F)m(or)14 b(an)o(y)f Fu(")f Ft(2)f(f)p Fu(")520 1550 y Fr(1)539 1544 y Fu(;)c(")577 1550 y Fr(2)596 1544 y Fu(;)g(:)g(:)g(:)t Ft(g)p Fv(,)13 b(let)h(us)h(denote)738 1632 y Fu(\022)757 1638 y Fn(")787 1632 y Fv(=)d(max)n Ft(f)p Fu(t)f Ft(2)g Fu(")p Fs(Z)l Fv(:)g Fu(t)h Ft(\024)f Fu(T)6 b Ft(g)p Fu(:)374 b Fv(\(3.15\))257 1720 y(Then)15 b Fd(P)400 1705 y Fm(\003)400 1732 y Fn("\022)432 1736 y Ff(")450 1720 y Fu(\026)475 1726 y Fn(")504 1720 y Fv(=)d Fu(\026)573 1726 y Fn(")591 1720 y Fv(.)18 b(W)m(e)13 b(rewrite)i(this)f(relation)f(as)h(follo)o(ws:)582 1813 y Fu(\026)607 1819 y Fn(")637 1813 y Fv(=)680 1780 y Fq(\000)699 1813 y Fd(P)733 1796 y Fm(\003)733 1824 y Fn("\022)765 1828 y Ff(")793 1813 y Ft(\000)9 b Fd(P)868 1796 y Fm(\003)868 1824 y Fn(T)895 1780 y Fq(\001)914 1813 y Fu(\026)939 1819 y Fn(")966 1813 y Fv(+)g Fd(P)1041 1796 y Fm(\003)1041 1824 y Fn(T)1068 1813 y Fv(\()p Fu(\026)1109 1819 y Fn(")1136 1813 y Ft(\000)g Fu(\025)p Fv(\))h(+)f Fd(P)1302 1796 y Fm(\003)1302 1824 y Fn(T)1329 1813 y Fu(\025:)218 b Fv(\(3.16\))257 1902 y(W)m(e)14 b(claim)e(that)h (\(3.16\))g(implies)f(the)j(equalit)o(y)886 1990 y Fd(P)920 1973 y Fm(\003)920 2000 y Fn(T)946 1990 y Fu(\025)d Fv(=)g Fu(\025:)521 b Fv(\(3.17\))257 2078 y(If)16 b(\(3.17\))11 b(is)h(pro)o(v)o(ed,)g(then)h Fu(\025)f Fv(=)g Fu(\026)p Fv(.)17 b(Indeed,)d(let)e Fu(u)p Fv(\()p Fu(t)p Fv(\))g(b)q(e)h(a)f (solution)g(for)g(\(0.2\))f(suc)o(h)i(that)257 2128 y Ft(D)q Fv(\()p Fu(u)p Fv(\(0\)\))g(=)g Fu(\025)p Fv(.)21 b(Then)15 b Ft(D)q Fv(\()p Fu(u)p Fv(\()p Fu(k)q(T)6 b Fv(\)\))13 b(=)g Fu(\025)h Fv(for)h Fu(k)e Ft(2)g Fs(Z)1064 2134 y Fr(+)1089 2128 y Fv(.)20 b(Since)15 b Fu(\026)1255 2134 y Fn(")1271 2138 y Ff(n)1306 2128 y Fu(*)d(\025)p Fv(,)i(inequalit)o(y)g(\(3.9\))257 2177 y(implies)h(that)i(the)g (measure)g Fu(\025)g Fv(has)g(a)f(\014nite)h(second)h(momen)o(t.)24 b(So)16 b(\(2.1\))g(applies)h(and)257 2227 y Ft(D)q Fv(\()p Fu(u)p Fv(\()p Fu(k)q(T)6 b Fv(\)\))12 b Fu(*)f(\026)p Fv(.)18 b(Hence,)d Fu(\026)d Fv(=)g Fu(\025)p Fv(.)320 2277 y(T)m(o)h(pro)o(v)o(e)h(\(3.17\))o(,)f(it)h(su\016ces)h(to)f(c)o (hec)o(k)g(that)645 2365 y(\()p Fd(P)695 2348 y Fm(\003)695 2375 y Fn(T)722 2365 y Fu(\025;)7 b(f)t Fv(\))12 b(=)g(\()p Fu(\025;)7 b(f)t Fv(\))42 b(for)14 b(an)o(y)41 b Fu(f)16 b Ft(2)11 b(X)6 b Fu(;)288 b Fv(\(3.18\))257 2453 y(where)16 b Ft(X)h(\032)12 b Fu(C)499 2459 y Fn(b)516 2453 y Fv(\()p Fu(H)s Fv(\))i(is)g(formed)f(b)o(y)h(all)f(functions)h Fu(f)19 b Fv(whic)o(h)14 b(extend)h(to)f(Lipsc)o(hitz)g(func-)257 2503 y(tions)e Fu(f)e Fv(:)h Fu(H)448 2488 y Fm(\000)p Fr(3)504 2503 y Ft(!)g Fs(R)6 b Fv(satisfying)11 b(the)i(conditions)f Ft(j)p Fu(f)t Ft(j)f(\024)h Fv(1)g(and)g(Lip)o(\()p Fu(f)t Fv(\))h Ft(\024)f Fv(1.)17 b(Indeed,)c(the)953 2628 y(12)p eop %%Page: 13 13 13 12 bop 257 262 a Fv(set)14 b Ft(X)k Fv(con)o(tains)12 b(all)g(b)q(ounded)h(Lipsc)o(hitz)g(cylindrical)1120 246 y Fr(2)1150 262 y Fv(functions)f(on)g Fu(H)k Fv(\(m)o(ultiplied)10 b(b)o(y)257 311 y(a)i(suitable)g(constan)o(t\),)h(and)f(therefore)h (\(3.18\))e(implies)g(that)h(all)f(cylindrical)g(pro)r(jections)257 361 y(of)j(the)g(t)o(w)o(o)g(measures)g(coincide.)k(So)c(\(3.17\))f (follo)o(ws.)320 411 y(T)m(o)h(pro)o(v)o(e)h(\(3.18\))o(,)g(w)o(e)g(in) o(tegrate)g Fu(f)20 b Fv(against)14 b(the)i(signed)f(measure)g(in)f (the)i(left-)f(and)257 461 y(righ)o(t-hand)f(sides)g(of)j(\(3.16\))c (and)g(next)i(send)g Fu(")c Fv(=)h Fu(")1094 467 y Fn(n)1131 461 y Fv(to)i(zero.)19 b(Then:)297 544 y(\(i\))h(The)14 b(term)g(\()p Fu(\026)587 550 y Fn(")603 554 y Ff(n)625 544 y Fu(;)7 b(f)t Fv(\))14 b(con)o(v)o(erges)h(to)f(\()p Fu(\025;)7 b(f)t Fv(\);)285 627 y(\(ii\))20 b(since)13 b(the)h(map)c Fd(P)656 612 y Fm(\003)656 638 y Fn(T)694 627 y Fv(:)i Ft(P)s Fv(\()p Fu(H)s Fv(\))f Ft(!)h(P)s Fv(\()p Fu(H)s Fv(\))g(is)h(con)o(tin)o(uous)f(with)g(resp)q(ect)j(to)d (the)i(w)o(eak)361 677 y(con)o(v)o(ergence,)h(w)o(e)f(ha)o(v)o(e)g(\()p Fd(P)809 662 y Fm(\003)809 688 y Fn(T)836 677 y Fv(\()p Fu(\026)877 683 y Fn(")893 687 y Ff(n)924 677 y Ft(\000)c Fu(\025)p Fv(\))p