This is a multi-part message in MIME format. ---------------0105281329634 Content-Type: text/plain; name="01-193.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-193.comments" To appear: Nonlinear Diff Eqns and Applications (NoDEA) ---------------0105281329634 Content-Type: text/plain; name="01-193.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-193.keywords" random dynamical systems, determining functionals, stochastic Navier Stokes equation ---------------0105281329634 Content-Type: application/postscript; name="df_fin.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="df_fin.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: df_fin.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips df_fin -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.28:1311 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (df_fin.dvi) @start %DVIPSBitmapFont: Fa lasy10 10 1 /Fa 1 51 df<003FB712FEB9FCA300F0C9120FB3B3A4B9FCA4303079B43E>50 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb msbm7 7 1 /Fb 1 70 df69 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy5 5 3 /Fc 3 77 df0 D48 D76 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmr5 5 14 /Fd 14 94 df<13301360EA01C0EA038013001206120E5AA25AA35AA312F0AB1270A37E A37EA27E12067E1380EA01C0EA006013300C297B9E16>40 D<12C0126012387E120C7E12 07EA0380A2EA01C0A3EA00E0A313F0AB13E0A3EA01C0A3EA0380A2EA070012065A121C5A 12605A0C297C9E16>I<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D48 D<1360EA01E0120F12FF12F11201B3A3387FFF80A2111C7B9B1C>IIII<001C13E0EA1FFF14C0140013FC0018C7FCA513FCEA1BFF381F07C0381C01 E01218EB00F0C7FC14F8A2127012F8A214F01301006013E0387003C0383C0F80380FFF00 EA03F8151D7D9B1C>II56 D61 D<12FFA312E0B3B112FFA308297A9E11>91 D<12FFA31207B3B112FFA308297E9E11>93 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmmi5 5 11 /Fe 11 117 df<007F130C48131E120FA2001E133CA2147814F05AEB01E0EB03C0EB0780 38781E00137CEA79F0EA7FC048C7FC12F017127C911E>23 D<0003B5FC000F14805A4814 00D8701CC7FCEAC01812001338A35BA313F0A3485A120019127D911C>28 D<0003B612E0A239003E0007ED01C01500A25B15C0A2140101F8EB8000140390B5FCA226 01F007C7FC80A216C03A03E00601801400ED0300A248481306150E5D15FCB6FC5D231C7C 9B2A>69 D<3A03FFF03FFFA23A003E0003E0A449EB07C0A449EB0F80A290B6FCA23A01F0 001F00A44848133EA448485BA43AFFFC0FFFC0A2281C7C9B2E>72 D86 D<137F3801FFC0EA07C3380F03E0381C07C0EA3C0348C7FCA25AA500701340007813 E0383C03C0381FFF00EA07F813127C911B>99 D106 DI<3A0F01F807E03A3F87FE1FF83A33CE1F387C3A63D80F 603CD8C3F013C001E01380D803C01300A22607801E5BA3EEF04048484814C0ED01E0EEE1 8016E3001E90397800FF00000C0130137C2A127D9133>109 D<137E3801FF80EA038138 0703C0380E0780EB0300EA0F80EA07F86CB4FC6C1380EA000FEA3003127812F8EB0700EA F00EEA7FFCEA1FF012127C911C>115 D<13C0EA01E0A3EA03C0A4EAFFFCA2EA0780A2EA 0F00A4121EA31304EA3C0CA213181370EA1FE0EA0F800E1A7D9917>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmex10 10 16 /Ff 16 112 df<1430147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B 1203A2485AA3485AA3121F90C7FCA25AA3123EA2127EA6127C12FCB3A2127C127EA6123E A2123FA37EA27F120FA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C013 01EB00E0147014301462738226>0 D<12C07E12707E123C7E7EA26C7E6C7EA26C7E7F12 007F1378137CA27FA37FA31480130FA214C0A31307A214E0A6130314F0B3A214E01307A6 14C0A2130FA31480A2131F1400A3133EA35BA2137813F85B12015B485AA2485A48C7FCA2 121E5A12385A5A5A14627C8226>I<151E153E157C15F8EC01F0EC03E01407EC0FC0EC1F 8015005C147E5CA2495A495AA2495AA2495AA2495AA249C7FCA2137EA213FE5B12015BA2 12035BA21207A25B120FA35B121FA45B123FA548C8FCA912FEB3A8127FA96C7EA5121F7F A4120F7FA312077FA21203A27F1201A27F12007F137EA27FA26D7EA26D7EA26D7EA26D7E A26D7E6D7EA2147E80801580EC0FC0EC07E01403EC01F0EC00F8157C153E151E1F947182 32>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C7E137EA27F6D7EA26D7EA26D7EA2 6D7EA26D7EA26D7EA280147E147F80A21580141FA215C0A2140F15E0A3140715F0A41403 15F8A5EC01FCA9EC00FEB3A8EC01FCA9EC03F8A515F01407A415E0140FA315C0141FA215 80A2143F1500A25C147E14FE5CA2495AA2495AA2495AA2495AA2495AA249C7FC137EA25B 485A5B1203485A485A5B48C8FC123E5A5A5A1F947D8232>I<160F161F163E167C16F8ED 01F0ED03E0ED07C0150FED1F801600153E157E5D4A5A5D14034A5A5D140F4A5AA24AC7FC 143E147E5CA2495AA2495AA2495AA2130F5CA2495AA2133F91C8FCA25B137E13FEA25B12 01A25B1203A35B1207A35B120FA35BA2121FA45B123FA690C9FC5AAA12FEB3AC127FAA7E 7FA6121F7FA4120FA27FA312077FA312037FA312017FA212007FA2137E137F7FA280131F A26D7EA2801307A26D7EA26D7EA26D7EA2147E143E143F6E7EA26E7E1407816E7E140181 6E7E157E153E811680ED0FC01507ED03E0ED01F0ED00F8167C163E161F160F28C66E823D >I<12F07E127C7E7E6C7E6C7E6C7E7F6C7E1200137C137E7F6D7E130F806D7E1303806D 7EA26D7E147C147E80A26E7EA26E7EA26E7EA2811403A26E7EA2811400A281157E157FA2 811680A2151F16C0A3150F16E0A3150716F0A31503A216F8A4150116FCA6150016FEAA16 7FB3AC16FEAA16FC1501A616F81503A416F0A21507A316E0150FA316C0151FA31680153F A216005DA2157E15FE5DA214015DA24A5AA214075DA24A5AA24A5AA24AC7FCA2147E147C 14FC495AA2495A5C1307495A5C131F49C8FC137E137C5B1201485A5B485A485A48C9FC12 3E5A5A5A28C67E823D>I<161E167EED01FE1507ED0FF8ED3FE0ED7FC0EDFF80913801FE 004A5A4A5A5D140F4A5A5D143F5D147F92C7FCA25C5CB3B3B3A313015CA3495AA213075C 495AA2495A495A137F49C8FC485A485AEA07F0EA1FE0485AB4C9FC12FCA2B4FCEA3FC06C 7EEA07F0EA03FC6C7E6C7E6D7E133F6D7E6D7EA26D7E801303A26D7EA3801300B3B3B3A3 8080A281143F81141F816E7E1407816E7E6E7E913800FF80ED7FC0ED3FE0ED0FF8ED07FE 1501ED007E161E27C675823E>26 D<12F012FCB4FC13C0EA3FE0EA0FF86C7E6C7EC67E6D 7E6D7E131F806D7E1307801303801301A2801300B3B3B3A38080A36E7EA281141F6E7EA2 6E7E6E7E816E7E6E7EED7F80ED1FC0ED0FF0ED07F8ED01FEED007EA2ED01FEED07F8ED0F F0ED1FC0ED7F80EDFF004A5A4A5A5D4A5A4A5AA24A5A143F5DA24AC7FCA35C5CB3B3B3A3 13015CA213035C13075C130F495A5C133F495A49C8FCEA03FE485A485AEA3FE0B45A90C9 FC12FC12F027C675823E>I40 D<12F012FCB4FC7FEA3FE06C7E6C7EEA03FC6C7E6C7E6D7E6D7E80131F6D7E8013076D7E A2801301A26D7EA46E7EB3B3B3B281143FA381141FA26E7EA21407811403816E7E140081 6F7E6F7E6F7E6F7E6F7E6F7EED00FE167FEE3FC0160FA2163FEE7F0016FEED03FC4B5A4B 5A4B5A4B5A4B5A4BC7FC5D14014A5A5D14075D140FA24A5AA2143F5DA3147F5DB3B3B3B2 4AC8FCA4495AA213035CA2495A130F5C495A133F5C495A49C9FC485A485AEA0FF8485A48 5AEAFF8090CAFC12FC12F02AF8748243>I80 D<167F923801FFC0923803C0 F0923807803892380F007892381F01FC151E153EA2157E92387C0070170015FCA44A5AA8 1403A45DA41407A94A5AAA4A5AA95DA4143FA492C8FCA7143E147EA4147C123800FE13FC 5CA2495A5CEA7803387007C0383C0F80D80FFEC9FCEA03F82E5C7C7F27>82 D88 D90 D110 D<12F012FE6C7E13E0EA3FF0EA0FFCEA03FE6C7E 6C6C7E6D7E6D7EA26D7E1307A2801303B3B3A76D7EA28013008080816E7E6E7E6E7E6E7E EC01FC6EB4FCED3FC0150FA2153FEDFF00EC01FCEC07F84A5A4A5A4A5A4A5A92C7FC5C5C 13015CA2495AB3B3A713075CA2130F495AA2495A495A4848C8FC485AEA0FFCEA3FF0B45A 138048C9FC12F02294768237>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmbx10 10 34 /Fg 34 122 df<913803FFC0027F13F00103B512FC010FEB00FED93FF8133FD97FE0EBFF 8049485A5A1480484A13C04A6C1380A36F1300167E93C7FCA592383FFFC0B8FCA4000390 C7FCB3ABB5D8FC3F13FFA4303A7EB935>12 D46 D<49B4FC010F13E0017F13FC9038FF83FE4848C67E48 48EB7F804848EB3FC04848EB1FE0A2001F15F0A24848EB0FF8A3007F15FCA500FF15FEB3 007F15FCA4003F15F8A26D131F001F15F0A2000F15E06D133F000715C06C6CEB7F806C6C EBFF003900FF83FE6DB45A011F13F0010190C7FC27387CB630>48 D<141E143E14FE1307133FB5FCA313CFEA000FB3B3A6007FB61280A4213779B630>IIII<001C15C0D81F801307 01F8137F90B61280A216005D5D15F05D15804AC7FC14F090C9FCA8EB07FE90383FFFE090 B512F89038FC07FC9038E003FFD98001138090C713C0120EC813E0157F16F0A216F8A212 06EA3F80EA7FE012FF7FA44914F0A26C4813FF90C713E0007C15C06C5B6C491380D9C007 1300390FF01FFE6CB512F8000114E06C6C1380D90FF8C7FC25387BB630>II<123C123EEA3FE090B71280A41700485D5E5E5EA2 5E007CC7EA0FC000784A5A4BC7FC00F8147E48147C15FC4A5A4A5AC7485A5D140F4A5A14 3F92C8FC5C147E14FE1301A2495AA31307A2130F5CA2131FA5133FA96D5A6D5A6D5A293A 7BB830>I<49B47E010F13F0013F13FC9038FE01FF3A01F8007F804848EB3FC04848EB1F E0150F485AED07F0121FA27FA27F7F01FEEB0FE0EBFF809138E01FC06CEBF03F02FC1380 9138FF7F006C14FC6C5C7E6C14FE6D7F6D14C04914E048B612F0EA07F848486C13F8261F E01F13FC383FC007EB8001007F6D13FE90C7123F48140F48140715031501A21500A216FC 7E6C14016D14F86C6CEB03F06D13076C6CEB0FE0D80FFEEB7FC00003B61200C614FC013F 13F00103138027387CB630>II65 D67 DI76 D82 D<003FB91280A4D9F800EBF003D87FC09238007FC049161F007EC7150FA2007C1707A200 781703A400F818E0481701A4C892C7FCB3AE010FB7FCA43B387DB742>84 D97 D<903801FFC0010F13FC017F13FFD9FF8013802603FE0013C048485AEA 0FF8121F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA27F123FED01E06C7E15 036C6CEB07C06C6C14806C6C131FC69038C07E006DB45A010F13F00101138023257DA42A >99 DI<9038 03FF80011F13F0017F13FC3901FF83FE3A03FE007F804848133F484814C0001FEC1FE05B 003FEC0FF0A2485A16F8150712FFA290B6FCA301E0C8FCA4127FA36C7E1678121F6C6C14 F86D14F000071403D801FFEB0FE06C9038C07FC06DB51200010F13FC010113E025257DA4 2C>I<161FD907FEEBFFC090387FFFE348B6EAEFE02607FE07138F260FF801131F48486C 138F003F15CF4990387FC7C0EEC000007F81A6003F5DA26D13FF001F5D6C6C4890C7FC39 07FE07FE48B512F86D13E0261E07FEC8FC90CAFCA2123E123F7F6C7E90B512F8EDFF8016 E06C15F86C816C815A001F81393FC0000F48C8138048157F5A163FA36C157F6C16006D5C 6C6C495AD81FF0EB07FCD807FEEB3FF00001B612C06C6C91C7FC010713F02B377DA530> 103 D<13FFB5FCA412077EAFED7FC0913803FFF8020F13FE91381F03FFDA3C0113801478 4A7E4A14C05CA25CA291C7FCB3A3B5D8FC3F13FFA4303A7DB935>II< 13FFB5FCA412077EAF92380FFFE0A4923803FC0016F0ED0FE0ED1F804BC7FC157E5DEC03 F8EC07E04A5A141FEC7FE04A7E8181A2ECCFFEEC0FFF496C7F806E7F6E7F82157F6F7E6F 7E82150F82B5D8F83F13F8A42D3A7EB932>107 D<13FFB5FCA412077EB3B3ACB512FCA4 163A7DB91B>I<01FED97FE0EB0FFC00FF902601FFFC90383FFF80020701FF90B512E0DA 1F81903983F03FF0DA3C00903887801F000749DACF007F00034914DE6D48D97FFC6D7E4A 5CA24A5CA291C75BB3A3B5D8FC1FB50083B512F0A44C257DA451>I<01FEEB7FC000FF90 3803FFF8020F13FE91381F03FFDA3C011380000713780003497E6D4814C05CA25CA291C7 FCB3A3B5D8FC3F13FFA430257DA435>I<903801FFC0010F13F8017F13FFD9FF807F3A03 FE003FE048486D7E48486D7E48486D7EA2003F81491303007F81A300FF1680A9007F1600 A3003F5D6D1307001F5DA26C6C495A6C6C495A6C6C495A6C6C6CB45A6C6CB5C7FC011F13 FC010113C029257DA430>I<9038FE03F000FFEB0FFEEC3FFF91387C7F809138F8FFC000 075B6C6C5A5CA29138807F80ED3F00150C92C7FC91C8FCB3A2B512FEA422257EA427> 114 D<130FA55BA45BA25B5BA25A1207001FEBFFE0B6FCA3000390C7FCB21578A815F86C EB80F014816CEBC3E090383FFFC06D1380903803FE001D357EB425>116 D119 D121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmsy7 7 10 /Fh 10 108 df0 D<1238127C12FEA3127C123807077A9114>I< 1406140EB3B812E0A3C7000EC8FCB1B812E0A32B2B7CA834>6 D<137F3801FFC0000713 F0380FC1F8381F007C003C131E0038130E0078130F00707F00F01480481303A56C130700 70140000785B0038130E003C131E001F137C380FC1F86CB45A000113C06C6CC7FC19197C 9A22>14 D<176017F01770A217781738173C171C171E83717E717E717EEF00F8BAFC1980 1900CB12F8EF01E04D5A4D5A4DC7FC171E171C173C173817781770A217F01760391F7C9D 42>33 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F13C0A3EA1F80A21300A25A123EA3 5AA3127812F8A25A12100D1E7D9F13>48 D<017F157F2601FFE0903803FFC0000701F890 380FF1F0260F83FC90381F0038261E00FF013C7F001890263F8078130C4890261FC0E07F 007090260FE1C07F0060EB07E3913803F780486DB4C7EA01806E5A157E157F81824B7E00 60DAF7E0EB0300913801E3F0DBC3F85B6C90260381FC13066C90260F00FE5B001C011E90 387F803C6C017C90381FE0F82607C7F86DB45A2601FFE0010313C06C6CC86CC7FC391B7C 9942>I<49B5FC130F133F01FFC7FCEA01F8EA03E0EA078048C8FC121E121C123C123812 781270A212F05AA2B7FCA300E0C8FCA27E1270A212781238123C121C121E7E6C7EEA03E0 EA01F86CB4FC013FB5FC130F130120277AA12D>I76 D<38C00180EAE003B3B3B3 A3EAC001113B78AB22>107 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi msbm10 10 4 /Fi 4 91 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmti10 10 63 /Fj 63 124 df<04FFEB03F003039038E00FFC923A0FC0F01F1E923A3F00783E0F923A7E 01F87C3FDB7C03EBFC7F03FC14F8DA01F813F905F1137EDC01E1133C913B03F00003F000 A314074B130760A3140F4B130F60A3010FB812C0A3903C001F80001F8000A3023F143F92 C790C7FCA44A5C027E147EA402FE14FE4A5CA413014A13015FA313034A13035FA313074A 495AA44948495AA44948495AA3001CD9038090C8FC007E90380FC03F013E143E00FE011F 5B133C017C5C3AF8780F01E0D878F0EB07C0273FE003FFC9FC390F8000FC404C82BA33> 11 DI< 150C151C153815F0EC01E0EC03C0EC0780EC0F00141E5C147C5C5C495A1303495A5C130F 49C7FCA2133EA25BA25BA2485AA212035B12075BA2120F5BA2121FA290C8FCA25AA2123E A2127EA2127CA412FC5AAD1278A57EA3121C121EA2120E7EA26C7E6C7EA212001E5274BD 22>40 D<140C140E80EC0380A2EC01C015E0A2140015F0A21578A4157C153CAB157CA715 FCA215F8A21401A215F0A21403A215E0A21407A215C0140F1580A2141F1500A2143EA25C A25CA2495AA2495A5C1307495A91C7FC5B133E133C5B5B485A12035B48C8FC120E5A1278 5A12C01E527FBD22>I44 D<387FFFF8A2B5FCA214 F0150579941E>I<120EEA3F80127F12FFA31300127E123C0909778819>I<151815381578 15F0140114031407EC0FE0141F147FEB03FF90383FEFC0148FEB1C1F13001580A2143FA2 1500A25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA2 5CA2133FA291C7FC497EB61280A31D3877B72A>49 DI I<16E0ED01F01503A3150716E0A3150F16C0A2151F1680A2ED3F00A3157EA2157C15FC5D 14015D14035D14075D140F5D141F92C7FC143EA25CECF81C153E903801F07EEB03E014C0 90380780FE130F49485A133EEB7C01137801F05BEA01E03803C003EA0FFE391FFFC3F048 13FB267C01FF13403AF0003FFFE000601307C71400EC0FE05DA3141F5DA3143F92C7FCA4 143E141C24487DB72A>I<010314186E13F8903907F007F091B512E016C01600495B15F8 010E13E0020CC7FC011EC8FC131CA3133C1338A313781370A2147F9038F3FFC09038EF83 E09038FC01F0496C7E485A497F49137CC8FC157EA315FEA41401000C5C123F5A1403485C 5A4A5A12F800E05C140F4A5A5D6C49C7FC0070137E00785B387C01F8383E07F0381FFFC0 6C90C8FCEA01F8253A77B72A>I<157F913803FFC0020F13E0EC3F8191387E00F002F813 70903903F003F0903807E007EB0FC0EB1F80020013E04914C0017E90C7FC13FE5B485AA2 1203485AA2380FE07E9038E1FF809038E783E0391FCE01F09038DC00F813F84848137C5B 157E5B485AA390C712FE5A5AA214015D5AA214035DA348495A5D140F5D4A5A6C49C7FC12 7C147C6C485A6C485A6CB45A6C1380D801FCC8FC243A76B72A>II57 D<133C137E13FF5AA313FE13FCEA00701300B2120EEA3F80127F12FFA313 00127E123C102477A319>I65 D<0107B612FCEFFF8018C0903B000FF0001FF04BEB07F81703021F15FC17014B14FEA202 3F1400A24B1301A2147F18FC92C7120318F84A140718F04AEC0FE0EF1FC00101ED3F80EF 7F004AEB01FEEE07F849B612E05F9139F80007F0EE01FC01076E7E177F4AEC3F80A2010F 16C0171F5CA2131F173F5CA2133FEF7F805C1800017F5D4C5A91C7485A5F49140FEE1FE0 494A5A00014AB45AB748C7FC16F816C037397BB83A>II<0103B612FEEFFFC018F0903B0007F8 000FF84BEB03FCEF00FE020F157FF03F804B141F19C0021F150F19E05D1807143F19F05D A2147FA292C8FCA25C180F5CA2130119E04A151FA2130319C04A153FA201071780187F4A 1600A2010F16FEA24A4A5A60011F15034D5A4A5D4D5A013F4B5A173F4A4AC7FC17FC017F EC03F84C5A91C7EA1FC04949B45A007F90B548C8FCB712F016803C397CB83F>I<0107B8 FCA3903A000FF000034BEB007F183E141F181E5DA2143FA25D181C147FA29238000380A2 4A130718004A91C7FC5E13015E4A133E167E49B512FEA25EECF8000107147C163C4A1338 A2010F147818E04A13701701011F16C016004A14031880013F150718004A5CA2017F151E 173E91C8123C177C4915FC4C5A4914070001ED7FF0B8FCA25F38397BB838>I<0107B712 FEA3903A000FF000074B1300187C021F153CA25DA2143FA25D1838147FA292C8FCEE0380 4A130718004A91C7FCA201015CA24A131E163E010314FE91B5FC5EA2903807F800167C4A 1378A2130FA24A1370A2011F14F0A24A90C8FCA2133FA25CA2137FA291CAFCA25BA25B48 7EB6FCA337397BB836>II<0103B5D8F80FB512E0A390260007F8C7381FE0004B5D A2020F153F615DA2021F157F96C7FC5DA2023F5D605DA2027F14016092C7FCA24A140360 5CA249B7FC60A202FCC712070103150F605CA20107151F605CA2010F153F605CA2011F15 7F95C8FC5CA2013F5D5F5CA2017F14015F91C7FC491403007FD9FE01B512F8B55BA24339 7CB83E>I<0103B512F8A390390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147F A292C7FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133F A25CA2137FA291C8FC497EB6FCA25C25397CB820>I<0207B512F0A391390007FC006F5A A215075EA3150F5EA3151F5EA3153F5EA3157F93C7FCA35D5DA314015DA314035DA31407 A25DA2140FA2003F5C5A141F485CA24A5A12FC00E049C8FC14FE00705B495A6C485A381E 0FC06CB4C9FCEA01F82C3B78B82C>I<0107B512FCA25E9026000FF8C7FC5D5D141FA25D A2143FA25DA2147FA292C8FCA25CA25CA21301A25CA21303A25CA21307A25CA2130F170C 4A141CA2011F153C17384A1478A2013F157017F04A14E01601017F140317C091C7120716 0F49EC1F80163F4914FF000102071300B8FCA25E2E397BB834>76 D<902607FFF8923807FFF0614F13E0D9000FEFF0004F5AA2021F167FF1EFC0141DDA1CFC EC01CF023C16DF9538039F800238ED071FA20278ED0E3F97C7FC0270151CA202F04B5AF0 707E14E0037E14E0010117FE4D485A02C0EC0380A20103ED0701610280140EA20107ED1C 0305385B14006F137049160705E05B010EEC01C0A2011E913803800F61011CEC0700A201 3C020E131F4C5C1338ED1FB80178163F04F091C8FC01705CA201F04A5B187E00015DD807 F816FEB500C09039007FFFFC151E150E4C397AB84A>I<902603FFF891B512E0A281D900 07923807F8006F6E5A61020F5E81DA0E7F5DA2021E6D1307033F92C7FC141C82DA3C1F5C 70130EEC380FA202786D131E0307141C147082DAF003143C70133814E0150101016E1378 030014705C8201036E13F0604A1480163F010715C1041F5B91C7FC17E149EC0FE360010E 15F31607011E15FF95C8FC011C80A2013C805F1338160013785F01F8157CEA03FC267FFF E0143CB51538A243397CB83E>I<0107B612F817FF1880903B000FF0003FE04BEB0FF0EF 03F8141FEF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15F817074A15F0EF0FE0 1301EF1FC04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD907F8C9FCA25CA2130F A25CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512FCA337397BB838>80 D<0103B612F017FEEFFF80903B0007F8003FC04BEB0FF01707020FEC03F8EF01FC5DA202 1F15FEA25DA2143FEF03FC5DA2027FEC07F818F092C7120F18E04AEC1FC0EF3F004A14FE EE01F80101EC0FE091B6128004FCC7FC9138FC003F0103EC0F80834A6D7E8301071403A2 5C83010F14075F5CA2011F140FA25CA2133F161F4AECE007A2017F160F180E91C7FC4902 0F131C007F01FE153CB5913807F078040313F0CAEAFFE0EF3F80383B7CB83D>82 D<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F7C027CEB0FF84A1307 49481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A36E90C7FCA2806D7E14 FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007FFC150F15031501A215 00A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A6D49C7FC6D133ED8F9 F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7ABA2F>I<0007B812E0 A25AD9F800EB001F01C049EB07C0485AD900011403121E001C5C003C1780140312380078 5C00701607140700F01700485CA2140FC792C7FC5DA2141FA25DA2143FA25DA2147FA292 C9FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CEB3FF0007FB512F8B6FC A2333971B83B>I<003FB539800FFFFEA326007F80C7EA7F8091C8EA3F00173E49153CA2 491538A20001167817705BA2000316F05F5BA2000715015F5BA2000F15035F5BA2001F15 0794C7FC5BA2003F5D160E5BA2007F151E161C90C8FCA2163C4815385A16781670A216F0 4B5A5E1503007E4A5A4BC8FC150E6C143E6C6C5B15F0390FC003E03907F01FC00001B5C9 FC38007FFCEB1FE0373B70B83E>III<14F8EB07FE90381F871C90383E03FE13 7CEBF801120148486C5A485A120FEBC001001F5CA2EA3F801403007F5C1300A21407485C 5AA2140F5D48ECC1C0A2141F15831680143F1587007C017F1300ECFF076C485B9038038F 8E391F0F079E3907FE03FC3901F000F0222677A42A>97 D<133FEA1FFFA3C67E137EA313 FE5BA312015BA312035BA31207EBE0F8EBE7FE9038EF0F80390FFC07C013F89038F003E0 13E0D81FC013F0A21380A2123F1300A214075A127EA2140F12FE4814E0A2141F15C05AEC 3F80A215005C147E5C387801F8007C5B383C03E0383E07C0381E1F80D80FFEC7FCEA01F0 1C3B77B926>I<147F903803FFC090380FC1E090381F0070017E13784913383901F801F8 3803F003120713E0120FD81FC013F091C7FC485AA2127F90C8FCA35A5AA45AA315301538 1578007C14F0007EEB01E0003EEB03C0EC0F806CEB3E00380F81F83803FFE0C690C7FC1D 2677A426>II<147F903803FFC090380FC1E090383F00F0017E13785B485A485A48 5A120F4913F8001F14F0383F8001EC07E0EC1F80397F81FF00EBFFF891C7FC90C8FC5A5A A55AA21530007C14381578007E14F0003EEB01E0EC03C06CEB0F806CEB3E00380781F838 03FFE0C690C7FC1D2677A426>III< EB03F0EA01FFA3EA00075CA3130F5CA3131F5CA3133F91C8FCA35B90387E07F0EC1FFCEC 783E9038FFE01F02C01380EC800F1400485A16C05B49EB1F8012035BA2153F000715005B A25D000F147E5B15FE5D121FD98001131C15F8163C003F01031338010013F0A216704814 E0007E15F016E0EDE1C000FE903801E38048903800FF000038143C263B7BB92A>II<150E153F157F A3157E151C1500ABEC1F80EC7FC0ECF1F0EB01C090380380F813071401130F130E131EEB 1C03133C013813F0A2EB0007A215E0A2140FA215C0A2141FA21580A2143FA21500A25CA2 147EA214FEA25CA21301A25CA213035C121C387E07E0A238FE0FC05C49C7FCEAF83EEA78 7CEA3FF0EA0FC0204883B619>IIIII<147F903803FFC090380FC1F090381F00F8017E137C5B4848137E4848133E 0007143F5B120F485AA2485A157F127F90C7FCA215FF5A4814FEA2140115FC5AEC03F8A2 EC07F015E0140F007C14C0007EEB1F80003EEB3F00147E6C13F8380F83F03803FFC0C648 C7FC202677A42A>I<9039078007C090391FE03FF090393CF0787C903938F8E03E903878 7FC00170497EECFF00D9F0FE148013E05CEA01E113C15CA2D80003143FA25CA20107147F A24A1400A2010F5C5E5C4B5A131F5EEC80035E013F495A6E485A5E6E48C7FC017F133EEC 70FC90387E3FF0EC0F8001FEC9FCA25BA21201A25BA21203A25B1207B512C0A3293580A4 2A>II<3903C003F0390FF01FFC391E783C 0F381C7C703A3C3EE03F8038383FC0EB7F800078150000701300151CD8F07E90C7FCEAE0 FE5BA2120012015BA312035BA312075BA3120F5BA3121F5BA3123F90C9FC120E212679A4 23>I<14FE903807FF8090380F83C090383E00E04913F00178137001F813F00001130313 F0A215E00003EB01C06DC7FC7FEBFFC06C13F814FE6C7F6D13807F010F13C01300143F14 1F140F123E127E00FE1480A348EB1F0012E06C133E00705B6C5B381E03E06CB45AD801FE C7FC1C267AA422>II<13 F8D803FEEB01C0D8078FEB03E0390E0F8007121E121C0038140F131F007815C01270013F 131F00F0130000E015805BD8007E133FA201FE14005B5D120149137EA215FE120349EBFC 0EA20201131E161C15F813E0163CD9F003133814070001ECF07091381EF8F03A00F83C78 E090393FF03FC090390FC00F00272679A42D>I<01F0130ED803FC133FD8071EEB7F80EA 0E1F121C123C0038143F49131F0070140FA25BD8F07E140000E08013FEC6485B150E1201 5B151E0003141C5BA2153C000714385B5DA35DA24A5A140300035C6D48C7FC0001130E38 00F83CEB7FF8EB0FC0212679A426>I<01F01507D803FC903903801F80D8071E903907C0 3FC0D80E1F130F121C123C0038021F131F49EC800F00701607A249133FD8F07E168000E0 ED000313FEC64849130718000001147E5B03FE5B0003160E495BA2171E00070101141C01 E05B173C1738A217781770020314F05F0003010713016D486C485A000190391E7C078028 00FC3C3E0FC7FC90393FF81FFE90390FE003F0322679A437>I<903907E007C090391FF8 1FF89039787C383C9038F03E703A01E01EE0FE3803C01F018013C0D8070014FC48148000 0E1570023F1300001E91C7FC121CA2C75AA2147EA214FEA25CA21301A24A1370A2010314 F016E0001C5B007E1401010714C000FEEC0380010F1307010EEB0F0039781CF81E903838 7C3C393FF03FF03907C00FC027267CA427>I<13F0D803FCEB01C0D8071EEB03E0D80E1F 1307121C123C0038140F4914C01270A249131FD8F07E148012E013FEC648133F16001201 5B5D0003147E5BA215FE00075C5BA214015DA314035D14070003130FEBF01F3901F87FE0 38007FF7EB1FC7EB000F5DA2141F003F5C48133F92C7FC147E147C007E13FC387001F8EB 03E06C485A383C1F80D80FFEC8FCEA03F0233679A428>I<903903C0038090380FF007D9 1FF81300496C5A017F130E9038FFFE1E9038F83FFC3901F007F849C65A495B1401C7485A 4A5A4AC7FC141E5C5C5C495A495A495A49C8FC131E5B49131C5B4848133C484813384913 78000714F8390FF801F0391FFF07E0383E1FFFD83C0F5B00785CD8700790C7FC38F003FC 