Fu(;)d(f)t Fv(\))12 b Ft(!)f Fv(0.)257 760 y(Hence,)k(to)f(pro)o(v)o(e)g(\(3.18\))o(,)f(it)h(remains)f(to)h(c) o(hec)o(k)g(that)274 843 y(\(iii\))19 b(\()p Fd(P)411 828 y Fm(\003)411 854 y Fn("\022)443 858 y Ff(")461 843 y Fu(\026)486 849 y Fn(")504 843 y Fu(;)7 b(f)t Fv(\))12 b Ft(!)f Fv(0)j(as)g Fu(")d Fv(=)h Fu(")807 849 y Fn(n)842 843 y Ft(!)f Fv(0.)320 926 y(Let)j(us)g(consider)h(Eq.)f(\(0.2\))f (with)g(a)h(deterministic)f(righ)o(t-hand)h(side:)364 1017 y(_)-19 b Fu(u)8 b Fv(+)i Fu(\027)s(Lu)e Fv(+)i Fu(B)r Fv(\()p Fu(u;)d(u)p Fv(\))k(=)h Fu(@)766 1023 y Fn(t)781 1017 y Fu(\020)s Fv(\()p Fu(t;)7 b(x)p Fv(\))p Fu(;)47 b(\020)15 b Ft(2)c Fu(C)s Fv(\([0)p Fu(;)c(T)f Fv(];)h Fu(H)1223 1000 y Fr(1)1239 1017 y Fv(\))p Fu(;)48 b(\020)s Fv(\(0\))12 b(=)g(0)p Fu(;)105 b Fv(\(3.19\))880 1079 y Fu(u)p Fv(\(0\))12 b(=)g Fu(u)1037 1085 y Fr(0)1055 1079 y Fu(:)516 b Fv(\(3.20\))257 1171 y(Substituting)14 b Fu(u)d Fv(=)h Fu(\020)h Fv(+)c Fu(v)16 b Fv(in)o(to)d(\(3.19\))o(,)g (w)o(e)h(get)h(for)e Fu(v)j Fv(the)e(equation)567 1262 y(_)-18 b Fu(v)11 b Fv(+)e Fu(Lv)i Fv(+)f Fu(B)r Fv(\()p Fu(v)h Fv(+)f Fu(\020)s(;)d(v)j Fv(+)g Fu(\020)s Fv(\))i(=)f Ft(\000)p Fu(L\020)s(;)49 b(v)q Fv(\(0\))12 b(=)g Fu(u)1356 1268 y Fr(0)1375 1262 y Fu(:)196 b Fv(\(3.21\))257 1353 y(Due)11 b(to)f(the)g(basic)g(prop)q(erties)i(of)d(the)i(2D)e(NS)i (system)f(\(see)h([CF88)o(]\),)f(the)h(problem)d(\(3.21\))257 1403 y(has)i(a)f(unique)h(solution)f Fu(v)k Ft(2)e Fu(C)s Fv(\([0)p Fu(;)c(T)f Fv(];)h Fu(H)s Fv(\))q Ft(\\)p Fu(L)986 1388 y Fr(2)1002 1403 y Fv(\([0)p Fu(;)g(T)f Fv(];)h Fu(H)1169 1388 y Fr(1)1186 1403 y Fv(\).)16 b(W)m(e)10 b(call)e(the)j(corresp)q(onding)257 1453 y(function)g Fu(u)g Fv(=)h Fu(v)5 b Fv(+)s Fu(\020)14 b Fv(a)d Fg(we)n(ak)g (solution)k Fv(for)10 b(\(3.19\))o(,)h(\(3.20\))f(and)g(denote)i(it)f (b)o(y)f Fu(u)h Fv(=)h Fu(U)5 b Fv(\()p Fu(\020)s Fv(\))12 b(=)257 1503 y Fu(U)5 b Fv(\()p Fu(\020)s(;)i(u)370 1509 y Fr(0)388 1503 y Fv(\).)27 b(It)17 b(is)g(kno)o(wn)f(\(see)i ([Kry95]\))e(that)h(solutions)f(for)g(\(0.2\))g(and)h(\(3.1\))f(\\can)h (b)q(e)257 1553 y(treated)e(path-wise,")f(that)g(is,)540 1644 y Fu(u)p Fv(\()p Fu(t)p Fv(;)7 b Fu(u)638 1650 y Fr(0)655 1644 y Fv(\))12 b(=)g Fu(U)5 b Fv(\()p Fu(\020)s Fv(;)i Fu(u)840 1650 y Fr(0)858 1644 y Fv(\)\()p Fu(t)p Fv(\))p Fu(;)48 b(u)1005 1650 y Fn(")1023 1644 y Fv(\()p Fu(t)p Fv(;)7 b Fu(u)1097 1650 y Fr(0)1115 1644 y Fv(\))12 b(=)f Fu(U)5 b Fv(\()p Fu(\020)1253 1650 y Fn(")1271 1644 y Fv(;)i Fu(u)1314 1650 y Fr(0)1332 1644 y Fv(\)\()p Fu(t)p Fv(\))p Fu(:)176 b Fv(\(3.22\))320 1735 y(T)m(o)14 b(v)o(erify)h(\(iii\),)f(w)o(e)i(need)g(the)g(follo)o(wing)d(lemma,)f (whic)o(h)j(is)g(pro)o(v)o(ed)h(at)f(the)h(end)g(of)257 1785 y(this)e(subsection:)257 1868 y Fi(Lemma)i(3.2.)21 b Fg(L)n(et)f Ft(j)p Fu(u)635 1874 y Fr(0)653 1868 y Ft(j)g(\024)i Fu(R)p Fg(,)f(let)e Fv(max)946 1874 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fr(1)1052 1868 y Ft(k)p Fu(\020)1091 1874 y Fn(i)1105 1868 y Fv(\()p Fu(t)p Fv(\))p Ft(k)i(\024)g Fu(N)k Fg(for)20 b Fu(i)h Fv(=)h(1)p Fu(;)7 b Fv(2)p Fg(,)20 b(and)h(let)257 1918 y Fv(max)335 1924 y Fr(0)p Fm(\024)p Fn(t)p Fm(\024)p Fr(1)441 1918 y Ft(k)p Fu(\020)480 1924 y Fr(1)499 1918 y Fv(\()p Fu(t)p Fv(\))10 b Ft(\000)g Fu(\020)616 1924 y Fr(2)635 1918 y Fv(\()p Fu(t)p Fv(\))p Ft(k)k(\024)f Fu(")h(<)g Fv(1)p Fg(.)22 b(We)16 b(set)g Fu(u)1063 1924 y Fn(i)1090 1918 y Fv(=)e Fu(U)5 b Fv(\()p Fu(\020)1203 1924 y Fn(i)1217 1918 y Fv(;)i Fu(u)1260 1924 y Fr(0)1277 1918 y Fv(\))p Fg(.)23 b(Then)16 b(for)f Fv(0)f Ft(\024)f Fu(t)h Ft(\024)f Fu(T)257 1968 y Fg(we)i(have)730 2059 y Ft(j)p Fu(u)766 2065 y Fr(1)784 2059 y Fv(\()p Fu(t)p Fv(\))p Ft(j)p Fu(;)7 b Ft(j)p Fu(u)898 2065 y Fr(2)915 2059 y Fv(\()p Fu(t)p Fv(\))p Ft(j)12 b(\024)f Fu(F)1056 2065 y Fr(1)1075 2059 y Fv(\()p Fu(R;)c(N)e Fv(\))p Fu(;)382 b Fv(\(3.23\))721 2121 y Ft(j)p Fu(u)757 2127 y Fr(1)775 2121 y Fv(\()p Fu(t)p Fv(\))9 b Ft(\000)h Fu(u)897 2127 y Fr(2)915 2121 y Fv(\()p Fu(t)p Fv(\))p Ft(j)i(\024)f Fu("F)1075 2127 y Fr(2)1094 2121 y Fv(\()p Fu(R;)c(N)e Fv(\))p Fu(;)363 b Fv(\(3.24\))257 2213 y Fg(wher)n(e)15 b Fu(F)402 2219 y Fr(1)431 2213 y Ft(\025)d Fv(1)j Fg(and)g Fu(F)618 2219 y Fr(2)648 2213 y Ft(\025)d Fv(1)j Fg(ar)n(e)f(\014xe)n(d)i(c)n(ontinuous)g(functions.)257 2296 y Fi(Corollary)f(3.3.)21 b Fg(Conver)n(genc)n(e)d Fv(\(3.7\))c Fg(holds)h(for)g(any)g Fu(u)1184 2302 y Fr(0)1218 2296 y Fg(and)g(any)h Fu(t)11 b Ft(2)g Fv([0)p Fu(;)c(T)f Fv(])p Fg(.)p 257 2333 573 2 v 304 2359 a Fc(2)321 2371 y Fk(A)16 b(function)d Fb(f)19 b Fk(on)c Fb(H)j Fk(is)d(called)f Fa(cylindric)n(al)21 b Fk(if)15 b Fb(f)t Fk(\()p Fb(u)p Fk(\))g(dep)q(ends)e(only)h(on)h(\014nitely)e (man)o(y)h(F)m(ourier)257 2411 y(co)q(e\016cien)o(ts)c(of)h Fb(u)p Fk(.)953 2628 y Fv(13)p eop %%Page: 14 14 14 13 bop 257 262 a Fg(Pr)n(o)n(of.)20 b Fv(It)e(is)f(su\016cien)o(t)h (to)f(v)o(erify)h(\(3.7\))e(for)h(an)o(y)g(sequence)j Fu(")1279 268 y Fn(n)1319 262 y Ft(!)d Fv(0.)28 b(Due)17 b(to)g(\(3.14\))257 311 y(and)d(\(3.24\))f(with)h Fu(N)i Fv(=)c Fu(K)s(T)714 296 y Fr(1)p Fn(=)p Fr(3)766 311 y Fv(,)318 365 y Fq(\014)318 389 y(\014)331 400 y Fu(U)5 b Fv(\()p Fu(\020)401 383 y Fm(0)398 410 y Fn(")414 414 y Ff(n)437 400 y Fv(\)\()p Fu(t)p Fv(\))k Ft(\000)h Fu(U)5 b Fv(\()p Fu(\020)s Fv(\)\()p Fu(t)p Fv(\))684 365 y Fq(\014)684 389 y(\014)710 400 y Ft(\024)754 366 y Fq(\000)773 400 y Fu(d)795 406 y Fn(n)817 400 y Fv(\()p Fu(!)q Fv(\))10 b(+)f Fu(K)s(")984 383 y Fr(1)p Fn(=)p Fr(3)984 410 y Fn(n)1037 366 y Fq(\001)1063 400 y Fu(F)1090 406 y Fr(2)1108 400 y Fv(\()p Ft(j)p Fu(u)1160 406 y Fr(0)1178 400 y Ft(j)p Fu(;)e(K)s(T)1277 383 y Fr(1)p Fn(=)p Fr(3)1329 400 y Fv(\))41 b(for)h Fu(!)13 b Ft(2)e Fv(\012)1586 406 y Fn(K)1618 400 y Fu(;)257 497 y Fv(where)16 b Fu(d)400 503 y Fn(n)434 497 y Ft(!)c Fv(0)i(a.s.)k(Hence,)e Fu(U)5 b Fv(\()p Fu(\020)808 482 y Fm(0)805 507 y Fn(")821 511 y Ff(n)843 497 y Fv(\)\()p Fu(t)p Fv(\))13 b Ft(!)e Fu(U)5 b Fv(\()1025 486 y(^)1021 497 y Fu(\020)t Fv(\)\()p Fu(t)p Fv(\))14 b(a.s.)g(on)g(the)g(set)i(\012)1420 503 y Fn(K)1466 497 y Fv(for)e(ev)o(ery)h Fu(K)s Fv(.)257 547 y(Since)h Fs(P)-5 b Fv(\(\012)442 553 y Fn(K)471 547 y Fv(\))14 b Ft(!)f Fv(1)i(as)g Fu(K)j Ft(!)13 b(1)p Fv(,)i(w)o(e)g(ha)o(v)o(e)g Fu(U)5 b Fv(\()p Fu(\020)1050 532 y Fm(0)1047 557 y Fn(")1063 561 y Ff(n)1086 547 y Fv(\)\()p Fu(t)p Fv(\))14 b Ft(!)f Fu(U)5 b Fv(\()p Fu(\020)s Fv(\)\()p Fu(t)p Fv(\))16 b(a.s.)22 b(It)15 b(remains)g(to)257 605 y(note)f(that)e(the)i(pro)q (cesses)h Fu(\020)707 590 y Fm(0)704 615 y Fn(")720 619 y Ff(n)755 605 y Fv(and)839 594 y(^)835 605 y Fu(\020)h Fv(ha)o(v)o(e)c(the)i(same)e(distributions)g(as)h Fu(\020)1446 611 y Fn(")1462 615 y Ff(n)1497 605 y Fv(and)g Fu(\020)s Fv(,)g(and)257 655 y(therefore)j(\(3.7\))d(follo)o(ws)f(from)g (\(3.22\))o(.)p 1659 655 2 29 v 1661 629 25 2 v 1661 655 V 1686 655 2 29 v 257 737 a Fi(Corollary)j(3.4.)21 b Fg(L)n(et)d(the)g(assumptions)h(of)f(L)n(emma)g(3.2)h(b)n(e)f (satis\014e)n(d,)h(let)f Fv(0)f Ft(\024)h Fu(t)1621 743 y Fr(1)1657 737 y Ft(\024)257 787 y Fu(t)272 793 y Fr(2)303 787 y Ft(\024)11 b Fu(t)361 793 y Fr(1)389 787 y Fv(+)f Fu(\016)j Ft(\024)f Fu(T)6 b Fg(,)15 b(and)g(let)g Ft(k)p Fu(\020)742 793 y Fr(1)760 787 y Fv(\()p Fu(t)791 793 y Fr(1)810 787 y Fv(\))9 b Ft(\000)h Fu(\020)895 793 y Fr(1)914 787 y Fv(\()p Fu(t)945 793 y Fr(2)963 787 y Fv(\))p Ft(k)i Fv(=)g Fu(C)1086 793 y Fn(\016)1103 787 y Fg(.)19 b(Then)600 840 y Fq(\015)600 865 y(\015)623 876 y Fu(u)647 882 y Fr(1)665 876 y Fv(\()p Fu(t)696 882 y Fr(1)715 876 y Fv(\))9 b Ft(\000)h Fu(u)806 882 y Fr(1)824 876 y Fv(\()p Fu(t)855 882 y Fr(2)874 876 y Fv(\))890 840 y Fq(\015)890 865 y(\015)913 892 y Fm(\000)p Fr(3)969 876 y Ft(\024)i Fu(C)1043 882 y Fn(\016)1070 876 y Fv(+)e Fu(\016)r(C)s(F)1198 859 y Fr(2)1192 886 y(1)1215 876 y Fv(\()p Fu(R;)d(N)e Fv(\))p Fu(:)235 b Fv(\(3.25\))257 970 y Fg(Pr)n(o)n(of.)20 b Fv(Since)15 b Fu(u)520 976 y Fr(1)552 970 y Fv(is)f(a)f(w)o(eak)h(solution)f(for)h (\(3.19\))o(,)f(\(3.20\))g(with)h Fu(\020)g Fv(=)e Fu(\020)1393 976 y Fr(1)1412 970 y Fv(,)h(w)o(e)i(ha)o(v)o(e)332 1087 y