38E000F021267BA422>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmr7 7 17 /Fk 17 127 df<1306130C13181330136013E0EA01C0EA0380A2EA07005A120E121EA212 1C123CA35AA512F85AAB7E1278A57EA3121C121EA2120E120F7EEA0380A2EA01C0EA00E0 136013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01C013 E0120013F0A213701378A3133CA5133E131EAB133E133CA51378A3137013F0A213E01201 13C0EA0380A2EA0700120E120C5A5A5A5A0F3B7DAB1A>I<140EB3A2B812E0A3C7000EC8 FCB3A22B2B7DA333>43 D48 D<13381378EA01F8121F12FE12E01200B3AB48 7EB512F8A215267BA521>I<13FF000313E0380E03F0381800F848137C48137E00787F12 FC6CEB1F80A4127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870 018013E0EA0180390300030012065A001FB5FC5A485BB5FCA219267DA521>I<13FF0003 13E0380F01F8381C007C0030137E003C133E007E133FA4123CC7123E147E147C5C495AEB 07E03801FF8091C7FC380001E06D7E147C80143F801580A21238127C12FEA21500485B00 78133E00705B6C5B381F01F03807FFC0C690C7FC19277DA521>I<1438A2147814F81301 A2130313071306130C131C131813301370136013C012011380EA03005A120E120C121C5A 12305A12E0B612E0A2C7EAF800A7497E90383FFFE0A21B277EA621>I<0018130C001F13 7CEBFFF85C5C1480D819FCC7FC0018C8FCA7137F3819FFE0381F81F0381E0078001C7F00 18133EC7FC80A21580A21230127C12FCA3150012F00060133E127000305B001C5B380F03 E03803FFC0C648C7FC19277DA521>II<137F3803FFE0380781F8380E007C48131E5A801278A3127C 007E131EEA3F80EBE03C6C6C5A380FFCF03807FFC06C5BC613E0487F38079FFC380F07FE EA1E0348C67E48133FEC1F8048130FA21407A315001278140E6C5B6C5B380F80F03803FF E0C66CC7FC19277DA521>56 D61 D91 D93 D<380F07C038FF1FF0EB38F8EA1F71EA0F6113C1EBC0F014005BAF487EEA FFFCA2151A7E991A>114 D<13C0A41201A312031207120F121FB512E0A23807C000AC14 30A73803E060A23801F0C03800FF80EB3F0014257FA31A>116 D<380F8010381FF03838 3FFFF04813E038E07FC038400F8015067BA621>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmsy10 10 29 /Fl 29 115 df<007FB81280B912C0A26C17803204799641>0 D<121C127FEAFF80A5EA 7F00121C0909799917>I<0060150600F8150F6C151F007E153F6C157E6C6C14FC6C6CEB 01F86C6CEB03F06C6CEB07E06C6CEB0FC06C6CEB1F80017EEB3F006D137E6D6C5A90380F C1F8903807E3F0903803F7E06DB45A6D5B6EC7FCA24A7E497F903803F7E0903807E3F090 380FC1F890381F80FC90383F007E017E7F49EB1F804848EB0FC04848EB07E04848EB03F0 4848EB01F84848EB00FC48C8127E007E153F48151F48150F00601506282874A841>I10 D14 D<007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912 FCA26C17F8CCFCAE007FB812F8B912FCA26C17F836287BA841>17 D20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03FE90 3800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0EE 0FF8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FC ED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC048 48CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB81280B912C0A26C17803244 79B441>I24 D<020FB6128091B712C01303010F1680D91FF8C9FCEB7F8001FECAFCEA01F8485A485A48 5A5B48CBFCA2123EA25AA2127812F8A25AA87EA21278127CA27EA27EA26C7E7F6C7E6C7E 6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F1580323279AD41>26 D<007FB512FCB712C016F06C15FCC8EA07FE9238007F80EE1FC0EE07E0707E707E707E17 7C83A283A2EF0F80A2170718C0A21703A81707A21880170FA2EF1F00A2173EA25F17FC4C 5A4C5A4C5AEE1FC0EE7F80DB07FEC7FC007FB65AB712F016C06C02FCC8FC323279AD41> I<181EA4181F84A285180785727EA2727E727E85197E85F11F80F10FC0F107F0007FBA12 FCBCFCA26C19FCCCEA07F0F10FC0F11F80F13F00197E61614E5A4E5AA24E5A61180F96C7 FCA260181EA4482C7BAA53>33 D49 D<91381FFFFE91B6FC1303010F14FED91FF0C7FCEB7F8001FEC8FCEA01F8485A485A485A 5B48C9FCA2123EA25AA2127812F8A25AA2B712FE16FFA216FE00F0C9FCA27EA21278127C A27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF06DB512FE010314FF1300021F13 FE283279AD37>I<387FFFF8B6FC15C06C14F0C7EA0FF8EC01FEEC007FED1F80ED0FC0ED 07E0ED03F01501ED00F8A2167CA2163EA2161E161FA2160FA2007FB7FCB8FCA27EC9120F A2161FA2161E163EA2167CA216F8A2ED01F01503ED07E0ED0FC0ED1F80ED7F00EC01FEEC 0FF8007FB55AB612C092C7FC6C13F8283279AD37>I<0238EB07FC02F890383FFF800103 91B512C0010F010314E0011FEB0F81017B90391E003FF09026E3F078131F010349130FEC F1E0902607F3C0130714F7DAFF8014E092C7FC18C04A140F49481580EF1F004A141E5F4A 5CEE01E0011F4A5A4A010FC7FC163E9138C001F8ED0FFC013F90383FFF804AB57E028114 F0DA83017F91C7EA3FFC496E7E1607017E6E7E8201FE6E1380A249157FA2173F12015BA2 1800485AA2177E4848157CA25F48484A5A01C75D019F4A5A261FBF80495A496C011EC7FC 003F01F0137C9138FC03F0D87E3FB512C0D87C1F91C8FCD8780713F8D8E00113C0343D7E BA37>66 D<0307B612FE033FEDFF804AB812C0140791260F807EC7FC91263C00FEEC3F00 4A161E4A491418010194C7FC495A01071301A2D90FC05B148014000118130390C75BA34B 5AA3150F5EA34B5AA293B512FC4B5C604B14C0037ECAFCA25DA25D1401A24A5AA25D1407 5D140F5D141F92CBFC5C0006133E003E137E007E137CB413FC6D5AEBC1F0EBF1E06CB45A 6C90CCFC6C5AEA07F0423C7EB83C>70 D76 D78 D92 D<14034A7E4A7EA24A7EA34A7E A2EC7CF8A2ECF87CA2ECF03C0101133EA249487EA249486C7EA249486C7EA2EC00034980 A2013E6D7EA2496D7EA20178147801F8147CA2484880A2484880A24848EC0F80A2491407 000F16C0A248C8EA03E0A2003EED01F0A2003C1500007C16F8A248167CA248163C006016 182E347CB137>94 D102 D<12FCEAFFC0EA07F0EA01 FCEA007E7F80131F80130FB3A7801307806D7E6D7EEB007EEC1FF0EC07F8EC1FF0EC7E00 495A495A495A5C130F5CB3A7131F5C133F91C7FC137E485AEA07F0EAFFC000FCC8FC1D53 7ABD2A>I<14C0EB01E01303A214C01307A21480130FA2EB1F00A2131E133EA25BA21378 13F8A2485AA25B1203A25B1207A2485AA290C7FC5AA2123EA2123C127CA2127812F8A412 78127CA2123C123EA27EA27E7FA26C7EA212037FA212017FA26C7EA21378137CA27FA213 1E131FA2EB0F80A2130714C0A2130314E0A21301EB00C0135278BD20>I<126012F07EA2 1278127CA2123C123EA27EA27E7FA26C7EA212037FA26C7EA212007FA21378137CA27FA2 131E131FA2EB0F80A2130714C0A2130314E0A414C01307A21480130FA2EB1F00A2131E13 3EA25BA2137813F8A25B1201A2485AA25B1207A2485AA290C7FC5AA2123EA2123C127CA2 127812F8A25A126013527CBD20>I<126012F0B3B3B3B3A91260045377BD17>I<0070131C 00F0131EB3B3B3B3A80070131C175277BD2A>I112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi7 7 40 /Fm 40 123 df11 D14 D18 D21 D23 D<48B512F8000714FC4814F84814F0D83C07C7FC1270EAC006130E12 00A3131E131CA2133CA35BA313F8A3485AA26C5A1E1A7D981F>28 D<16E00003140148EC03F0120E000C1401001C14005A003015701660481330147014F048 15C0A25C0101EB018014C015031600D8E0035B903807E00E39F01FF03E39FFFEFFFC6C48 6C5AD83FF85B391FE03FE0390F800F80241B7E992A>33 DI<1238127C12FE A3127C123807077A8614>58 D<1238127C12FE12FFA2127F123B1203A31206A3120C1218 12381270122008127A8614>I61 D64 D<4B7E1503150782150F151FA2153FA2156F15CF82EC0187140315071406140E140C0218 7FA2EC30031460A214C013011480D903007F91B5FC5B90380C0001A25B13380130805B01 E013005B12011203000F4A7ED8FFF890381FFFE0A22B2A7DA932>I<013FB512F816FF90 3A01FC001FC04AEB07E0EE03F001031401A24A14F8A2130717F04A130317E0010F1407EE 0FC04AEB1F80EE7E00011F495A91B512F0A291388001FC013FEB007E8291C7EA1F80160F 4915C0A2137EA213FEEE1F805BEE3F000001153E16FE49EB01F84B5A0003EC1FC0B7C7FC 15F82D287DA732>I<013FB512FC16FF903A01FC001FC04AEB03E0EE01F801031400177C 4A80A2010781A25CA2130F18805CA2011F1600A24A5CA2133F173E91C8127E177C4915FC 5F017E14015F01FE4A5AA2494A5A4C5A00014BC7FC167C495CED03E00003EC1FC0B600FE C8FC15F031287DA736>68 D<013FB612FCA2903901FC00014AEB007C173C010315381718 5CA21307A24A13C0A2010F010113005E14C01503011F130F91B5C7FCA2EC800F013F7F15 061400A249010E13E0030C13C0017E90C7FC160101FEEC0380A249EC0700A20001150E16 1E495C16FC0003EC07F8B7FC5E2E287DA731>I<903B3FFFF01FFFF8A2D901FCC7EAFE00 4A5CA2010314015F5CA2010714035F5CA2010F14075F5CA2011F140F91B65AA291388000 0F013F141F5F91C7FCA249143F94C7FC137EA201FE5C167E5BA2000115FE5E5BA2000314 01B539C07FFFE0A235287DA736>72 D<90383FFFF8A2D901FCC7FC5CA21303A25CA21307 A25CA2130FA25CA2131FA25CA2133FA291C8FCA249141C1618137E163801FE1430167049 146016E000011401ED03C0491307ED0F800003147FB7FC160026287DA72E>76 D78 D<4AB4FC021F13E091387E01F8903901F8007ED907E0131FD90F80EB0F8049C7EA07C013 7E49EC03E0485A4915F0484814011207485A4915F8121F90C8FC5A17F0007E1503A4007C ED07E012FC17C0160F1780161F007C1600163E007E157E003E017C5BD901FE5B3A1F0387 01F09039070387C03A0F86018F80D807C6019FC7FCD803F613FC3900FF03F090393FFFC0 06EB07FDD90001130E6F5A163C6F5AEDFFF85E6E5B5E6F5A033EC7FC2D347DA834>81 D86 DI<903B3FFFE00FFFC0A2010190390001FC006D4814F017C0027F495A4C C7FC6E130E6F5A021F5B6F5A5E91380FE1C0EDE380DA07F7C8FC15FE6E5A5D6E7EA28114 03EC077F140E4A7E02187FEC301F02607F14C049486C7EEB030001066D7E5B01386D7E5B 01F06D7E485AD80FF0497ED8FFFC90381FFFE0A232287DA736>II99 D<15F8141FA2EC01F0A21403A215E0A21407 A215C0A2140FEB1F8F90387FCF80EBF0EF3803C03FEA0780390F001F00A2001E5B123E00 3C133E127C147E5A147CA214FC5AECF830A3903801F060A2EA7803010E13C0393C1CF980 381FF07F3907C01E001D297CA723>I<130E131F5BA2133E131C90C7FCA7EA03E0487EEA 0C78EA187C1230A212605B12C0A2EA01F0A3485AA2485AA2EBC180EA0F81A2381F0300A2 13066C5A131CEA07F06C5A11287DA617>105 D<1407EC0F80141FA21500140E91C7FCA7 EB03E0EB07F8EB0C3C1318EB303E136013C0A248485AA2C7FCA25CA4495AA4495AA4495A A4495AA21238D87C1FC7FC12FC133E485AEA70F8EA7FE0EA1F80193380A61B>I<133EEA 07FEA2EA007CA213FCA25BA21201A25BA21203EC07809038E01FC0EC38600007EB61E014 C3EBC187EBC307D80FC613C09038CC038001B8C7FC13E0487E13FEEB3F80EB0FC0486C7E 1303003E1460A2127EECC0C0127CECC18012FC903801E30038F800FE0070137C1B297CA7 23>I<137CEA0FFCA2EA00F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380 A2121FA21300A25AA2123EA2127EA2EA7C18A3EAF830A21320EA786013C0EA3F80EA0F00 0E297EA715>I<3B07801FC007E03B0FE07FF01FF83B18F0E0F8783C3B30F1807CE03E90 3AFB007D801ED860FEEB3F005B49133E00C14A133E5B1201A24848495BA35F4848485A18 30EE01F0A23C0F8003E003E060A218C0933801E180271F0007C013E3933800FF00000E6D 48137C341B7D993B>I<3907801FC0390FE07FF03918F0E0F83930F1807CEBFB00D860FE 133C5B5B00C1147C5B1201A248485BA34A5AEA07C01660EC03E0A23A0F8007C0C0A2EDC1 80913803C300D81F0013C7EC01FE000EEB00F8231B7D9929>II<9038F007C0 3901FC1FF039031E78780006EBE03C90381FC01C000CEB801E14005B0018141F133E1200 137E153E137CA213FC157C5B1578000114F0A2EC01E0EC03C03903FC07809038FE1F00EB E7FCEBE1F0D807E0C7FCA25BA2120FA25B121FEAFFF8A22025809922>II<3807803E390FE0FF803818F3C13930F703C0EBFE0738 60FC0F13F8158039C1F0070091C7FC1201A2485AA4485AA4485AA448C8FCA2120E1A1B7D 991F>II<131C133EA25BA45BA4485AB512E0A23801F0 00485AA4485AA4485AA448C7FC1460A214C0123EEB0180EB0300EA1E06EA1F1CEA0FF8EA 03E013267EA419>I<90387C03C03901FF0FF03907079C30390E03B078000CEBF0F80018 13E1123015F0396007C0E015001200A2495AA449C7FC15301238007C1460EAFC3E15C0EA F87E39F06F03803970C70700383F83FE381F01F81D1B7D9926>120 D<013E13C0137F9038FF818048EBC3004813FF380701FE3806000C00045BC75A5C5CEB03 800106C7FC5B5B5B5B9038C00180EA038039060003005C380FF81E381FFFFE38383FFC38 601FF86D5A38C007C01A1B7D9920>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmmi10 10 67 /Fn 67 123 df11 D<1403EC3FF891387FFF80D901E313C014800103133F9138001F80ED07 0092C7FC80A280A2808013018080130080147F81143F8149B47E130790380F8FF0EB3E0F 496C7E13F83801F003D803E07F1207380FC0011380121FEA3F0014005A127EA212FE5D48 1301A35DA24813035D6C13075D127C4A5A6C91C7FC5C6C133E6C6C5A3807C0F03801FFE0 D8003FC8FC223D7DBB25>14 DI17 DI<133F14C0EB07F06D7E801301A26D7EA3 147FA36E7EA36E7EA36E7EA36E7EA36E7EA36E7EA26E7EA214014A7E5C4A7E91381E3F80 143C14784A6C7E1301EB03E049486C7EEB0F80EB1F00496D7E137E5B48486D7E485A485A 000F6E7E485A485A48C87E12FE167F4816800070151F293B7CB930>21 DI<017E1438D83FFE147E16FEA2D801FC14FC12000001140116F85BED03F0120315 074914E0150F000715C0ED1F805BED3F00000F147EA2495B4A5A001F495A5D49485A4A5A 003F49C7FC143EEB00F8495A48485AEB0F80D87E3EC8FC13F8EAFFE0138000F8C9FC2725 7CA429>I<15FE913803FF8091380F83E091383E01F091387C00F85C494813FC0103147C 4948137E5C130F495AA249C7FC16FE5B137EA2150113FE4914FCA20001140316F85BED07 F01203ED0FE04914C0151F000715806DEB3F00157E6D5B390FEE01F09038E707E09038C3 FF80D9C0FCC7FC001F90C8FCA25BA2123FA290C9FCA25AA2127EA212FEA25AA212702737 7EA42B>26 D<027FB512C00103B612E0130F5B017F15C09026FF81FEC7FC3901FC007E48 487F485A497F484880485AA248C7FCA2127EA2153F00FE92C7FC5AA25D157E5A5DA24A5A A24A5A007C495A5D003C495A003E013FC8FC6C137C380F81F83803FFE0C66CC9FC2B257D A32F>I<013FB512FE90B7FC5A5A4815FE260F801CC7FCEA1E005A00385B5A5A481378C7 FC147014F0A4495AA31303A3495AA3130FA25C131FA3133FA291C8FC131E28257EA324> I31 D<0140151E01E0153F0001 5E484816805B120790C9123F000E161F170F5A1707481700A2003014C014010070010314 061260A2170E00E04948130C5A171C92C7FC5FA26C495C4A14F04A7E6C017F495A4A6C48 5A3AF801F7E00F3BFE0FF3F83F80267FFFE3B5FC02C191C7FC6C01815B02005BD80FFCEB 7FF0D803F0EB0FC031267FA434>33 DI 39 D<121C127FEAFF80A5EA7F00121C0909798817>58 D<121C127FEAFF80A213C0A312 7F121C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>II<150C151E153EA2153C157CA2 157815F8A215F01401A215E01403A215C01407A21580140FA215005CA2141E143EA2143C 147CA2147814F8A25C1301A25C1303A2495AA25C130FA291C7FC5BA2131E133EA2133C13 7CA2137813F8A25B1201A25B1203A25B1207A25B120FA290C8FC5AA2121E123EA2123C12 7CA2127812F8A25A12601F537BBD2A>I<126012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38 007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07 FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007F80EF1FC0A2EF7F80933801FF00EE 07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948 C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0EA7FC048CBFC12FC1270323279AD41 >I64 D<1760177017F01601A21603A21607160FA24C7EA21633167316 6316C3A2ED0183A2ED0303150683150C160115181530A21560A215C014011580DA03007F A202061300140E140C5C021FB5FC5CA20260C7FC5C83495A8349C8FC1306A25BA25B1338 5B01F01680487E000716FFB56C013F13FF5EA2383C7DBB3E>I<0103B77E4916F018FC90 3B0007F80003FE4BEB00FFF07F80020FED3FC0181F4B15E0A2141FA25DA2143F19C04B14 3F1980027F157F190092C812FE4D5A4A4A5AEF0FF04AEC1FC005FFC7FC49B612FC5F02FC C7B4FCEF3FC00103ED0FE0717E5C717E1307844A1401A2130F17035CA2131F4D5A5C4D5A 133F4D5A4A4A5A4D5A017F4BC7FC4C5A91C7EA07FC49EC3FF0B812C094C8FC16F83B397D B83F>I<9339FF8001C0030F13E0037F9038F80380913A01FF807E07913A07F8000F0FDA 1FE0EB079FDA3F80903803BF0002FFC76CB4FCD901FC80495A4948157E495A495A494815 3E017F163C49C9FC5B1201484816385B1207485A1830121F4993C7FCA2485AA3127F5BA3 12FF90CCFCA41703A25F1706A26C160E170C171C5F6C7E5F001F5E6D4A5A6C6C4A5A1607 6C6C020EC8FC6C6C143C6C6C5C6CB4495A90393FE00FC0010FB5C9FC010313FC9038007F C03A3D7CBA3B>I<0103B7FC4916E018F8903B0007F80007FE4BEB00FFF03F80020FED1F C0180F4B15E0F007F0021F1503A24B15F81801143F19FC5DA2147FA292C8FCA25C18035C A2130119F84A1507A2130319F04A150FA2010717E0181F4A16C0A2010FEE3F80A24AED7F 00187E011F16FE4D5A4A5D4D5A013F4B5A4D5A4A4A5A057FC7FC017F15FEEE03FC91C7EA 0FF049EC7FC0B8C8FC16FC16C03E397DB845>I<0103B812E05BA290260007F8C7123F4B 140FF003C0140F18015DA2141FA25D1980143FA25D1760027F14E095C7FC92C75AA24A13 01A24A495A16070101141F91B6FC94C8FCA2903903FC001F824A130EA21307A24A130CA2 010F141CA24A90C9FCA2131FA25CA2133FA25CA2137FA291CBFC497EB612C0A33B397DB8 35>70 D<0103B5D8F803B512F8495DA290260007F8C73807F8004B5DA2020F150F615DA2 021F151F615DA2023F153F615DA2027F157F96C7FC92C8FCA24A5D605CA249B7FC60A202 FCC7120101031503605CA201071507605CA2010F150F605CA2011F151F605CA2013F153F 605CA2017F157F95C8FC91C8FC496C4A7EB690B6FCA345397DB845>72 D<0107B512FCA216F890390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292 C7FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25C A2137FA291C8FC497EB6FCA326397DB824>I<0103B500F8903807FFFC5BA290260007F8 C813804BEDFC0019F0020F4B5AF003804B4AC7FC180E021F1538604B5CEF0380023F4AC8 FC170E4B133C1770027F5C4C5ADB0007C9FC160E4A5B167E4A13FE4B7E01015B92380E7F 80ECFC1CED383F010301E07FECFDC04A486C7EECFF00D907FC6D7E5C4A130783130F707E 5C1601011F81A24A6D7EA2013F6F7EA24A143F84137F717E91C8123F496C81B60107B512 C0A26146397DB847>75 D<0103B6FC5B5E90260007FCC8FC5D5D140FA25DA2141FA25DA2 143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA2130718404A15C0A2010F 150118804A1403A2011F16005F4A1406170E013F151E171C4A143C177C017F5D160391C7 120F49EC7FF0B8FCA25F32397DB839>I<902603FFF893383FFF80496081D900079438FF 80000206DC01BFC7FCA2020E4C5A1A7E020C1606190CDA1C7E16FE4F5A02181630A20238 166162023016C1F00181DA703F158395380303F002601506A202E0ED0C076202C0151818 3001016D6C140F06605B028015C0A20103923801801FDD03005B140092380FC00649173F 4D91C8FC01065DA2010E4B5B4D137E130C6F6C5A011C17FEDCE1805B011802E3C7FCA201 3802E6130104EC5C1330ED03F8017016034C5C01F05CD807FC4C7EB500E0D9C007B512F0 1680150151397CB851>I<902603FFF891381FFFF8496D5CA2D90007030113006FEC007C 02061678DA0EFF157081020C6D1460A2DA1C3F15E0705CEC181F82023815016F6C5C1430 150702706D1303030392C7FC02607FA2DAE0015C701306ECC0008201016E130EEF800C5C 163F0103EDC01C041F131891C713E0160F49EDF03818300106140717F8010E02031370EF FC60130CEE01FE011C16E004005B011815FF177F1338600130153FA20170151F95C8FC01 F081EA07FCB512E01706A245397DB843>I<0103B7FC4916E018F8903B0007F80007FC4B EB00FE187F020FED3F80F01FC05DA2021F16E0A25DA2143FF03FC05DA2027FED7F80A292 C8130018FE4A4A5A604AEC07F04D5A0101ED3FC04CB4C7FC91B612FC17E0D903FCCAFCA2 5CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CBFC497EB6FCA33B397D B835>80 D<4BB4FC031F13F09238FE01FC913903F0007EDA07C0EB1F80DA1F80EB0FC002 3EC7EA07E002FCEC03F0495A4948EC01F8495A4948EC00FC495A013F16FE49C9FC13FE18 7F485A12035B12075B120F4916FF121FA2485AA34848ED01FEA448C9EA03FCA3EF07F8A2 18F0170F18E0171F18C0EF3F807EEF7F0017FEDA07C05B6C90391FF001F8903980383803 001F496C485A9139E00C0FE0260FC0C0EB1F80D807E1D90E3FC7FC0280137ED803F1EB07 F8D801F95C3A007FC00FC0903A3FE07F0003903807FFFE0100018F5BDA000F1306170E17 1E705A177CEEC1F816FF5FA25F5F6F5B6F48C7FCED00F8384B7CBA42>I<0103B612F849 EDFF8018E0903B0007F8001FF84BEB03FCEF00FE020F157FA24BEC3F80A2021F16C0A25D A2143FF07F805DA2027FEDFF006092C7485A4D5A4A4A5A4D5A4AEC1F80057FC7FC0101EC 07F891B612E094C8FC9139FC000FC00103EC03F0707E4A6D7E831307177E5C177F010F5D 5F5CA2011F1401A25CA2133F16034A4A1360A2017F17E019C091C71401496C01011480B6 1503933900FE0700EF7E0ECAEA1FFCEF07F03B3B7DB83F>I<92391FE00380DBFFFC1300 02036D5A91390FE01F8F91393F0007DF027EEB01FE02F81300495A4948147E177C494814 3C495AA2011F153891C8FCA3491530A28094C7FC80806D7E14FEECFFE06D13FE6DEBFFC0 6D14F06D806D80021F7F02037FEC003F03037F1500167F163F161FA3120C160FA2001C15 1F94C7FCA3003C153EA25E003E5D127E007F4A5A6D495A6DEB0FC0D8F9F0495AD8F0FE01 FEC8FC39E03FFFF8010F13E0D8C00190C9FC313D7CBA33>I<0003B812FEA25A903AF800 3FC00101C0913880007E4848163C90C7007F141C121E001C92C7FCA2485CA200305C0070 17180060130112E0485CA21403C716005DA21407A25DA2140FA25DA2141FA25DA2143FA2 5DA2147FA292C9FCA25CA25CA21301A25CA21303A25CEB0FFC003FB6FC5AA237397EB831 >I<003FB56C48B51280485DA226007F80C7381FF00091C8EA07C0604993C7FCA2491506 A20001160E170C5BA20003161C17185BA20007163817305BA2000F167017605BA2001F16 E05F5BA2003F15015F5BA2007F150394C8FC90C8FCA25E4815065A160E160C161C161816 385E127E5E4B5A6C4A5A4BC9FC6C6C131E6C6C5B6C6C13F83903F807E06CB55A6C6C48CA FCEB0FF0393B7BB839>I<267FFFFC91383FFFC0B55DA2000390C83807FC006C48ED03E0 6060000094C7FC5F17065FA25F6D5DA26D5D17E05F4C5AA24CC8FC6E1306A2013F5C161C 16185EA25E6E5BA2011F495A150393C9FC1506A25D6E5AA2010F5B157015605DA2ECE180 02E3CAFC14F3EB07F614FE5C5CA25C5CA26D5AA25C91CBFC3A3B7CB830>I<277FFFFC01 B500F890B51280B5FC60000390C7D807FCC7380FF80001FC4BEC03E000016204035E98C7 FC621A0604075DA2040F5DA2041B5D6216336D02735D1663000003C34A5A83DB01834AC8 FC04815CDB0301140603075D1506030C5DA203185D1970033015606115606D01E04A5A15 C090267F01804AC9FC17FEDA030014060400130E0206150C020E5D140C4A5DA24A5D18E0 4A5D715A5C4A92CAFCA26DC85AA2013E157C1778133C1770133801301560513B7CB84E> I<49B500F890387FFFF095B5FC1AE0D90003018090380FFC004BC713E00201ED07804EC7 FC6E6C140E606F5C705B606F6C485A4D5A031F91C8FCEEE0065F6F6C5A5F03075B705A16 F96FB45A94C9FC6F5AA36F7EA34B7FED037F9238063FC0150E4B6C7E1538ED700F03E07F 15C04A486C7EEC0300020613034A805C4A6D7E14704A1300494880495A49C86C7E130E01 1E153F017E4B7ED803FF4B7E007F01E0011FEBFFC0B5FC6144397EB845>II<147E903803FF8090390FC1C38090391F00EFC0017E137F49 133F485A4848EB1F8012075B000F143F48481400A2485A5D007F147E90C7FCA215FE485C 5AA214015D48150CA21403EDF01C16181407007C1538007E010F1330003E131F027B1370 6C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F0026267DA42C>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0FCEBE3FF9038E707 C0390FFE03E09038F801F001F013F8EBE000485A15FC5BA2123F90C7FCA214015A127EA2 140312FE4814F8A2140715F05AEC0FE0A215C0EC1F80143F00781400007C137E5C383C01 F86C485A380F07C06CB4C7FCEA01FC1E3B7CB924>II<163FED1FFFA3ED007F167EA216 FEA216FCA21501A216F8A21503A216F0A21507A2027E13E0903803FF8790380FC1CF9038 1F00EF017EEB7FC049133F485A4848131F000715805B000F143F485A1600485A5D127F90 C7127EA215FE5A485CA21401A248ECF80CA21403161CEDF0181407007C1538007E010F13 30003E131F027B13706C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F00283B7DB9 2B>II<16F8ED03FEED0F8792381F0F80ED3E3F167F157CA215FC1700161C4A48 C7FCA414035DA414075DA20107B512F0A39026000FE0C7FC5DA4141F5DA4143F92C8FCA4 5C147EA514FE5CA413015CA4495AA45C1307A25C121E123F387F8F80A200FF90C9FC131E 12FEEA7C3CEA7878EA1FF0EA07C0294C7CBA29>II I<14E0EB03F8A21307A314F0EB01C090C7FCAB13F8EA03FEEA070F000E1380121C121812 381230EA701F1260133F00E0130012C05BEA007EA213FE5B1201A25B12035BA200071318 13E01438000F133013C01470EB806014E014C01381EB838038078700EA03FEEA00F81539 7EB71D>I<150FED3F80A2157FA31600151C92C7FCABEC0F80EC3FE0ECF0F0903801C0F8 49487E14005B130E130C131CEB1801133801305BA2EB0003A25DA21407A25DA2140FA25D A2141FA25DA2143FA292C7FCA25CA2147EA214FEA25CA21301001E5B123F387F83F0A238 FF87E0495A00FE5BD87C1FC8FCEA707EEA3FF8EA0FC0214981B722>IIIII<90390F8003F0 90391FE00FFC903939F03C1F903A70F8700F80903AE0FDE007C09038C0FF80030013E000 01491303018015F05CEA038113015CA2D800031407A25CA20107140FA24A14E0A2010F14 1F17C05CEE3F80131FEE7F004A137E16FE013F5C6E485A4B5A6E485A90397F700F80DA38 3FC7FC90387E1FFCEC07E001FEC9FCA25BA21201A25BA21203A25B1207B512C0A32C3583 A42A>112 D<02FC13C0903803FF0190380F838390383F01C790397E00EF8049137F485A 4848133F000715005B485A001F5C157E485AA2007F14FE90C75AA3481301485CA3140348 5CA314075D140F127C141F007E495A003E137F381F01EF380F839F3903FF1F80EA00FC13 00143F92C7FCA35C147EA314FE5C130190387FFFF0A322357DA425>I<3903E001F83907 F807FE390E3C1E07391C3E381F3A183F703F800038EBE07F0030EBC0FF00705B00601500 EC007E153CD8E07F90C7FCEAC07EA2120013FE5BA312015BA312035BA312075BA3120F5B A3121F5B0007C9FC21267EA425>I<14FF010313C090380F80F090383E00380178131C15 3C4913FC0001130113E0A33903F000F06D13007F3801FFE014FC14FF6C14806D13C0011F 13E013039038003FF014071403001E1301127FA24814E0A348EB03C012F800E0EB078000 70EB0F006C133E001E13F83807FFE0000190C7FC1E267CA427>II<13F8D803FE1438D8070F147C000E6D13FC121C 1218003814011230D8701F5C12601503EAE03F00C001005B5BD8007E1307A201FE5C5B15 0F1201495CA2151F120349EC80C0A2153F1681EE0180A2ED7F0303FF130012014A5B3A00 F8079F0E90397C0E0F1C90393FFC07F8903907F001F02A267EA430>I<01F8EB03C0D803 FEEB07E0D8070F130F000E018013F0121C12180038140700301403D8701F130112601500 