Fu(u)356 1093 y Fr(1)374 1087 y Fv(\()p Fu(t)405 1093 y Fr(2)424 1087 y Fv(\))9 b Ft(\000)h Fu(u)515 1093 y Fr(1)533 1087 y Fv(\()p Fu(t)564 1093 y Fr(1)583 1087 y Fv(\))i(=)655 1053 y Fq(\000)674 1087 y Fu(\020)692 1093 y Fr(1)710 1087 y Fv(\()p Fu(t)741 1093 y Fr(2)760 1087 y Fv(\))d Ft(\000)h Fu(\020)845 1093 y Fr(1)864 1087 y Fv(\()p Fu(t)895 1093 y Fr(1)914 1087 y Fv(\))930 1053 y Fq(\001)958 1087 y Ft(\000)999 1031 y Fq(Z)1041 1041 y Fn(t)1054 1045 y Fo(2)1022 1125 y Fn(t)1035 1129 y Fo(1)1072 1053 y Fq(\000)1091 1087 y Fu(Lu)1143 1093 y Fr(1)1161 1087 y Fv(\()p Fu(s)p Fv(\))g(+)g Fu(B)r Fv(\()p Fu(u)1337 1093 y Fr(1)1356 1087 y Fv(\()p Fu(s)p Fv(\))p Fu(;)d(u)1450 1093 y Fr(1)1469 1087 y Fv(\()p Fu(s)p Fv(\)\))1536 1053 y Fq(\001)1562 1087 y Fu(ds:)257 1206 y Fv(It)16 b(follo)o(ws)e(from)g(\(3.23\))h(that)h Ft(k)p Fu(Lu)829 1212 y Fr(1)847 1206 y Ft(k)868 1212 y Fm(\000)p Fr(2)927 1206 y Ft(\024)f Fu(F)1001 1212 y Fr(1)1035 1206 y Fv(and)g Ft(k)p Fu(B)r Fv(\()p Fu(u)1211 1212 y Fr(1)1230 1206 y Fu(;)7 b(u)1273 1212 y Fr(1)1291 1206 y Fv(\))p Ft(k)1328 1212 y Fm(\000)p Fr(3)1387 1206 y Ft(\024)15 b Fu(C)1464 1212 y Fr(1)1482 1206 y Fu(F)1515 1191 y Fr(2)1509 1216 y(1)1533 1206 y Fv(.)23 b(Hence,)257 1256 y(the)15 b Fu(H)367 1240 y Fm(\000)p Fr(3)411 1256 y Fv(-norm)e(of)g(the)h(in)o(tegral)g(is)f(b)q(ounded)i(b)o(y)e Fu(\016)r(C)s(F)1158 1240 y Fr(2)1152 1266 y(1)1176 1256 y Fv(,)h(and)f(\(3.25\))g(follo)o(ws.)p 1659 1256 V 1661 1229 25 2 v 1661 1256 V 1686 1256 2 29 v 320 1338 a(The)g(transition)f (function)g Fu(P)780 1344 y Fn("t)810 1338 y Fv(\()p Fu(v)q(;)7 b Fv(\000\))13 b(w)o(as)f(de\014ned)i(for)e Fu(t)g Ft(2)f Fu(")p Fs(Z)1319 1344 y Fr(+)1343 1338 y Fv(.)18 b(W)m(e)12 b(no)o(w)g(extend)i(it)257 1388 y(to)g Fu(t)e Ft(2)f Fs(R)407 1394 y Fr(+)445 1388 y Fv(b)o(y)j(the)g(relation)723 1476 y Fu(P)750 1482 y Fn("t)780 1476 y Fv(\()p Fu(v)q(;)7 b Fv(\000\))k(=)h Fs(P)962 1443 y Fq(\010)983 1476 y Fu(u)1007 1482 y Fn(")1024 1476 y Fv(\()p Fu(t)p Fv(;)7 b Fu(v)q Fv(\))12 b Ft(2)f Fv(\000)1188 1443 y Fq(\011)1213 1476 y Fu(:)257 1565 y Fv(Then)k(for)e(an)o(y)h Fu(f)i Ft(2)11 b(X)20 b Fv(w)o(e)14 b(ha)o(v)o(e)413 1639 y Fq(\000)432 1672 y Fv(\()p Fd(P)482 1655 y Fm(\003)482 1683 y Fn("\022)514 1687 y Ff(")541 1672 y Ft(\000)c Fd(P)617 1655 y Fm(\003)617 1683 y Fn("T)659 1672 y Fv(\))p Fu(\026)700 1678 y Fn(")718 1672 y Fu(;)d(f)761 1639 y Fq(\001)792 1672 y Fv(=)836 1616 y Fq(Z)859 1710 y Fn(H)897 1672 y Fs(E)929 1639 y Fq(\000)951 1672 y Fu(f)t Fv(\()p Fu(u)1015 1678 y Fn(")1033 1672 y Fv(\()p Fu(\022)1068 1678 y Fn(")1086 1672 y Fv(;)g Fu(v)q Fv(\))j Ft(\000)f Fu(f)t Fv(\()p Fu(u)1257 1678 y Fn(")1276 1672 y Fv(\(1;)e Fu(v)q Fv(\)\))1385 1639 y Fq(\001)1404 1672 y Fu(\026)1429 1678 y Fn(")1447 1672 y Fv(\()p Fu(dv)q Fv(\))p Fu(:)257 1784 y Fv(T)m(o)j(estimate)g(the)i(righ)o(t-hand)e (side)h(of)f(this)h(relation,)f(w)o(e)h(replace)g Fu(u)1330 1790 y Fn(")1348 1784 y Fv(\()p Fu(\022)1383 1790 y Fn(")1401 1784 y Fv(;)c Fu(v)q Fv(\))k(and)g Fu(u)1570 1790 y Fn(")1587 1784 y Fv(\()p Fu(T)6 b Fv(;)h Fu(v)q Fv(\))257 1834 y(b)o(y)15 b Fu(U)5 b Fv(\()p Fu(\020)383 1840 y Fn(")401 1834 y Fv(;)i Fu(v)q Fv(\)\()p Fu(\022)492 1840 y Fn(")511 1834 y Fv(\))13 b(=:)h Fu(u)622 1819 y Fm(0)622 1844 y Fn(")639 1834 y Fv(\()p Fu(\022)674 1840 y Fn(")693 1834 y Fv(\))h(and)g Fu(U)5 b Fv(\()p Fu(\020)873 1840 y Fn(")891 1834 y Fv(;)i Fu(v)q Fv(\)\()p Fu(T)f Fv(\))14 b(=:)f Fu(u)1104 1819 y Fm(0)1104 1844 y Fn(")1121 1834 y Fv(\()p Fu(T)6 b Fv(\),)16 b(resp)q(ectiv)o(ely)m(.)22 b(Since)16 b Fu(f)i Ft(2)13 b(X)6 b Fv(,)257 1884 y(w)o(e)14 b(conclude)h(that)f(the)h(righ)o(t-hand)e(side)h(is)g(b)q(ounded)g(b)o (y)399 1991 y(2)p Fu(\026)445 1997 y Fn(")462 1957 y Fq(\010)487 1991 y Ft(j)p Fu(v)q Ft(j)d Fu(>)h(R)619 1957 y Fq(\011)652 1991 y Fv(+)693 1935 y Fq(Z)716 2029 y Fm(j)p Fn(v)q Fm(j\024)p Fn(R)814 1991 y Fs(E)846 1957 y Fq(\000)868 1991 y Ft(k)p Fu(u)913 1974 y Fm(0)913 2001 y Fn(")930 1991 y Fv(\()p Fu(\022)965 1997 y Fn(")983 1991 y Fv(\))e Ft(\000)f Fu(u)1074 1974 y Fm(0)1074 2001 y Fn(")1092 1991 y Fv(\()p Fu(T)d Fv(\))p Ft(k)1175 1997 y Fm(\000)p Fr(3)1228 1991 y Ft(^)j Fv(2)1286 1957 y Fq(\001)1312 1991 y Fu(\026)1337 1997 y Fn(")1355 1991 y Fv(\()p Fu(dv)q Fv(\))p Fu(:)141 b Fv(\(3.26\))257 2113 y(By)14 b(\(3.9\),)f(the)h(\014rst)g(term)e(is)h(ma)r(jorised)f(b) o(y)h Fu(R)1007 2098 y Fm(\000)p Fr(2)1052 2113 y Fu(B)1083 2119 y Fr(0)1102 2113 y Fv(.)18 b(T)m(o)12 b(estimate)h(the)h(in)o (tegral,)e(let)h(us)257 2163 y(note)i(that)f(for)f Fu(")f Fv(=)g Fu(")597 2169 y Fn(n)634 2163 y Fv(and)h Fu(!)g Ft(2)e Fv(\012)822 2169 y Fn(K)868 2163 y Fv(\(see)k(\(3.13\)\),)e(due) h(to)g(\(3.14\))o(,)g(w)o(e)g(ha)o(v)o(e)333 2256 y Ft(k)p Fu(\020)375 2238 y Fm(0)372 2266 y Fn(")389 2256 y Fv(\()p Fu(\022)424 2262 y Fn(")443 2256 y Fv(\))9 b Ft(\000)h Fu(\020)531 2238 y Fm(0)528 2266 y Fn(")545 2256 y Fv(\()p Fu(T)c Fv(\))p Ft(k)12 b(\024)g(k)p Fu(\020)726 2238 y Fm(0)723 2266 y Fn(")741 2256 y Fv(\()p Fu(\022)776 2262 y Fn(")794 2256 y Fv(\))d Ft(\000)864 2245 y Fv(^)861 2256 y Fu(\020)s Fv(\()p Fu(\022)917 2262 y Fn(")935 2256 y Fv(\))p Ft(k)h Fv(+)f Ft(k)1048 2245 y Fv(^)1044 2256 y Fu(\020)s Fv(\()p Fu(\022)1100 2262 y Fn(")1118 2256 y Fv(\))h Ft(\000)1189 2245 y Fv(^)1185 2256 y Fu(\020)s Fv(\()p Fu(T)c Fv(\))p Ft(k)k Fv(+)f Ft(k)1365 2245 y Fv(^)1361 2256 y Fu(\020)s Fv(\()p Fu(T)d Fv(\))k Ft(\000)f Fu(\020)1516 2238 y Fm(0)1513 2266 y Fn(")1531 2256 y Fv(\()p Fu(T)d Fv(\))p Ft(k)640 2326 y(\024)12 b Fv(2)p Fu(d)727 2332 y Fn(n)749 2326 y Fv(\()p Fu(!)q Fv(\))e(+)f(3)p Fu(K)s(")937 2309 y Fr(1)p Fn(=)p Fr(3)990 2326 y Fu(:)257 2415 y Fv(Applying)k(Corollary)g(3.4,)f(w)o(e)i(\014nd)g(that)329 2503 y Ft(k)p Fu(u)374 2486 y Fm(0)374 2513 y Fn(")391 2503 y Fv(\()p Fu(\022)426 2509 y Fn(")445 2503 y Fv(\))9 b Ft(\000)h Fu(u)536 2486 y Fm(0)536 2513 y Fn(")553 2503 y Fv(\()p Fu(T)c Fv(\))p Ft(k)12 b(\024)692 2469 y Fq(\000)711 2503 y Fv(2)p Fu(d)754 2509 y Fn(n)776 2503 y Fv(\()p Fu(!)q Fv(\))d(+)h(3)p Fu(K)s(")964 2486 y Fr(1)p Fn(=)p Fr(3)1016 2469 y Fq(\001)1045 2503 y Fv(+)f Fu(C)h(")1145 2509 y Fn(n)1168 2503 y Fu(F)1195 2509 y Fr(1)1213 2503 y Fv(\()p Fu(R;)d(N)e Fv(\))41 b(for)g Fu(!)13 b Ft(2)e Fv(\012)1574 2509 y Fn(K)1606 2503 y Fu(;)953 2628 y Fv(14)p eop %%Page: 15 15 15 14 bop 257 262 a Fv(where)15 b Fu(N)i Fv(=)12 b Fu(K)s(T)539 246 y Fr(1)p Fn(=)p Fr(3)591 262 y Fv(.)18 b(Therefore,)359 353 y(\(3.26\))10 b Ft(\024)i Fu(R)552 336 y Fm(\000)p Fr(2)596 353 y Fu(B)627 359 y Fr(0)656 353 y Fv(+)d(2)p Fs(E)g Fu(d)772 359 y Fn(n)806 353 y Fv(+)h(3)p Fu(K)s(")926 336 y Fr(1)p Fn(=)p Fr(3)926 363 y Fn(n)987 353 y Fv(+)g Fu(C)f(")1087 359 y Fn(n)1110 353 y Fu(F)1137 359 y Fr(1)1156 353 y Fv(\()p Fu(R;)e(N)e Fv(\))j(+)i(2)1348 319 y Fq(\000)1367 353 y Fv(1)f Ft(\000)g Fs(P)-6 b Fv(\(\012)1512 359 y Fn(K)1542 353 y Fv(\))1558 319 y Fq(\001)1577 353 y Fu(:)257 444 y Fv(Sending)14 b(to)g Ft(1)f Fv(\014rst)i Fu(R)f Fv(and)f Fu(K)k Fv(and)d(then)h Fu(n)p Fv(,)e(w)o(e)h(get)580 502 y Fq(\000)599 535 y Fv(\()p Fd(P)649 518 y Fm(\003)649 546 y Fn("\022)681 550 y Ff(")709 535 y Ft(\000)9 b Fd(P)784 518 y Fm(\003)784 546 y Fn("T)826 535 y Fv(\))p Fu(\026)867 541 y Fn(")885 535 y Fu(;)e(f)928 502 y Fq(\001)959 535 y Ft(!)k Fv(0)41 b(as)h Fu(")12 b Fv(=)g Fu(")1247 541 y Fn(n)1281 535 y Ft(!)f Fv(0)p Fu(:)257 627 y Fv(Hence,)k(to)f(pro)o (v)o(e)g(\(iii\),)e(it)i(remains)f(to)g(c)o(hec)o(k)i(that)537 718 y Fu(I)555 724 y Fn(")585 718 y Fv(:=)640 685 y Fq(\000)659 718 y Fv(\()p Fd(P)709 701 y Fm(\003)709 728 y Fn("T)761 718 y Ft(\000)9 b Fd(P)836 701 y Fm(\003)836 728 y Fn(T)863 718 y Fv(\))p Fu(\026)904 724 y Fn(")922 718 y Fu(;)e(f)965 685 y Fq(\001)995 718 y Ft(!)k Fv(0)42 b(as)f Fu(")12 b Fv(=)g Fu(")1283 724 y Fn(n)1317 718 y Ft(!)g Fv(0)p Fu(;)179 b Fv(\(3.27\))257 809 y(for)14 b(an)o(y)f Fu(f)k Ft(2)11 b(X)6 b Fv(.)18 b(T)m(o)13 b(pro)o(v)o(e)h(the)g(con)o(v)o (ergence,)h(w)o(e)f(write)g Ft(j)p Fu(I)1223 815 y Fn(")1241 809 y Ft(j)f Fv(as)593 922 y Ft(j)p Fu(I)623 928 y Fn(")641 922 y Ft(j)e Fv(=)707 874 y Fq(\014)707 899 y(\014)707 924 y(\014)721 