D8E03F14E000C090C7FC5BEA007E16C013FE5B1501000115805B150316001203495B1506 150E150C151C151815385D00015C6D485A6C6C485AD97E0FC7FCEB1FFEEB07F024267EA4 28>I<01F816F0D803FE9138E001F8D8070F903801F003000ED9800314FC121C12180038 020713010030EDE000D8701F167C1260030F143CD8E03F163800C001005B5BD8007E131F 183001FE5C5B033F1470000117604991C7FCA218E000034A14C049137E17011880170318 005F03FE1306170E000101015C01F801BF5B3B00FC039F8070903A7E0F0FC0E0903A1FFC 03FFC0902703F0007FC7FC36267EA43B>I<903907E001F090391FF807FC9039783E0E0F 9039E01F1C1FD801C09038383F803A03800FF07F0100EBE0FF5A000E4A1300000C157E02 1F133C001C4AC7FC1218A2C7123FA292C8FCA25CA2147EA214FEA24A130CA20101141C00 1E1518003F5BD87F81143801835C00FF1560010714E03AFE0E7C01C0D87C1C495A277838 3E0FC7FC391FF00FFC3907C003F029267EA42F>I<13F8D803FE1470D8070F14F8000EEB 8001121C121800381403003015F0EA701F1260013F130700E0010013E012C05BD8007E13 0F16C013FE5B151F000115805BA2153F000315005BA25D157EA315FE5D14010001130338 00F80790387C1FF8EB3FF9EB0FE1EB00035DA2000E1307D83F805B007F495AA24A5A92C7 FCEB003E007C5B00705B6C485A381E07C06CB4C8FCEA01FC25367EA429>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmbx12 14.4 32 /Fo 32 121 df45 D<157815FC14031407141F14FF130F0007B5 FCB6FCA2147F13F0EAF800C7FCB3B3B3A6007FB712FEA52F4E76CD43>49 DI<91380FFFC091B512FC0107ECFF80011F15E09026 3FF8077F9026FF800113FC4848C76C7ED803F86E7E491680D807FC8048B416C080486D15 E0A4805CA36C17C06C5B6C90C75AD801FC1680C9FC4C13005FA24C5A4B5B4B5B4B13C04B 5BDBFFFEC7FC91B512F816E016FCEEFF80DA000713E0030113F89238007FFE707E701380 7013C018E07013F0A218F8A27013FCA218FEA2EA03E0EA0FF8487E487E487EB57EA318FC A25E18F891C7FC6C17F0495C6C4816E001F04A13C06C484A1380D80FF84A13006CB44A5A 6CD9F0075BC690B612F06D5D011F1580010302FCC7FCD9001F1380374F7ACD43>I<171F 4D7E4D7EA24D7EA34C7FA24C7FA34C7FA34C7FA24C7FA34C8083047F80167E8304FE804C 7E03018116F8830303814C7E03078116E083030F814C7E031F81168083033F8293C77E4B 82157E8403FE824B800201835D840203834B800207835D844AB87EA24A83A3DA3F80C880 92C97E4A84A2027E8202FE844A82010185A24A820103854A82010785A24A82010F855C01 1F717FEBFFFCB600F8020FB712E0A55B547BD366>65 D68 D73 D 78 D82 D<91260FFF80130791B500F85B010702FF5B011FEDC03F49EDF07F9026FFFC006D5A4801 E0EB0FFD4801800101B5FC4848C87E48488149150F001F824981123F4981007F82A28412 FF84A27FA26D82A27F7F6D93C7FC14C06C13F014FF15F86CECFF8016FC6CEDFFC017F06C 16FC6C16FF6C17C06C836C836D826D82010F821303010082021F16801400030F15C0ED00 7F040714E01600173F050F13F08383A200788200F882A3187FA27EA219E07EA26CEFFFC0 A27F6D4B13806D17006D5D01FC4B5A01FF4B5A02C04A5A02F8EC7FF0903B1FFFC003FFE0 486C90B65AD8FC0393C7FC48C66C14FC48010F14F048D9007F90C8FC3C5479D24B>I<00 3FBC1280A59126C0003F9038C0007F49C71607D87FF8060113C001E08449197F49193F90 C8171FA2007E1A0FA3007C1A07A500FC1BE0481A03A6C994C7FCB3B3AC91B912F0A55351 7BD05E>I97 D<913801FFF8021FEBFF8091B612F0010315FC010F9038C00FFE903A1FFE0001FFD97FFC 491380D9FFF05B4817C048495B5C5A485BA2486F138091C7FC486F1300705A4892C8FC5B A312FFAD127F7FA27EA2EF03E06C7F17076C6D15C07E6E140F6CEE1F806C6DEC3F006C6D 147ED97FFE5C6D6CEB03F8010F9038E01FF0010390B55A01001580023F49C7FC020113E0 33387CB63C>99 D<4DB47E0407B5FCA5EE001F1707B3A4913801FFE0021F13FC91B6FC01 0315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F4849143F484980485B83 485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E6C6D5C6C6D49B5 FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101ECFE0FD9003F13 F8020301C049C7FC41547CD24B>I<913803FFC0023F13FC49B6FC010715C04901817F90 3A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18C0488191C7FC48 17E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218E06CEE01F06E14 037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE903A0F FFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>IIII<137F497E000313 E0487FA2487FA76C5BA26C5BC613806DC7FC90C8FCADEB3FF0B5FCA512017EB3B3A6B612 E0A51B547BD325>I107 DIII<913801FFE0021F13FE91B612C0010315F0010F9038807F FC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F48496D7F4A147F48834890C86C7E A24883A248486F7EA3007F1880A400FF18C0AC007F1880A3003F18006D5DA26C5FA26C5F 6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE011F90C7FC903A0FFF807FFC 6D90B55A010015C0023F91C8FC020113E03A387CB643>I<903A3FF001FFE0B5010F13FE 033FEBFFC092B612F002F301017F913AF7F8007FFE0003D9FFE0EB1FFFC602806D7F92C7 6C7F4A824A6E7F4A6E7FA2717FA285187F85A4721380AC1A0060A36118FFA2615F616E4A 5BA26E4A5B6E4A5B6F495B6F4990C7FC03F0EBFFFC9126FBFE075B02F8B612E06F148003 1F01FCC8FC030313C092CBFCB1B612F8A5414D7BB54B>I<912601FFE0EB0780021F01F8 130F91B500FE131F0103ECFF80010F9039F03FC03F499039800FE07F903A7FFE0003F049 48903801F8FF4849EB00FD4849147F4A805A4849805A4A805AA291C87E5AA35B12FFAC6C 7EA37EA2806C5EA26C6D5CA26C6D5C6C6D5C6C93B5FC6C6D5B6D6C5B6DB4EB0FEF010F90 38C07FCF6D90B5120F010114FED9003F13F80203138091C8FCB1040FB61280A5414D7CB5 47>I<90397FE003FEB590380FFF80033F13E04B13F09238FE1FF89139E1F83FFC0003D9 E3E013FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8EE0FF04A90C7FCA55CB3 AAB612FCA52F367CB537>I<903903FFF00F013FEBFE1F90B7FC120348EB003FD80FF813 07D81FE0130148487F4980127F90C87EA24881A27FA27F01F091C7FC13FCEBFFC06C13FF 15F86C14FF16C06C15F06C816C816C81C681013F1580010F15C01300020714E0EC003F03 0713F015010078EC007F00F8153F161F7E160FA27E17E07E6D141F17C07F6DEC3F8001F8 EC7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F807148048C601F8C7FC2C387C B635>I<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A000F90B6FCB8 FCA426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E6DEBF0FC6D EBFFF86D6C5B021F5B020313802A4D7ECB34>III<007FB500F090387FFFFEA5C66C48C700 0F90C7FC6D6CEC07F86D6D5C6D6D495A6D4B5A6F495A6D6D91C8FC6D6D137E6D6D5B9138 7FFE014C5A6E6C485A6EEB8FE06EEBCFC06EEBFF806E91C9FCA26E5B6E5B6F7E6F7EA26F 7F834B7F4B7F92B5FCDA01FD7F03F87F4A486C7E4A486C7E020F7FDA1FC0804A486C7F4A 486C7F02FE6D7F4A6D7F495A49486D7F01076F7E49486E7E49486E7FEBFFF0B500FE49B6 12C0A542357EB447>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmr10 10 86 /Fp 86 128 df<1506150FA24B7EA24B7EA24B7EA2EDDFF0A29138018FF8A291380307FC A291380603FEA291380E01FF140CDA1C007F141802386D7E143002706D7E146002E06D7E 5C01016E7E5C01036E7E91C7FC496E7E1306010E6E7E130C011C6E7F131801386F7E1330 01706F7E136001E06F7E5B170F484882170748C97F17030006831701488383481880001F B9FC4818C0A24818E0A2BA12F0A23C3C7CBB45>1 D<003FB712FCA60030C9120C007016 0EA200601606A4CBFCA701C01403A490B7FCA601C0C71203A490CAFCA900C01603A56C16 07A200601606007FB712FEA630397DB837>4 D6 D10 DIIII<133C13 7EA213FE1201EA03FC13F0EA07E0EA0FC0EA1F80EA1E005A5A5A12C00F0F6FB92A>19 D22 D<14FF010713C090381F83F090383F00FC017E137E49137F 4848EB3F80A24848131F16C0A7ED3F80A2ED7F00157E5D4A5AEC07E000FFEB7F80A2EC07 E00003EB01F0EC00FC157E811680151FED0FC0A216E0150716F0A3ED03F8AA16F0ECF007 EBF1F816E0ED0FC0A200079038F01F803AFFF0E03F00EC707CEC3FF0C7EA0FC0253C7EBA 2A>25 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E 5A5A5A12600A1979B917>39 D<146014E0EB01C0EB0380EB0700130E131E5B5BA25B485A A2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3121EA2121F7EA2 7F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E01460135278BD20 >I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C133E131EA2131F 7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C137C1378 A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<15301578B3A6007FB8 12F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41>43 D<121C127FEAFF80A213C0 A3127F121C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>II<121C127FEAFF80A5EA7F00121C0909798817>I48 D III<1538A2157815F8 A2140114031407A2140F141F141B14331473146314C313011483EB030313071306130C13 1C131813301370136013C01201EA038013005A120E120C5A123812305A12E0B712F8A3C7 3803F800AB4A7E0103B512F8A325397EB82A>I<0006140CD80780133C9038F003F890B5 FC5D5D158092C7FC14FC38067FE090C9FCABEB07F8EB3FFE9038780F803907E007E09038 8003F0496C7E12066E7EC87EA28181A21680A4123E127F487EA490C71300485C12E00060 5C12700030495A00385C6C1303001E495A6C6C485A3907E03F800001B5C7FC38007FFCEB 1FE0213A7CB72A>II<12301238123E003FB612E0A3 16C05A168016000070C712060060140E5D151800E01438485C5D5DC712014A5A92C7FC5C 140E140C141C5CA25CA214F0495AA21303A25C1307A2130FA3495AA3133FA5137FA96DC8 FC131E233B7BB82A>III<121C12 7FEAFF80A5EA7F00121CC7FCB2121C127FEAFF80A5EA7F00121C092479A317>I<121C12 7FEAFF80A5EA7F00121CC7FCB2121C127F5A1380A4127F121D1201A412031300A25A1206 A2120E5A121812385A1260093479A317>I<007FB812F8B912FCA26C17F8CCFCAE007FB8 12F8B912FCA26C17F836167B9F41>61 D<1538A3157CA315FEA34A7EA34A6C7EA202077F EC063FA2020E7FEC0C1FA2021C7FEC180FA202387FEC3007A202707FEC6003A202C07F15 01A2D901807F81A249C77F167FA20106810107B6FCA24981010CC7121FA2496E7EA3496E 7EA3496E7EA213E0707E1201486C81D80FFC02071380B56C90B512FEA3373C7DBB3E>65 DI<913A01FF800180020FEBE003027F13F8903A01FF807E07903A03 FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01FE153F12014848151F48 48150FA248481507A2485A1703123F5B007F1601A35B00FF93C7FCAD127F6DED0180A312 3F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618017F15386D6C5CD91FE0 5C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC020F13F002011380313D 7BBA3C>III< B812F8A30001903880001F6C90C71201EE00FC177C173C171CA2170CA4170E1706A2ED01 80A21700A41503A21507151F91B5FCA3EC001F15071503A21501A692C8FCAD4813C0B612 C0A32F397DB836>III I<013FB512E0A39039001FFC00EC07F8B3B3A3123FEA7F80EAFFC0A44A5A1380D87F005B 0070131F6C5C6C495A6C49C7FC380781FC3801FFF038007F80233B7DB82B>III< B5933807FFF86E5DA20001F0FC002600DFC0ED1BF8A2D9CFE01533A3D9C7F01563A3D9C3 F815C3A2D9C1FCEC0183A3D9C0FEEC0303A2027F1406A36E6C130CA36E6C1318A26E6C13 30A36E6C1360A26E6C13C0A3913901FC0180A3913900FE0300A2ED7F06A3ED3F8CA2ED1F D8A3ED0FF0A3486C6D5A487ED80FFC6D48497EB500C00203B512F8A2ED018045397DB84C >I III82 D I<003FB812E0A3D9C003EB001F273E0001FE130348EE01F00078160000701770A3006017 30A400E01738481718A4C71600B3B0913807FF80011FB612E0A335397DB83C>IIII89 D<003FB7FCA39039FC0001 FE01C0130349495A003EC7FC003C4A5A5E0038141F00784A5A12704B5A5E006014FF4A90 C7FCA24A5A5DC712074A5AA24A5A5D143F4A5AA24A5A92C8FC5B495AA2495A5C130F4948 EB0180A2495A5C137F495A16034890C7FC5B1203485AEE0700485A495C001F5D48485C5E 4848495A49130FB8FCA329397BB833>II93 D97 DIIII<147E903803FF8090380FC1E0EB1F8790 383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387FFFF8 A31C3B7FBA19>IIII< EB01C0EB07F0EB0FF8A5EB07F0EB01C090C7FCAAEB01F813FFA313071301B3B3A2123C12 7E00FF13F01303A214E038FE07C0127C383C0F00EA0FFEEA03F8154984B719>III<2703F00FF0EB1FE000FFD93FFCEB7FF8913AF03F01E0 7E903BF1C01F83803F3D0FF3800FC7001F802603F70013CE01FE14DC49D907F8EB0FC0A2 495CA3495CB3A3486C496CEB1FE0B500C1B50083B5FCA340257EA445>I<3903F00FF000 FFEB3FFCECF03F9039F1C01F803A0FF3800FC03803F70013FE496D7EA25BA35BB3A3486C 497EB500C1B51280A329257EA42E>II<3903F01FE000FFEB7FF89038 F1E07E9039F3801F803A0FF7000FC0D803FEEB07E049EB03F04914F849130116FC150016 FEA3167FAA16FEA3ED01FCA26DEB03F816F06D13076DEB0FE001F614C09039F7803F0090 38F1E07E9038F0FFF8EC1FC091C8FCAB487EB512C0A328357EA42E>II<3807E01F00 FFEB7FC09038E1E3E09038E387F0380FE707EA03E613EE9038EC03E09038FC0080491300 A45BB3A2487EB512F0A31C257EA421>II<1318A51338A31378A313F8120112031207001FB5FCB6FC A2D801F8C7FCB215C0A93800FC011580EB7C03017E13006D5AEB0FFEEB01F81A347FB220 >IIIIII<003FB512FCA2EB8003D83E0013F8003CEB07F00038EB0FE012300070EB1FC0 EC3F800060137F150014FE495AA2C6485A495AA2495A495A495AA290387F000613FEA248 5A485A0007140E5B4848130C4848131CA24848133C48C7127C48EB03FC90B5FCA21F247E A325>II126 D<001C131C007F137F39FF80FF 80A5397F007F00001C131C190978B72A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmr8 8 27 /Fq 27 122 df<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A 5A126009157A8714>44 D48 D<130C133C137CEA03FC12FFEA FC7C1200B3B113FE387FFFFEA2172C7AAB23>I51 D<1230123C003FB512F8A215F05A15E039700001C000601480140348EB0700140E140CC7 121C5C143014705C495AA2495AA249C7FCA25B130E131EA2133EA3133C137CA413FCA913 781D2E7CAC23>55 D68 D75 D77 D85 D<13FF000713C0380F01F0381C00F8003F137C80A2143F001E7FC7FC A4EB07FF137F3801FE1FEA07F0EA1FC0EA3F80EA7F00127E00FE14065AA3143F7E007E13 7F007FEBEF8C391F83C7FC390FFF03F83901FC01E01F207D9E23>97 D99 D<15F8141FA214011400ACEB0FE0EB7FF83801F81E3803E007 3807C003380F8001EA1F00481300123E127EA25AA9127C127EA2003E13017EEB8003000F 13073903E00EFC3A01F03CFFC038007FF090391FC0F800222F7EAD27>III104 DI107 DI<2607C07FEB07F03B FFC3FFC03FFC903AC783F0783F3C0FCE01F8E01F803B07DC00F9C00F01F8D9FF8013C049 90387F000749137EA249137CB2486C01FEEB0FE03CFFFE0FFFE0FFFEA2371E7E9D3C>I< 3807C0FE39FFC3FF809038C703E0390FDE01F0EA07F8496C7EA25BA25BB2486C487E3AFF FE1FFFC0A2221E7E9D27>II<3807C0FE39FFC7FF80 9038CF03E0390FDC01F03907F800FC49137E49133E49133FED1F80A3ED0FC0A8151F1680 A2ED3F00A26D137E6D137C5D9038FC01F09038CE07E09038C7FF80D9C1FCC7FC01C0C8FC A9487EEAFFFEA2222B7E9D27>I<380781F838FF87FEEB8E3FEA0F9CEA07B813B0EBF01E EBE000A45BB0487EB5FCA2181E7E9D1C>114 D<3801FE183807FFB8381E01F8EA3C0048 1378481338A21418A27E7EB41300EA7FF06CB4FC6C13C06C13F0000113F838001FFC1301 38C0007E143EA26C131EA27EA26C133CA26C137838FF01F038E3FFC000C0130017207E9E 1C>I<1360A413E0A312011203A21207121FB512F0A23803E000AF1418A714383801F030 14703800F860EB3FE0EB0F80152A7FA81B>I<3AFFFC01FFC0A23A0FE0007E000007147C 15380003143015706C6C1360A26C6C5BA390387C0180A26D48C7FCA2EB3F07EB1F06A2EB 0F8CA214DCEB07D8A2EB03F0A36D5AA26D5A221E7F9C25>118 D<3AFFFC01FFC0A23A0F E0007E000007147C1538000314306D137000011460A26C6C5BA2EBFC01017C5BEB7E0301 3E90C7FCA2EB1F06A2148EEB0F8CA2EB07D8A2EB03F0A36D5AA26D5AA2495AA2130391C8 FC1278EAFC06A25B131CEA7838EA7070EA3FE0EA0F80222B7F9C25>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmsy6 6 1 /Fr 1 4 df<136013701360A20040132000E0137038F861F0387E67E0381FFF803807FE 00EA00F0EA07FE381FFF80387E67E038F861F038E060700040132000001300A213701360 14157B9620>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmti9 9 12 /Fs 12 121 df97 D100 DI105 D<133FEA07FF5A13FEEA007EA3137CA213FCA213F8A21201A213F0A21203A213E0A21207 A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2127C1318EAFC1C133CEAF838 A21378137012F013F0EAF8E01279EA3FC0EA0F00103579B314>108 D<2703C003F8137F3C0FF00FFE01FFC03C1E783C1F07C1E03C1C7CF00F8F01F03B3C3DE0 079E0026383FC001FC7FD97F805B007001005B5E137ED8F0FC90380FC00100E05FD860F8 148012000001021F130360491400A200034A13076049013E130FF081800007027EEC83C0 051F138049017C1403A2000F02FC1407053E130049495CEF1E0E001F01015D183C010049 EB0FF0000E6D48EB03E03A227AA03F>I<3903C007F0390FF01FFC391E787C1E391C7CF0 1F393C3DE00F26383FC01380EB7F8000781300EA707EA2D8F0FC131F00E01500EA60F812 0000015C153E5BA20003147E157C4913FCEDF8180007153C0201133801C013F0A2000F15 78EDE070018014F016E0001FECE1C015E390C7EAFF00000E143E26227AA02B>I<14FCEB 07FF90381F07C090383E03E09038FC01F0EA01F83903F000F8485A5B120F484813FCA248 C7FCA214014814F8127EA2140300FE14F05AA2EC07E0A2007CEB0FC01580141FEC3F006C 137E5C381F01F0380F83E03803FF80D800FCC7FC1E2278A027>I<011E137C90387F81FF 9039F3C387C09039E3EF03E03901E1FE01D9C1FC13F0EBC3F8000313F0018314F814E0EA 07871307000313C01200010F130316F01480A2011F130716E01400A249EB0FC0A2013EEB 1F80A2017EEB3F00017F133E5D5D9038FF81F09038FDC3E09038F8FF80027EC7FC000190 C8FCA25BA21203A25BA21207A25BB5FCA325307FA027>I<3903C00FC0390FF03FF0391E 78F078391C7DE03C393C3FC0FC00381380EB7F00007814F8D8707E13701500EAF0FC12E0 EA60F812001201A25BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1E227A A020>114 D<1303EB0F80A3131FA21400A25BA2133EA2137EA2137C387FFFF8A2B5FC38 00F800A21201A25BA21203A25BA21207A25BA2120FA25B1460001F13F014E01300130114 C01303001E1380EB07005BEA0F1EEA07F8EA01E015307AAE19>116 D<011F137C90387FC1FF3A01E1E787803A03C0F703C0903880FE0FEA07004813FC000E15 80001E9038F80700001C91C7FC1301003C5B1218120013035CA31307A25C1506010F130F 150E14800038141ED87C1F131C00FC143C1538013F5B39F07FC0E03970F3C3C0393FE1FF 80260F807EC7FC22227CA023>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr9 9 38 /Ft 38 123 df12 D<123C127EB4FCA21380A2127F123D1201A412031300A25A1206 120E120C121C5A5A126009177A8715>44 DI<123C127E12FFA4 127E123C08087A8715>I50 D66 D68 D73 D78 D<90381FE00390387FFC0748 B5FC3907F01FCF390F8003FF48C7FC003E80814880A200788000F880A46C80A27E92C7FC 127F13C0EA3FF013FF6C13F06C13FF6C14C06C14F0C680013F7F01037F9038003FFF1403 02001380157F153FED1FC0150F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E6C14 7E01C05B39F1FC03F800E0B512E0011F138026C003FEC7FC22377CB42B>83 D<007FB712FEA390398007F001D87C00EC003E0078161E0070160EA20060160600E01607 A3481603A6C71500B3AB4A7E011FB512FCA330337DB237>I87 D 97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003 BF3907E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B 6C6C4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F 83809039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C 380EFFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F00 01003EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007 E0C6B51280D91FFCC7FC22337EA126>IIIIII<2703F01FE013FF00FF90267FF80313C0903BF1E07C0F 03E0903BF3803E1C01F02807F7003F387FD803FE1470496D486C7EA2495CA2495CB3486C 496C487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000FFEB7FF09038F1E0FC90 38F3807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7FFFCA326217EA02B> II<3903F03F8000FFEBFFE0 9038F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0ED07F0A3ED03F8A915 0716F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381F89038F1FFE0D9F07F C7FC91C8FCAA487EB512C0A325307EA02B>I<903807F00390383FFC07EBFC0F3901F803 8F3807E001000F14DF48486CB4FC497F123F90C77E5AA25A5AA9127FA36C6C5B121F6D5B 000F5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE090C7FCAAED7F8091380FFFFCA3 26307DA029>I<3803E07C38FFE1FF9038E38F809038E71FC0EA07EEEA03ECA29038FC0F 8049C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A212031207381FFFFEB5FCA23803F000AF 1403A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0182F7FAD1E>III II<3A7FFF807FF8A33A07F8001FC00003EC0F800001EC070015066C6C5BA26D131C 017E1318A26D5BA2EC8070011F1360ECC0E0010F5BA2903807E180A214F3010390C7FC14 FBEB01FEA26D5AA31478A21430A25CA214E05CA2495A1278D8FC03C8FCA21306130EEA70 1CEA7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2EB000F003C14E00038EB1FC000 30EB3F800070137F1500006013FE495A13035CC6485A495AA2495A495A49C7FC153013FE 485A12035B48481370485A001F14604913E0485A387F000348130F90B5FCA21C207E9F22 >I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmbx9 9 7 /Fu 7 117 df65 D97 DI<903807FF80013F13F090B512FC3903FE01FE4848 487EEA0FF8EA1FF0EA3FE0A2007F6D5A496C5A153000FF91C7FCA9127F7FA2003FEC0780 7F6C6C130F000FEC1F00D807FE133E3903FF80FCC6EBFFF8013F13E0010790C7FC21217D A027>I<3901F81F8000FFEB7FF0ECFFF89038F9E3FC9038FBC7FE380FFF876C1307A213 FEEC03FCEC01F8EC0060491300B1B512F0A41F217EA024>114 D<9038FFE1C0000713FF 5A383F803F387E000F14075A14037EA26C6CC7FC13FCEBFFE06C13FC806CEBFF80000F14 C06C14E0C6FC010F13F0EB007F140F00F0130714037EA26C14E06C13076CEB0FC09038C0 1F8090B5120000F913FC38E03FE01C217DA023>I<133CA5137CA313FCA21201A2120312 07001FB51280B6FCA3D807FCC7FCB0EC03C0A79038FE078012033901FF0F006C13FEEB3F FCEB0FF01A2F7EAE22>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmsy8 8 1 /Fv 1 4 df<130C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F0038 07CCF83801FFE038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000 FCEB0FC039781E078000601301000090C7FCA5130C1A1D7C9E23>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr12 12 49 /Fw 49 128 df25 D<121EEA7F8012FF13C0A213E0A3127FEA 1E601200A413E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>44 DI<14FF010713E090381F81F890383E007C01FC133F4848EB1F 8049130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A2003F15FCA390C8FC48 15FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB03F0A36C6CEB07E0 000315C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F8903807FFE0010090C7FC 28447CC131>48 D<143014F013011303131F13FFB5FC13E713071200B3B3B0497E497E00 7FB6FCA3204278C131>II<49B4FC01 0F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1FE0120EED0FF0EA0F E0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3FC01680ED7F0015FE 4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0ED07 F816FC150316FEA2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC00701407 003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE0039007FFF F8010F13E0010190C7FC28447CC131>II<000615C0D807C0130701FCEB7F80 90B612005D5D5D15E0158026063FFCC7FC90C9FCAE14FF010713C090381F01F090383800 FC01F0137ED807C07F49EB1F8016C090C7120F000615E0C8EA07F0A316F81503A216FCA5 123E127F487EA416F890C712075A006015F0A20070140F003015E00038EC1FC07E001EEC 3F806CEC7F006C6C13FE6C6C485A3901F807F039007FFFE0011F90C7FCEB07F826447BC1 31>II<121CA2EA1F8090B712C0A3481680A217005E 0038C8120C0030151C00705D0060153016705E5E4814014B5A4BC7FCC81206150E5D1518 15385D156015E04A5AA24A5A140792C8FC5CA25C141E143EA2147E147CA214FCA21301A3 495AA41307A6130FAA6D5AEB01C02A457BC231>I<14FF010713E0011F13F890387F00FE 01FC133FD801F0EB1F804848EB0FC049EB07E00007EC03F048481301A290C713F8481400 A47FA26D130116F07F6C6CEB03E013FC6C6CEB07C09039FF800F806C9038C01F006CEBF0 3EECF87839007FFEF090383FFFC07F01077F6D13F8497F90381E7FFFD97C1F1380496C13 C02601E00313E048486C13F000079038007FF84848EB3FFC48C7120F003EEC07FE150148 140016FF167F48153FA2161FA56C151E007C153EA2007E153C003E157C6C15F86DEB01F0 6C6CEB03E06C6CEB07C0D803F8EB1F80C6B4EBFF0090383FFFFC010F13F0010113802844 7CC131>I<16C04B7EA34B7EA34B7EA34B7EA3ED19FEA3ED30FFA203707FED607FA203E0 7FEDC03FA2020180ED801FA2DA03007F160FA20206801607A24A6D7EA34A6D7EA34A6D7E A20270810260147FA202E08191B7FCA249820280C7121FA249C87F170FA20106821707A2 496F7EA3496F7EA3496F7EA201788313F8486C83D80FFF03037FB500E0027FEBFFC0A342 477DC649>65 DIII70 DI73 D<010FB512FEA3D9000313806E130080B3B3AB123F487E487EA44A 5A13801300006C495A00705C6C13076C5C6C495A6CEB1F802603E07FC7FC3800FFFCEB1F E027467BC332>I76 DI<49B41303010FEBE007013F13F89039FE00FE0F D801F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E82127C00FC81 A282A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE06C14FC6C14 FF6C15C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F13C0167F16 3F161F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D6D147ED8FB C05CD8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C0003FF02B48 7BC536>83 D<003FB912F8A3903BF0001FF8001F01806D481303003EC7150048187C0078 183CA20070181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B77EA33F447DC3 46>II97 DII<167FED3FFFA315018182B3EC7F80903803FFF090380FC07C 90383F000E017E1307496D5AD803F87F48487F5B000F81485AA2485AA2127FA290C8FC5A AB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C6C010E13C0013F01 