866 y(Z)744 960 y Fn(H)783 922 y Fs(E)814 888 y Fq(\000)836 922 y Fu(f)t Fv(\()p Fu(u)900 905 y Fm(0)900 932 y Fn(")919 922 y Fv(\()p Fu(T)6 b Fv(\)\))k Ft(\000)f Fu(f)t Fv(\()s(^)-24 b Fu(u)q Fv(\()p Fu(T)6 b Fv(\)\))1191 888 y Fq(\001)1210 922 y Fu(\026)1235 928 y Fn(")1253 922 y Fv(\()p Fu(dv)q Fv(\))1328 874 y Fq(\014)1328 899 y(\014)1328 924 y(\014)1342 922 y Fu(;)257 1045 y Fv(where)15 b Fu(u)401 1030 y Fm(0)401 1056 y Fn(")419 1045 y Fv(\()p Fu(T)6 b Fv(\))12 b(=)g Fu(U)5 b Fv(\()p Fu(\020)607 1030 y Fm(0)604 1056 y Fn(")622 1045 y Fv(;)i Fu(v)q Fv(\)\()p Fu(T)f Fv(\))14 b(and)i(^)-23 b Fu(u)o Fv(\()p Fu(T)6 b Fv(\))12 b(=)g Fu(U)5 b Fv(\()1029 1034 y(^)1025 1045 y Fu(\020)s Fv(;)i Fu(v)q Fv(\)\()p Fu(T)f Fv(\).)19 b(Then)407 1158 y Ft(j)p Fu(I)437 1164 y Fn(")455 1158 y Ft(j)11 b(\024)h Fv(2)p Fu(\026)568 1164 y Fn(")585 1124 y Fq(\010)609 1158 y Ft(j)p Fu(v)q Ft(j)g Fu(>)f(R)741 1124 y Fq(\011)775 1158 y Fv(+)816 1101 y Fq(Z)839 1196 y Fm(j)p Fn(v)q Fm(j\024)p Fn(R)937 1158 y Fs(E)968 1124 y Fq(\000)990 1158 y Ft(k)p Fu(u)1035 1141 y Fm(0)1035 1168 y Fn(")1053 1158 y Fv(\()p Fu(T)6 b Fv(\))j Ft(\000)j Fv(^)-23 b Fu(u)o Fv(\()p Fu(T)6 b Fv(\))p Ft(k)1272 1164 y Fm(\000)p Fr(3)1326 1158 y Ft(^)j Fv(2)1384 1124 y Fq(\001)1410 1158 y Fu(\026)1435 1164 y Fn(")1453 1158 y Fv(\()p Fu(dv)q Fv(\))p Fu(:)257 1282 y Fv(The)17 b(\014rst)g(term)f(on)g(the)g(righ)o(t-hand)g(side)h (is)f(b)q(ounded)g(b)o(y)g Fu(R)1268 1267 y Fm(\000)p Fr(2)1313 1282 y Fu(B)1344 1288 y Fr(0)1363 1282 y Fv(.)25 b(T)m(o)15 b(estimate)h(the)257 1332 y(second)f(term,)e(w)o(e)h(note)g (that,)f(b)o(y)h(Lemma)d(3.2)h(and)i(inequalit)o(y)f(\(3.14\))o(,)g (for)h Ft(j)p Fu(v)q Ft(j)d(\024)h Fu(R)h Fv(w)o(e)257 1382 y(ha)o(v)o(e)488 1397 y Fq(\014)488 1421 y(\014)502 1432 y Fu(u)526 1415 y Fm(0)526 1442 y Fn(")542 1446 y Ff(n)564 1432 y Fv(\()p Fu(T)6 b Fv(\))j Ft(\000)j Fv(^)-23 b Fu(u)o Fv(\()p Fu(T)6 b Fv(\))762 1397 y Fq(\014)762 1421 y(\014)788 1432 y Ft(\024)832 1398 y Fq(\000)851 1432 y Fu(d)873 1438 y Fn(n)904 1432 y Fv(+)k Fu(K)s(")1003 1415 y Fr(1)p Fn(=)p Fr(3)1003 1442 y Fn(n)1056 1398 y Fq(\001)1082 1432 y Fu(F)1109 1438 y Fr(2)1127 1432 y Fv(\()p Fu(R;)d(N)e Fv(\))p Fu(;)47 b(!)13 b Ft(2)e Fv(\012)1415 1438 y Fn(K)1447 1432 y Fu(:)257 1507 y Fv(Therefore,)436 1598 y Ft(j)p Fu(I)466 1604 y Fn(")482 1608 y Ff(n)504 1598 y Ft(j)g(\024)h Fu(R)603 1581 y Fm(\000)p Fr(2)647 1598 y Fu(B)678 1604 y Fr(0)707 1598 y Fv(+)d(2)769 1564 y Fq(\000)788 1598 y Fv(1)g Ft(\000)h Fs(P)-6 b Fv(\(\012)934 1604 y Fn(K)963 1598 y Fv(\))979 1564 y Fq(\001)1007 1598 y Fv(+)1049 1564 y Fq(\000)1068 1598 y Fs(E)9 b Fu(d)1122 1604 y Fn(n)1156 1598 y Fv(+)h Fu(K)s(")1255 1581 y Fr(1)p Fn(=)p Fr(3)1255 1608 y Fn(n)1307 1564 y Fq(\001)1333 1598 y Fu(F)1360 1604 y Fr(2)1379 1598 y Fv(\()p Fu(R;)d(N)e Fv(\))p Fu(;)257 1689 y Fv(and,)14 b(for)f(the)i(same)e(reason)h(as)g(ab)q(o)o(v)o(e,)f Ft(j)p Fu(I)928 1695 y Fn(")944 1699 y Ff(n)966 1689 y Ft(j)e(!)g Fv(0)j(as)g Fu(n)d Ft(!)g(1)p Fv(.)320 1739 y(Th)o(us,)i(con)o(v)o(ergence)j(\(iii\))d(is)g(established,)i(and)e (Theorem)h(3.1)f(is)g(pro)o(v)o(ed.)257 1822 y Fg(Pr)n(o)n(of)i(of)g(L) n(emma)g(3.2.)20 b Fv(T)m(o)15 b(pro)o(v)o(e)g(\(3.23\))o(,)g(w)o(e)h (write)g Fu(u)1157 1828 y Fn(i)1184 1822 y Fv(=)e Fu(\020)1248 1828 y Fn(i)1273 1822 y Fv(+)c Fu(v)1335 1828 y Fn(i)1349 1822 y Fu(;)d(i)14 b Fv(=)g(1)p Fu(;)7 b Fv(2,)14 b(and)h(note)257 1872 y(that)g(b)q(oth)f Fu(v)466 1878 y Fr(1)499 1872 y Fv(and)f Fu(v)599 1878 y Fr(2)632 1872 y Fv(satisfy)h(Eqs.)g (\(3.21\))o(,)g(where)h Ft(k)p Fu(\020)s Ft(k)d(\024)g Fu(N)5 b Fv(.)19 b(Multiplying)12 b(the)i(equa-)257 1922 y(tion)g(b)o(y)f Fu(v)j Fv(in)d Fu(H)s Fv(,)g(w)o(e)i(get)578 2004 y(1)p 578 2023 21 2 v 578 2061 a(2)616 2004 y Fu(d)p 609 2023 37 2 v 609 2061 a(dt)650 2032 y Ft(j)p Fu(v)q Ft(j)695 2015 y Fr(2)723 2032 y Fv(+)9 b Ft(k)p Fu(v)q Ft(k)827 2015 y Fr(2)855 2032 y Fv(+)897 1999 y Fq(\000)916 2032 y Fu(B)r Fv(\()p Fu(v)i Fv(+)f Fu(\020)s(;)d(\020)s Fv(\))p Fu(;)g(v)1155 1999 y Fq(\001)1186 2032 y Ft(\024)k(k)p