1C13FE90380FC0F8903803FFE09026007F0013002F467DC436>III III<143C14FFA2491380A46D1300A2143C91C7FCADEC7F80EB 3FFFA31300147F143FB3B3AA123E127F39FF807F00A2147EA25C6C485A383C01F06C485A 3807FF80D801FEC7FC195785C21E>I108 DI<3901FC01FE00FF903807FFC091381E07F091383801F8000701707F 0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CECFF80B5D8F83F13FEA3 2F2C7DAB36>II<3901FC03FC00FF90380F FF8091383C07E091387001F83A07FDE000FE00030180137FD801FFEC3F8091C7EA1FC049 15E049140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8A217F0160F6D15E0EE 1FC06D143F17806EEB7E00D9FDC05B9039FCF003F891383C0FE091381FFF80DA03FCC7FC 91C9FCAE487EB512F8A32F3F7DAB36>I<91387F8003903903FFE00790380FE07890393F 801C0F90387E000E496D5AD803F8EB039F0007EC01BF4914FF48487F121F5B003F81A248 5AA348C8FCAB6C7EA3123F7F121F6D5C120F6D5B12076C6C5B6C6C497E6C6C130E013F13 1C90380FC0F8903803FFE09038007F0091C7FCAEEEFF80033F13FEA32F3F7DAB33>I<39 03F803F000FFEB1FFCEC3C3EEC707F0007EBE0FF3803F9C000015B13FBEC007E153C01FF 13005BA45BB3A748B4FCB512FEA3202C7DAB26>I<90383FE0183901FFFC383907E01F78 390F0003F8001E1301481300007C1478127800F81438A21518A27EA27E6C6C13006C7E13 FC383FFFE06C13FC6C13FF6C14C06C14E0C614F0011F13F81300EC0FFC140300C0EB01FE 1400157E7E153EA27EA36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F8039 E07FFE0038C00FF01F2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE1201120700 1FB512F0B6FCA2C648C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC070 903807E0E0903801FFC09038007F001E3E7EBC26>III121 D123 D<001EEB0780007FEB0FE039FF801FF0EBC03FA4EB801F39 7F000FE0001EEB07801C0A76C231>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmr17 17.28 19 /Fx 19 118 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 749 872 a Fx(Determining)45 b(functionals)g(for)d(random)h (partial)1393 1054 y(di\013eren)l(tial)k(equations)1685 1295 y Fw(Igor)32 b(Ch)m(uesho)m(v)2295 1259 y Fv(\003)1146 1411 y Fw(Institute)g(f)s(\177)-51 b(ur)32 b(Dynamisc)m(he)g(Systeme,)i (FB3)1573 1528 y(Univ)m(ersit\177)-49 b(at)32 b(Bremen)1410 1644 y(D-28334)e(Bremen,)j(German)m(y)1705 1809 y(Jinqiao)e(Duan)1206 1925 y(Departmen)m(t)h(of)g(Applied)g(Mathematics)1338 2041 y(Illinois)d(Institute)k(of)f(T)-8 b(ec)m(hnology)1474 2158 y(Chicago,)32 b(IL)h(60616,)e(USA)1635 2323 y(Bj\177)-49 b(orn)33 b(Sc)m(hmalfu\031)1307 2439 y(Departmen)m(t)g(of)f(Applied)f (Sciences)993 2555 y(Univ)m(ersit)m(y)j(of)e(T)-8 b(ec)m(hnology)33 b(and)g(Applied)e(Sciences)1668 2671 y(Geusaer)i(Stra\031e)1342 2787 y(D{06217)e(Merseburg,)j(German)m(y)1663 2983 y(Marc)m(h)g(15,)e (2001)1822 3507 y Fu(Abstract)837 3608 y Ft(Determining)22 b(functionals)h(are)f(to)r(ols)h(to)f(describ)r(e)h(the)e(\014nite)h (dimensional)g(long-)722 3700 y(term)29 b(dynamics)g(of)h(in\014nite)f (dimensional)h(dynamical)f(systems.)h(There)g(also)h(exist)722 3791 y(sev)n(eral)j(applications)g(to)f(in\014nite)g(dimensional)g Fs(r)l(andom)h Ft(dynamical)e(systems.)h(In)722 3882 y(these)23 b(applications)h(the)e(con)n(v)n(ergence)g(condition)h(of)g (the)f(tra)t(jectories)j(of)e(an)f(in\014nite)722 3973 y(dimensional)28 b(random)f(dynamical)g(system)g(with)h(resp)r(ect)g (to)f(a)h(\014nite)g(set)g(of)g(linear)722 4065 y(functionals)j(is)g (assumed)e(to)h(b)r(e)g(either)g(in)g(mean)f(or)h Fs(exp)l(onential)h Ft(with)f(resp)r(ect)h(to)722 4156 y(the)37 b(con)n(v)n(ergence)g (almost)g(surely)-6 b(.)37 b(In)f(con)n(trast)i(to)f(these)g(ideas)g(w) n(e)h(in)n(tro)r(duce)f(a)722 4247 y(con)n(v)n(ergence)18 b(concept)g(whic)n(h)g(is)h(based)f(on)g(the)f(con)n(v)n(ergence)h(in)g (probabilit)n(y)-6 b(.)18 b(By)g(this)722 4339 y(ansatz)24 b(w)n(e)g(get)f(rid)g(of)h(the)f(assumption)f(of)i(exp)r(onen)n(tial)f (con)n(v)n(ergence.)h(In)e(addition,)722 4430 y(setting)k(the)g(random) f(terms)f(to)i(zero)h(w)n(e)f(obtain)g(usual)g(deterministic)f (results.)722 4521 y(W)-6 b(e)26 b(apply)f(our)h(results)g(to)g(the)g (2D)f(Na)n(vier)h(-)f(Stok)n(es)h(equations)g(forced)h(b)n(y)d(a)i (white)722 4613 y(noise.)p 515 4654 1182 4 v 606 4708 a Fr(\003)642 4731 y Fq(on)38 b(lea)n(v)n(e)h(from)d(Departmen)n(t)i (of)f(Mec)n(hanics)i(and)f(Mathematics,)g(Khark)n(o)n(v)g(Univ)n(ersit) n(y)-6 b(,)38 b(310077)515 4810 y(Khark)n(o)n(v,)24 b(Ukraine)1970 5059 y Fp(1)p eop %%Page: 2 2 2 1 bop 515 523 a Fo(1)134 b(In)l(tro)t(duction)515 705 y Fp(The)23 b(question)f(of)h(the)g(n)n(um)n(b)r(er)g(of)g(parameters)e (that)i(are)f(necessary)g(for)g(the)h(description)g(of)515 805 y(the)g(long-term)f(b)r(eha)n(viour)f(of)i(solutions)f(to)g (nonlinear)g(partial)g(di\013eren)n(tial)h(equations)f(w)n(as)515 904 y(\014rst)31 b(discussed)h(b)n(y)f(F)-7 b(oias)32 b(and)f(Pro)r(di)g([14)o(])h(and)g(b)n(y)g(Ladyzhensk)-5 b(a)n(y)n(a)29 b([20)o(])j(for)g(the)g(deter-)515 1004 y(ministic)27 b(2D)g(Na)n(vier-Stok)n(es)e(equations.)h(They)h(pro)n(v) n(ed)f(that)h(the)g(asymptotic)g(b)r(eha)n(viour)515 1103 y(of)j(the)h(solutions)e(is)i(completely)f(determined)g(b)n(y)h (the)f(dynamics)g(of)h(the)f(\014rst)g Fn(N)40 b Fp(F)-7 b(ourier)515 1203 y(mo)r(des,)30 b(if)h Fn(N)39 b Fp(is)30 b(su\016cien)n(tly)g(large.)f(After)h([14])g(and)g([20)o(])g(similar)g (results)g(w)n(ere)f(obtained)515 1303 y(for)35 b(other)g(parameters)g (and)g(other)h(deterministic)g(equations)f(and)h(a)f(general)f(approac) n(h)515 1402 y(to)27 b(the)h(problem)f(of)g(the)h(existence)f(of)h(a)f (\014nite)h(n)n(um)n(b)r(er)f(of)h(determining)f(parameters)f(w)n(as) 515 1502 y(dev)n(elop)r(ed)h(\(see)g([7,)h(8)o(,)g(16)o(])g(and)f(the)h (literature)f(quoted)g(therein\).)704 1647 y(Assume)33 b(that)g(w)n(e)g(ha)n(v)n(e)f(a)h(dynamical)f(system)h(with)h(the)g (phase)e(state)h Fn(H)40 b Fp(and)33 b(the)515 1747 y(ev)n(olution)c (op)r(erator)f Fn(S)1265 1759 y Fm(t)1294 1747 y Fp(.)i(Roughly)f(sp)r (eaking)g(the)h(general)e(problem)i(on)f(the)h(existence)g(of)515 1846 y(\014nite)39 b(sets)f(of)h(determining)f(parameters)f (\(functionals\))i(can)f(b)r(e)h(stated)f(\(cf.)i([7)o(,)f(8)o(]\))g (as)515 1946 y(follo)n(ws:)30 b(\014nd)i(the)f(conditions)g(on)g(a)g (\014nite)g(set)h Fl(f)p Fn(l)2139 1958 y Fm(j)2202 1946 y Fp(:)d Fn(j)34 b Fp(=)29 b(1)p Fn(;)14 b(:::;)g(N)9 b Fl(g)30 b Fp(of)h(functionals)g(on)g Fn(H)515 2046 y Fp(whic)n(h)c(guaran)n(tees)f(that)i(the)g(con)n(v)n(ergence)d(\(in)j (certain)f(sense\))968 2217 y(max)1030 2270 y Fm(j)1136 2217 y Fl(j)p Fn(l)1184 2229 y Fm(j)1219 2217 y Fp(\()p Fn(S)1302 2229 y Fm(t)1332 2217 y Fn(u)1380 2229 y Fk(1)1435 2217 y Fl(\000)18 b Fn(S)1569 2229 y Fm(t)1598 2217 y Fn(u)1646 2229 y Fk(2)1683 2217 y Fp(\))p Fl(j)23 b(!)g Fp(0)83 b(when)g Fn(t)23 b Fl(!)g Fp(+)p Fl(1)p Fn(;)37 b(u)2679 2229 y Fk(1)2715 2217 y Fn(;)14 b(u)2800 2229 y Fk(2)2860 2217 y Fl(2)23 b Fn(H)515 2428 y Fp(implies)36 b(that)g Fn(S)1044 2440 y Fm(t)1073 2428 y Fn(u)1121 2440 y Fk(1)1182 2428 y Fl(\000)24 b Fn(S)1322 2440 y Fm(t)1351 2428 y Fn(u)1399 2440 y Fk(2)1473 2428 y Fl(!)37 b Fp(0)e(in)h(some)g(top)r(ology)e(of)i(the)h(phase)e(space)g Fn(H)7 b Fp(.)36 b(Besides)515 2527 y(from)23 b(applied)h(p)r(oin)n(t)g (of)g(view)f(it)h(is)g(also)f(imp)r(ortan)n(t)g(to)h(\014nd)g(b)r (ounds)g(for)f(the)h(n)n(um)n(b)r(er)g Fn(N)33 b Fp(of)515 2627 y(determining)22 b(functionals)g(\(in)g(the)h(sense)f(ab)r(o)n(v)n (e\))f(and)h(to)g(describ)r(e)f(families)i(of)f(functionals)515 2727 y(with)28 b(minimal)g Fn(N)9 b Fp(.)704 2872 y(The)27 b(deterministic)g(theory)g(of)g(determining)g(functionals)g(w)n(as)f (dev)n(elop)r(ed)h(b)n(y)g(man)n(y)515 2971 y(authors)22 b(\(see,)i(e.g.)g([7)o(,)g(8])g(and)f(the)i(references)d(therein\).)j (Similar)e(problems)g(for)g(sto)r(c)n(hastic)515 3071 y(systems)i(w)n(ere)f(also)h(discussed)g(in)h([6)o(,)g(4)o(,)g(12)o(,)g (9)o(].)g(In)g(pap)r(ers)e([6,)i(4)o(,)g(12)o(])f Fn(!)s Fp(-wise)g(approac)n(h)f(to)515 3171 y(construction)k(of)i(determining) f(functionals)g(w)n(ere)g(dev)n(elop)r(ed.)g(Ho)n(w)n(ev)n(er)e(in)j (these)f(pap)r(ers)515 3270 y(it)f(w)n(as)g(assumed)g(that)g(either)g (\(see)h([6)o(,)g(4)o(,)g(12)o(]\))f(the)h(nonlinear)e(term)i(in)f(the) h(equation)f(is)g(a)515 3370 y(globally)19 b(Lipsc)n(hitz)h(mapping)h (or)e(\(see)i([4)o(,)g(12)o(]\))g(one)f(of)h(initial)f(data)g(b)r (elongs)g(to)h(the)g(random)515 3469 y(attractor.)k(The)j(mo)r(de)f(of) g(con)n(v)n(ergence)e(for)i(functionals)g(and)g(tra)5 b(jectories)25 b(is)i(the)h(con)n(v)n(er-)515 3569 y(gence)33 b(almost)g(surely)g(in)i(these)f(pap)r(ers.)f(Moreo)n(v)n(er)e(in)j (the)g(pap)r(ers)g([4)o(,)g(12)o(])g(the)h(authors)515 3669 y(assume)26 b(that)i(the)g(functionals)f(of)h(the)g(di\013erence)f (of)g(t)n(w)n(o)g(solutions)g(go)f(to)i(zero)e(exp)r(onen-)515 3768 y(tially)34 b(fast.)g(Then)h(they)f(pro)n(v)n(e)f(that)i(some)f (norm)f(of)i(the)f(di\013erence)h(of)f(these)g(solutions)515 3868 y(tends)c(to)f(zero)f(with)i(exp)r(onen)n(tial)f(sp)r(eed)h(whic)n (h)g(is)f(less)g(than)h(the)g(sp)r(eed)g(of)f(con)n(v)n(ergence)515 3968 y(of)h(the)h(functionals.)g(On)f(the)h(other)f(hand)h(the)g (approac)n(h)e(presen)n(ted)g(in)i([9])g(do)r(es)f(not)h(as-)515 4067 y(sume)36 b(these)g(conditions,)g(and)h(it)f(relies)g(on)g(some)g (estimates)g(exp)r(onen)n(tial)g(momen)n(ts)g(of)515 4167 y(solutions)30 b(and)h(deals)g(with)h(con)n(v)n(ergence)d(in)i (mean.)h(The)f(sp)r(eed)g(of)h(con)n(v)n(ergence)c(to)k(zero)515 4266 y(of)27 b(the)h(functionals)g(and)f(of)h(the)g(norms)e(are)h(the)h (same)f(as)g(in)h([9)o(].)704 4412 y(In)d(con)n(trast)g(to)g([6,)g(4,)g (12)o(,)h(9])f(w)n(e)g(consider)g(determinig)g(functionals)g(with)h (resp)r(ect)g(to)515 4511 y(the)19 b(con)n(v)n(ergence)d(in)i (probabilit)n(y)-7 b(.)18 b(Using)g(suc)n(h)g(determining)h (functionals)f(w)n(e)g(can)g(a)n(v)n(oid)f(the)515 4611 y(assumption)25 b(that)h(the)f(images)g(\(under)h(linear)e (functionals\))i(of)f(the)h(tra)5 b(jectories)24 b(con)n(v)n(erge)515 4710 y(exp)r(onen)n(tially)32 b(fast.)h(In)g(particular,)f(our)g (approac)n(h)f(reco)n(v)n(ers)f(the)j(deterministic)g(results)515 4810 y(when)22 b(w)n(e)f(remo)n(v)n(e)f(the)i(sto)r(c)n(hastic)f (terms.)g(Another)h(adv)-5 b(an)n(tage)20 b(is)i(that)g(w)n(e)f(do)g (not)h(ha)n(v)n(e)f(to)1970 5059 y(2)p eop %%Page: 3 3 3 2 bop 515 523 a Fp(assume)22 b(that)g(one)h(tra)5 b(jectory)20 b(m)n(ust)j(b)r(e)g(con)n(tained)f(in)h(the)g(random)e(attractor.)g (Finally)-7 b(,)23 b(w)n(e)515 623 y(men)n(tion)k(that)g(the)g(con)n(v) n(ergence)d(in)k(probabilit)n(y)d(is)i(quite)g(natural)f(for)h(RDS,)g (see)g([11)o(,)g(2)o(].)704 770 y(In)19 b(Section)g(2)f(w)n(e)h (consider)e(an)i(abstract)f(setting)h(of)f(random)g(dissipativ)n(e)g (systems)h(and)515 870 y(pro)n(v)n(e)28 b(t)n(w)n(o)g(existence)h (theorem)g(of)h(\014nite)g(sets)f(of)h(determining)f(functionals)g(in)h (the)g(sense)515 969 y(of)g(the)g(de\014nition)h(giv)n(en)e(b)r(elo)n (w)h(for)g(arbitrary)e(initial)i(data.)g(These)g(theorems)f(sho)n(w)g (t)n(w)n(o)515 1069 y(di\013eren)n(t)20 b(approac)n(hes)e(to)i(the)g (construction)g(of)g(determining)g(functionals.)g(In)g(Section)g(3)g(w) n(e)515 1168 y(apply)27 b(the)h(results)e(of)i(Section)f(2)g(to)g(2D)h (Na)n(vier)e(-)h(Stok)n(es)g(equations)f(sub)5 b(ject)28 b(to)f(additiv)n(e)515 1268 y(white)34 b(noise.)f(W)-7 b(e)35 b(pro)n(v)n(e)d(the)i(existence)g(of)g(determining)g (functionals)f(for)h(this)g(problem)515 1368 y Fj(without)28 b Fp(the)g(assuming)e(that)i(one)f(of)h(solutions)f(b)r(elongs)g(to)g (the)h(attractor.)515 1641 y Fo(2)134 b(The)45 b(existence)h(of)f (determining)h(functionals)515 1823 y Fp(W)-7 b(e)34 b(consider)g(a)g Fj(r)l(andom)i(dynamic)l(al)i(system)c Fp(\(REDS\))i(whic)n(h)e(consists)f(of)i(t)n(w)n(o)e(comp)r(o-)515 1923 y(nen)n(ts.)22 b(The)g(\014rst)f(comp)r(onen)n(t)h(is)g(a)f Fj(metric)k(dynamic)l(al)h(system)c Fp(\(\012)p Fn(;)14 b Fl(F)8 b Fn(;)14 b Fi(P)p Fn(;)g(\022)r Fp(\))21 b(as)h(a)f(mo)r(del) h(for)515 2022 y(a)27 b(noise,)f(where)h(\(\012)p Fn(;)14 b Fl(F)8 b Fn(;)14 b Fi(P)p Fp(\))27 b(is)g(a)g(probabilit)n(y)f(space) h(and)g Fn(\022)j Fp(is)d(a)g Fl(F)f(\012)17 b(B)s Fp(\()p Fi(R)p Fp(\))p Fn(;)d Fl(F)41 b Fp(measurable)515 2122 y(\015o)n(w:)27 b(w)n(e)g(ha)n(v)n(e)1373 2222 y Fn(\022)1412 2234 y Fk(0)1473 2222 y Fp(=)22 b(id)p Fn(;)180 b(\022)1871 2234 y Fm(t)p Fk(+)p Fm(\034)2012 2222 y Fp(=)22 b Fn(\022)2138 2234 y Fm(t)2186 2222 y Fl(\016)c Fn(\022)2285 2234 y Fm(\034)2350 2222 y Fp(=:)k Fn(\022)2499 2234 y Fm(t)2529 2222 y Fn(\022)2568 2234 y Fm(\034)515 2368 y Fp(for)36 b Fn(t;)28 b(\034)48 b Fl(2)39 b Fi(R)p Fp(.)k(The)36 b(measure)g Fi(P)g Fp(is)g(supp)r(osed)h(to)g(b)r(e)g(ergo)r(dic)f (with)h(resp)r(ect)g(to)f Fn(\022)r Fp(.)i(The)515 2467 y(second)27 b(comp)r(onen)n(t)g(of)g(a)g(random)g(dynamical)g(system)g (is)g(a)g Fl(B)s Fp(\()p Fi(R)2645 2437 y Fk(+)2706 2467 y Fp(\))18 b Fl(\012)g(F)26 b(\012)18 b(B)s Fp(\()p Fn(H)7 b Fp(\))p Fn(;)14 b Fl(B)s Fp(\()p Fn(H)7 b Fp(\)-)515 2567 y(measurable)26 b(mapping)h Fn(')h Fp(satisfying)f(the)h Fj(c)l(o)l(cycle)h Fp(prop)r(ert)n(y)995 2744 y Fn(')p Fp(\()p Fn(t)19 b Fp(+)f Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))23 b(=)g Fn(')p Fp(\()p Fn(t;)14 b(\022)1765 2756 y Fm(\034)1807 2744 y Fn(!)s(;)g(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\)\))p Fn(;)181 b(')p Fp(\(0)p Fn(;)14 b(!)s(;)g(x)p Fp(\))23 b(=)g Fn(x;)515 2921 y Fp(where)h(the)i(phase)e(space)g Fn(H)32 b Fp(is)25 b(a)f(separable)g (metric)h(space)f(and)h Fn(x)g Fp(is)g(c)n(hosen)f(arbitrarily)f(in)515 3021 y Fn(H)7 b Fp(.)27 b(W)-7 b(e)28 b(will)g(denote)g(this)f(RDS)i(b) n(y)e(sym)n(b)r(ol)g Fn(')p Fp(.)515 3121 y(A)j(standard)f(mo)r(del)h (for)f(suc)n(h)g(a)h(noise)f Fn(\022)j Fp(is)d(the)i(t)n(w)n(osided)e Fj(Br)l(ownian)j(motion)p Fp(:)f(Let)f Fn(U)38 b Fp(b)r(e)515 3220 y(a)27 b(separable)f(Hilb)r(ert)i(space.)f(W)-7 b(e)28 b(consider)f(the)h(probabilit)n(y)e(space)1512 3397 y(\()p Fn(C)1603 3409 y Fk(0)1641 3397 y Fp(\()p Fi(R)p Fn(;)14 b(U)9 b Fp(\))p Fn(;)14 b Fl(B)s Fp(\()p Fn(C)2048 3409 y Fk(0)2091 3397 y Fp(\()p Fi(R)p Fn(;)g(U)9 b Fp(\)\))p Fn(;)14 b Fi(P)p Fp(\))515 3575 y(where)21 b Fn(C)808 3587 y Fk(0)846 3575 y Fp(\()p Fi(R)p Fn(;)14 b(U)9 b Fp(\))28 b(is)22 b(the)g(F)-7 b(r)n(\023)-39 b(ec)n(het)21 b(space)g(of)h(con)n(tin)n(uous)g(functions)g(on)g Fi(R)28 b Fp(whic)n(h)22 b(are)f(zero)g(at)515 3674 y(zero)j(and)i Fl(B)s Fp(\()p Fn(C)997 3686 y Fk(0)1034 3674 y Fp(\()p Fi(R)p Fn(;)14 b(U)9 b Fp(\)\))32 b(is)25 b(the)i(corresp)r(onding)c (Borel)i Fn(\033)s Fp(-algebra.)f(Supp)r(ose)i(that)f(w)n(e)h(ha)n(v)n (e)515 3774 y(a)j(co)n(v)-5 b(ariance)27 b(op)r(erator)g Fn(Q)j Fp(on)f Fn(U)9 b Fp(.)29 b(Then)g Fi(P)f Fp(denotes)h(the)h Fj(Wiener)i(me)l(asur)l(e)d Fp(with)h(resp)r(ect)515 3873 y(to)d Fn(Q)p Fp(.)h(Note)f(that)h Fi(P)f Fp(is)g(ergo)r(dic)f (with)j(resp)r(ect)e(to)g(the)h(\015o)n(w)1196 4051 y Fn(\022)1235 4063 y Fm(t)1264 4051 y Fn(!)e Fp(=)c Fn(!)s Fp(\()p Fl(\001)d Fp(+)f Fn(t)p Fp(\))g Fl(\000)g Fn(!)s Fp(\()p Fn(t)p Fp(\))p Fn(;)181 b Fp(for)27 b Fn(!)e Fl(2)f Fn(C)2499 4063 y Fk(0)2536 4051 y Fp(\()p Fi(R)q Fn(;)14 b(U)9 b Fp(\))p Fn(:)515 4228 y Fp(F)-7 b(or)22 b(detailed)g(presen)n(tation)g(of)h(random)e(dynamical)h(systems)h(w)n (e)f(refer)g(to)g(the)h(monograph)515 4327 y(b)n(y)k(L.)h(Arnold)f ([1].)515 4475 y(On)35 b(a)f Fn(V)55 b Fl(\032)35 b Fn(H)42 b Fl(\032)35 b Fn(V)1218 4444 y Fh(0)1277 4475 y Fp(rigged)f(Hilb)r (ert)h(space)g(with)g(compact)g(em)n(b)r(edding)g Fn(V)55 b Fl(\032)35 b Fn(H)42 b Fp(and)515 4574 y(dualit)n(y)27 b(mapping)g Fl(h\001)p Fn(;)14 b Fl(\001i)29 b Fp(w)n(e)e(in)n(v)n (estigate)f(RDS)i Fn(')g Fp(generated)f(b)n(y)g(the)h(ev)n(olution)f (equation)1382 4738 y Fn(du)p 1382 4776 91 4 v 1384 4852 a(d)14 b(t)1501 4795 y Fp(+)k Fn(A)c(u)23 b Fp(=)g Fn(F)12 b Fp(\()p Fn(u;)i(\022)2040 4807 y Fm(t)2068 4795 y Fn(!)s Fp(\))p Fn(;)97 b(u)p Fp(\(0\))23 b(=)g Fn(x;)751 b Fp(\(1\))1970 5059 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fp(o)n(v)n(er)29 b(some)i(metric)h(dynamical)f (system)g(\(\012)p Fn(;)14 b Fl(F)8 b Fn(;)14 b Fi(P)p Fn(;)g(\022)r Fp(\).)31 b(Here)g Fn(A)h Fp(is)f(a)g(p)r(ositiv)n(e)g (self-adjoin)n(t)515 623 y(op)r(erator)h(in)j Fn(H)41 b Fp(suc)n(h)34 b(that)g Fn(D)r Fp(\()p Fn(A)1615 593 y Fk(1)p Fm(=)p Fk(2)1720 623 y Fp(\))g(=)g Fn(V)19 b Fp(,)34 b(where)g Fn(D)r Fp(\()p Fn(B)t Fp(\))h(denotes)f(the)g(domain) g(of)g(the)515 722 y(op)r(erator)c Fn(B)t Fp(.)i(W)-7 b(e)33 b(supp)r(ose)f(that)g Fn(V)51 b Fp(is)32 b(equipp)r(ed)g(with)h (the)f(norm)g Fl(k)21 b(\001)g(k)2883 734 y Fm(V)2971 722 y Fp(=)30 b Fl(k)p Fn(A)3170 692 y Fk(1)p Fm(=)p Fk(2)3295 722 y Fl(\001)22 b(k)3382 734 y Fm(H)3444 722 y Fp(.)515 822 y(W)-7 b(e)33 b(also)g(assume)f(that)i(the)g(nonlinear)e (mapping)h Fn(F)45 b Fp(from)33 b Fn(V)42 b Fl(\002)21 b Fp(\012)34 b(in)n(to)f Fn(H)40 b Fp(is)33 b(suc)n(h)g(that)515 922 y Fn(F)12 b Fp(\()p Fn(u;)i(!)s Fp(\))25 b(is)h(measurable)e(for)i (an)n(y)f(\014xed)g Fn(u)e Fl(2)g Fn(V)45 b Fp(and)26 b(sub)r(ordinate)f(\(in)h(the)g(sense)g(of)f(\(3\)\))i(to)515 1021 y(the)g(op)r(erator)e Fn(A)p Fp(.)i(W)-7 b(e)27 b(supp)r(ose)g(that)g(the)g(solution)f Fn(u)p Fp(\()p Fn(t;)14 b(!)s Fp(\))27 b(of)f(the)h(problem)g(\(1\))g(is)f(unique)515 1121 y(and)h(dep)r(ends)h(measurably)e(on)i(\()p Fn(t;)14 b(!)s(;)g(x)p Fp(\).)28 b(Then)g(the)g(op)r(erator)1371 1302 y(\()p Fn(t;)14 b(!)s(;)g(x)p Fp(\))24 b Fl(!)f Fn(u)p Fp(\()p Fn(t;)14 b(!)s Fp(\))p Fn(;)180 b(u)p Fp(\(0)p Fn(;)14 b(!)s Fp(\))22 b(=)g Fn(x)515 1484 y Fp(de\014nes)k(a)f(random)g(dynamical)g(system)h(\(co)r(cycle\))g Fn(')p Fp(.)g(In)g(addition,)g(this)g(random)f(dynam-)515 1583 y(ical)i(system)g(is)h(supp)r(osed)f(to)h(b)r(e)g(con)n(tin)n (uous)e(whic)n(h)i(means)f(that)1740 1765 y Fn(x)d Fl(!)f Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)p Fp(\))515 1946 y(is)25 b(con)n(tin)n(uous)f(for)g(an)n(y)h(\()p Fn(t;)14 b(!)s Fp(\).)25 b(The)g(tra)5 b(jectories)23 b(of)i(this)h(random)e (dynamical)g(system)h(has)515 2046 y(to)i(b)r(e)h(con)n(tained)f(in)h Fn(L)1259 2058 y Fk(2)p Fm(;loc)1399 2046 y Fp(\(0)p Fn(;)14 b Fl(1)p Fp(;)g Fn(V)19 b Fp(\))f Fl(\\)h Fn(C)6 b Fp(\([0)p Fn(;)14 b Fl(1)p Fp(\);)g Fn(H)7 b Fp(\).)704 2245 y(In)26 b(the)h(follo)n(wing)e(w)n(e)h(assume)g(that)h Fn(')f Fp(is)h Fj(dissip)l(ative)p Fp(.)h(It)f(means)f(that)h(there)f (exists)g(a)515 2345 y Fj(c)l(omp)l(act)58 b Fp(random)26 b(set)i Fn(B)f Fl(\032)c Fn(V)46 b Fp(whic)n(h)28 b(is)f(forw)n(ard)f (in)n(v)-5 b(arian)n(t:)1444 2526 y Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(B)t Fp(\()p Fn(!)s Fp(\)\))24 b Fl(\032)e Fn(B)t Fp(\()p Fn(\022)2156 2538 y Fm(t)2186 2526 y Fn(!)s Fp(\))p Fn(;)37 b(t)23 b(>)g Fp(0)p Fn(;)515 2708 y Fp(and)g(whic)n(h)h(is)f (absorbing:)f(for)h(an)n(y)g Fn(")g(>)f Fp(0)i(and)f(for)g(an)n(y)g (random)g(v)-5 b(ariable)22 b Fn(x)p Fp(\()p Fn(!)s Fp(\))i Fl(2)f Fn(H)31 b Fp(there)515 2807 y(exists)c(a)g Fn(t)843 2819 y Fm(";x)959 2807 y Fn(>)c Fp(0)k(suc)n(h)g(that)h(if)g Fn(t)23 b Fl(\025)g Fn(t)1730 2819 y Fm(";x)1591 2989 y Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)p Fp(\()p Fn(!)s Fp(\)\))24 b Fl(2)g Fn(B)t Fp(\()p Fn(\022)2275 3001 y Fm(t)2304 2989 y Fn(!)s Fp(\))515 3170 y(with)k(probabilit)n(y)f(1)19 b Fl(\000)f Fn(")p Fp(.)28 b(Note)g(that)g Fn(B)k Fp(is)c(absorbing)f (with)h(probabilit)n(y)f(one,)h(due)g(to)g(the)515 3270 y(forw)n(ard)e(in)n(v)-5 b(ariance.)515 3370 y(A)28 b(random)e(v)-5 b(ariable)27 b Fn(x)c Fl(\025)g Fp(0)k(is)h(called)f(temp)r(ered)h(if) 1533 3605 y(lim)1486 3655 y Fm(t)p Fh(!\0061)1719 3549 y Fp(log)1827 3512 y Fk(+)1896 3549 y Fn(x)p Fp(\()p Fn(\022)2014 3561 y Fm(t)2043 3549 y Fn(!)s Fp(\))p 1719 3586 412 4 v 1887 3662 a Fl(j)p Fn(t)p Fl(j)2163 3605 y Fp(=)23 b(0)83 b(a.s.)515 3831 y(Note)27 b(that)h(the)g(only)f (alternativ)n(e)g(to)g(this)h(prop)r(ert)n(y)f(is)g(that)1432 4066 y(lim)14 b(sup)1454 4132 y Fm(t)p Fh(!