Fu(\020)s Ft(k)c(k)p Fu(v)q Ft(k)p Fu(:)257 2140 y Fv(Since)14 b Ft(k)p Fu(\020)s Ft(k)d(\024)h Fu(N)5 b Fv(,)13 b(w)o(e)h(ha)o(v)o(e) f(\()p Fu(B)r Fv(\()p Fu(v)d Fv(+)f Fu(\020)s(;)e(\020)s Fv(\))p Fu(;)g(v)q Fv(\))p Ft(j)k(\024)h Fu(C)s(N)1109 2125 y Fr(2)1127 2140 y Ft(k)p Fu(v)q Ft(k)c Fv(+)h Fu(C)s(N)c Ft(j)p Fu(v)q Ft(j)i(k)p Fu(v)q Ft(k)p Fv(,)12 b(and)h(therefore)621 2225 y Fu(d)p 614 2244 V 614 2282 a(dt)655 2253 y Ft(j)p Fu(v)q Ft(j)700 2236 y Fr(2)728 2253 y Fv(+)c Ft(k)p Fu(v)q Ft(k)832 2236 y Fr(2)862 2253 y Ft(\024)j Fu(C)936 2259 y Fr(1)954 2220 y Fq(\000)974 2253 y Fu(N)1012 2236 y Fr(2)1039 2253 y Fv(+)e Fu(N)1119 2236 y Fr(4)1146 2253 y Fv(+)g Fu(N)1226 2236 y Fr(2)1244 2253 y Ft(j)p Fu(v)q Ft(j)1289 2236 y Fr(2)1308 2220 y Fq(\001)1327 2253 y Fu(:)244 b Fv(\(3.28\))257 2356 y(This)14 b(implies)e(the)j (inequalit)o(y)575 2453 y Ft(j)p Fu(v)q Fv(\()p Fu(t)p Fv(\))p Ft(j)667 2436 y Fr(2)697 2453 y Ft(\024)d Fu(e)760 2436 y Fn(C)784 2440 y Fo(1)800 2436 y Fn(N)829 2423 y Fo(2)846 2436 y Fn(t)860 2419 y Fq(\000)879 2453 y Fu(R)911 2436 y Fr(2)939 2453 y Fv(+)e Fu(C)1011 2459 y Fr(2)1029 2453 y Fu(tN)1082 2436 y Fr(4)1100 2419 y Fq(\001)1119 2453 y Fu(;)48 b Fv(0)12 b Ft(\024)f Fu(t)h Ft(\024)g Fu(T)s(;)218 b Fv(\(3.29\))953 2628 y(15)p eop %%Page: 16 16 16 15 bop 257 262 a Fv(whic)o(h)16 b(pro)o(v)o(es)h(\(3.23\))o(.)25 b(Besides,)18 b(in)o(tegrating)e(\(3.28\))f(from)g(0)g(to)h(1)g(and)g (using)h(\(3.29\))o(,)257 311 y(w)o(e)d(obtain)764 367 y Fq(Z)805 377 y Fn(T)787 461 y Fr(0)838 423 y Ft(k)p Fu(v)q Ft(k)901 406 y Fr(2)920 423 y Fu(dt)d Ft(\024)h Fu(F)6 b Fv(\()p Fu(R;)h(N)e Fv(\))p Fu(;)405 b Fv(\(3.30\))257 535 y(where)15 b Fu(F)20 b Fv(is)13 b(a)h(con)o(tin)o(uous)g(function.) 320 585 y(W)m(e)h(no)o(w)h(set)h Fu(v)g Fv(=)e Fu(v)653 591 y Fr(1)683 585 y Ft(\000)c Fu(v)746 591 y Fr(2)781 585 y Fv(and)k Fu(\020)k Fv(=)c Fu(\020)965 591 y Fr(1)995 585 y Ft(\000)c Fu(\020)1056 591 y Fr(2)1075 585 y Fv(.)24 b(Subtracting)17 b(the)f(equation)g(for)g Fu(v)1671 591 y Fr(2)257 635 y Fv(from)c(the)j(equation)e(for)h Fu(v)680 641 y Fr(1)713 635 y Fv(and)f(m)o(ultiplying)e(the)j(result)h(b)o(y)f Fu(v)h Fv(in)e Fu(H)s Fv(,)h(w)o(e)g(deriv)o(e)489 718 y(1)p 489 736 21 2 v 489 774 a(2)527 718 y Fu(d)p 519 736 37 2 v 519 774 a(dt)561 746 y Ft(j)p Fu(v)q Ft(j)606 729 y Fr(2)633 746 y Fv(+)c Ft(k)p Fu(v)q Ft(k)738 729 y Fr(2)766 746 y Fv(+)807 712 y Fq(\000)827 746 y Fu(B)r Fv(\()p Fu(u)900 752 y Fr(1)919 746 y Fu(;)d(u)962 752 y Fr(1)979 746 y Fv(\))j Ft(\000)f Fu(B)r Fv(\()p Fu(u)1119 752 y Fr(2)1138 746 y Fu(;)e(u)1181 752 y Fr(2)1199 746 y Fv(\))p Fu(;)g(v)1255 712 y Fq(\001)1286 746 y Fv(=)12 b(\()p Fu(Lv)q(;)7 b(\020)s Fv(\))p Fu(:)257 851 y Fv(Since)k Ft(k)p Fu(\020)s Ft(k)g(\024)h Fu(")p Fv(,)e(then)g(ev)o(oking)f(the)h (usual)g(estimates)f(for)g(the)i(nonlinearit)o(y)d(\(see)j([CF88)o (]\),)257 901 y(w)o(e)j(get)352 957 y Fq(\014)352 982 y(\014)366 958 y(\000)385 992 y Fu(B)r Fv(\()p Fu(u)458 998 y Fr(1)477 992 y Fu(;)7 b(u)520 998 y Fr(1)538 992 y Fv(\))i Ft(\000)h Fu(B)r Fv(\()p Fu(u)678 998 y Fr(2)697 992 y Fu(;)d(u)740 998 y Fr(2)758 992 y Fv(\))p Fu(;)g(v)814 958 y Fq(\001)833 957 y(\014)833 982 y(\014)858 992 y Ft(\024)12 b Fu(C)e(")d Ft(k)p Fu(v)q Ft(k)1031 958 y Fq(\000)1050 992 y Ft(k)p Fu(u)1095 998 y Fr(1)1113 992 y Ft(k)i Fv(+)h Ft(k)p Fu(u)1230 998 y Fr(2)1248 992 y Ft(k)1269 958 y Fq(\001)1297 992 y Fv(+)f Fu(C)h Ft(k)p Fu(v)q Ft(k)d(j)p Fu(v)q Ft(j)g(k)p Fu(u)1545 998 y Fr(2)1563 992 y Ft(k)p Fu(:)257 1083 y Fv(It)14 b(follo)o(ws)f(that)482 1158 y Fu(d)p 474 1176 V 474 1214 a(dt)516 1186 y Ft(j)p Fu(v)q Ft(j)561 1169 y Fr(2)588 1186 y Fv(+)d Ft(k)p Fu(v)q Ft(k)693 1169 y Fr(2)723 1186 y Ft(\024)i Fu(C)797 1192 y Fr(1)815 1152 y Fq(\000)834 1186 y Fu(")853 1169 y Fr(2)872 1186 y Fv(\()p Ft(k)p Fu(u)933 1192 y Fr(1)952 1186 y Ft(k)d Fv(+)g Ft(k)p Fu(u)1068 1192 y Fr(2)1086 1186 y Ft(k)p Fv(\))1123 1169 y Fr(2)1151 1186 y Fv(+)g Ft(j)p Fu(v)q Ft(j)1237 1169 y Fr(2)1256 1186 y Ft(k)p Fu(u)1301 1192 y Fr(2)1319 1186 y Ft(k)1340 1169 y