\0061)1709 4010 y Fp(log)1817 3973 y Fk(+)1886 4010 y Fn(x)p Fp(\()p Fn(\022)2004 4022 y Fm(t)2034 4010 y Fn(!)s Fp(\))p 1709 4047 V 1877 4123 a Fl(j)p Fn(t)p Fl(j)2154 4066 y Fp(=)22 b Fl(1)83 b Fp(a.s.)p Fn(;)515 4300 y Fp(see)28 b(Arnold)g([1],)g(page) g(164)f(f.)i(W)-7 b(e)29 b(also)e(assume)h(that)h Fn(B)k Fp(is)28 b(temp)r(ered)h(whic)n(h)f(means)g(that)515 4399 y(the)g(mapping)1680 4499 y Fn(!)e Fl(!)76 b Fp(sup)1864 4573 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))2109 4499 y Fl(k)p Fn(x)p Fl(k)2240 4511 y Fm(H)515 4710 y Fp(is)27 b(temp)r(ered.)704 4810 y(W)-7 b(e)28 b(no)n(w)f(giv)n(e)f (our)h(basic)g(de\014nition:)1970 5059 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fg(De\014nition)31 b(2.1)40 b Fj(A)d(set)g Fl(L)g Fp(=)g Fl(f)p Fn(l)1623 535 y Fm(j)1657 523 y Fn(;)51 b(j)42 b Fp(=)37 b(1)p Fn(;)14 b Fl(\001)g(\001)g(\001)f Fn(;)h(k)s Fl(g)36 b Fj(of)j(line)l(ar)f(c)l(ontinuous)e(and)i(line)l (arly)515 623 y(indep)l(endent)28 b(functionals)g(on)f Fn(V)47 b Fj(is)28 b(c)l(al)t(le)l(d)g(asymptotic)l(al)t(ly)i (determining)e(in)g(pr)l(ob)l(ability)i(if)994 845 y Fp(\()p Fi(P)p Fp(\))k(lim)1124 895 y Fm(t)p Fh(!1)1295 732 y Ff(Z)1378 753 y Fm(t)p Fk(+1)1341 921 y Fm(t)1505 845 y Fp(max)1567 897 y Fm(j)1673 845 y Fl(j)p Fn(l)1721 857 y Fm(j)1756 845 y Fp(\()p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)2091 857 y Fk(1)2129 845 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)2566 857 y Fk(2)2604 845 y Fp(\)\))p Fl(j)2691 811 y Fk(2)2728 845 y Fn(d\034)33 b Fl(!)24 b Fp(0)515 1055 y Fj(for)30 b(two)g(initial)h(c)l(onditions)g Fn(x)1490 1067 y Fk(1)1528 1055 y Fn(;)c(x)1625 1067 y Fk(2)1686 1055 y Fl(2)d Fn(H)36 b Fj(implies)1258 1223 y Fp(\()p Fi(P)p Fp(\))f(lim)1388 1272 y Fm(t)p Fh(!1)1559 1223 y Fl(k)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1893 1235 y Fk(1)1930 1223 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(x)2356 1235 y Fk(2)2393 1223 y Fp(\))p Fl(k)2467 1235 y Fm(H)2553 1223 y Fl(!)23 b Fp(0)p Fn(:)515 1419 y Fp(As)38 b(in)h([7)o(,)g(8)o(]) f(w)n(e)g(use)g(the)h(concept)f(of)g(the)h Fj(c)l(ompleteness)h(defe)l (ct)f Fp(for)f(a)g(description)g(of)515 1519 y(sets)f(of)h(determining) f(functionals.)g(Assume)h(that)g Fn(X)44 b Fp(and)37 b Fn(Y)56 b Fp(are)37 b(Banac)n(h)f(spaces)h(and)515 1619 y Fn(X)45 b Fp(con)n(tin)n(uously)38 b(and)g(densely)h(em)n(b)r (edded)g(in)n(to)g Fn(Y)18 b Fp(.)39 b(Let)g Fl(L)j Fp(=)g Fl(f)p Fn(l)2744 1631 y Fm(j)2820 1619 y Fp(:)f Fn(j)47 b Fp(=)41 b(1)p Fn(;)14 b(:::;)g(k)s Fl(g)38 b Fp(b)r(e)515 1718 y(a)d(\014nite)i(set)f(of)g(linearly)f(indep)r(enden)n(t)i(con)n (tin)n(uous)e(functionals)h(on)g Fn(X)7 b Fp(.)35 b(W)-7 b(e)37 b(de\014ne)f(the)515 1818 y(completeness)d(defect)i Fn(")1308 1830 y Fh(L)1358 1818 y Fp(\()p Fn(X)r(;)14 b(Y)k Fp(\))34 b Fl(\021)g Fn(")1768 1830 y Fh(L)1852 1818 y Fp(of)g(the)g(set)h Fl(L)f Fp(with)h(resp)r(ect)e(to)h(the)h (pair)e(of)h(the)515 1917 y(spaces)26 b Fn(X)34 b Fp(and)28 b Fn(Y)46 b Fp(b)n(y)27 b(the)h(form)n(ula)963 2085 y Fn(")1002 2097 y Fh(L)1075 2085 y Fp(=)23 b(sup)p Fl(fk)p Fn(w)r Fl(k)1475 2097 y Fm(Y)1578 2085 y Fp(:)g Fn(w)j Fl(2)d Fn(X)r(;)37 b(l)1943 2097 y Fm(j)1977 2085 y Fp(\()p Fn(w)r Fp(\))25 b(=)d(0)p Fn(;)37 b(l)2341 2097 y Fm(j)2399 2085 y Fl(2)23 b(L)p Fn(;)37 b Fl(k)p Fn(w)r Fl(k)2739 2097 y Fm(X)2825 2085 y Fl(\024)23 b Fp(1)p Fl(g)p Fn(:)515 2253 y Fp(The)h(v)-5 b(alue)23 b Fn(\017)926 2265 y Fh(L)1000 2253 y Fp(is)h(pro)n(v)n(ed)e(to)i(b)r(e)h(v)n(ery)d(useful)j(for)e(c)n (haracterization)e(of)j(sets)g(of)g(determining)515 2353 y(functionals)35 b(\(see,)h(e.g.,)g([7)o(,)g(8])g(and)f(the)h (references)f(therein\).)h(One)g(can)f(sho)n(w)g(that)h(the)515 2452 y(completeness)e(defect)h Fn(")1309 2464 y Fh(L)1359 2452 y Fp(\()p Fn(X)r(;)14 b(Y)k Fp(\))36 b(is)e(the)h(b)r(est)g(p)r (ossible)g(global)e(error)g(of)h(appro)n(ximation)515 2563 y(in)41 b Fn(Y)59 b Fp(of)41 b(elemen)n(ts)g Fn(u)k Fl(2)g Fn(X)i Fp(b)n(y)41 b(elemen)n(ts)f(of)h(the)g(form)g Fn(u)2505 2575 y Fh(L)2600 2563 y Fp(=)2709 2501 y Ff(P)2797 2522 y Fm(k)2797 2588 y(j)s Fk(=1)2930 2563 y Fn(l)2955 2575 y Fm(j)2990 2563 y Fp(\()p Fn(u)p Fp(\))p Fn(')3156 2575 y Fm(j)3191 2563 y Fp(,)g(where)515 2663 y Fl(f)p Fn(')611 2675 y Fm(j)669 2663 y Fp(:)23 b Fn(j)28 b Fp(=)22 b(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(k)s Fl(g)25 b Fp(is)f(an)h(arbitrary)e (set)i(in)g Fn(X)7 b Fp(.)24 b(The)h(smallness)f(of)h Fn(")2753 2675 y Fh(L)2803 2663 y Fp(\()p Fn(X)r(;)14 b(Y)k Fp(\))26 b(is)e(the)i(main)515 2763 y(condition)33 b(\(see)g(the)g(results)g(presen)n(ted)f(b)r(elo)n(w\))h(that)h(guaran) n(tee)d(the)j(prop)r(ert)n(y)e(of)h(a)g(set)515 2862 y(of)d(functionals)g(to)f(b)r(e)i(asymptotically)e(determining.)h(The)g (so-called)e(mo)r(des,)i(no)r(des)g(and)515 2962 y(lo)r(cal)25 b(v)n(olume)g(a)n(v)n(erages)e(\(the)k(description)e(of)h(these)g (functionals)f(can)h(b)r(e)g(found)g(in)g([8],)g(for)515 3061 y(instance\))k(are)f(the)h(main)g(examples)g(of)g(sets)f(of)h (functionals)g(with)h(a)e(small)h(completeness)515 3161 y(defect.)d(F)-7 b(or)26 b(further)h(discussions)e(and)i(for)f(other)g (prop)r(erties)g(of)h(the)g(completeness)f(defect)515 3261 y(w)n(e)h(refer)g(to)g([7,)h(8)o(].)g(Here)f(w)n(e)g(only)h(p)r (oin)n(t)f(out)h(the)g(follo)n(wing)e(estimate)1113 3428 y Fl(k)p Fn(u)p Fl(k)1245 3440 y Fm(Y)1324 3428 y Fl(\024)c Fn(")1450 3440 y Fh(L)1519 3428 y Fl(\001)c(k)p Fn(u)p Fl(k)1692 3440 y Fm(X)1772 3428 y Fp(+)g Fn(C)1914 3440 y Fh(L)1983 3428 y Fl(\001)67 b Fp(max)2025 3483 y Fm(j)s Fk(=1)p Fm(;:::;k)2289 3428 y Fl(j)p Fn(l)2337 3440 y Fm(j)2372 3428 y Fp(\()p Fn(u)p Fp(\))p Fl(j)p Fn(;)97 b(u)22 b Fl(2)i Fn(X)r(;)491 b Fp(\(2\))515 3636 y(where)27 b Fn(C)814 3648 y Fh(L)887 3636 y Fn(>)c Fp(0)k(is)h(a)f(constan)n(t)g (dep)r(ending)h(on)f Fl(L)p Fp(.)704 3780 y(W)-7 b(e)29 b(are)f(no)n(w)h(in)g(a)g(p)r(osition)f(to)h(pro)n(v)n(e)f(the)h (\014rst)g(main)g(theorem)g(for)f(systems)h(in)n(tro-)515 3879 y(duced)h(in)g(\(1\).)g(T)-7 b(o)29 b(do)h(this)g(w)n(e)g(will)g (use)f(the)i(completeness)e(defect)h Fn(")2780 3891 y Fh(L)2857 3879 y Fl(\021)c Fn(")2987 3891 y Fh(L)3037 3879 y Fp(\()p Fn(X)r(;)14 b(Y)19 b Fp(\))30 b(with)515 3979 y Fn(H)g Fp(=)22 b Fn(Y)5 b(;)28 b(V)42 b Fp(=)22 b Fn(X)7 b Fp(.)515 4133 y Fg(Theorem)30 b(2.2)41 b Fj(L)l(et)26 b Fl(L)d Fp(=)g Fl(f)p Fn(l)1459 4145 y Fm(j)1516 4133 y Fp(:)g Fn(j)28 b Fp(=)23 b(1)p Fn(;)14 b(:::;)g(k)s Fl(g)25 b Fj(b)l(e)h(a)h(set)f(of)i(line)l(ar)f(c)l(ontinuous)e(and)i (line)l(arly)515 4233 y(indep)l(endent)32 b(functionals)h(on)e Fn(V)19 b Fj(.)33 b(We)e(assume)h(that)g(we)g(have)h(a)f(forwar)l(d)i (absorbing)f(and)515 4332 y(forwar)l(d)f(invariant)g(set)e Fn(B)35 b Fj(in)30 b Fn(V)50 b Fj(such)31 b(that)f Fp(sup)2093 4353 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))2342 4332 y Fl(k)p Fn(x)p Fl(k)2473 4302 y Fk(2)2473 4355 y Fm(V)2561 4332 y Fj(is)h(b)l(ounde)l(d)g(by)g(a)g(temp)l(er)l(e)l(d) 515 4443 y(r)l(andom)24 b(variable)h(and)f Fn(t)f Fl(!)g Fp(sup)1554 4463 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(\022)1748 4471 y Fe(t)1774 4463 y Fm(!)r Fk(\))1862 4443 y Fl(k)p Fn(x)p Fl(k)1993 4413 y Fk(2)1993 4466 y Fm(V)2074 4443 y Fj(is)g(lo)l(c)l(al)t(ly)i(inte)l(gr)l(able.)g(Supp)l(ose)e(ther)l(e) h(exist)515 4543 y(a)35 b(c)l(onstant)g Fn(c)e(>)g Fp(0)i Fj(and)g(a)h(me)l(asur)l(able)g(function)f Fn(l)g Fl(\025)e Fp(0)h Fj(such)i(that)f(for)h Fn(x)2967 4555 y Fk(1)3005 4543 y Fp(\()p Fn(!)s Fp(\))p Fn(;)28 b(x)3222 4555 y Fk(2)3260 4543 y Fp(\()p Fn(!)s Fp(\))33 b Fl(2)515 4642 y Fn(B)t Fp(\()p Fn(!)s Fp(\))d Fj(we)g(have)1167 4810 y Fl(h\000)p Fn(A)p Fp(\()p Fn(x)1405 4822 y Fk(1)1461 4810 y Fl(\000)18 b Fn(x)1591 4822 y Fk(2)1629 4810 y Fp(\))h(+)f Fn(F)12 b Fp(\()p Fn(x)1907 4822 y Fk(1)1945 4810 y Fn(;)i(!)s Fp(\))k Fl(\000)g Fn(F)12 b Fp(\()p Fn(x)2314 4822 y Fk(2)2352 4810 y Fn(;)i(!)s Fp(\))p Fn(;)g(x)2560 4822 y Fk(1)2616 4810 y Fl(\000)k Fn(x)2746 4822 y Fk(2)2783 4810 y Fl(i)546 b Fp(\(3\))1970 5059 y(5)p eop %%Page: 6 6 6 5 bop 1226 523 a Fl(\024)23 b(\000)p Fn(c)p Fl(k)p Fn(x)1504 535 y Fk(1)1559 523 y Fl(\000)18 b Fn(x)1689 535 y Fk(2)1727 523 y Fl(k)1769 489 y Fk(2)1769 544 y Fm(V)1844 523 y Fp(+)g Fn(l)r Fp(\()p Fn(x)2033 535 y Fk(1)2071 523 y Fn(;)c(x)2155 535 y Fk(2)2192 523 y Fn(;)g(!)s Fp(\))p Fl(k)p Fn(x)2405 535 y Fk(1)2461 523 y Fl(\000)k Fn(x)2591 535 y Fk(2)2629 523 y Fl(k)2671 489 y Fk(2)2671 544 y Fm(H)2733 523 y Fn(:)515 664 y Fj(Assume)29 b(that)926 831 y Fp(1)p 910 868 73 4 v 910 944 a Fn(m)993 887 y Fi(E)1056 745 y Ff(\()1243 887 y Fp(sup)1129 961 y Fm(x)1167 969 y Fd(1)1199 961 y Fm(;x)1257 969 y Fd(2)1289 961 y Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))1496 774 y Ff(Z)1579 795 y Fm(m)1542 963 y Fk(0)1656 887 y Fn(l)r Fp(\()p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)2007 899 y Fk(1)2044 887 y Fp(\))p Fn(;)g(')p Fp(\()p Fn(t;)g(!)s(;)g(x)2405 899 y Fk(2)2443 887 y Fp(\))p Fn(;)g(\022)2551 899 y Fm(t)2581 887 y Fn(!)s Fp(\))p Fn(dt)2741 745 y Ff(\))2831 887 y Fn(<)22 b(c")2993 852 y Fh(\000)p Fk(2)2993 911 y Fh(L)3361 887 y Fp(\(4\))515 1126 y Fj(for)39 b(some)g Fn(m)f(>)h Fp(0)p Fj(.)f(Then)h Fl(L)g Fj(is)f(a)h(set)f(of)h (asymptotic)l(al)t(ly)i(determining)e(functionals)g(in)515 1226 y(pr)l(ob)l(ability)32 b(for)e(RDS)f Fp(\()p Fn(\022)r(;)14 b(')p Fp(\))p Fj(.)704 1381 y(Pr)l(o)l(of.)31 b Fp(1\))e(Without)h (loss)f(of)g(generalit)n(y)-7 b(,)28 b(w)n(e)h(only)g(consider)g(the)g (case)g Fn(m)d Fp(=)g(1.)j(That)515 1480 y(is,)36 b(w)n(e)h(assume)f (that)h(\(4\))g(is)f(ful\014lled)i(for)e Fn(m)i Fp(=)g(1.)f(Since)g(w)n (e)f(in)n(tend)h(to)g(pro)n(v)n(e)e(con)n(v)n(er-)515 1580 y(gence)c(in)i(probabilit)n(y)e(w)n(e)g(can)h(supp)r(ose)g(that)g (the)h(random)e(v)-5 b(ariables)31 b Fn(x)2913 1592 y Fk(1)2950 1580 y Fp(\()p Fn(!)s Fp(\))p Fn(;)d(x)3167 1592 y Fk(2)3205 1580 y Fp(\()p Fn(!)s Fp(\))33 b(are)515 1680 y(con)n(tained)g(in)h Fn(B)t Fp(\()p Fn(!)s Fp(\).)g(Suc)n(h)f (random)g(v)-5 b(ariables)32 b(exist)i(b)r(ecause)f Fn(B)38 b Fp(is)33 b(a)h(random)e(set,)i(see)515 1779 y(Caistaing)28 b(and)h(V)-7 b(aladier)28 b([5])h(Chapter)g(I)r(I)r(I.)h(Indeed,)f Fn(B)34 b Fp(is)29 b Fj(forwar)l(d)i Fp(absorbing)c(suc)n(h)i(that)515 1879 y Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)807 1891 y Fm(i)835 1879 y Fp(\()p Fn(!)s Fp(\)\))23 b Fl(2)h Fn(B)t Fp(\()p Fn(\022)1226 1891 y Fm(t)1255 1879 y Fn(!)s Fp(\))g(with)g (probabilit)n(y)f(1)11 b Fl(\000)g Fn(")21 b Fp(for)j(an)n(y)e Fn(")h(>)g Fp(0)g(if)h Fn(t)g Fp(is)f(su\016cien)n(tly)h(large.)704 2078 y(Let)g Fn(w)r Fp(\()p Fn(t;)14 b(!)s Fp(\))24 b(b)r(e)g (de\014ned)g(b)n(y)f Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1914 2090 y Fk(1)1952 2078 y Fp(\()p Fn(!)s Fp(\)\))d Fl(\000)g Fn(')p Fp(\()p Fn(t;)j(!)s(;)g(x)2482 2090 y Fk(2)2519 2078 y Fp(\()p Fn(!)s Fp(\)\).)24 b(Since)g Fl(k)11 b(\001)g(k)3059 2090 y Fm(V)3138 2078 y Fp(=)23 b Fl(k)p Fn(A)3330 2048 y Fk(1)p Fm(=)p Fk(2)3444 2078 y Fl(\001)515 2178 y(k)557 2190 y Fm(H)619 2178 y Fp(,)28 b(w)n(e)f(obtain)g(b)n(y)h(\(3\):)852 2344 y Fn(d)p Fl(k)p Fn(w)r Fl(k)1040 2313 y Fk(2)1040 2366 y Fm(H)p 852 2381 251 4 v 940 2457 a Fn(dt)1131 2400 y Fp(+)18 b(2)p Fn(c)p Fl(k)p Fn(w)r Fl(k)1437 2365 y Fk(2)1437 2420 y Fm(V)1517 2400 y Fl(\024)k Fp(2)p Fn(l)r Fp(\()p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1997 2412 y Fk(1)2034 2400 y Fp(\()p Fn(!)s Fp(\)\))p Fn(;)g(')p Fp(\()p Fn(t;)g(!)s(;)g(x)2514 2412 y Fk(2)2552 2400 y Fp(\()p Fn(!)s Fp(\)\))p Fn(;)g(\022)2779 2412 y Fm(t)2809 2400 y Fn(!)s Fp(\))p Fl(k)p Fn(w)r Fl(k)3041 2365 y Fk(2)3041 2420 y Fm(H)3118 2400 y Fn(:)515 2597 y Fp(W)-7 b(e)28 b(ha)n(v)n(e)e(b)n(y)h(\(2\))1084 2765 y Fl(k)p Fn(w)r Fl(k)1229 2731 y Fk(2)1229 2786 y Fm(V)1309 2765 y Fl(\025)c Fp(\(1)18 b(+)g Fn(\016)s Fp(\))1644 2731 y Fh(\000)p Fk(1)1733 2765 y Fn(")1772 2730 y Fh(\000)p Fk(2)1772 2790 y Fh(L)1861 2765 y Fl(k)p Fn(w)r Fl(k)2006 2731 y Fk(2)2006 2786 y Fm(H)2087 2765 y Fl(\000)g Fn(C)2229 2777 y Fm(\016)o(;)p Fh(L)2390 2765 y Fp(max)2342 2820 y Fm(j)s Fk(=1)p Fm(;)p Fh(\001\001\001)p Fm(;k)2606 2765 y Fl(j)p Fn(l)2654 2777 y Fm(j)2689 2765 y Fp(\()p Fn(w)r Fp(\))p Fl(j)2837 2731 y Fk(2)2876 2765 y Fn(:)515 2974 y Fp(for)26 b(an)n(y)g Fn(\016)g(>)d Fp(0)j(with)h(appropriate)e (p)r(ositiv)n(e)i(constan)n(t)f Fn(C)2351 2986 y Fm(\016)o(;)p Fh(L)2449 2974 y Fp(.)h(This)g(allo)n(ws)e(us)i(to)g(write)f(the)515 3074 y(follo)n(wing)g(inequalit)n(y:)795 3296 y Fl(k)p Fn(w)r Fp(\()p Fn(t)p Fp(\))p Fl(k)1034 3262 y Fk(2)1034 3316 y Fm(H)1120 3296 y Fl(\024)d(k)p Fn(w)r Fp(\(0\))p Fl(k)1459 3262 y Fk(2)1459 3316 y Fm(H)1522 3296 y Fn(e)1561 3191 y Ff(R)1616 3211 y Fe(t)1600 3287 y Fd(0)1655 3251 y Fm(q)r Fk(\()p Fm(s;!)r Fk(\))p Fm(ds)1950 3296 y Fp(+)18 b Fn(C)2092 3308 y Fm(\016)o(;)p Fh(L)2210 3296 y Fl(\001)2251 3183 y Ff(Z)2334 3203 y Fm(t)2297 3372 y Fk(0)2377 3296 y Fn(e)2416 3191 y Ff(R)2471 3211 y Fe(t)2455 3287 y(s)2510 3251 y Fm(q)r Fk(\()p Fm(\034)s(;!)r Fk(\))p Fm(d\034)2768 3296 y Fn(\021)2809 3308 y Fh(L)2859 3296 y Fp(\()p Fn(s;)c(!)s Fp(\))p Fn(ds)28 b(;)174 b Fp(\(5\))515 3517 y(where)27 b Fn(\021)796 3529 y Fh(L)846 3517 y Fp(\()p Fn(s;)14 b(!)s Fp(\))23 b(=)g(max)1306 3529 y Fm(j)1355 3517 y Fl(j)p Fn(l)1403 3529 y Fm(j)1438 3517 y Fp(\()p Fn(w)r Fp(\()p Fn(s;)14 b(!)s Fp(\)\))p Fl(j)1781 3487 y Fk(2)1847 3517 y Fp(and)844 3686 y Fn(q)s Fp(\()p Fn(t;)g(!)s Fp(\))23 b(=)f(2)p Fn(l)r Fp(\()p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1573 3698 y Fk(1)1610 3686 y Fp(\()p Fn(!)s Fp(\)\))p Fn(;)g(')p Fp(\()p Fn(t;)g(!)s(;)g(x)2090 3698 y Fk(2)2129 3686 y Fp(\()p Fn(!)s Fp(\)\))p Fn(;)g(\022)2356 3698 y Fm(t)2385 3686 y Fn(!)s Fp(\))19 b Fl(\000)f Fp(2)p Fn(c)p Fp(\(1)f(+)h Fn(\016)s Fp(\))2898 3651 y Fh(\000)p Fk(1)2988 3686 y Fn(")3027 3650 y Fh(\000)p Fk(2)3027 3710 y Fh(L)3116 3686 y Fn(:)515 3854 y Fp(Let)701 4062 y Fn(Q)p Fp(\()p Fn(!)s Fp(\))23 b(=)137 b(sup)996 4135 y Fm(x)1034 4143 y Fd(1)1066 4135 y Fm(;x)1124 4143 y Fd(2)1156 4135 y Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))1364 3949 y Ff(Z)1447 3969 y Fk(1)1410 4138 y(0)1498 4062 y Fp(2)p Fn(l)r Fp(\()p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1891 4074 y Fk(1)1928 4062 y Fp(\))p Fn(;)g(')p Fp(\()p Fn(t;)g(!)s(;)g(x)2289 4074 y Fk(2)2327 4062 y Fp(\))p Fn(;)g(\022)2435 4074 y Fm(t)2464 4062 y Fn(!)s Fp(\))p Fn(dt)19 b Fl(\000)f Fp(2)c Fn(c)p Fp(\(1)j(+)h Fn(\016)s Fp(\))3064 4028 y Fh(\000)p Fk(1)3154 4062 y Fn(")3193 4026 y Fh(\000)p Fk(2)3193 4086 y Fh(L)515 4298 y Fp(with)28 b Fn(\016)e(>)d Fp(0)k(c)n(hosen)f(suc)n(h)i(that)g Fi(E)8 b Fn(Q)29 b(<)22 b Fp(0.)27 b(This)h(is)g(p)r(ossible)f(b)r(ecause)g(of)h(\(4\).) 704 4497 y(2\))d(Since)h Fn(B)k Fp(is)c(forw)n(ard)e(in)n(v)-5 b(arian)n(t)24 b(and)i Fn(q)s Fp(\()p Fn(t;)14 b(!)s Fp(\))23 b Fl(\025)f(\000)p Fp(2)p Fn(c)p Fp(\(1)14 b(+)g Fn(\016)s Fp(\))2720 4467 y Fh(\000)p Fk(1)2809 4497 y Fn(")2848 4462 y Fh(\000)p Fk(2)2848 4522 y Fh(L)2937 4497 y Fp(,)26 b(the)g(\014rst)g(term)515 4597 y(on)h(the)h(righ)n(t)f (hand)g(side)h(of)g(\(5\))f(can)g(b)r(e)h(estimated)g(b)n(y)1366 4810 y Fl(k)p Fn(w)r Fp(\(0\))p Fl(k)1617 4776 y Fk(2)1617 4831 y Fm(H)1679 4810 y Fn(e)1718 4706 y Ff(P)1806 4726 y Fd([)p Fe(t)p Fd(])1806 4793 y Fe(j)r Fd(=0)1919 4762 y Fm(Q)p Fk(\()p Fm(\022)2029 4770 y Fe(j)2060 4762 y Fm(!)r Fk(\))2134 4810 y Fn(e)2173 4776 y Fk(2)p Fm(c)p Fk(\(1+)p Fm(\016)r Fk(\))2404 4751 y Fc(\000)p Fd(1)2481 4776 y Fm(")2512 4749 y Fc(\000)p Fd(2)2512 4797 y Fc(L)2594 4810 y Fn(:)1970 5059 y Fp(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fp(Since)28 b Fi(E)8 b Fn(Q)29 b(<)22 b Fp(0,)27 b(b)n(y)h(the)g(ergo)r(dic)e(theorem)h(w)n(e)h(ha)n(v)n(e)e (for)h Fn(t)c Fl(!)g(1)1427 681 y Fk([)p Fm(t)p Fk(])1399 714 y Ff(X)1401 890 y Fm(j)s Fk(=0)1533 792 y Fn(Q)p Fp(\()p Fn(\022)1670 804 y Fm(j)1704 792 y Fn(!)s Fp(\))g Fl(\030)g Fp(\([)p Fn(t)p Fp(])c(+)f(1\))p Fi(E)8 b Fn(Q)29 b Fl(!)23 b(\0001)777 b Fp(\(6\))515 1058 y(whic)n(h)27 b(sho)n(ws)g(the)h(con)n(v)n(ergence)d(assertion)h(for)h(the)h(\014rst) f(term.)515 1158 y(W)-7 b(e)25 b(no)n(w)e(in)n(v)n(estigate)h(the)g (second)g(term)h(in)f(\(5\).)h(Since)g Fn(B)j Fp(is)d(forw)n(ard)d(in)n (v)-5 b(arian)n(t,)24 b(this)h(term)515 1257 y(can)i(b)r(e)h(estimated) g(b)n(y)934 1500 y Fn(C)993 1512 y Fm(\016)o(;)p Fh(L)1110 1500 y Fl(\001)1152 1387 y Ff(Z)1235 1407 y Fk([)p Fm(t)p Fk(]+1)1198 1576 y(0)1400 1500 y Fn(e)1439 1384 y Ff(R)1494 1405 y Fd([)p Fe(t)p Fd(]+1)1478 1481 y([)p Fe(s)p Fd(])1637 1445 y Fm(q)r Fk(\()p Fm(\034)s(;!)r Fk(\))p Fm(d\034)1895 1500 y Fn(\021)1936 1512 y Fh(L)1986 1500 y Fp(\()p Fn(s;)14 b(!)s Fp(\))p Fn(ds)g(e)2316 1466 y Fk(4)p Fm(c)p Fk(\(1+)p Fm(\016)r Fk(\))2547 1441 y Fc(\000)p Fd(1)2624 1466 y Fm(")2655 1439 y Fc(\000)p Fd(2)2655 1487 y Fc(L)1100 1772 y Fl(\024)23 b Fn(C)1247 1784 y Fm(\016)o(;)p Fh(L)1364 1772 y Fl(\001)c Fn(e)1445 1737 y Fk(4)p Fm(c)p Fk(\(1+)p Fm(\016)r Fk(\))1676 1712 y Fc(\000)p Fd(1)1753 1737 y Fm(")1784 1710 y Fc(\000)p Fd(2)1784 1759 y Fc(L)1908 1661 y Fk([)p Fm(t)p Fk(])1879 1693 y Ff(X)1882 1870 y Fm(j)s Fk(=0)2013 1772 y Fn(e)2052 1667 y Ff(P)2139 1688 y Fd([)p Fe(t)p Fd(])2139 1755 y Fe(j)2165 1743 y Fc(0)2189 1755 y Fd(=)p Fe(j)2274 1723 y Fm(Q)p Fk(\()p Fm(\022)2384 1739 y Fe(j)2410 1727 y Fc(0)2437 1723 y Fm(!)r Fk(\))2525 1659 y Ff(Z)2608 1679 y Fk(1)2571 1847 y(0)2659 1772 y Fn(\021)2700 1784 y Fh(L)2750 1772 y Fp(\()p Fn(s)g Fp(+)f Fn(j;)c(!)s Fp(\))p Fn(ds)28 b(:)515 2042 y Fp(W)-7 b(e)22 b(use)h(here)e(that)i Fn(l)r Fp(\()p Fn(x)1246 2054 y Fk(1)1283 2042 y Fn(;)14 b(x)1367 2054 y Fk(2)1405 2042 y Fn(;)g(!)s Fp(\))22 b(is)g(a)g(nonnegativ)n(e)f (function.)i(Th)n(us)f(w)n(e)g(ha)n(v)n(e)f(to)i(pro)n(v)n(e)d(that) 1037 2305 y(\()p Fi(P)p Fp(\))54 b(lim)1180 2360 y Fm(k)q Fh(!1)1405 2202 y Fm(k)1363 2227 y Ff(X)1366 2403 y Fm(j)s Fk(=0)1497 2188 y Ff(\022)1558 2305 y Fn(e)1597 2201 y Ff(P)1684 2222 y Fe(k)1684 2288 y(j)1710 2276 y Fc(0)1734 2288 y Fd(=)p Fe(j)1819 2257 y Fm(Q)p Fk(\()p Fm(\022)1929 2273 y Fe(j)1955 2261 y Fc(0)1982 2257 y Fm(!)r Fk(\))2070 2192 y Ff(Z)2153 2213 y Fk(1)2116 2381 y(0)2204 2305 y Fn(\021)2245 2317 y Fh(L)2295 2305 y Fp(\()p Fn(s)19 b Fp(+)f Fn(j;)c(!)s Fp(\))p Fn(ds)2708 2188 y Ff(\023)2793 2305 y Fp(=)22 b(0)p Fn(:)416 b Fp(\(7\))515 2571 y(W)-7 b(e)28 b(no)n(w)f(replace)f Fn(!)31 b Fp(b)n(y)c Fn(\022)1349 2583 y Fh(\000)p Fm(k)1442 2571 y Fn(!)j Fp(in)e(the)g(relation)e (under)i(the)g(limit)g(sign.)f(It)h(giv)n(es)1241 2727 y Fk(0)1198 2751 y Ff(X)1173 2930 y Fm(j)s Fk(=)p Fh(\000)p Fm(k)1357 2830 y Fn(e)1396 2726 y Ff(P)1483 2747 y Fd(0)1483 2813 y Fe(j)1509 2801 y Fc(0)1533 2813 y Fd(=)p Fe(j)1618 2782 y Fm(Q)p Fk(\()p Fm(\022)1728 2797 y Fe(j)1754 2785 y Fc(0)1781 2782 y Fm(!)r Fk(\))1869 2717 y Ff(Z)1952 2738 y Fk(1)1915 2906 y(0)2003 2830 y Fn(\021)2044 2842 y Fh(L)2094 2830 y Fp(\()p Fn(s)19 b Fp(+)f Fn(k)j Fp(+)d Fn(j;)c(\022)2524 2842 y Fh(\000)p Fm(k)2617 2830 y Fn(!)s Fp(\))p Fn(ds;)515 3098 y Fp(whic)n(h)27 b(is)h(equal)f(to)1155 3250 y Fk(0)1111 3275 y Ff(X)1071 3452 y Fm(j)s Fk(=)p Fh(\0001)1285 3354 y Fn(e)1324 3250 y Ff(P)1411 3270 y Fd(0)1411 3337 y Fe(j)1437 3325 y Fc(0)1461 3337 y Fd(=)p Fe(j)1546 3306 y Fm(Q)p Fk(\()p Fm(\022)1656 3321 y Fe(j)1682 3309 y Fc(0)1709 3306 y Fm(!)r Fk(\))1783 3354 y Fn(\037)1835 3366 y Fm(k)1876 3354 y Fp(\()p Fn(j)5 b Fp(\))1993 3241 y Ff(Z)2076 3261 y Fk(1)2039 3430 y(0)2127 3354 y Fn(\021)2168 3366 y Fh(L)2219 3354 y Fp(\()p Fn(s)18 b Fp(+)g Fn(k)k Fp(+)c Fn(j;)c(\022)2649 3366 y Fh(\000)p Fm(k)2742 3354 y Fn(!)s Fp(\))p Fn(ds)515 3624 y Fp(where)27 b Fn(\037)807 3636 y Fm(k)848 3624 y Fp(\()p Fn(j)5 b Fp(\))23 b(=)g(1)k(if)h Fn(j)g Fl(\025)23 b(\000)p Fn(k)30 b Fp(and)d(0)g(otherwise.)515 3724 y(3\))20 b(Since)h(w)n(e)g(can)f (assume)g(that)h Fn(x)1580 3736 y Fm(i)1608 3724 y Fp(\()p Fn(!)s Fp(\))i Fl(2)h Fn(B)t Fp(\()p Fn(!)s Fp(\))d(there)f(exists)h(a) f(temp)r(ered)h(random)f(v)-5 b(ariable)515 3823 y Fn(b)32 b Fp(suc)n(h)g(that)g Fn(\021)1000 3835 y Fh(L)1051 3823 y Fp(\()p Fn(s;)14 b(!)s Fp(\))31 b Fl(\024)f Fn(b)p Fp(\()p Fn(\022)1479 3835 y Fm(s)1515 3823 y Fn(!)s Fp(\))i(where)g Fn(s)f Fl(!)f Fn(b)p Fp(\()p Fn(\022)2169 3835 y Fm(s)2205 3823 y Fn(!)s Fp(\))i(is)g(lo)r(cally)g(in)n(tegrable.)f(Since)i Fn(s)d Fl(!)515 3923 y Fn(b)p Fp(\()p Fn(\022)622 3935 y Fm(s)657 3923 y Fn(!)s Fp(\))e(is)f(temp)r(ered)1065 4164 y Fn(j)h Fl(!)23 b Fn(\037)1285 4176 y Fm(k)1326 4164 y Fp(\()p Fn(j)5 b Fp(\))1443 4051 y Ff(Z)1527 4072 y Fk(1)1490 4240 y(0)1578 4164 y Fn(\021)1619 4176 y Fh(L)1669 4164 y Fp(\()p Fn(s)19 b Fp(+)f Fn(k)j Fp(+)d Fn(j;)c(\022)2099 4176 y Fh(\000)p Fm(k)2192 4164 y Fn(!)s Fp(\))23 b Fl(\024)2390 4051 y Ff(Z)2473 4072 y Fk(1)2436 4240 y(0)2524 4164 y Fn(b)p Fp(\()p Fn(\022)2631 4176 y Fm(s)p Fk(+)p Fm(j)2748 4164 y Fn(!)s Fp(\))p Fn(ds)515 4392 y Fp(has)d(a)g(sub)r(exp)r(onen)n(tial)h(gro)n(wth)e(for)h(an)n(y) g Fn(k)26 b Fl(\025)c Fp(0.)f(W)-7 b(e)21 b(consider)e(the)i(\014nite)h (measure)d Fn(\026)3228 4362 y Fm(!)3276 4392 y Fp(\()p Fn(j)5 b Fp(\))24 b(=)515 4532 y Fn(e)564 4461 y Fd(1)p 563 4470 29 4 v 563 4504 a(2)613 4428 y Ff(P)701 4448 y Fd(0)701 4515 y Fe(j)727 4503 y Fc(0)750 4515 y Fd(=)p Fe(j)835 4484 y Fm(Q)p Fk(\()p Fm(\022)945 4499 y Fe(j)971 4487 y Fc(0)998 4484 y Fm(!)r Fk(\))1072 4532 y Fn(\016)1109 4544 y Fm(j)1172 4532 y Fp(on)j Fi(Z)1348 4502 y Fh(\000)1426 4532 y Fp(where)g Fn(\016)1703 4544 y Fm(j)1766 4532 y Fp(are)f(Dirac)h(measures)g(on)g Fn(j)5 b Fp(.)28 b(Set)909 4776 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))23 b(:=)f Fn(\037)1417 4788 y Fm(k)1458 4776 y Fp(\()p Fn(j)5 b Fp(\))p Fn(e)1610 4706 y Fd(1)p 1610 4715 V 1610 4748 a(2)1660 4672 y Ff(P)1747 4692 y Fd(0)1747 4759 y Fe(j)1773 4747 y Fc(0)1797 4759 y Fd(=)p Fe(j)1882 4728 y Fm(Q)p Fk(\()p Fm(\022)1992 4743 y Fe(j)2018 4731 y Fc(0)2045 4728 y Fm(!)r Fk(\))2133 4663 y Ff(Z)2216 4683 y Fk(1)2179 4852 y(0)2267 4776 y Fn(\021)2308 4788 y Fh(L)2358 4776 y Fp(\()p Fn(s)19 b Fp(+)f Fn(k)j Fp(+)d Fn(j;)c(\022)2788 4788 y Fh(\000)p Fm(k)2881 4776 y Fn(!)s Fp(\))p Fn(ds:)1970 5059 y Fp(7)p eop %%Page: 8 8 8 7 bop 515 524 a Fp(Since)28 b Fn(j)g Fl(!)900 457 y Ff(R)955 477 y Fk(1)939 553 y(0)1006 524 y Fn(\021)1047 536 y Fh(L)1097 524 y Fp(\()p Fn(s)19 b Fp(+)f Fn(k)j Fp(+)d Fn(j;)c(\022)1527 536 y Fh(\000)p Fm(k)1620 524 y Fn(!)s Fp(\))p Fn(ds)28 b Fp(has)f(a)g(sub)r(exp)r(onen)n(tial)h(gro) n(wth)e(and)1628 727 y Fn(j)i Fl(!)23 b Fn(e)1845 657 y Fd(1)p 1845 666 29 4 v 1845 699 a(2)1895 623 y Ff(P)1982 644 y Fd(0)1982 710 y Fe(j)2008 698 y Fc(0)2032 710 y Fd(=)p Fe(j)2117 679 y Fm(Q)p Fk(\()p Fm(\022)2227 695 y Fe(j)2253 683 y Fc(0)2280 679 y Fm(!)r Fk(\))515 882 y Fp(go)r(es)j(to)g(zero)g(exp)r(onen)n(tially)g(fast)h(\(see)f (\(6\)\),)i(there)e(exists)h(a)f(constan)n(t)g Fn(n)h Fp(dep)r(ending)g(only)515 982 y(on)g Fn(!)j Fp(suc)n(h)e(that)1204 1137 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))22 b Fl(\024)h Fn(n)p Fp(\()p Fn(!)s Fp(\))166 b(for)27 b(an)n(y)101 b Fl(\000)18 b Fn(j;)28 b(k)e Fl(2)d Fi(Z)2707 1103 y Fk(+)2756 1137 y Fn(:)582 b Fp(\(8\))515 1305 y(The)24 b(term)878 1238 y Ff(R)933 1259 y Fk(1)917 1335 y(0)984 1305 y Fn(\021)1025 1317 y Fh(L)1076 1305 y Fp(\()p Fn(s)13 b Fp(+)g Fn(k)h Fp(+)f Fn(j;)h(!)s Fp(\))p Fn(ds)24 b Fp(tends)h(to)g(zero)e(in)i(probabilit)n(y)f(for)g Fn(k)i Fl(!)d(1)h Fp(and)h(\014xed)f Fn(j)5 b Fp(.)515 1427 y(Hence,)30 b(also)955 1360 y Ff(R)1010 1380 y Fk(1)994 1456 y(0)1062 1427 y Fn(\021)1103 1439 y Fh(L)1153 1427 y Fp(\()p Fn(s)20 b Fp(+)f Fn(k)k Fp(+)c Fn(j;)14 b(\022)1588 1439 y Fh(\000)p Fm(k)1681 1427 y Fn(!)s Fp(\))p Fn(ds)30 b Fp(tends)g(to)f(zero)g(in)h(probabilit)n(y)e(for)h Fn(k)h Fl(!)c(1)k Fp(and)515 1544 y(\014xed)e Fn(j)5 b Fp(.)28 b(Let)g Fn(\025)p Fp(\()p Fn(j)5 b Fp(\))25 b(=)e Fn(e)1268 1491 y Fd(1)p 1268 1500 V 1268 1534 a(4)1306 1514 y Fb(E)p Fm(Qj)1432 1544 y Fn(\016)1469 1556 y Fm(j)1504 1544 y Fn(;)28 b(j)h Fl(2)24 b Fi(Z)1758 1514 y Fh(\000)1836 1544 y Fp(b)r(e)29 b(a)e(\014nite)i(measure)e(on)h Fi(Z)2733 1514 y Fh(\000)2811 1544 y Fp(and)g(\004)3028 1556 y Fm(N)3091 1544 y Fp(\()p Fn(!)s Fp(\))h(b)r(e)f(the)515 1643 y(indicator)f(function)h(of)f(the)h(set)1197 1817 y Fl(f)p Fn(!)d Fp(:)e Fn(e)1411 1747 y Fd(1)p 1411 1756 V 1411 1789 a(2)1460 1713 y Ff(P)1548 1734 y Fd(0)1548 1800 y Fe(j)1574 1788 y Fc(0)1597 1800 y Fd(=)p Fe(j)1682 1769 y Fm(Q)p Fk(\()p Fm(\022)1792 1785 y Fe(j)1818 1773 y Fc(0)1846 1769 y Fm(!)r Fk(\))1943 1817 y Fl(\024)f Fn(N)9 b(e)2155 1761 y Fd(1)p 2155 1770 V 2155 1803 a(4)2193 1783 y Fb(E)p Fm(Qj)2342 1817 y Fp(for)27 b Fn(j)h Fl(2)23 b Fi(Z)2671 1783 y Fh(\000)2721 1817 y Fl(g)p Fn(;)515 1972 y Fp(where)28 b(\004)811 1984 y Fm(N)903 1972 y Fp(tends)h(increasingly)f(to)g(one)h(for)f Fn(N)34 b Fl(!)25 b(1)p Fp(.)k(W)-7 b(e)30 b(set)e Fn(\026)2636 1942 y Fm(!)2636 1995 y(N)2725 1972 y Fp(=)c(\004)2869 1984 y Fm(N)2933 1972 y Fp(\()p Fn(!)s Fp(\))p Fn(\026)3102 1942 y Fm(!)3150 1972 y Fp(.)29 b(F)-7 b(or)28 b(the)515 2072 y(asserted)e(con)n(v)n(ergence)f(w)n(e)j(consider)821 2265 y Fi(P)p Fp(\()905 2152 y Ff(Z)1001 2265 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))p Fn(d\026)1417 2230 y Fm(!)1465 2265 y Fp(\()p Fn(j)5 b Fp(\))23 b Fn(>)g Fp(3)p Fn(\016)s Fp(\))821 2482 y(=)g Fi(P)p Fp(\()993 2369 y Ff(Z)1075 2482 y Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))k Fl(^)h Fn(N)9 b Fp(\))p Fn(d\026)1723 2448 y Fm(!)1723 2503 y(N)1786 2482 y Fp(\()p Fn(j)c Fp(\))19 b(+)1991 2369 y Ff(Z)2088 2482 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))k Fl(\000)g Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))19 b Fl(^)g Fn(N)9 b Fp(\))p Fn(d\026)3161 2448 y Fm(!)3161 2503 y(N)3224 2482 y Fp(\()p Fn(j)c Fp(\))987 2700 y(+)1066 2587 y Ff(Z)1162 2700 y Fn(f)k Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))p Fn(d)p Fp(\()p Fn(\026)1610 2666 y Fm(!)1677 2700 y Fl(\000)k Fn(\026)1810 2666 y Fm(!)1810 2721 y(N)1873 2700 y Fp(\)\()p Fn(j)5 b Fp(\))24 b Fn(>)f Fp(3)p Fn(\016)s Fp(\))821 2918 y Fl(\024)g Fi(P)p Fp(\()993 2805 y Ff(Z)1075 2918 y Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))k Fl(^)h Fn(N)9 b Fp(\))p Fn(d\026)1723 2884 y Fm(!)1723 2938 y(N)1786 2918 y Fp(\()p Fn(j)c Fp(\))24 b Fn(>)e(\016)s Fp(\))987 3136 y(+)p Fi(P)p Fp(\()1136 3023 y Ff(Z)1231 3136 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))19 b Fl(\000)f Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))k Fl(^)h Fn(N)9 b Fp(\))p Fn(d\026)2304 3101 y Fm(!)2352 3136 y Fp(\()p Fn(j)c Fp(\))24 b Fn(>)f(\016)s Fp(\))987 3353 y(+)p Fi(P)p Fp(\()1136 3240 y Ff(Z)1231 3353 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))p Fn(d)p Fp(\()p Fn(\026)1679 3319 y Fm(!)1746 3353 y Fl(\000)k Fn(\026)1879 3319 y Fm(!)1879 3374 y(N)1942 3353 y Fp(\)\()p Fn(j)5 b Fp(\))24 b Fn(>)f(\016)s Fp(\))p Fn(:)515 3551 y Fp(Note)29 b(that)h(b)n(y)g(\(8\))g(the)g (second)f(term)g(on)h(the)g(righ)n(t)f(hand)g(side)h(is)f(less)g(than)h Fn(")g Fp(uniformly)515 3650 y(in)c Fn(k)j Fp(if)d Fn(N)35 b Fp(is)26 b(su\016cien)n(tly)g(large)e(uniformly)i(in)g Fn(k)s Fp(.)g(The)g(in)n(tegral)f(in)h(the)h(third)f(term)g(can)f(b)r (e)515 3750 y(estimated)i(b)n(y)592 3830 y Ff(Z)689 3943 y Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))p Fn(d)p Fp(\()p Fn(\026)1137 3908 y Fm(!)1204 3943 y Fl(\000)k Fn(\026)1337 3908 y Fm(!)1337 3963 y(N)1400 3943 y Fp(\)\()p Fn(j)5 b Fp(\))24 b Fl(\024)e Fn(n)p Fp(\()p Fn(!)s Fp(\)\()p Fn(\026)1897 3908 y Fm(!)1964 3943 y Fl(\000)c Fn(\026)2097 3908 y Fm(!)2097 3963 y(N)2160 3943 y Fp(\)\()p Fi(Z)2286 3908 y Fh(\000)2336 3943 y Fp(\))23 b(=)g Fn(n)p Fp(\()p Fn(!)s Fp(\)\(1)18 b Fl(\000)g Fp(\004)2878 3955 y Fm(N)2942 3943 y Fp(\()p Fn(!)s Fp(\)\))p Fn(\026)3143 3908 y Fm(!)3191 3943 y Fp(\()p Fi(Z)3285 3908 y Fh(\000)3335 3943 y Fp(\))p Fn(:)515 4135 y Fp(Therefore)j(this)i(third)f(term)g(is)h(less)f(than)g Fn(")g Fp(for)g(large)f Fn(N)9 b Fp(.)22 b(T)-7 b(o)22 b(see)g(that)h(the)g(\014rst)f(term)g(tends)515 4235 y(to)k(zero)e(in)j(probabilit)n(y)d(for)i Fn(k)g Fl(!)d(1)j Fp(and)g(an)n(y)f Fn(N)9 b Fp(,)26 b(w)n(e)f(note)h(that)g(b)n(y)g(the) g(de\014nition)g(of)g(the)515 4335 y(metric)h(of)h(the)g(con)n(v)n (ergence)d(in)j(probabilit)n(y)681 4539 y Fi(E)818 4415 y Ff(R)873 4481 y Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))k Fl(^)h Fn(N)9 b Fp(\))p Fn(d\026)1521 4451 y Fm(!)1521 4504 y(N)1584 4481 y Fp(\()p Fn(j)c Fp(\))p 740 4520 1014 4 v 740 4597 a(1)24 b(+)889 4530 y Ff(R)944 4597 y Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))19 b Fl(^)g Fn(N)9 b Fp(\))p Fn(d\026)1593 4568 y Fm(!)1593 4621 y(N)1656 4597 y Fp(\()p Fn(j)c Fp(\))1793 4539 y Fl(\024)22 b Fi(E)1943 4426 y Ff(Z)2032 4539 y Fp(\()p Fn(f)9 b Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))19 b Fl(^)g Fn(N)9 b Fp(\))p Fn(d\026)2681 4505 y Fm(!)2681 4560 y(N)2744 4539 y Fp(\()p Fn(j)c Fp(\))847 4774 y(=)934 4661 y Ff(Z)1031 4774 y Fi(E)k Fp(\()p Fn(f)g Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))24 b Fl(^)19 b Fn(N)9 b Fp(\))p Fn(d\026)1735 4740 y Fm(!)1735 4794 y(N)1798 4774 y Fp(\()p Fn(j)c Fp(\))24 b Fl(\024)e Fn(N)9 b Fp(\()p Fn(N)28 b Fp(+)18 b(1\))2386 4661 y Ff(Z)2564 4718 y Fi(E)8 b Fp(\()p Fn(f)h Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))25 b Fl(^)18 b Fn(N)9 b Fp(\))p 2492 4755 754 4 v 2492 4831 a(1)18 b(+)g Fi(E)8 b Fp(\()q Fn(f)h Fp(\()p Fn(k)s(;)14 b(j;)g(!)s Fp(\))24 b Fl(^)19 b Fn(N)9 b Fp(\))3256 4774 y Fn(d\025)p Fp(\()p Fn(j)c Fp(\))p Fn(;)1970 5059 y Fp(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fp(where)33 b(the)i(righ)n(t)e(hand)i(side)f(tends)g (to)g(zero)f(for)h(an)n(y)f Fn(N)43 b Fl(\025)34 b Fp(0)g(b)n(y)g(Leb)r (esgue's)f(theorem.)515 623 y(Th)n(us)27 b(the)h(asserted)e(con)n(v)n (ergence)g(\(7\))h(follo)n(ws.)1356 b Fa(2)704 822 y Fp(No)n(w)37 b(w)n(e)h(presen)n(t)f(another)g(v)n(ersion)g(of)h(the)g (theorem)g(on)g(the)g(existence)g(of)g(\014nite)515 922 y(n)n(um)n(b)r(er)h(of)g(determining)g(functionals)g(whic)n(h)g(can)g (b)r(e)h(easily)e(applied)i(to)f(the)h(random)515 1021 y(squeezing)26 b(prop)r(ert)n(y)h(in)n(tro)r(duced)g(b)n(y)h(Flandoli)f (and)g(Langa)g([12)o(].)515 1195 y Fg(Theorem)j(2.3)41 b Fj(L)l(et)36 b Fn(')i Fj(b)l(e)f(RDS)f(whose)i(phase)h(sp)l(ac)l(e)e (is)h(a)f(Banach)i(sp)l(ac)l(e)f Fn(H)44 b Fj(with)37 b(the)515 1295 y(norm)g Fl(k)23 b(\001)h(k)p Fj(.)37 b(Supp)l(ose)g(that)g(this)g(RDS)f(is)i(dissip)l(ative)h(in)e Fn(H)44 b Fj(with)37 b(a)h(forwar)l(d)h(invariant)515 1394 y(absorbing)c(r)l(andom)f(set)f Fn(B)t Fp(\()p Fn(!)s Fp(\))h Fj(such)f(that)h(the)g(r)l(andom)g(variable)h Fn(\032)p Fp(\()p Fn(!)s Fp(\))30 b(=)g(sup)3088 1415 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))3337 1394 y Fl(k)p Fn(x)p Fl(k)515 1502 y Fj(is)h(temp)l(er)l(e)l(d)g(and)g Fn(\032)p Fp(\()p Fn(\022)1229 1514 y Fm(t)1258 1502 y Fn(!)s Fp(\))25 b Fl(2)g Fn(L)1507 1462 y Fm(p)1507 1527 y(loc)1595 1502 y Fp(\()p Fi(R)p Fp(\))37 b Fj(for)32 b(some)f Fn(p)24 b Fl(\025)h Fp(1)30 b Fj(and)h(al)t(l)h Fn(!)c Fl(2)d Fp(\012)p Fj(.)31 b(Assume)f(that)h(for)515 1602 y(e)l(ach)f Fn(!)c Fl(2)e Fp(\012)29 b Fj(RDS)g Fn(')h Fj(p)l(ossesses)g(the)g(fol)t(lowing)i(pr)l(op)l(erties:)1200 1776 y Fl(k)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1534 1788 y Fk(1)1571 1776 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(x)1997 1788 y Fk(2)2035 1776 y Fp(\))p Fl(k)23 b(\024)f Fn(M)9 b Fp(\()p Fn(!)s Fp(\))p Fl(k)p Fn(x)2517 1788 y Fk(1)2573 1776 y Fl(\000)18 b Fn(x)2703 1788 y Fk(2)2741 1776 y Fl(k)578 b Fp(\(9\))515 1950 y Fj(for)30 b(al)t(l)h Fn(t)23 b Fl(2)h Fp([0)p Fn(;)14 b Fp(1])p Fn(;)36 b(x)1171 1962 y Fk(1)1208 1950 y Fn(;)14 b(x)1292 1962 y Fk(2)1353 1950 y Fl(2)23 b Fn(B)t Fp(\()p Fn(!)s Fp(\))30 b Fj(and)913 2124 y Fl(k)p Fn(')p Fp(\(1)p Fn(;)14 b(!)s(;)g(x)1259 2136 y Fk(1)1296 2124 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(x)1734 2136 y Fk(2)1771 2124 y Fp(\))p Fl(k)83 b(\024)f(N)12 b Fp(\()p Fn(')p Fp(\(1)p Fn(;)i(!)s(;)g(x)2491 2136 y Fk(1)2529 2124 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(x)2967 2136 y Fk(2)3004 2124 y Fp(\)\))1928 2289 y(+)82 b Fn(e)2114 2184 y Ff(R)2169 2204 y Fd(1)2153 2280 y(0)2213 2245 y Fm(r)r Fk(\()p Fm(\022)2304 2253 y Fe(\034)2341 2245 y Fm(!)r Fk(\))p Fm(d\034)2505 2289 y Fl(\001)19 b(k)p Fn(x)2636 2301 y Fk(1)2691 2289 y Fl(\000)f Fn(x)2821 2301 y Fk(2)2859 2289 y Fl(k)419 b Fp(\(10\))515 2463 y Fj(for)34 b(al)t(l)g Fn(x)820 2475 y Fk(1)858 2463 y Fn(;)14 b(x)942 2475 y Fk(2)1009 2463 y Fl(2)30 b Fn(B)t Fp(\()p Fn(!)s Fp(\))p Fj(.)k(Her)l(e)f Fn(M)9 b Fp(\()p Fn(!)s Fp(\))33 b Fj(is)g(a)h(temp)l(er)l(e)l(d)f(and)h(\014nite)f (almost)h(sur)l(ely)f(r)l(andom)515 2563 y(variable,)40 b Fl(N)12 b Fp(\()p Fl(\001)p Fp(\))39 b Fj(is)f(a)g(p)l(ositive)i(c)l (ontinuous)d(sc)l(alar)i(function)e(on)h Fn(H)45 b Fj(such)38 b(that)g Fl(N)12 b Fp(\()p Fn(x)p Fp(\))39 b Fl(\024)515 2662 y Fn(C)32 b Fl(\001)26 b Fp(\(1)g(+)f Fl(k)p Fn(x)p Fl(k)976 2632 y Fm(p)1014 2662 y Fp(\))41 b Fj(and)f Fn(r)r Fp(\()p Fn(!)s Fp(\))h Fj(is)g(a)f(r)l(andom)h(variable)h(with)f (\014nite)f(exp)l(e)l(ctation)g(such)g(that)515 2762 y Fi(E)8 b Fn(r)32 b(<)23 b Fp(0)p Fj(.)29 b(Then)i(the)f(c)l(ondition) 1226 2936 y Fp(\()p Fi(P)p Fp(\))68 b(lim)1356 2986 y Fm(n)p Fh(!)p Fk(+)p Fh(1)1594 2936 y Fl(N)12 b Fp(\()p Fn(')p Fp(\()p Fn(n;)i(!)s(;)g(x)2018 2948 y Fk(1)2056 2936 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(n;)c(!)s(;)g(x)2502 2948 y Fk(2)2539 2936 y Fp(\)\))24 b(=)e(0)564 b(\(11\))515 3141 y Fj(for)30 b(some)h Fn(x)907 3153 y Fk(1)944 3141 y Fn(;)14 b(x)1028 3153 y Fk(2)1089 3141 y Fl(2)23 b Fn(H)37 b Fj(implies)31 b(that)1274 3315 y Fp(\()p Fi(P)p Fp(\))59 b(lim)1404 3365 y Fm(t)p Fh(!)p Fk(+)p Fh(1)1625 3315 y Fl(k)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1959 3327 y Fk(1)1996 3315 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(x)2422 3327 y Fk(2)2460 3315 y Fp(\))p Fl(k)22 b Fp(=)h(0)p Fn(:)611 b Fp(\(12\))704 3525 y Fj(Pr)l(o)l(of.)53 b Fp(As)28 b(ab)r(o)n(v)n(e)f(w)n(e)g(can)h(assume)f (that)i Fn(x)2123 3537 y Fm(i)2151 3525 y Fp(\()p Fn(!)s Fp(\))24 b Fl(2)g Fn(B)t Fp(\()p Fn(!)s Fp(\).)k(Using)g(the)g(co)r (cycle)g(prop-)515 3625 y(ert)n(y)f Fn(')p Fp(\()p Fn(m;)14 b(!)s Fp(\))23 b(=)g Fn(')p Fp(\(1)p Fn(;)14 b(\022)1284 3637 y Fm(m)p Fh(\000)p Fk(1)1432 3625 y Fn(!)s(;)g(')p Fp(\()p Fn(m)k Fl(\000)g Fp(1)p Fn(;)c(!)s Fp(\)\))28 b(and)f(relation)g(\(10\))g(w)n(e)g(obtain)g(that)1175 3840 y Fn(d)1218 3852 y Fm(m)1282 3840 y Fp(\()p Fn(!)s Fp(\))c Fl(\024)g(N)12 b Fp(\()p Fn(m;)i(!)s Fp(\))k(+)g Fn(e)1961 3728 y Ff(R)2016 3749 y Fe(m)2000 3824 y(m)p Fc(\000)p Fd(1)2140 3789 y Fm(r)r Fk(\()p Fm(\022)2231 3797 y Fe(\034)2267 3789 y Fm(!)r Fk(\))p Fm(d\034)2432 3840 y Fl(\001)g Fn(d)2516 3852 y Fm(m)p Fh(\000)p Fk(1)2665 3840 y Fp(\()p Fn(!)s Fp(\))p Fn(;)515 4014 y Fp(where)1168 4114 y Fl(N)12 b Fp(\()p Fn(n;)i(!)s Fp(\))24 b(=)e Fl(N)12 b Fp(\()p Fn(')p Fp(\()p Fn(n;)i(!)s(;)g(x)1989 4126 y Fk(1)2028 4114 y Fp(\()p Fn(!)s Fp(\)\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(n;)c(!)s(;)g(x)2593 4126 y Fk(2)2630 4114 y Fp(\()p Fn(!)s Fp(\)\)\))515 4258 y(and)1255 4358 y Fn(d)1298 4370 y Fm(t)1327 4358 y Fp(\()p Fn(!)s Fp(\))24 b(=)e Fl(k)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1891 4370 y Fk(1)1929 4358 y Fp(\()p Fn(!)s Fp(\)\))k Fl(\000)h Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)2474 4370 y Fk(2)2511 4358 y Fp(\()p Fn(!)s Fp(\)\))p Fl(k)p Fn(:)515 4502 y Fp(After)28 b(iterations)e(w)n(e)i(\014nd)g(that)898 4734 y Fn(d)941 4746 y Fm(m)1004 4734 y Fp(\()p Fn(!)s Fp(\))23 b Fl(\024)g Fn(d)1277 4746 y Fk(0)1314 4734 y Fp(\()p Fn(!)s Fp(\))p Fn(e)1472 4629 y Ff(R)1528 4649 y Fe(m)1512 4725 y Fd(0)1594 4689 y Fm(r)r Fk(\()p Fm(\022)1685 4697 y Fe(\034)1722 4689 y Fm(!)r Fk(\))p Fm(d\034)1886 4734 y Fp(+)1969 4630 y Fm(m)p Fh(\000)p Fk(1)1981 4655 y Ff(X)1983 4832 y Fm(j)s Fk(=0)2127 4734 y Fl(N)12 b Fp(\()p Fn(m)19 b Fl(\000)f Fn(j;)c(!)s Fp(\))p Fn(e)2611 4621 y Ff(R)2666 4641 y Fe(m)2650 4717 y(m)p Fc(\000)p Fe(j)2788 4681 y Fm(r)r Fk(\()p Fm(\022)2879 4689 y Fe(\034)2915 4681 y Fm(!)r Fk(\))p Fm(d\034)3061 4734 y Fn(:)1970 5059 y Fp(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fp(Applying)34 b(no)n(w)f(the)h(same)f(argumen)n(ts) g(as)g(in)h(the)g(pro)r(of)f(of)h(Theorem)f(2.2)g(w)n(e)g(\014nd)h (that)515 623 y(\(11\))44 b(implies)i(that)f(\()p Fi(P)p Fp(\))27 b(lim)1462 635 y Fm(m)p Fh(!)p Fk(+)p Fh(1)1723 623 y Fn(d)1766 635 y Fm(m)1829 623 y Fp(\()p Fn(!)s Fp(\))52 b(=)g(0.)45 b(F)-7 b(rom)44 b(\(9\))h(w)n(e)g(ha)n(v)n(e)f (that)h Fn(d)3202 635 y Fm(t)3232 623 y Fp(\()p Fn(!)s Fp(\))52 b Fl(\024)515 722 y Fn(M)9 b Fp(\()p Fn(\022)676 737 y Fk([)p Fm(t)p Fk(])742 722 y Fn(!)s Fp(\))p Fn(d)872 737 y Fk([)p Fm(t)p Fk(])939 722 y Fp(\()p Fn(!)s Fp(\).)37 b(Th)n(us)f(w)n(e)h(should)f(pro)n(v)n(e)f(that)h(\()p Fi(P)p Fp(\))27 b(lim)2421 734 y Fm(n)p Fh(!)p Fk(+)p Fh(1)2663 722 y Fn(M)9 b Fp(\()p Fn(\022)2824 734 y Fm(n)2869 722 y Fn(!)s Fp(\))p Fn(d)2999 734 y Fm(n)3045 722 y Fp(\()p Fn(!)s Fp(\))38 b(=)f(0.)f(It)515 822 y(follo)n(ws)27 b(from)i(\()p Fi(P)p Fp(\))e(lim)1244 834 y Fm(n)p Fh(!)p Fk(+)p Fh(1)1486 822 y Fn(M)9 b Fp(\()p Fn(!)s Fp(\))p Fn(d)1738 834 y Fm(n)1783 822 y Fp(\()p Fn(\022)1854 834 y Fh(\000)p Fm(n)1952 822 y Fn(!)s Fp(\))24 b(=)h(0.)j(The)h(last)f (relation)f(follo)n(ws)h(from)g(the)515 922 y(con)n(v)n(ergence)d(\()p Fi(P)p Fp(\))i(lim)1232 934 y Fm(m)p Fh(!)p Fk(+)p Fh(1)1492 922 y Fn(d)1535 934 y Fm(m)1599 922 y Fp(\()p Fn(\022)1670 934 y Fh(\000)p Fm(m)1785 922 y Fn(!)s Fp(\))c(=)g(0)k(and)g(the)h (prop)r(erties)f(of)g Fn(M)9 b Fp(\()p Fn(!)s Fp(\).)332 b Fa(2)704 1121 y Fp(No)n(w)29 b(follo)n(wing)f(Flandoli)h(and)g(Langa) f([12)o(])i(w)n(e)e(in)n(tro)r(duce)h(the)h(concept)f(of)h(random)515 1220 y(squeezing)c(prop)r(ert)n(y)-7 b(.)515 1386 y Fg(De\014nition)31 b(2.4)40 b Fj(L)l(et)31 b Fn(')h Fj(b)l(e)g(RDS)e(whose)j(phase)g(sp)l (ac)l(e)f(is)g(a)g(sep)l(ar)l(able)h(Hilb)l(ert)f(sp)l(ac)l(e)g Fn(H)7 b Fj(.)515 1486 y(We)29 b(say)g(that)g(RDS)f Fp(\()p Fn(\022)r(;)14 b(')p Fp(\))30 b Fj(satis\014es)f(a)h Fp(random)c(squeezing)g(prop)r(ert)n(y)h Fj(\(RSP\))i(on)g(the)g(r)l (an-)515 1586 y(dom)c(set)g Fn(B)t Fp(\()p Fn(!)s Fp(\))g Fj(if)h(ther)l(e)e(exist)h(a)g(\014nite-dimensional)h(pr)l(oje)l(ctor)g Fn(P)37 b Fj(and)25 b(a)g(r)l(andom)g(variable)515 1685 y Fn(r)r Fp(\()p Fn(!)s Fp(\))35 b Fj(with)g(\014nite)f(exp)l(e)l (ctation)h(such)g(that)f Fi(E)9 b Fn(r)40 b(<)31 b Fp(0)j Fj(and)h(for)h(almost)f(al)t(l)g Fn(!)g Fl(2)d Fp(\012)i Fj(we)h(have)515 1785 y(either)835 1968 y Fl(k)p Fp(\()p Fn(I)25 b Fl(\000)18 b Fn(P)12 b Fp(\))p Fn(')p Fp(\(1)p Fn(;)i(!)s(;)g(x)1454 1980 y Fk(1)1491 1968 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(x)1929 1980 y Fk(2)1966 1968 y Fp(\))p Fl(k)23 b(\024)g(k)p Fn(P)12 b(')p Fp(\(1)p Fn(;)i(!)s(;)g(x)2562 1980 y Fk(1)2599 1968 y Fp(\))k Fl(\000)g Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(x)3036 1980 y Fk(2)3074 1968 y Fp(\))p Fl(k)515 2150 y Fj(or)1058 2270 y Fl(k)p Fn(')p Fp(\(1)p Fn(;)g(!)s(;)g(x)1404 2282 y Fk(1)1440 2270 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(x)1878 2282 y Fk(2)1915 2270 y Fp(\))p Fl(k)23 b(\024)g Fn(e)2139 2165 y Ff(R)2194 2185 y Fd(1)2178 2261 y(0)2238 2225 y Fm(r)r Fk(\()p Fm(\022)2329 2233 y Fe(\034)2365 2225 y Fm(!)r Fk(\))p Fm(d\034)2529 2270 y Fl(\001)c(k)p Fn(x)2660 2282 y Fk(1)2715 2270 y Fl(\000)g Fn(x)2846 2282 y Fk(2)2883 2270 y Fl(k)515 2420 y Fj(for)30 b(al)t(l)h Fn(x)813 2432 y Fk(1)851 2420 y Fn(;)14 b(x)935 2432 y Fk(2)996 2420 y Fl(2)23 b Fn(B)t Fp(\()p Fn(!)s Fp(\))p Fj(.)515 2586 y Fp(In)i(deterministic)h(case)e(a)h(similar)g (prop)r(ert)n(y)f(is)h(w)n(ell-kno)n(wn)f(for)h(dissipativ)n(e)f (systems)h(with)515 2685 y(\014nite-dimensional)31 b(long-time)g(b)r (eha)n(viour)f(\(see,)i(e.g.)f([24)o(])h(and)f(the)h(references)e (therein\).)515 2785 y(Flandoli)37 b(and)h(Langa)e([12])h(ha)n(v)n(e)g (pro)n(v)n(ed)f(random)h(squeezing)g(prop)r(ert)n(y)g(for)g(a)h(class)f (of)515 2884 y(sto)r(c)n(hastic)22 b(reaction-di\013usion)g(equations)g (and)h(for)g(sto)r(c)n(hastic)f(2)p Fn(D)j Fp(Na)n(vier)d(-)h(Stok)n (es)f(equa-)515 2984 y(tions)27 b(with)h(p)r(erio)r(dic)g(b)r(oundary)f (condition.)515 3134 y(No)n(w)g(w)n(e)g(are)g(in)h(p)r(osition)f(to)g (state)h(corollaries)d(from)i(Theorem)g(2.3.)515 3300 y Fg(Corollary)k(2.5)41 b Fj(Assume)25 b(that)h(RDS)e Fn(')i Fj(with)h(Hilb)l(ert)f(phase)h(sp)l(ac)l(e)f Fn(H)33 b Fj(is)26 b(dissip)l(ative)i(with)515 3399 y(a)g(forwar)l(d)i (invariant)e(absorbing)i(r)l(andom)e(set)g Fn(B)t Fp(\()p Fn(!)s Fp(\))g Fj(satisfying)h(the)f(hyp)l(otheses)i(of)f(The)l(o-)515 3499 y(r)l(em)h(2.3.)i(Supp)l(ose)f(that)g Fn(')g Fj(p)l(ossesses)g(pr) l(op)l(erty)g(\(9\))g(and)g(satis\014es)g(RSP)f(with)h(an)g(ortho)l(g-) 515 3598 y(onal)f(pr)l(oje)l(ctor)h Fn(P)12 b Fj(.)30 b(Then)h(the)e(pr)l(op)l(erty)711 3781 y Fp(\()p Fi(P)p Fp(\))67 b(lim)841 3831 y Fm(n)p Fh(!)p Fk(+)p Fh(1)1078 3781 y Fl(f)o Fp(\()p Fn(')p Fp(\()p Fn(n;)14 b(!)s(;)g(x)1463 3793 y Fk(1)1501 3781 y Fp(\))p Fn(;)g(e)1609 3793 y Fm(i)1637 3781 y Fp(\))1669 3793 y Fm(H)1751 3781 y Fl(\000)k Fp(\()p Fn(')p Fp(\()p Fn(n;)c(!)s(;)g(x)2178 3793 y Fk(2)2215 3781 y Fp(\))p Fn(;)g(e)2323 3793 y Fm(i)2351 3781 y Fp(\))2383 3793 y Fm(H)2446 3781 y Fl(g)23 b Fp(=)f(0)p Fn(;)99 b(i)22 b Fp(=)h(1)p Fn(;)14 b Fp(2)p Fn(;)g(:)g(:)g(:)e(;)i(d;) 515 4000 y Fj(for)26 b(some)f Fn(x)897 4012 y Fk(1)935 4000 y Fn(;)14 b(x)1019 4012 y Fk(2)1079 4000 y Fl(2)24 b Fn(H)32 b Fj(implies)26 b(\(12\).)g(Her)l(e)f Fl(f)p Fn(e)2016 4012 y Fm(i)2066 4000 y Fp(:)e Fn(i)g Fp(=)g(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(d)p Fl(g)24 b Fj(is)i(a)f(b)l(asis)h(in)f (the)g(subsp)l(ac)l(e)515 4099 y Fn(P)12 b(H)7 b Fj(.)515 4265 y(Pr)l(o)l(of.)55 b Fp(It)29 b(is)f(clear)g(that)h(RSP)f(implies)h (\(10\))f(with)h Fl(N)12 b Fp(\()p Fn(u)p Fp(\))25 b(=)g(2)p Fl(k)p Fn(P)12 b(u)p Fl(k)p Fp(.)26 b(Th)n(us)i(w)n(e)h(can)f(apply)515 4365 y(Theorem)f(2.3.)2409 b Fa(2)704 4564 y Fp(This)19 b(result)g(on)g(determining)g(mo)r(des)h(extends)f(in)g(some)g(sense)g (the)h(result)f(b)n(y)g(Flandoli)515 4664 y(and)27 b(Langa)f([12)o(])i (for)f(the)h(case)f Fn(k)f Fp(=)d(0.)1950 5059 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fg(Corollary)31 b(2.6)41 b Fj(Assume)24 b(that)g(RDS)g Fn(')h Fj(satis\014es)f(the)h(hyp)l(otheses)h(of)g(Cor)l (ol)t(lary)h(2.5.)f(Sup-)515 623 y(p)l(ose)37 b(that)g(ther)l(e)g (exists)f(a)i(Banach)g(sp)l(ac)l(e)g Fn(W)48 b Fj(such)37 b(that)g Fn(H)44 b Fj(c)l(ontinuously)36 b(and)i(densely)515 722 y(emb)l(e)l(dde)l(d)43 b(into)f Fn(W)54 b Fj(and)43 b(the)f(pr)l(oje)l(ctor)h Fn(P)54 b Fj(c)l(an)42 b(b)l(e)g(extende)l(d) g(to)g(c)l(ontinuous)f(op)l(er)l(ator)515 822 y(fr)l(om)h Fn(W)53 b Fj(into)42 b Fn(H)48 b Fj(such)41 b(that)h Fl(k)p Fn(P)12 b(u)p Fl(k)1734 834 y Fm(H)1839 822 y Fl(\024)44 b Fn(a)1992 834 y Fk(0)2029 822 y Fl(k)p Fn(u)p Fl(k)2161 834 y Fm(W)2277 822 y Fj(with)e(a)f(p)l(ositive)i(c)l (onstant)e Fn(a)3250 834 y Fk(0)3287 822 y Fj(.)h(L)l(et)515 922 y Fl(L)30 b Fp(=)f Fl(f)p Fn(l)763 934 y Fm(j)827 922 y Fp(:)h Fn(j)35 b Fp(=)29 b(1)p Fn(;)14 b(:::;)g(k)s Fl(g)33 b Fj(b)l(e)g(a)h(set)f(of)i(line)l(arly)f(indep)l(endent)h(c)l (ontinuous)d(functionals)i(on)515 1021 y Fn(H)i Fj(with)30 b(the)g(c)l(ompleteness)g(defe)l(ct)g Fn(")1706 1033 y Fh(L)1756 1021 y Fp(\()p Fn(H)r(;)14 b(W)e Fp(\))30 b Fj(with)g(r)l(esp)l(e)l(ct)g(to)f(the)h(p)l(air)g(of)h(the)f(sp)l(ac) l(es)g Fn(H)515 1121 y Fj(and)g Fn(W)12 b Fj(.)30 b(If)962 1312 y Fp(2)p Fn(a)1048 1324 y Fk(0)1085 1312 y Fn(")1124 1324 y Fh(L)1174 1312 y Fp(\()p Fn(H)r(;)14 b(W)e Fp(\))24 b Fn(<)e Fp(1)85 b Fj(and)g Fi(E)8 b Fn(r)28 b Fp(+)18 b(log)2505 1256 y(1)p 2218 1293 618 4 v 2218 1369 a(1)g Fl(\000)g Fp(2)p Fn(a)2447 1381 y Fk(0)2483 1369 y Fn(")2522 1381 y Fh(L)2572 1369 y Fp(\()p Fn(H)r(;)c(W)e Fp(\))2868 1312 y Fn(<)22 b Fp(0)p Fn(;)515 1513 y Fj(then)29 b(the)h(pr)l(op)l (erty)841 1670 y Fp(\()p Fi(P)p Fp(\))68 b(lim)971 1720 y Fm(n)p Fh(!)p Fk(+)p Fh(1)1209 1670 y Fl(f)o Fn(l)1275 1682 y Fm(i)1303 1670 y Fp(\()p Fn(')p Fp(\()p Fn(n;)14 b(!)s(;)g(x)1647 1682 y Fk(1)1685 1670 y Fp(\)\))19 b Fl(\000)f Fn(l)1876 1682 y Fm(i)1903 1670 y Fp(\()p Fn(')p Fp(\()p Fn(n;)c(!)s(;)g(x)2247 1682 y Fk(2)2285 1670 y Fp(\)\))p Fl(g)23 b Fp(=)g(0)84 b Fn(i)23 b Fp(=)f(1)p Fn(;)14 b Fp(2)p Fn(;)g(:)g(:)g(:)f(;)h(k)s(;)515 1863 y Fj(for)30 b(some)h Fn(x)907 1875 y Fk(1)944 1863 y Fn(;)14 b(x)1028 1875 y Fk(2)1089 1863 y Fl(2)23 b Fn(H)37 b Fj(implies)31 b(\(12\).)515 2009 y(Pr)l(o)l(of.)51 b Fp(As)26 b(ab)r(o)n(v)n(e)f(RSP)i(implies)f(\(10\))g(with)h Fl(N)12 b Fp(\()p Fn(u)p Fp(\))24 b(=)f(2)p Fl(k)p Fn(P)12 b(u)p Fl(k)2514 2021 y Fm(H)2574 2009 y Fp(.)27 b(Ho)n(w)n(ev)n(er)d (using)i(\(2\))h(with)515 2108 y Fn(X)i Fp(=)23 b Fn(H)34 b Fp(and)28 b Fn(Y)41 b Fp(=)23 b Fn(W)40 b Fp(w)n(e)27 b(ha)n(v)n(e)1007 2265 y(2)p Fl(k)p Fn(P)12 b(u)p Fl(k)1246 2277 y Fm(H)1330 2265 y Fl(\024)23 b Fp(2)p Fn(a)1504 2277 y Fk(0)1540 2265 y Fl(k)p Fn(u)p Fl(k)1672 2277 y Fm(W)1770 2265 y Fl(\024)f Fp(2)p Fn(a)1943 2277 y Fk(0)1980 2265 y Fn(")2019 2277 y Fh(L)2069 2265 y Fp(\()p Fn(H)r(;)14 b(W)e Fp(\))19 b Fl(\001)f(k)p Fn(u)p Fl(k)2523 2277 y Fm(H)2603 2265 y Fp(+)g Fn(C)2745 2277 y Fh(L)2802 2265 y Fp(\026)-48 b Fn(\021)s Fp(\()p Fn(u)p Fp(\))p Fn(;)515 2422 y Fp(where)33 b(\026)-48 b Fn(\021)796 2434 y Fh(L)846 2422 y Fp(\()p Fn(u)p Fp(\))23 b(=)g(max)p Fl(fj)p Fn(l)1314 2434 y Fm(j)1348 2422 y Fp(\()p Fn(u)p Fp(\))p Fl(j)37 b Fp(:)23 b Fn(j)28 b Fp(=)23 b(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(k)s Fl(g)p Fp(.)27 b(Therefore)f(from)h(\(10\))h(w)n (e)f(ha)n(v)n(e)547 2625 y Fl(k)p Fn(')p Fp(\(1)p Fn(;)14 b(!)s(;)g(x)893 2637 y Fk(1)930 2625 y Fp(\))83 b Fl(\000)f Fn(')p Fp(\(1)p Fn(;)14 b(!)s(;)g(x)1496 2637 y Fk(2)1534 2625 y Fp(\))p Fl(k)1608 2637 y Fm(H)1694 2625 y Fl(\024)2045 2569 y Fn(C)2104 2581 y Fh(L)p 1791 2606 V 1791 2682 a Fp(1)k Fl(\000)g Fp(2)p Fn(a)2020 2694 y Fk(0)2057 2682 y Fn(")2096 2694 y Fh(L)2146 2682 y Fp(\()p Fn(H)r(;)c(W)e Fp(\))2437 2625 y Fl(\001)24 b Fp(\026)-48 b Fn(\021)s Fp(\()p Fn(')p Fp(\(1)p Fn(;)14 b(!)s(;)g(x)2858 2637 y Fk(1)2896 2625 y Fp(\))19 b Fl(\000)f Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(x)3334 2637 y Fk(2)3371 2625 y Fp(\)\))1045 2866 y(+)82 b(exp)1333 2749 y Ff(\032)1395 2753 y(Z)1478 2773 y Fk(1)1442 2941 y(0)1530 2866 y Fn(r)r Fp(\()p Fn(\022)1640 2878 y Fm(\034)1682 2866 y Fn(!)s Fp(\))p Fn(d\034)28 b Fp(+)18 b(log)2378 2809 y(1)p 2090 2846 V 2090 2923 a(1)g Fl(\000)g Fp(2)p Fn(a)2319 2935 y Fk(0)2356 2923 y Fn(")2395 2935 y Fh(L)2445 2923 y Fp(\()p Fn(H)r(;)c(W)e Fp(\))2717 2749 y Ff(\033)2798 2866 y Fl(\001)19 b(k)p Fn(x)2929 2878 y Fk(1)2984 2866 y Fl(\000)f Fn(x)3114 2878 y Fk(2)3152 2866 y Fl(k)3194 2878 y Fm(H)3256 2866 y Fn(:)515 3068 y Fp(Th)n(us)27 b(w)n(e)g(can)g(apply)h(Theorem)f(2.3.) 