Fr(2)1368 1186 y Fv(+)g Fu(")1428 1169 y Fr(2)1447 1152 y Fq(\001)1466 1186 y Fu(:)257 1291 y Fv(Since)15 b Fu(v)q Fv(\(0\))d(=)g(0,)h (application)f(of)i(the)g(Gron)o(w)o(all)e(inequalit)o(y)h(results)i (in)402 1411 y Ft(j)p Fu(v)q Fv(\()p Fu(t)p Fv(\))p Ft(j)494 1394 y Fr(2)524 1411 y Ft(\024)d Fu(C)598 1417 y Fr(1)616 1411 y Fu(")635 1394 y Fr(2)661 1355 y Fq(Z)702 1365 y Fn(t)684 1449 y Fr(0)724 1411 y Fv(exp)787 1365 y Fq(\020)812 1411 y Fu(C)842 1417 y Fr(2)867 1355 y Fq(Z)909 1365 y Fn(t)890 1449 y(s)930 1411 y Ft(k)p Fu(u)975 1417 y Fr(2)993 1411 y Ft(k)1014 1394 y Fr(2)1033 1411 y Fu(d\022)1075 1365 y Fq(\021)1107 1378 y(\000)1126 1411 y Fv(1)d(+)g(\()p Ft(k)p Fu(u)1258 1417 y Fr(1)1277 1411 y Ft(k)g Fv(+)g Ft(k)p Fu(u)1393 1417 y Fr(2)1411 1411 y Ft(k)p Fv(\))1448 1394 y Fr(2)1467 1378 y Fq(\001)1493 1411 y Fu(ds:)257 1526 y Fv(Using)15 b(\(3.30\))e(and)g(\(3.23\))o(,)h(w)o(e)g(obtain)f (\(3.24\))o(.)p 1659 1526 2 29 v 1661 1500 25 2 v 1661 1526 V 1686 1526 2 29 v 257 1663 a Fw(References)257 1754 y Fv([Bil99])56 b(P)m(.)20 b(Billingsley)m(,)g Fg(Conver)n(genc)n (e)i(of)f(Pr)n(ob)n(ability)f(Me)n(asur)n(es)p Fv(,)i(John)f(Wiley)e(&) 432 1804 y(Sons,)14 b(New)g(Y)m(ork,)f(1999.)257 1887 y([BKL00])22 b(J.)c(Bricmon)o(t,)f(A.)g(Kupiainen,)h(and)g(R.)e(Lefev)o (ere,)k Fg(Pr)n(ob)n(abilistic)d(estimates)432 1937 y(for)12 b(the)g(two-dimensional)h(sto)n(chastic)f(Navier{Stokes)h(e)n(quations) p Fv(,)f(J.)f(Statist.)432 1987 y(Ph)o(ys.)j Fi(100)f Fv(\(2000\),)g(743{756.)257 2070 y([BKL02])22 b(J.)c(Bricmon)o(t,)f(A.) h(Kupiainen,)g(and)g(R.)f(Lefev)o(ere,)j Fg(Exp)n(onential)g(mixing)f (of)432 2120 y(the)e(2D)h(sto)n(chastic)f(Navier-Stokes)g(dynamics)p Fv(,)h(Comm.)13 b(Math.)j(Ph)o(ys.)g Fi(230)432 2169 y Fv(\(2002\),)d(no.)g(1,)g(87{132.)257 2252 y([CF88])52 b(P)m(.)18 b(Constan)o(tin)g(and)g(C.)g(F)m(oias,)g Fg(Navier-Stokes)h (Equations)p Fv(,)h(Univ)o(ersit)o(y)e(of)432 2302 y(Chicago)13 b(Press,)i(Chicago,)e(1988.)257 2385 y([EMS01])20 b(W.)14 b(E,)h(J.)g(C.)g(Mattingly)m(,)f(and)h(Y)m(a.)f(G.)g(Sinai,)g Fg(Gibbsian)i(dynamics)h(and)g(er-)432 2435 y(go)n(dicity)h(for)f(the)h (sto)n(chastic)n(al)r(ly)f(for)n(c)n(e)n(d)g(Navier{Stokes)h(e)n (quation)p Fv(,)g(Comm.)432 2485 y(Math.)13 b(Ph)o(ys.)h Fi(224)g Fv(\(2001\),)e(83{106.)953 2628 y(16)p eop %%Page: 17 17 17 16 bop 257 262 a Fv([Kry95])39 b(N.)12 b(V.)f(Krylo)o(v,)h Fg(Intr)n(o)n(duction)h(to)g(the)g(The)n(ory)g(of)g(Di\013usion)i(Pr)n (o)n(c)n(esses)p Fv(,)c(AMS)432 311 y(T)m(ranslations)k(of)g (Mathematical)f(Monographs,)h(v)o(ol.)f(142,)h(Pro)o(vidence,)h(RI,)432 361 y(1995.)257 444 y([KS00])54 b(S.)10 b(B.)h(Kuksin)g(and)f(A.)g (Shiriky)o(an,)g Fg(Sto)n(chastic)i(dissip)n(ative)g(PDE's)g(and)g (Gibbs)432 494 y(me)n(asur)n(es)p Fv(,)h(Comm.)e(Math.)i(Ph)o(ys.)h Fi(213)f Fv(\(2000\),)g(291{330.)257 577 y([KS01])54 b(S.)13 b(B.)g(Kuksin)g(and)g(A.)g(Shiriky)o(an,)f Fg(Er)n(go)n(dicity) h(for)h(the)g(r)n(andomly)g(for)n(c)n(e)n(d)g(2D)432 627 y(Navier{Stokes)e(e)n(quations)p Fv(,)g(Math.)f(Ph)o(ys.)g(Anal.)f (Geom.)e Fi(4)j Fv(\(2001\),)g(147{195.)257 710 y([KS02])54 b(S.)10 b(B.)g(Kuksin)h(and)f(A.)g(Shiriky)o(an,)g Fg(Coupling)h(appr)n (o)n(ach)i(to)e(white-for)n(c)n(e)n(d)g(non-)432 760 y(line)n(ar)j(PDE's)p Fv(,)g(J.)g(Math.)f(Pures)i(Appl.)e Fi(81)h Fv(\(2002\),)f(567{602.)257 843 y([Kuk02])32 b(S.)12 b(B.)g(Kuksin,)g Fg(Er)n(go)n(dic)h(the)n(or)n(ems)f(for)h(2D)h (statistic)n(al)d(hydr)n(o)n(dynamics)p Fv(,)i(Rev.)432 892 y(Math.)g(Ph)o(ys.)h Fi(14)g Fv(\(2002\),)f(no.)g(6,)g(585{600.)257 976 y([Mey66])31 b(P)m(.-A.)13 b(Mey)o(er,)i Fg(Pr)n(ob)n(ability)g (and)h(Potentials)p Fv(,)e(Balisdell,)f(W)m(altham-T)l(oron)o(to-)432 1025 y(London,)g(1966.)257 1108 y([Shi02])51 b(A.)29 b(Shiriky)o(an,)k Fg(A)o(nalyticity)c(of)g(solutions)h(for)f(r)n (andomly)g(for)n(c)n(e)n(d)g(two-)432 1158 y(dimensional)f (Navier{Stokes)f(e)n(quations)p Fv(,)k(Russian)c(Math.)g(Surv)o(eys)i Fi(57)432 1208 y Fv(\(2002\),)13 b(no.)g(4,)g(785{799.)953 2628 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0212060413476--