1696 b Fa(2)515 3324 y Fg(Remark)31 b(2.7)40 b Fp(The)g(space)f Fn(W)52 b Fp(with)41 b(the)f(prop)r(erties)f(listed)i(in)f(Corollary)d (2.6)j(can)f(b)r(e)515 3424 y(easily)f(constructed)g(in)h(the)g(follo)n (wing)e(situation.)i(Assume)g(that)g Fn(A)g Fp(is)f(a)g(p)r(ositiv)n(e) h(self-)515 3524 y(adjoin)n(t)e(op)r(erator)f(in)h Fn(H)45 b Fp(with)37 b(compact)g(resolv)n(en)n(t.)f(Let)i(0)h Fn(<)g(\025)2685 3536 y Fk(1)2762 3524 y Fl(\024)g Fn(\025)2914 3536 y Fk(2)2991 3524 y Fl(\024)g Fn(:)14 b(:)g(:)37 b Fp(b)r(e)h(the)515 3623 y(corresp)r(onding)25 b(eigen)n(v)-5 b(alues.)26 b(If)h Fn(P)39 b Fp(is)27 b(orthogonal)e(pro)5 b(jector)26 b(on)h(the)g(\014rst)g Fn(k)j Fp(eigen)n(v)n(ectors)515 3723 y(of)e Fn(A)p Fp(,)h(then)g(w)n(e)f(ha)n(v)n(e)f Fl(k)p Fn(P)12 b(u)p Fl(k)1426 3735 y Fm(H)1512 3723 y Fl(\024)24 b Fn(\025)1649 3693 y Fm(s)1649 3746 y(k)q Fk(+1)1775 3723 y Fl(k)p Fn(A)1879 3693 y Fh(\000)p Fm(s=)p Fk(2)2033 3723 y Fn(u)p Fl(k)2123 3735 y Fm(H)2213 3723 y Fp(for)k(an)n(y)g Fn(s)c(>)g Fp(0.)k(Th)n(us)h(w)n(e)f(can)g(c)n(ho)r (ose)515 3835 y Fn(W)37 b Fp(as)24 b(a)h(completion)g(of)g Fn(H)32 b Fp(with)25 b(resp)r(ect)g(to)g(the)h(norm)e Fl(k)p Fn(A)2437 3805 y Fh(\000)p Fm(s=)p Fk(2)2605 3835 y Fl(\001)13 b(k)2683 3847 y Fm(H)2771 3835 y Fp(for)24 b(some)h(p)r(ositiv)n(e)g Fn(s)p Fp(.)515 3992 y Fg(Remark)31 b(2.8)40 b Fp(W)-7 b(e)43 b(p)r(oin)n(t)g(out)f(the)h(essen)n(tial)f (di\013erence)h(b)r(et)n(w)n(een)f(Theorem)g(2.2)g(and)515 4092 y(Corollary)29 b(2.6.)j(This)g(corollary)d(relies)j(on)f(the)i (random)e(squeezing)g(prop)r(ert)n(y)-7 b(.)31 b(F)-7 b(or)32 b(prob-)515 4192 y(lems)f(lik)n(e)g(\(1\))g(this)h(prop)r(ert)n (y)e(is)h(usually)g(pro)n(v)n(ed)e(in)j(the)f(main)h(space)e Fn(H)7 b Fp(.)31 b(Therefore)f(the)515 4291 y(corollary)g(men)n(tioned) k(is)f(applied)g(to)g(functionals)g(on)g Fn(H)40 b Fp(only)-7 b(.)33 b(Ho)n(w)n(ev)n(er)e(in)j(the)f(case)g(of)515 4391 y(Theorem)h(2.2)f(the)i(functionals)g(are)f(de\014ned)h(on)f Fn(V)54 b Fp(with)35 b Fn(V)53 b Fl(\032)35 b Fn(H)7 b Fp(.)34 b(Th)n(us)h(Theorem)e(2.2)515 4490 y(admits)c(more)f (singular)g(functionals)h(in)g(comparison)e(with)j(Corollary)d(2.6.)h (On)h(the)g(other)515 4590 y(hand)d(Corollary)d(2.6)i(requires)g(con)n (v)n(ergence)e(of)j(functionals)g(on)f(the)i(discrete)e(sequence)h(of) 515 4690 y(times)h Fn(t)766 4702 y Fm(n)834 4690 y Fp(=)c Fn(n)p Fp(.)k(W)-7 b(e)27 b(note)g(that)g(as)f(in)h(the)g (deterministic)g(case)f(\(see)h([8]\))g(it)g(is)g(also)f(p)r(ossible) 515 4789 y(to)h(consider)g(more)g(general)f(sequences)h Fl(f)p Fn(t)1883 4801 y Fm(n)1927 4789 y Fl(g)p Fp(.)1950 5059 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fo(3)134 b(Application)42 b(to)f(the)g(2D)g(sto)t (c)l(hastic)g(Na)l(vier-Stok)l(es)716 672 y(equations)515 854 y Fp(W)-7 b(e)28 b(consider)e(the)i(sto)r(c)n(hastic)f(Na)n (vier-Stok)n(es)e(equations)1464 1037 y Fn(dv)h Fp(=)d(\()p Fl(\000)p Fn(\027)5 b(Av)22 b Fp(+)2030 1016 y(~)2011 1037 y Fn(F)12 b Fp(\()p Fn(v)s Fp(\)\))p Fn(dt)20 b Fp(+)e Fn(dw)r(;)802 b Fp(\(13\))515 1220 y(where)1241 1338 y Fn(A)24 b Fp(=)e Fl(\000)1489 1281 y Fp(1)p 1489 1318 42 4 v 1489 1395 a(2)1540 1338 y(\001)p Fn(;)1748 1317 y Fp(~)1729 1338 y Fn(F)12 b Fp(\()p Fn(v)s Fp(\))24 b(=)f Fl(\000)2088 1281 y Fn(\027)p 2088 1318 47 4 v 2090 1395 a Fp(2)2144 1338 y(\001)18 b Fl(\000)g Fp(\()p Fn(v)s(;)c Fl(r)p Fp(\))p Fn(v)23 b Fp(+)18 b Fn(f)t(;)515 1515 y Fp(as)34 b(an)h(ev)n(olution)f(equation)h(on)f(the)i(rigged)e (Hilb)r(ert)h(space)g Fn(V)54 b Fl(\032)35 b Fn(H)43 b Fl(\032)35 b Fn(V)2995 1485 y Fh(0)3053 1515 y Fp(where)g Fn(V)54 b Fp(=)515 1636 y Fl(f)p Fn(u)28 b Fl(2)752 1570 y Fh(\016)718 1644 y Fn(W)808 1606 y Fk(1)808 1657 y(2)845 1636 y Fp(\()p Fn(D)r Fp(\))p Fn(;)h Fp(div)15 b Fn(u)29 b Fp(=)g(0)p Fl(g)i Fp(and)g Fn(H)36 b Fp(=)p 1810 1569 67 4 v 29 w Fn(V)1877 1583 y Fm(L)1923 1591 y Fd(2)1955 1583 y Fk(\()p Fm(D)r Fk(\))2099 1636 y Fp(is)31 b(the)h(closure)f(of)g Fn(V)51 b Fp(in)31 b Fn(L)2969 1606 y Fk(2)3006 1636 y Fp(\()p Fn(D)r Fp(\).)h(Here)f Fn(D)515 1736 y Fp(is)k(a)f(b)r (ounded)i(domain)e(with)i(su\016cien)n(tly)f(smo)r(oth)f(b)r(oundary)h Fn(@)5 b(D)37 b Fp(in)e Fi(R)2946 1705 y Fk(2)2989 1736 y Fp(,)g Fn(f)44 b Fl(2)36 b Fn(H)42 b Fp(and)515 1835 y Fn(\027)35 b(>)30 b Fp(0)i(is)f(a)h(constan)n(t.)f(W)-7 b(e)33 b(supplemen)n(t)f(the)g(Na)n(vier-Stok)n(es)e(equations)h(with)h (no-slip)f(or)515 1935 y(zero)25 b(Diric)n(hlet)h(b)r(oundary)g (condition)g Fn(v)s Fl(j)1832 1947 y Fm(@)t(D)1954 1935 y Fp(=)d(0.)j(W)-7 b(e)26 b(supp)r(ose)g Fn(w)j Fp(is)d(a)g(Wiener)g (pro)r(cess)f(in)515 2034 y(the)g(space)g Fn(H)951 2004 y Fk(2)1014 2034 y Fp(with)g(co)n(v)-5 b(ariance)24 b Fn(Q)h Fp(suc)n(h)g(that)g(tr)2118 2049 y Fm(H)2176 2033 y Fd(2)2214 2034 y Fn(Q)d(<)h Fl(1)p Fp(.)i(Here)g(and)h(b)r(elo)n(w)f (w)n(e)g(denote)515 2134 y(b)n(y)k Fn(H)708 2104 y Fm(s)773 2134 y Fp(the)i(domain)e(of)h(the)g(op)r(erator)e Fn(A)1857 2104 y Fm(s=)p Fk(2)1960 2134 y Fp(,)i Fn(s)d(>)f Fp(0.)k(W)-7 b(e)30 b(ob)n(viously)f(ha)n(v)n(e)f Fn(V)46 b Fp(=)26 b Fn(H)3234 2104 y Fk(1)p Fm(=)p Fk(2)3338 2134 y Fp(.)k(In)515 2234 y(the)e(space)f Fn(V)46 b Fp(w)n(e)27 b(will)h(use)g(the)g(norm)f Fl(k)18 b(\001)g(k)1900 2246 y Fm(V)1980 2234 y Fp(:=)23 b Fl(kr)18 b(\001)h(k)2304 2246 y Fm(H)2389 2234 y Fp(=)2477 2165 y Fl(p)p 2546 2165 42 4 v 69 x Fp(2)p Fl(k)p Fn(A)2692 2204 y Fk(1)p Fm(=)p Fk(2)2814 2234 y Fl(\001)g(k)2898 2246 y Fm(H)2960 2234 y Fp(.)515 2333 y(F)-7 b(or)27 b(di\013eren)n(t)g(ideas)g(to)h(treat)f(this)h(problem,)f(one)g(can)h (\014nd)g(in)f([11],)g([10)o(],)h([15)o(],)g([3],)f([25].)515 2433 y(W)-7 b(e)36 b(no)n(w)f(transform)g(this)h(sto)r(c)n(hastic)f (equation)g(to)g(a)h(random)f(equation)g(as)g(in)h(\(1\).)g(T)-7 b(o)515 2533 y(do)26 b(this)i(w)n(e)e(need)h(a)g(stationary)e(Ornstein) i(-)f(Uhlen)n(b)r(ec)n(k)h(pro)r(cess)f Fn(z)t Fp(.)g(This)h(pro)r (cess)f(will)h(b)r(e)515 2632 y(generated)f(b)n(y)h(the)h(sto)r(c)n (hastic)f(di\013eren)n(tial)h(equation)1523 2815 y Fn(dz)22 b Fp(+)c(2\()p Fn(k)k Fp(+)c(1\))p Fn(\027)5 b(Az)17 b(dt)23 b Fp(=)g Fn(dw)864 b Fp(\(14\))515 2997 y(for)36 b(a)g(p)r(ositiv)n(e)f(su\016cien)n(tly)i(large)e(constan)n(t)g Fn(k)s Fp(.)i(It)f(is)h(kno)n(wn)e(\(see,)i(e.g.)f(Da)g(Prato)f(and)515 3097 y(Zab)r(czyk)23 b([21)o(])h(Chapter)f(5\))h(that)g(there)f(exists) h(a)f(temp)r(ered)h(random)f(v)-5 b(ariable)23 b Fn(z)k Fp(in)d Fn(V)43 b Fp(suc)n(h)515 3197 y(that)1716 3296 y Fi(R)29 b Fl(3)23 b Fn(t)g Fl(!)h Fn(z)t Fp(\()p Fn(\022)2151 3308 y Fm(t)2179 3296 y Fn(!)s Fp(\))515 3446 y(solv)n(es)f(\(14\).)h (This)g(Ornstein)g(-)g(Uhlen)n(b)r(ec)n(k)h(pro)r(cess)e Fn(z)t Fp(\()p Fn(\022)2321 3458 y Fm(t)2350 3446 y Fn(!)s Fp(\))h(has)g(tra)5 b(jectories)23 b(in)h(the)h(space)515 3545 y Fn(L)572 3515 y Fk(2)572 3569 y Fm(loc)659 3545 y Fp(\()p Fi(R)p Fp(;)14 b Fn(H)858 3515 y Fk(3)901 3545 y Fp(\).)32 b(The)g(constan)n(t)e Fn(k)35 b Fp(ma)n(y)30 b(b)r(e)i(considered)f(as)f(a)h(con)n(trol)f(parameter.)g(W)-7 b(e)32 b(no)n(w)515 3645 y(consider)26 b(the)i(nonautonomous)e (di\013eren)n(tial)i(equation)1273 3810 y Fn(du)p 1273 3847 91 4 v 1275 3923 a(d)14 b(t)1392 3866 y Fp(+)k Fn(\027)5 b(A)14 b(u)23 b Fp(=)f Fn(F)12 b Fp(\()p Fn(u;)i(\022)1976 3878 y Fm(t)2005 3866 y Fn(!)s Fp(\))p Fn(;)97 b(u)p Fp(\(0\))23 b(=)f Fn(x)i Fl(2)f Fn(H)r(;)601 b Fp(\(15\))515 4072 y(where)992 4172 y Fn(F)12 b Fp(\()p Fn(u;)i(!)s Fp(\))22 b(=)h Fl(\000)p Fn(\027)5 b(Au)18 b Fl(\000)g Fp(\()p Fn(u)g Fl(\001)h(r)p Fp(\))p Fn(u)f Fl(\000)g Fp(\()p Fn(z)t Fp(\()p Fn(!)s Fp(\))h Fl(\001)f(r)p Fp(\))p Fn(u)h Fl(\000)f Fp(\()p Fn(u)g Fl(\001)g(r)p Fp(\))p Fn(z)t Fp(\()p Fn(!)s Fp(\))1386 4321 y Fl(\000)p Fp(\()p Fn(z)t Fp(\()p Fn(!)s Fp(\))g Fl(\001)h(r)p Fp(\))p Fn(z)t Fp(\()p Fn(!)s Fp(\))f(+)g(2)p Fn(\027)5 b(k)s(Az)t Fp(\()p Fn(!)s Fp(\))18 b(+)g Fn(f)t(:)515 4471 y Fp(The)32 b(idea)h(of)f(this) h(transformation)e(can)h(b)r(e)h(found)g(in)g(Crauel)f(and)g(Flandoli)h ([11)o(].)f(Since)515 4570 y(the)h(co)r(e\016cien)n(ts)g(of)g(this)g (equation)f(ha)n(v)n(e)g(similar)g(prop)r(erties)g(as)g(the)i(co)r (e\016cien)n(ts)e(of)h(the)515 4670 y(original)26 b(2D)i(Na)n(vier)f(-) 56 b(Stok)n(es)27 b(equations,)h(this)g(equation)f(has)h(a)g(unique)g (solution.)f(More)515 4770 y(precisely)-7 b(,)27 b(w)n(e)g(ha)n(v)n(e) 1950 5059 y(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fg(Lemma)29 b(3.1)41 b Fj(The)29 b(solution)f(of)h(\(15\))g(de\014nes)f(a)g(c)l(ontinuous)f(r)l(andom)i (dynamic)l(al)h(system)515 623 y Fn(')g Fj(with)g(r)l(esp)l(e)l(ct)g (to)f Fn(\022)j Fj(on)e Fn(H)7 b Fj(.)30 b(L)l(et)1549 798 y Fn(x)23 b Fl(!)h Fn(T)12 b Fp(\()p Fn(!)s(;)i(x)p Fp(\))22 b(:=)h Fn(x)c Fl(\000)f Fn(z)t Fp(\()p Fn(!)s Fp(\))886 b(\(16\))515 974 y Fj(b)l(e)32 b(a)g(r)l(andom)g(home)l (omorphism)i(on)e Fn(H)7 b Fj(.)32 b(Then)g Fn(T)2148 944 y Fh(\000)p Fk(1)2237 974 y Fp(\()p Fn(\022)2308 986 y Fm(t)2337 974 y Fn(!)s(;)14 b(')p Fp(\()p Fn(t;)g(!)s(;)g(T)e Fp(\()p Fn(!)s(;)i(x)p Fp(\)\)\))26 b(=:)40 b(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)p Fp(\))515 1074 y Fj(de\014nes)30 b(a)g(r)l(andom)g(dynamic)l(al)i(system)d(with)h(r)l (esp)l(e)l(ct)g(to)g Fn(\022)r Fj(.)g(In)g(p)l(articular,)1749 1249 y Fn(t)23 b Fl(!)37 b Fp(~)-56 b Fn(')q Fp(\()p Fn(t;)14 b(!)s(;)g(x)p Fp(\))515 1425 y Fj(solves)30 b(\(13\).)515 1586 y Fp(Since)e Fn(z)t Fp(\()p Fn(!)s Fp(\))22 b Fl(2)h Fn(V)47 b Fp(the)28 b(mapping)f Fn(T)39 b Fp(can)27 b(b)r(e)h(considered)f(as)g(a)g(homeomorphism)f(on)i Fn(V)18 b Fp(.)515 1685 y(It)27 b(is)g(w)n(ell)g(kno)n(wn)g(that)h(the) f(random)f(dynamical)h(system)g Fn(')h Fp(has)e(a)h(random)f(compact)h (ab-)515 1785 y(sorbing)21 b(forw)n(ard)g(in)n(v)-5 b(arian)n(t)22 b(set)h Fn(B)k Fp(in)c Fn(V)c Fp(,)k(see)g(for)f(instance)h(Crauel)f (and)g(Flandoli)h([11)o(].)g(W)-7 b(e)515 1884 y(no)n(w)23 b(form)n(ulate)g(a)h(v)n(ersion)e(of)i(these)g(results)f(and)h(will)g (pro)n(v)n(e)e(some)h(additional)g(prop)r(erties.)515 2045 y Fg(Lemma)29 b(3.2)41 b Fj(The)c(r)l(andom)f(dynamic)l(al)i (system)e Fn(')g Fj(has)h(a)f(c)l(omp)l(act)h(forwar)l(d)h(invariant) 515 2145 y(absorbing)30 b(set)e Fn(B)33 b Fj(in)28 b Fn(H)7 b Fj(.)29 b(This)h(absorbing)g(set)e(is)h(c)l(ontaine)l(d)g(in)f (the)h(close)l(d)h(b)l(al)t(l)f(in)g Fn(H)35 b Fj(with)515 2244 y(c)l(enter)29 b(zer)l(o)h(and)g(with)h(squar)l(e)e(r)l(adius)1024 2475 y Fn(R)1088 2440 y Fk(2)1125 2475 y Fp(\()p Fn(!)s Fp(\))24 b(=)e(\(1)c(+)h Fn(")p Fp(\))1616 2362 y Ff(Z)1698 2382 y Fk(0)1661 2550 y Fh(\0001)1798 2475 y Fn(m)p Fp(\()p Fn(\022)1942 2487 y Fm(\034)1983 2475 y Fn(!)s Fp(\))p Fn(e)2109 2430 y Fm(\027)t(\025)2185 2438 y Fd(1)2218 2430 y Fm(\034)7 b Fk(+)2318 2408 y Fd(8)p 2316 2417 33 4 v 2316 2450 a Fe(\027)2370 2369 y Ff(R)2425 2390 y Fd(0)2409 2466 y Fe(\034)2469 2430 y Fh(k)p Fm(z)r Fk(\()p Fm(\022)2595 2438 y Fe(s)2627 2430 y Fm(!)r Fk(\))p Fh(k)2731 2405 y Fd(2)2731 2447 y Fe(V)2780 2430 y Fm(ds)2850 2475 y Fn(d\034)e(;)363 b Fp(\(17\))515 2699 y Fj(wher)l(e)36 b Fn(")e(>)f Fp(0)j Fj(is)g(arbitr)l(ary,)h Fn(\025)1525 2711 y Fk(1)1598 2699 y Fj(is)f(the)g(\014rst)f(eigenvalue)i(of)g(the)f (op)l(er)l(ator)g Fl(\000)p Fp(\001)g Fj(with)g(the)515 2799 y(Dirichlet)31 b(b)l(oundary)f(c)l(ondition,)1092 3020 y Fn(m)p Fp(\()p Fn(!)s Fp(\))24 b(=)1408 2964 y(4)p 1405 3001 47 4 v 1405 3077 a Fn(\027)1475 2903 y Ff(\022)1568 2964 y Fp(2)p 1546 3001 86 4 v 1546 3077 a Fn(\025)1594 3089 y Fk(1)1642 3020 y Fl(k)p Fn(z)t Fp(\()p Fn(!)s Fp(\))p Fl(k)1888 2986 y Fk(4)1888 3040 y Fm(V)1963 3020 y Fp(+)18 b Fn(k)2092 2986 y Fk(2)2129 3020 y Fn(\027)2175 2986 y Fk(2)2212 3020 y Fl(k)p Fn(z)t Fp(\()p Fn(!)s Fp(\))p Fl(k)2458 2986 y Fk(2)2458 3040 y Fm(V)2533 3020 y Fp(+)g Fl(k)p Fn(f)9 b Fl(k)2750 2986 y Fk(2)2750 3040 y Fm(V)2802 3024 y Fc(0)2829 2903 y Ff(\023)3320 3020 y Fp(\(18\))515 3245 y Fj(and)30 b(the)g(p)l(ar)l(ameter)g Fn(k)j Fj(in)d(\(14\))g(is)g(chosen)h(such)f(that)1703 3463 y Fn(\025)1751 3475 y Fk(1)1812 3463 y Fn(>)1953 3407 y Fp(4)14 b(tr)2074 3419 y Fm(H)2137 3407 y Fn(Q)p 1909 3444 337 4 v 1909 3520 a Fp(\()p Fn(k)22 b Fp(+)c(1\))p Fn(\027)2209 3496 y Fk(3)2256 3463 y Fn(:)1041 b Fp(\(19\))515 3693 y Fj(In)34 b(addition,)i Fn(B)i Fj(is)d(temp)l(er)l(e)l(d)f(and)h Fn(t)c Fl(!)g Fp(sup)1994 3713 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(\022)2188 3721 y Fe(t)2215 3713 y Fm(!)r Fk(\))2302 3693 y Fl(k)p Fn(x)p Fl(k)2433 3663 y Fk(2)2433 3716 y Fm(H)2530 3693 y Fj(is)k(a)f(lo)l(c)l(al)t(ly)i(inte)l(gr)l(able)f (sta-)515 3793 y(tionary)30 b(pr)l(o)l(c)l(ess.)515 3953 y(Pr)l(o)l(of.)78 b Fp(W)-7 b(e)38 b(sk)n(etc)n(h)e(the)i(pro)r(of)e (of)i(this)f(lemma.)h(W)-7 b(e)37 b(obtain)g(b)n(y)g(T)-7 b(emam)37 b([23)o(])h(Lemma)515 4053 y(I)r(I)r(I.3.4)522 4247 y(2)p Fl(h)p Fn(F)12 b Fp(\()p Fn(u;)i(\022)817 4259 y Fm(t)846 4247 y Fn(!)s Fp(\))p Fn(;)g(u)p Fl(i)22 b(\024)1172 4191 y Fp(8)p 1170 4228 47 4 v 1170 4304 a Fn(\027)1226 4247 y Fl(k)p Fn(u)p Fl(k)1358 4213 y Fk(2)1358 4267 y Fm(H)1420 4247 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)1576 4259 y Fm(t)1604 4247 y Fn(!)s Fp(\))p Fl(k)1733 4213 y Fk(2)1733 4267 y Fm(V)1809 4247 y Fp(+)1947 4191 y(8)p 1902 4228 132 4 v 1902 4304 a Fn(\027)5 b(\025)1996 4316 y Fk(1)2044 4247 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)2200 4259 y Fm(t)2228 4247 y Fn(!)s Fp(\))p Fl(k)2357 4213 y Fk(4)2357 4267 y Fm(V)2432 4247 y Fp(+)2528 4191 y(4)p 2525 4228 47 4 v 2525 4304 a Fn(\027)2582 4247 y Fl(k)p Fn(f)k Fl(k)2716 4213 y Fk(2)2716 4267 y Fm(V)2768 4251 y Fc(0)2813 4247 y Fp(+)18 b(4)p Fn(k)2984 4213 y Fk(2)3020 4247 y Fn(\027)5 b Fl(k)p Fn(z)t Fp(\()p Fn(\022)3222 4259 y Fm(t)3251 4247 y Fn(!)s Fp(\))p Fl(k)3380 4213 y Fk(2)3380 4267 y Fm(V)3437 4247 y Fn(:)515 4468 y Fp(Let)23 b Fn(R)723 4438 y Fk(2)722 4489 y(0)783 4468 y Fp(b)r(e)g(the)h (stationary)d(solution)i(of)g(the)g(random)f(a\016ne)h(one-dimensional) f(di\013eren)n(tial)515 4568 y(equation)606 4726 y Fn(d\032)p 606 4763 87 4 v 612 4839 a(dt)721 4782 y Fp(+)c Fn(\027)5 b(\025)898 4794 y Fk(1)936 4782 y Fn(\032)23 b Fp(=)1102 4726 y(8)p 1099 4763 47 4 v 1099 4839 a Fn(\027)1155 4782 y(\032)p Fl(k)p Fn(z)t Fp(\()p Fn(\022)1354 4794 y Fm(t)1383 4782 y Fn(!)s Fp(\))p Fl(k)1512 4748 y Fk(2)1512 4803 y Fm(V)1587 4782 y Fp(+)1725 4726 y(8)p 1680 4763 132 4 v 1680 4839 a Fn(\027)5 b(\025)1774 4851 y Fk(1)1822 4782 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)1978 4794 y Fm(t)2006 4782 y Fn(!)s Fp(\))p Fl(k)2135 4748 y Fk(4)2135 4803 y Fm(V)2211 4782 y Fp(+)2306 4726 y(4)p 2304 4763 47 4 v 2304 4839 a Fn(\027)2360 4782 y Fl(k)p Fn(f)k Fl(k)2494 4748 y Fk(2)2494 4803 y Fm(V)2546 4786 y Fc(0)2591 4782 y Fp(+)18 b(4)p Fn(k)2762 4748 y Fk(2)2799 4782 y Fn(\027)5 b Fl(k)p Fn(z)t Fp(\()p Fn(\022)3001 4794 y Fm(t)3029 4782 y Fn(!)s Fp(\))p Fl(k)3158 4748 y Fk(2)3158 4803 y Fm(V)3216 4782 y Fn(:)81 b Fp(\(20\))1950 5059 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fp(This)27 b(stationary)f(solution)h Fn(R)1477 493 y Fk(2)1476 544 y(0)1514 523 y Fp(\()p Fn(!)s Fp(\))h(exists)g(and)f(it)h(is)f(exp)r(onen)n(tially)g (attracting)g(pro)n(vided)1146 709 y(8)p 1144 746 47 4 v 1144 822 a Fn(\027)1267 765 y Fp(lim)1214 815 y Fm(\034)7 b Fh(!\0001)1484 709 y Fp(1)p 1459 746 92 4 v 1459 822 a Fl(j)p Fn(\034)i Fl(j)1575 652 y Ff(Z)1658 673 y Fk(0)1621 841 y Fm(\034)1709 765 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)1865 777 y Fm(s)1899 765 y Fn(!)s Fp(\))p Fl(k)2028 731 y Fk(2)2028 786 y Fm(V)2085 765 y Fn(ds)24 b Fl(\021)2291 709 y Fp(8)p 2288 746 47 4 v 2288 822 a Fn(\027)2345 765 y Fi(E)8 b Fl(k)p Fn(z)t Fl(k)2521 731 y Fk(2)2521 786 y Fm(V)2606 765 y Fn(<)23 b(\027)5 b(\025)2788 777 y Fk(1)2825 765 y Fn(:)515 996 y Fp(A)43 b(Simple)h(calculation)e(sho)n (ws)g(that)i(this)f(relation)f(is)h(equiv)-5 b(alen)n(t)43 b(to)g(\(19\).)g(Moreo)n(v)n(er)515 1096 y Fn(R)579 1066 y Fk(2)616 1096 y Fp(\()p Fn(!)s Fp(\))33 b(=)g(\(1)23 b(+)f Fn(")p Fp(\))p Fn(R)1185 1066 y Fk(2)1184 1117 y(0)1222 1096 y Fp(\()p Fn(!)s Fp(\))34 b(has)f(the)i(form)e(\(17\).)h (The)g(temp)r(eredness)f(of)h Fn(R)2949 1066 y Fk(2)3020 1096 y Fp(follo)n(ws)f(from)515 1196 y(Flandoli)c(and)g(Sc)n (hmalfu\031)f([13])h(Lemma)g(7.2.)f(Since)h(the)h(solution)f Fn(R)2766 1165 y Fk(2)2765 1216 y(0)2803 1196 y Fp(\()p Fn(\022)2874 1208 y Fm(t)2903 1196 y Fn(!)s Fp(\))g(of)g(the)h(ab)r(o)n (v)n(e)515 1295 y(equation)d(is)g(con)n(tin)n(uous,)g(the)h(mapping) 1767 1478 y Fn(t)23 b Fl(!)g Fn(R)1990 1444 y Fk(2)2027 1478 y Fp(\()p Fn(\022)2098 1490 y Fm(t)2128 1478 y Fn(!)s Fp(\))515 1661 y(is)30 b(lo)r(cally)g(in)n(tegrable.)f(In)i(addition,)f (a)g(comparison)f(argumen)n(t)g(yields)h(that)h(the)g(random)515 1760 y(ball)c Fn(B)t Fp(\(0)p Fn(;)14 b(R)q Fp(\()p Fn(!)s Fp(\)\))28 b(is)f(forw)n(ard)f(in)n(v)-5 b(arian)n(t)27 b(and)g(forw)n(ard)f(absorbing.)g(Finally)-7 b(,)28 b(w)n(e)f(note)g (that)1110 1943 y Fn(B)t Fp(\()p Fn(!)s Fp(\))d(:=)p 1430 1871 939 4 v 22 w Fn(')p Fp(\(1)p Fn(;)14 b(\022)1634 1955 y Fh(\000)p Fk(1)1723 1943 y Fn(!)s(;)g(B)t Fp(\(0)p Fn(;)g(R)q Fp(\()p Fn(\022)2128 1955 y Fh(\000)p Fk(1)2217 1943 y Fn(!)s Fp(\)\)\))23 b Fl(\032)g Fn(B)t Fp(\(0)p Fn(;)14 b(R)q Fp(\()p Fn(!)s Fp(\)\))448 b(\(21\))515 2125 y(is)35 b(a)f(compact)h(forw)n(ard)f(in)n(v)-5 b(arian)n(t)34 b(and)h(forw)n(ard)e(absorbing)g(set)j(b)n(y)e(the)i(regularization)515 2225 y(prop)r(ert)n(y)26 b(of)i Fn(')p Fp(.)2379 b Fa(2)704 2424 y Fp(Ho)n(w)n(ev)n(er,)20 b(there)i(are)f(other)h(compact)f (absorbing)g(sets)g(de\014ned)i(b)n(y)f(a)g(ball)f Fn(B)t Fp(\(0)p Fn(;)14 b(R)q Fp(\()p Fn(!)s Fp(\)\))515 2524 y(with)20 b(random)g(radius)f Fn(R)q Fp(\()p Fn(!)s Fp(\),)h(see)g(for) g(instance)g(Flandoli)f(and)h(Langa)f([12)o(].)i(In)f(the)h(follo)n (wing)515 2624 y(w)n(e)k(prop)r(ose)f(another)g(metho)r(d)i(to)f (calculate)g(momen)n(ts)g(of)g(\(17\).)g(This)h(tec)n(hnique)f(is)g (based)515 2723 y(on)i(the)h(standard)f(densit)n(y)g(of)h(the)g (Girsano)n(v)d(theory)-7 b(.)515 2889 y Fg(Lemma)29 b(3.3)41 b Fj(L)l(et)29 b Fn(R)1224 2859 y Fk(2)1291 2889 y Fj(b)l(e)h(de\014ne) l(d)g(by)g(\(17\))g(then)g(if)g(we)h(cho)l(ose)g(a)f Fn(k)i Fj(such)e(that)1304 3114 y Fn(\025)1352 3126 y Fk(1)1413 3114 y Fn(>)1534 3058 y Fp(16)14 b(tr)1696 3070 y Fm(H)1759 3058 y Fn(Q)p 1511 3095 337 4 v 1511 3171 a Fp(\()p Fn(k)21 b Fp(+)d(1\))p Fn(\027)1810 3147 y Fk(3)1858 3114 y Fn(;)183 b(\025)2112 3126 y Fk(1)2173 3114 y Fl(\025)2292 3058 y Fp(256)14 b(tr)2495 3070 y Fm(H)2558 3058 y Fn(Q)p 2271 3095 375 4 v 2271 3171 a Fp(\()p Fn(k)21 b Fp(+)d(1\))2524 3147 y Fk(2)2561 3171 y Fn(\027)2607 3147 y Fk(3)2655 3114 y Fn(;)642 b Fp(\(22\))515 3351 y Fj(we)30 b(have)h Fi(E)22 b Fn(R)956 3321 y Fk(8)1022 3351 y Fn(<)h Fl(1)p Fj(.)515 3517 y(Pr)l(o)l(of.)52 b Fp(W)-7 b(e)28 b(rewrite)1217 3754 y Fn(R)1281 3720 y Fk(2)1341 3754 y Fp(=)23 b(\(1)18 b(+)g Fn(")p Fp(\))1689 3641 y Ff(Z)1772 3662 y Fk(0)1735 3830 y Fh(\0001)1871 3754 y Fn(m)p Fp(\()p Fn(\022)2015 3766 y Fm(\034)2057 3754 y Fn(!)s Fp(\))p Fn(e)2183 3710 y Fm(\027)t(\025)2259 3718 y Fd(1)2291 3710 y Fm(\034)7 b Fk(+)p Fm(c)2421 3649 y Ff(R)2476 3669 y Fd(0)2460 3745 y Fe(\034)2520 3710 y Fh(k)p Fm(z)r Fh(k)2622 3685 y Fd(2)2622 3726 y Fe(V)2676 3754 y Fn(d\034)515 3998 y Fp(for)29 b Fn(c)e Fp(=)811 3966 y Fk(8)p 809 3980 38 4 v 809 4027 a Fm(\027)886 3998 y Fp(and)j(some)f Fn(")e(>)g Fp(0.)j(W)-7 b(e)30 b(obtain)g(b)n(y)f(the)i(Cauc)n(h)n(y)e(-)h(Sc)n(h)n(w)n(arz)e (inequalit)n(y)h(for)h(an)515 4098 y(appropriate)c Fn(c)999 4110 y Fk(1)1059 4098 y Fn(>)c Fp(0)998 4364 y Fi(E)8 b Fn(R)1111 4330 y Fk(8)1237 4364 y Fl(\024)83 b Fn(c)1421 4376 y Fk(1)1458 4364 y Fp(\()p Fi(E)9 b Fn(m)1613 4330 y Fk(8)1656 4364 y Fp(\))1698 4307 y Fd(1)p 1698 4316 29 4 v 1698 4350 a(2)1754 4247 y Ff(\022)1815 4364 y Fi(E)1879 4251 y Ff(Z)1968 4271 y Fk(0)1931 4440 y Fh(\0001)2067 4364 y Fn(e)2106 4319 y Fk(4)p Fm(\027)t(\025)2215 4327 y Fd(1)2247 4319 y Fm(\034)e Fk(+8)p Fm(c)2410 4259 y Ff(R)2465 4279 y Fd(0)2449 4355 y Fe(\034)2509 4319 y Fh(k)p Fm(z)r Fh(k)2611 4294 y Fd(2)2611 4336 y Fe(V)2665 4364 y Fn(d\034)2753 4247 y Ff(\023)2825 4232 y Fd(1)p 2825 4241 V 2825 4274 a(2)1237 4629 y Fp(=)83 b Fn(c)1421 4641 y Fk(1)1458 4629 y Fp(\()p Fi(E)9 b Fn(m)1613 4594 y Fk(8)1656 4629 y Fp(\))1698 4572 y Fd(1)p 1698 4581 V 1698 4614 a(2)1754 4511 y Ff(\022)1815 4516 y(Z)1898 4536 y Fh(1)1862 4704 y Fk(0)1983 4629 y Fn(e)2022 4594 y Fh(\000)p Fk(4)p Fm(\027)t(\025)2183 4602 y Fd(1)2215 4594 y Fm(\034)2275 4629 y Fl(\001)19 b Fi(E)8 b Fn(e)2405 4584 y Fk(8)p Fm(c)2485 4523 y Ff(R)2540 4544 y Fe(\034)2524 4620 y Fd(0)2589 4584 y Fh(k)p Fm(z)r Fh(k)2691 4559 y Fd(2)2691 4600 y Fe(V)2745 4629 y Fn(d\034)2833 4511 y Ff(\023)2905 4506 y Fd(1)p 2905 4515 V 2905 4549 a(2)2961 4629 y Fn(:)1950 5059 y Fp(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fp(The)31 b(\014rst)h(factor)f(is)g(\014nite,)i (since)e Fn(z)k Fp(is)d(a)f(Gaussian)g(random)f(v)-5 b(ariable.)31 b(W)-7 b(e)32 b(no)n(w)f(restrict)515 623 y(ourselv)n(es)23 b(to)i(calculate)f Fi(E)f Fp(exp)5 b Fl(f)p Fp(8)p Fn(c)1636 556 y Ff(R)1690 576 y Fm(\034)1674 652 y Fk(0)1746 623 y Fl(k)p Fn(z)t Fl(k)1873 593 y Fk(2)1873 646 y Fm(V)1929 623 y Fl(g)p Fp(.)24 b(Ito's)h(form)n(ula)f(applied)h (to)g Fl(k)13 b(\001)g(k)3025 593 y Fk(2)3025 646 y Fm(H)3113 623 y Fp(for)25 b Fn(z)t Fp(\()p Fn(\022)3352 635 y Fm(t)3381 623 y Fn(!)s Fp(\))515 722 y(yields:)576 933 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)732 945 y Fm(\034)773 933 y Fn(!)s Fp(\))p Fl(k)902 899 y Fk(2)902 953 y Fm(H)982 933 y Fp(+)18 b(2\()p Fn(k)k Fp(+)c(1\))p Fn(\027)1421 820 y Ff(Z)1504 840 y Fm(\034)1467 1008 y Fk(0)1559 933 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)1715 945 y Fm(s)1749 933 y Fn(!)s Fp(\))p Fl(k)1878 899 y Fk(2)1878 953 y Fm(V)1935 933 y Fn(ds)24 b Fp(=)e Fl(k)p Fn(z)t Fp(\()p Fn(!)s Fp(\))p Fl(k)2374 899 y Fk(2)2374 953 y Fm(H)2454 933 y Fp(+)c(2)2593 820 y Ff(Z)2676 840 y Fm(\034)2639 1008 y Fk(0)2717 933 y Fp(\()p Fn(z)t(;)c(dw)r Fp(\))2965 945 y Fm(H)3047 933 y Fp(+)k Fn(\034)24 b Fp(tr)3254 945 y Fm(H)3317 933 y Fn(Q:)515 1143 y Fp(Hence)j(w)n(e)h(can)f(deriv)n (e)g(that)824 1334 y Fn(e)863 1290 y Fk(16)p Fm(c)970 1229 y Ff(R)1025 1250 y Fe(\034)1009 1326 y Fd(0)1074 1290 y Fh(k)p Fm(z)r Fh(k)1176 1265 y Fd(2)1176 1306 y Fe(V)1253 1334 y Fl(\024)c Fn(e)1452 1271 y Fd(8)p Fe(c)p 1389 1280 181 4 v 1389 1315 a Fd(\()p Fe(k)q Fd(+1\))p Fe(\027)1580 1294 y Fh(k)p Fm(z)r Fh(k)1682 1268 y Fd(2)1682 1310 y Fe(H)1758 1334 y Fl(\001)18 b Fn(e)1911 1271 y Fd(8)p Fe(c)p 1848 1280 V 1848 1315 a Fd(\()p Fe(k)q Fd(+1\))p Fe(\027)2038 1294 y Fm(\034)g Fk(tr)2139 1302 y Fe(H)2192 1294 y Fm(Q)2267 1334 y Fl(\001)h Fn(e)p Fp(\()p Fn(\034)9 b Fp(\))19 b Fl(\001)g Fn(e)2617 1263 y Fd(256)p Fe(c)2728 1247 y Fd(2)p 2566 1272 246 4 v 2566 1315 a(\()p Fe(k)q Fd(+1\))2713 1303 y(2)2747 1315 y Fe(\027)2780 1303 y Fd(2)2833 1225 y Ff(R)2888 1246 y Fe(\034)2872 1321 y Fd(0)2925 1286 y Fk(\()p Fm(Qz)r(;z)r Fk(\))3135 1334 y Fn(;)515 1503 y Fp(where)696 1703 y Fn(e)p Fp(\()p Fn(\034)9 b Fp(\))24 b Fl(\021)e Fn(e)p Fp(\()p Fn(\034)5 b(;)14 b(!)s Fp(\))23 b(=)g(exp)1442 1586 y Ff(\032)1605 1647 y Fp(16)p Fn(c)p 1515 1684 300 4 v 1515 1760 a Fp(\()p Fn(k)e Fp(+)d(1\))p Fn(\027)1838 1590 y Ff(Z)1921 1611 y Fm(\034)1884 1779 y Fk(0)1963 1703 y Fp(\()p Fn(z)t(;)c(dw)r Fp(\))2211 1715 y Fm(H)2293 1703 y Fl(\000)2474 1647 y Fp(256)p Fn(c)2636 1617 y Fk(2)p 2386 1684 375 4 v 2386 1760 a Fp(\()p Fn(k)21 b Fp(+)d(1\))2639 1736 y Fk(2)2676 1760 y Fn(\027)2722 1736 y Fk(2)2784 1590 y Ff(Z)2867 1611 y Fm(\034)2830 1779 y Fk(0)2908 1703 y Fp(\()p Fn(Qz)t(;)c(z)t Fp(\))3161 1586 y Ff(\033)3264 1703 y Fn(:)515 1948 y Fp(F)-7 b(rom)30 b(\(22\))h(w)n(e)f(ha)n(v)n(e)g(that)1442 1911 y Fk(256)p Fm(c)1571 1886 y Fd(2)1603 1911 y Fk(tr)1655 1919 y Fe(H)1708 1911 y Fm(Q)p 1428 1929 347 4 v 1428 1977 a Fk(\()p Fm(k)q Fk(+1\))1600 1960 y Fd(2)1633 1977 y Fm(\027)1670 1960 y Fd(2)1703 1977 y Fm(\025)1742 1985 y Fd(1)1813 1948 y Fl(\024)e Fp(8)p Fn(c)p Fp(.)j(Therefore)e(using)i(the)g(Cauc)n(h)n (y)f(-)61 b(Sc)n(h)n(w)n(arz)515 2057 y(inequalit)n(y)27 b(and)1366 2194 y(\()p Fn(Qz)t(;)14 b(z)t Fp(\))21 b Fl(\024)i Fp(tr)1793 2206 y Fm(H)1856 2194 y Fn(Q)p Fl(k)p Fn(z)t Fl(k)2049 2160 y Fk(2)2049 2215 y Fm(H)2133 2194 y Fl(\024)2230 2138 y Fp(tr)2295 2150 y Fm(H)2358 2138 y Fn(Q)p 2230 2175 194 4 v 2284 2251 a(\025)2332 2263 y Fk(1)2434 2194 y Fl(k)p Fn(z)t Fl(k)2561 2160 y Fk(2)2561 2215 y Fm(V)515 2371 y Fp(w)n(e)k(ha)n(v)n(e)984 2503 y Fi(E)8 b Fn(e)1072 2458 y Fk(8)p Fm(c)1152 2398 y Ff(R)1207 2418 y Fe(\034)1191 2494 y Fd(0)1256 2458 y Fh(k)p Fm(z)r Fh(k)1358 2433 y Fd(2)1358 2475 y Fe(V)1435 2503 y Fl(\024)1523 2411 y Ff(\020)1572 2503 y Fi(E)h Fn(e)1725 2440 y Fd(16)p Fe(c)p 1676 2449 181 4 v 1676 2484 a Fd(\()p Fe(k)q Fd(+1\))p Fe(\027)1867 2462 y Fh(k)p Fm(z)r Fh(k)1969 2437 y Fd(2)1969 2479 y Fe(H)2026 2411 y Ff(\021)2086 2406 y Fd(1)p 2086 2415 29 4 v 2086 2448 a(2)2142 2436 y Ff(\000)2180 2503 y Fi(E)g Fn(e)p Fp(\()p Fn(\034)g Fp(\))2379 2469 y Fk(2)2422 2436 y Ff(\001)2470 2430 y Fd(1)p 2470 2439 V 2470 2472 a(2)2526 2503 y Fn(e)2637 2440 y Fd(8)p Fe(c)p 2575 2449 181 4 v 2575 2484 a Fd(\()p Fe(k)q Fd(+1\))p Fe(\027)2765 2462 y Fm(\034)18 b Fk(tr)2866 2470 y Fe(H)2919 2462 y Fm(Q)2975 2503 y Fn(:)322 b Fp(\(23\))515 2669 y(W)-7 b(e)30 b(can)g(use)f(the)i(standard)e(argumen)n(ts)f(\(see,)i(e.g.,)g ([19)o(])g(and)g([18)o(]\))g(to)g(\014nd)g(the)h(estimate)515 2768 y Fi(E)8 b Fp([)p Fn(e)p Fp(\()p Fn(\034)i Fp(\))736 2738 y Fk(2)779 2768 y Fp(])29 b Fl(\024)f Fp(1)j(for)f(the)h(mean)g(v) -5 b(alue)31 b(of)g(Girsano)n(v's)e(densit)n(y)i Fn(e)p Fp(\()p Fn(\034)9 b Fp(\))2663 2738 y Fk(2)2701 2768 y Fp(.)31 b(The)g(v)-5 b(alue)31 b Fn(z)t Fp(\()p Fn(!)s Fp(\))f(is)h(a)515 2868 y(Gaussian)26 b(v)-5 b(ariable)27 b(in)h Fn(H)34 b Fp(with)28 b(the)g(zero)f(mean)g(and)h(with)g(the)g (co)n(v)-5 b(ariance)1242 3036 y Fi(E)9 b Fl(h)p Fn(z)t(;)14 b(h)1452 3048 y Fk(1)1494 3036 y Fl(ih)p Fn(z)t(;)g(h)1686 3048 y Fk(2)1723 3036 y Fl(i)23 b Fp(=)g Fl(h)1917 3015 y Fp(~)1898 3036 y Fn(Qh)2012 3048 y Fk(1)2049 3036 y Fn(;)14 b(h)2134 3048 y Fk(2)2171 3036 y Fl(i)p Fn(;)97 b(h)2371 3048 y Fk(1)2408 3036 y Fn(;)28 b(h)2507 3048 y Fk(2)2567 3036 y Fl(2)c Fn(H)r(;)515 3205 y Fp(where)1338 3309 y(~)1319 3330 y Fn(Q)f Fp(=)1495 3217 y Ff(Z)1578 3238 y Fh(1)1541 3406 y Fk(0)1662 3330 y Fn(e)1701 3296 y Fh(\000)p Fk(2)p Fm(t)p Fk(\()p Fm(k)q Fk(+1\))p Fm(\027)t(A)2075 3330 y Fn(Qe)2180 3296 y Fh(\000)p Fk(2)p Fm(t)p Fk(\()p Fm(k)q Fk(+1\))p Fm(\027)t(A)2567 3330 y Fn(dt:)515 3518 y Fp(Therefore)37 b(simple)h(calculation)g(\(see,)g(e.g.)g(Kuo)f([17)o (]\))i(P)n(age)e(105)f(sho)n(ws)i(that)g(the)h(\014rst)515 3617 y(factor)27 b(in)g(the)h(righ)n(t)f(hand)h(side)f(of)h(\(23\))f (is)h(\014nite)g(pro)n(vided)2541 3585 y Fk(16)p Fm(c)p 2484 3599 210 4 v 2484 3646 a Fk(\()p Fm(k)q Fk(+1\))p Fm(\027)2727 3617 y Fn(<)2903 3585 y Fk(1)p 2825 3599 191 4 v 2825 3655 a(2tr)2910 3663 y Fe(H)2978 3640 y Fk(~)2963 3655 y Fm(Q)3025 3617 y Fp(.)g(Moreo)n(v)n(er)1173 3889 y Fi(E)8 b Fn(e)1325 3826 y Fd(16)p Fe(c)p 1276 3835 181 4 v 1276 3869 a Fd(\()p Fe(k)q Fd(+1\))p Fe(\027)1467 3848 y Fh(k)p Fm(z)r Fh(k)1569 3823 y Fd(2)1569 3865 y Fe(H)1650 3889 y Fl(\024)22 b Fp(exp)1878 3747 y Ff(\()2155 3833 y Fp(16)p Fn(c)p Fp(tr)2339 3845 y Fm(H)2421 3812 y Fp(~)2402 3833 y Fn(Q)p 1955 3870 714 4 v 1955 3956 a Fp(\()p Fn(k)f Fp(+)d(1\))p Fn(\027)24 b Fl(\000)18 b Fp(32)p Fn(c)p Fp(tr)2540 3968 y Fm(H)2622 3935 y Fp(~)2603 3956 y Fn(Q)2678 3747 y Ff(\))2787 3889 y Fn(:)515 4146 y Fp(Ho)n(w)n(ev)n(er)30 b(it)k(is)e(easy)g(to)h(see)f(that)h(tr)1715 4158 y Fm(H)1797 4125 y Fp(~)1778 4146 y Fn(Q)e Fl(\024)2060 4109 y Fk(tr)2111 4117 y Fe(H)2165 4109 y Fm(Q)p 1981 4127 315 4 v 1981 4175 a Fk(2)p Fm(\025)2053 4183 y Fd(1)2086 4175 y Fk(\()p Fm(k)q Fk(+1\))p Fm(\027)2306 4146 y Fp(.)i(Therefore)e (from)h(the)h(second)g(as-)515 4260 y(sumption)28 b(of)f(\(22\))g(w)n (e)h(ha)n(v)n(e)e(that)1021 4478 y Fi(E)8 b Fn(e)1173 4415 y Fd(16)p Fe(c)p 1125 4424 181 4 v 1125 4459 a Fd(\()p Fe(k)q Fd(+1\))p Fe(\027)1315 4437 y Fh(k)p Fm(z)r Fh(k)1417 4412 y Fd(2)1417 4454 y Fe(H)1498 4478 y Fl(\024)23 b Fp(exp)1726 4361 y Ff(\032)2100 4422 y Fp(8)p Fn(c)p Fp(tr)2242 4434 y Fm(H)2305 4422 y Fn(Q)p 1799 4459 874 4 v 1799 4535 a(\025)1847 4547 y Fk(1)1884 4535 y Fp(\()p Fn(k)f Fp(+)c(1\))2138 4511 y Fk(2)2175 4535 y Fn(\027)2221 4511 y Fk(2)2277 4535 y Fl(\000)g Fp(16)p Fn(c)p Fp(tr)2544 4547 y Fm(H)2607 4535 y Fn(Q)2682 4361 y Ff(\033)2768 4478 y Fn(<)k Fl(1)p Fn(:)359 b Fp(\(24\))515 4710 y(Since)28 b(from)f(\(22\))g(w)n(e)g(also)g(ha)n(v)n(e)f(4)p Fn(\027)5 b(\025)1719 4722 y Fk(1)1780 4710 y Fn(>)1877 4673 y Fk(8)p Fm(c)p Fk(tr)1992 4681 y Fe(H)2045 4673 y Fm(Q)p 1877 4691 221 4 v 1882 4739 a Fk(\()p Fm(k)q Fk(+1\))p Fm(\027)2108 4710 y Fp(,)27 b(the)h(exp)r(ectation)g(of)f Fn(R)2907 4680 y Fk(8)2972 4710 y Fp(is)g(\014nite.)143 b Fa(2)1950 5059 y Fp(15)p eop %%Page: 16 16 16 15 bop 704 523 a Fp(The)34 b(follo)n(wing)g(lemma)h(allo)n(ws)e(us)h (to)h(conclude)f(the)h(existence)g(of)f(a)h(set)f Fl(L)h Fp(of)g(de-)515 623 y(termining)i(functionals)f(for)h(the)g(random)f (dynamical)h(system)50 b(~)-56 b Fn(')38 b Fp(generated)e(b)n(y)g (\(13\))h(if)515 722 y(the)i(random)e(dynamical)h(system)g Fn(')h Fp(generated)f(b)n(y)g(\(15\))g(has)g(the)h(set)f(of)h (determining)515 822 y(functionals)c Fl(L)p Fp(.)g(This)h(lemma)f(is)g (form)n(ulated)f(for)h(more)f(general)g(transformations)f(than)515 922 y(\(16\).)515 1088 y Fg(Lemma)c(3.4)41 b Fj(Supp)l(ose)28 b(that)f(the)h(r)l(andom)g(dynamic)l(al)i(systems)40 b Fp(~)-55 b Fn(')28 b Fj(and)g Fn(')g Fj(ar)l(e)g(c)l(onjugate)l(d)515 1187 y(by)i(a)h(r)l(andom)g(home)l(omorphism)h Fn(T)41 b Fj(on)31 b Fn(H)7 b Fj(,)30 b(i.e.)45 b Fp(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)p Fp(\()p Fn(!)s Fp(\)\))24 b(=)g Fn(T)2718 1157 y Fh(\000)p Fk(1)2806 1187 y Fp(\()p Fn(\022)2877 1199 y Fm(t)2906 1187 y Fn(!)s(;)14 b(')p Fp(\()p Fn(t;)g(!)s(;)g(x)p Fp(\()p Fn(!)s Fp(\)\))p Fj(,)515 1287 y(wher)l(e)26 b Fn(x)p Fp(\()p Fn(!)s Fp(\))e(=)f Fn(T)12 b Fp(\()p Fn(!)s(;)18 b Fp(~)-46 b Fn(x)o Fp(\()p Fn(!)s Fp(\)\))p Fj(.)27 b(Supp)l(ose)f(that)g Fn(')g Fj(has)h(a)f(c)l(omp)l(act)h (absorbing)g(set)e(and)i(forwar)l(d)515 1386 y(invariant)37 b(set)e Fn(B)t Fj(.)i(Then)50 b Fp(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)1660 1398 y Fk(1)1697 1386 y Fp(\()p Fn(!)s Fp(\)\))24 b Fl(\000)36 b Fp(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)2252 1398 y Fk(2)2289 1386 y Fp(\()p Fn(!)s Fp(\)\))37 b Fj(tends)f(to)g (zer)l(o)g(in)g(pr)l(ob)l(ability)515 1486 y(for)29 b Fn(t)23 b Fl(!)h(1)k Fj(if)i(and)g(only)f(if)h Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1709 1498 y Fk(1)1746 1486 y Fp(\()p Fn(!)s Fp(\)\))k Fl(\000)e Fn(')p Fp(\()p Fn(t;)e(!)s(;)g(x) 2288 1498 y Fk(2)2325 1486 y Fp(\()p Fn(!)s Fp(\)\))30 b Fj(tends)e(to)h(zer)l(o)g(in)g(pr)l(ob)l(ability)515 1586 y(for)h Fn(t)23 b Fl(!)g(1)p Fj(.)31 b(Her)l(e)e Fn(x)1190 1598 y Fm(i)1218 1586 y Fp(\()p Fn(!)s Fp(\))24 b(=)e Fn(T)12 b Fp(\()p Fn(!)s(;)18 b Fp(~)-46 b Fn(x)1680 1598 y Fm(i)1707 1586 y Fp(\()p Fn(!)s Fp(\)\))p Fj(.)515 1752 y(Pr)l(o)l(of.)50 b Fp(Supp)r(ose)25 b(that)g Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1574 1764 y Fk(1)1612 1752 y Fp(\()p Fn(!)s Fp(\)\))f Fl(\000)g Fn(')p Fp(\()p Fn(t;)h(!)s(;)g(x) 2146 1764 y Fk(2)2184 1752 y Fp(\()p Fn(!)s Fp(\)\))26 b(tends)f(to)g(zero)f(in)h(probabilit)n(y)f(for)515 1851 y Fn(t)f Fl(!)g(1)p Fp(.)i(By)f(the)h(absorbing)e(prop)r(ert)n(y)h(of)g Fn(B)29 b Fp(w)n(e)c(can)f(assume)g(that)h Fn(x)2745 1863 y Fk(1)2783 1851 y Fp(\()p Fn(!)s Fp(\))p Fn(;)j(x)3000 1863 y Fk(2)3037 1851 y Fp(\()p Fn(!)s Fp(\))c Fl(2)f Fn(B)t Fp(\()p Fn(!)s Fp(\).)515 1951 y(F)-7 b(or)24 b(an)n(y)f Fn(")g(>)g Fp(0)h(there)g(exists)h(a)f(compact)g(set)g Fn(C)2046 1963 y Fm(")2107 1951 y Fp(suc)n(h)g(that)h Fn(C)2527 1963 y Fm(")2586 1951 y Fl(\033)e Fn(B)t Fp(\()p Fn(!)s Fp(\))i(with)g(probabilit)n(y)515 2051 y(bigger)35 b(than)h(1)24 b Fl(\000)g Fn(")p Fp(.)36 b(Indeed,)g(this)h(follo)n(ws) e(b)n(y)h(the)h(regularization)d(prop)r(ert)n(y)h(of)h Fn(')h Fp(and)515 2150 y(b)n(y)30 b(the)h(construction)f(of)h Fn(B)k Fp(in)c(\(21\).)f Fn(T)1816 2120 y Fh(\000)p Fk(1)1904 2150 y Fp(\()p Fn(!)s Fp(\))h(is)g(uniformly)f(con)n(tin)n(uous)g(on)g Fn(C)3118 2162 y Fm(")3154 2150 y Fp(:)h(for)f(an)n(y)515 2250 y Fn(!)e Fl(2)d Fp(\012,)k Fn(\026)c(>)g Fp(0,)k Fn(y)1087 2262 y Fk(1)1124 2250 y Fn(;)e(y)1215 2262 y Fk(2)1277 2250 y Fl(2)f Fn(C)1417 2262 y Fm(")1482 2250 y Fp(there)i(exists)h(a)f Fn(\016)s Fp(\()p Fn(!)s Fp(\))e Fn(>)f Fp(0)j(suc)n(h)h(that)g(if)g Fl(k)p Fn(y)2871 2262 y Fk(1)2927 2250 y Fl(\000)19 b Fn(y)3052 2262 y Fk(2)3089 2250 y Fl(k)3131 2262 y Fm(H)3218 2250 y Fn(<)25 b(\016)s Fp(\()p Fn(!)s Fp(\))515 2350 y(then)34 b Fl(k)p Fn(T)813 2319 y Fh(\000)p Fk(1)901 2350 y Fp(\()p Fn(!)s(;)14 b(y)1066 2362 y Fk(1)1103 2350 y Fp(\))23 b Fl(\000)f Fn(T)1306 2319 y Fh(\000)p Fk(1)1395 2350 y Fp(\()p Fn(!)s(;)14 b(y)1560 2362 y Fk(2)1596 2350 y Fp(\))p Fl(k)1670 2362 y Fm(H)1767 2350 y Fn(<)34 b(\026)p Fp(.)g(On)g(the)h(other)e(hand)h (since)g Fn(\016)s Fp(\()p Fn(!)s Fp(\))g Fn(>)g Fp(0)g(there)515 2449 y(exists)27 b(a)g Fn(\016)850 2461 y Fm(")909 2449 y Fn(>)22 b Fp(0:)1605 2549 y Fi(P)p Fp(\()p Fn(\016)1726 2561 y Fm(")1783 2549 y Fn(<)h(\016)s Fp(\()p Fn(!)s Fp(\)\))g Fn(>)g Fp(1)18 b Fl(\000)g Fn(":)515 2698 y Fp(Hence)27 b(for)h(su\016cien)n(tly)f(large)f Fn(t)1537 2710 y Fm(")1600 2698 y Fp(w)n(e)i(ha)n(v)n(e)1028 2881 y Fi(P)p Fp(\()p Fl(k)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)1446 2893 y Fk(1)1482 2881 y Fp(\()p Fn(!)s Fp(\)\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(x)2027 2893 y Fk(2)2065 2881 y Fp(\()p Fn(!)s Fp(\)\))p Fl(k)2258 2893 y Fm(H)2344 2881 y Fn(>)22 b(\016)s Fp(\()p Fn(\022)2542 2893 y Fm(t)2572 2881 y Fn(!)s Fp(\)\))1194 3005 y Fl(\024)g Fn(")c Fp(+)g Fi(P)p Fp(\()p Fl(k)p Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(x)1839 3017 y Fk(1)1876 3005 y Fp(\()p Fn(!)s Fp(\)\))19 b Fl(\000)f Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(x)2421 3017 y Fk(2)2458 3005 y Fp(\()p Fn(!)s Fp(\)\))p Fl(k)2651 3017 y Fm(H)2737 3005 y Fn(>)23 b(\016)2862 3017 y Fm(")2898 3005 y Fp(\))g Fn(<)f Fp(2)p Fn(")515 3188 y Fp(if)28 b Fn(t)23 b Fl(\025)g Fn(t)762 3200 y Fm(")797 3188 y Fp(.)28 b(Hence)888 3371 y Fl(k)13 b Fp(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)1222 3383 y Fk(1)1260 3371 y Fp(\()p Fn(!)s Fp(\)\))19 b Fl(\000)31 b Fp(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)1805 3383 y Fk(2)1842 3371 y Fp(\()p Fn(!)s Fp(\)\))p Fl(k)2035 3383 y Fm(H)1054 3495 y Fp(=)23 b Fl(k)p Fn(T)1245 3461 y Fh(\000)p Fk(1)1333 3495 y Fp(\()p Fn(\022)1404 3507 y Fm(t)1433 3495 y Fn(!)s(;)14 b(')p Fp(\()p Fn(t;)g(!)s(;)g(x)1817 3507 y Fk(1)1855 3495 y Fp(\()p Fn(!)s Fp(\)\)\))19 b Fl(\000)f Fn(T)2201 3461 y Fh(\000)p Fk(1)2289 3495 y Fp(\()p Fn(\022)2360 3507 y Fm(t)2390 3495 y Fn(!)s(;)c(')p Fp(\()p Fn(t;)g(!)s(;)g(x)2774 3507 y Fk(2)2811 3495 y Fp(\()p Fn(!)s Fp(\)\)\))p Fl(k)3036 3507 y Fm(H)3122 3495 y Fn(<)23 b(\026)515 3678 y Fp(with)28 b(probabilit)n(y)e(bigger)h(than)g(1)18 b Fl(\000)g Fp(3)p Fn(")27 b Fp(for)g Fn(t)c(>)g(t)2120 3690 y Fm(")2156 3678 y Fp(.)534 3757 y(~)515 3778 y Fn(B)32 b Fp(=)c Fn(T)764 3747 y Fh(\000)p Fk(1)852 3778 y Fp(\()p Fn(B)t Fp(\))j(is)g(a)f(compact)g(forw)n(ard)f(in)n(v)-5 b(arian)n(t)29 b(absorbing)g(set)i(for)f(\(13\))g(if)h(and)g(only)f(if)515 3877 y Fn(B)36 b Fp(is)c(a)f(compact)h(forw)n(ard)e(in)n(v)-5 b(arian)n(t)31 b(absorbing)f(set)i(for)g(\(15\).)f(Therefore)g(w)n(e)h (can)f(sho)n(w)515 3977 y(the)d(second)f(direction)g(similarly)g(as)f (the)i(pro)r(of)f(ab)r(o)n(v)n(e)g(for)g(the)h(\014rst)f(direction.)313 b Fa(2)515 4259 y Fg(Corollary)31 b(3.5)41 b Fj(The)29 b(set)f(of)h(line)l(ar)g(functionals)g Fl(L)g Fj(on)f Fn(V)47 b Fj(is)29 b(determining)g(in)f(pr)l(ob)l(ability)515 4359 y(for)36 b(the)f(r)l(andom)h(dynamic)l(al)h(system)49 b Fp(~)-56 b Fn(')36 b Fj(gener)l(ate)l(d)g(by)f(\(13\))h(if)g(and)g (only)g(if)g Fl(L)g Fj(is)f(deter-)515 4458 y(mining)30 b(in)g(pr)l(ob)l(ability)i(for)e Fn(')g Fj(de\014ne)l(d)g(by)g(\(15\).) 515 4641 y(Pr)l(o)l(of.)52 b Fp(The)27 b(pro)r(of)g(is)g(based)g(on)g (the)h(fact)f(that)h(for)f(some)g Fn(l)d Fl(2)f(L)28 b Fp(the)g(limit)g(in)g(probabilit)n(y)515 4741 y(for)21 b Fn(t)i Fl(!)g(1)e Fp(of)h Fn(l)r Fp(\()13 b(~)-55 b Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)1339 4753 y Fk(1)1376 4741 y Fp(\()p Fn(!)s Fp(\)\))6 b Fl(\000)20 b Fp(~)-56 b Fn(')q Fp(\()p Fn(t;)14 b(!)s(;)19 b Fp(~)-47 b Fn(x)1897 4753 y Fk(2)1934 4741 y Fp(\()p Fn(!)s Fp(\)\)\))22 b(is)g(zero)e(if)i(and)f(only)g(if)h Fn(l)r Fp(\()p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)3208 4753 y Fk(1)3245 4741 y Fp(\()p Fn(!)s Fp(\)\))6 b Fl(\000)1950 5059 y Fp(16)p eop %%Page: 17 17 17 16 bop 515 523 a Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(x)807 535 y Fk(2)844 523 y Fp(\()p Fn(!)s Fp(\)\)\))29 b(tends)e(to)h(zero)e (in)h(probabilit)n(y)g(for)g Fn(t)c Fl(!)g(1)k Fp(whic)n(h)h(follo)n (ws)e(from)h(the)h(par-)515 623 y(ticular)f(shap)r(e)g(of)h Fn(T)12 b Fp(.)27 b(On)g(the)h(other)f(hand,)h(w)n(e)f(can)g(also)g (apply)g(the)h(last)f(lemma.)227 b Fa(2)704 822 y Fp(F)-7 b(or)27 b(the)h(follo)n(wing)e(w)n(e)h(need)h(t)n(w)n(o)f(a)g(priori)g (estimates)g(for)g Fn(')p Fp(:)515 974 y Fg(Lemma)i(3.6)41 b Fj(The)32 b(r)l(andom)h(dynamic)l(al)g(system)f Fn(')f Fj(satis\014es)h(the)g(fol)t(lowing)i(a)e(priori)i(es-)515 1074 y(timate)827 1276 y Fn(\027)72 b Fp(sup)887 1349 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))1132 1163 y Ff(Z)1215 1183 y Fm(t)1178 1351 y Fk(0)1258 1276 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)1677 1241 y Fk(2)1677 1296 y Fm(V)1735 1276 y Fn(d\034)33 b Fl(\024)22 b Fn(R)1998 1241 y Fk(2)2035 1276 y Fp(\()p Fn(!)s Fp(\))d(+)2268 1219 y(8)p 2266 1257 47 4 v 2266 1333 a Fn(\027)2336 1163 y Ff(Z)2419 1183 y Fm(t)2382 1351 y Fk(0)2462 1276 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)2618 1288 y Fm(\034)2658 1276 y Fn(!)s Fp(\))p Fl(k)2787 1241 y Fk(2)2787 1296 y Fm(H)2850 1276 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)3006 1288 y Fm(\034)3046 1276 y Fn(!)s Fp(\))p Fl(k)3175 1241 y Fk(2)3175 1296 y Fm(V)3233 1276 y Fn(d\034)912 1533 y Fp(+)989 1477 y(4)p 987 1514 V 987 1590 a Fn(\027)1042 1533 y(t)p Fl(k)p Fn(f)9 b Fl(k)1206 1499 y Fk(2)1206 1554 y Fm(V)1259 1537 y Fc(0)1304 1533 y Fp(+)18 b Fn(\027)5 b(k)1479 1499 y Fk(2)1530 1420 y Ff(Z)1613 1441 y Fm(t)1576 1609 y Fk(0)1656 1533 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)1812 1545 y Fm(\034)1852 1533 y Fn(!)s Fp(\))p Fl(k)1981 1499 y Fk(2)1981 1554 y Fm(V)2039 1533 y Fn(d\034)28 b Fp(+)18 b Fn(c)2265 1545 y Fm(E)2321 1533 y Fn(M)2424 1420 y Ff(Z)2507 1441 y Fm(t)2471 1609 y Fk(0)2550 1533 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)2706 1545 y Fm(\034)2747 1533 y Fn(!)s Fp(\))p Fl(k)2876 1499 y Fk(2)2876 1556 y Fm(H)2934 1539 y Fd(3)2971 1533 y Fn(d\034)912 1771 y Fp(+)987 1714 y Fn(c)1023 1726 y Fm(E)p 986 1751 93 4 v 987 1827 a Fn(M)1102 1658 y Ff(Z)1185 1678 y Fm(t)1148 1846 y Fk(0)1228 1771 y Fn(R)1292 1736 y Fk(4)1329 1771 y Fp(\()p Fn(\022)1400 1783 y Fm(\034)1442 1771 y Fn(!)s Fp(\))p Fn(d\034)5 b(:)515 1978 y Fj(wher)l(e)35 b Fn(M)43 b Fj(is)35 b(an)f(arbitr)l(arily)j(p)l(ositive)e(numb)l(er)f(and)h Fn(c)2300 1990 y Fm(E)2391 1978 y Fj(is)f(the)h(norm)f(of)i(the)e(emb)l (e)l(dding)515 2077 y(op)l(er)l(ator)d(of)h Fn(H)1017 2047 y Fk(3)1079 2077 y Fp(=)24 b Fn(D)r Fp(\()p Fn(A)1333 2047 y Fk(3)p Fm(=)p Fk(2)1438 2077 y Fp(\))31 b Fj(into)g(the)g(sp)l (ac)l(e)h Fn(W)2121 2047 y Fk(1)2109 2098 y Fh(1)2179 2077 y Fp(\()p Fn(D)r Fp(\))g Fj(of)f(two-dimensional)i(functions)d Fn(v)515 2177 y Fj(such)g(that)f Fn(v)s(;)14 b Fl(r)p Fn(v)27 b Fl(2)c Fn(L)1224 2147 y Fh(1)1294 2177 y Fp(\()p Fn(D)r Fp(\))p Fj(.)31 b(Similarly,)h(for)e(an)g(appr)l(opriate)i(p)l (olynomial)g Fn(p)714 2400 y Fp(2)p Fn(\027)71 b Fp(sup)815 2473 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))1060 2287 y Ff(Z)1143 2307 y Fm(t)1106 2475 y Fk(0)1186 2400 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)1605 2365 y Fk(2)1605 2420 y Fm(H)1668 2400 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)2087 2365 y Fk(2)2087 2420 y Fm(V)2145 2400 y Fn(d\034)33 b Fl(\024)22 b Fn(R)2408 2365 y Fk(4)2445 2400 y Fp(\()p Fn(!)s Fp(\))883 2658 y(+)962 2545 y Ff(Z)1045 2565 y Fm(t)1008 2733 y Fk(0)1088 2658 y Fn(p)p Fp(\()p Fl(k)p Fn(z)t Fp(\()p Fn(\022)1318 2670 y Fm(\034)1358 2658 y Fn(!)s Fp(\))p Fl(k)1487 2623 y Fk(2)1487 2678 y Fm(H)1550 2658 y Fn(;)14 b Fl(k)p Fn(z)t Fp(\()p Fn(\022)1743 2670 y Fm(\034)1783 2658 y Fn(!)s Fp(\))p Fl(k)1912 2623 y Fk(2)1912 2678 y Fm(V)1969 2658 y Fn(;)g Fl(k)p Fn(z)t Fp(\()p Fn(\022)2162 2670 y Fm(\034)2203 2658 y Fn(!)s Fp(\))p Fl(k)2332 2623 y Fk(3)2332 2678 y Fm(H)2394 2658 y Fn(;)g Fl(k)p Fn(f)9 b Fl(k)2565 2623 y Fk(2)2565 2678 y Fm(V)2617 2661 y Fc(0)2644 2658 y Fp(\))p Fn(d\034)28 b Fp(+)2866 2545 y Ff(Z)2949 2565 y Fm(t)2913 2733 y Fk(0)2992 2658 y Fn(R)3056 2623 y Fk(8)3093 2658 y Fp(\()p Fn(\022)3164 2670 y Fm(\034)3206 2658 y Fn(!)s Fp(\))p Fn(d\034)40 b(:)515 2875 y Fp(This)32 b(a)g(priori)f(estimate)h(is)g (based)g(on)g(the)h(calculation)e(of)h Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)2692 2844 y Fk(2)2692 2897 y Fm(H)2787 2875 y Fp(for)f(\(15\).)h(The)h(term)515 2974 y Fl(h)p Fp(\()p Fn(u)18 b Fl(\001)h(r)p Fp(\))p Fn(z)t Fp(\))p Fn(;)14 b(u)p Fl(i)27 b Fp(arising)f(in)i(the)g (calculation)f(can)g(b)r(e)h(estimated)g(b)n(y)f(the)h(Sob)r(olev)f (lemma:)1000 3182 y Fl(jh)p Fp(\()p Fn(u)19 b Fl(\001)f(r)p Fp(\))p Fn(z)t(;)c(u)p Fl(ij)23 b(\024)g Fn(c)1626 3194 y Fm(E)1682 3182 y Fl(k)p Fn(z)t Fl(k)1809 3197 y Fm(H)1867 3180 y Fd(3)1902 3182 y Fl(k)p Fn(u)p Fl(k)2034 3147 y Fk(2)2034 3202 y Fm(H)2119 3182 y Fl(\024)2216 3126 y Fn(c)2252 3138 y Fm(E)2308 3126 y Fn(M)p 2216 3163 182 4 v 2286 3239 a Fp(2)2408 3182 y Fl(k)p Fn(z)t Fl(k)2535 3147 y Fk(2)2535 3204 y Fm(H)2593 3187 y Fd(3)2647 3182 y Fp(+)2760 3126 y Fn(c)2796 3138 y Fm(E)p 2740 3163 132 4 v 2740 3239 a Fp(2)p Fn(M)2881 3182 y(R)2945 3147 y Fk(4)515 3380 y Fp(b)r(ecause)k Fn(x)c Fl(2)h Fn(B)t Fp(.)k(The)f(second)g(estimate)h(follo)n(ws)e(similarly)h(for)g Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)2825 3350 y Fk(4)2825 3403 y Fm(H)2887 3380 y Fp(.)515 3645 y Fg(Lemma)i(3.7)41 b Fj(Under)30 b(c)l(onditions)g(\(22\))h(the)f(fol)t(lowing)i(estimate) e(holds:)910 3835 y Fp(\006)970 3847 y Fm(k)1094 3835 y Fl(\021)22 b Fp(lim)15 b(sup)1436 3855 y Fm(m)p Fh(!1)1668 3802 y Fk(1)p 1655 3816 59 4 v 1655 3864 a Fm(m)1724 3835 y Fi(E)1787 3743 y Ff(n)1848 3835 y Fp(sup)1973 3855 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))2222 3768 y Ff(R)2278 3789 y Fm(m)2262 3864 y Fk(0)2355 3835 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)2774 3805 y Fk(2)2774 3858 y Fm(V)2831 3835 y Fn(d\034)2919 3743 y Ff(o)3320 3835 y Fp(\(25\))1201 3993 y Fl(\024)1289 3925 y Ff(\000)1355 3960 y Fk(4)p 1337 3974 70 4 v 1337 4021 a Fm(\027)1374 4005 y Fd(2)1417 3993 y Fl(k)p Fn(f)9 b Fl(k)1551 3963 y Fk(2)1551 4016 y Fm(V)1603 3999 y Fc(0)1648 3993 y Fp(+)18 b Fn(g)1771 4005 y Fm(k)1811 3993 y Fp(\()p Fn(\027)q(;)c(\025)1970 4005 y Fk(1)2008 3993 y Fn(;)g(Q)p Fp(\))2143 3925 y Ff(\001)2199 3993 y Fl(\001)19 b Fp(\(1)f(+)g Fn(h)2464 4005 y Fm(k)2505 3993 y Fp(\()p Fn(\027)q(;)c(A;)g(Q)p Fp(\)\))g Fn(;)515 4158 y Fj(wher)l(e)1178 4286 y Fn(g)1218 4298 y Fm(k)1258 4286 y Fp(\()p Fn(\027)q(;)g(\025)1417 4298 y Fk(1)1455 4286 y Fn(;)g(Q)p Fp(\))23 b(=)g Fn(a)1745 4298 y Fk(0)1792 4230 y Fp(tr)1856 4242 y Fm(H)1919 4230 y Fn(Q)p 1792 4267 194 4 v 1865 4343 a(\027)2013 4286 y Fl(\001)2055 4169 y Ff(\022)2116 4286 y Fn(k)e Fp(+)2366 4230 y Fn(a)2410 4242 y Fk(1)2447 4230 y Fp(tr)2512 4242 y Fm(H)2575 4230 y Fn(Q)p 2273 4267 460 4 v 2273 4343 a Fp(\()p Fn(k)h Fp(+)c(1\))2527 4319 y Fk(2)2564 4343 y Fn(\025)2612 4355 y Fk(1)2650 4343 y Fn(\027)2696 4319 y Fk(3)2743 4169 y Ff(\023)3320 4286 y Fp(\(26\))515 4471 y Fj(and)857 4626 y Fn(h)905 4638 y Fm(k)946 4626 y Fp(\()p Fn(\027)q(;)c(A;)g(Q)p Fp(\))23 b(=)g(2)p Fn(c)1443 4638 y Fm(E)1512 4509 y Ff(\022)2060 4570 y Fp([tr)2148 4582 y Fm(H)2211 4570 y Fn(QA)2339 4539 y Fk(2)2376 4570 y Fp(])2399 4539 y Fk(2)p 1583 4607 1330 4 v 1583 4683 a Fn(\027)1629 4659 y Fk(3)1667 4683 y Fn(\025)1715 4654 y Fk(3)1715 4705 y(1)1752 4683 y Fp(\()p Fn(k)f Fp(+)c(1\))g Fl(\001)h Fp([)p Fn(\027)2135 4659 y Fk(3)2172 4683 y Fn(\025)2220 4695 y Fk(1)2258 4683 y Fp(\()p Fn(k)j Fp(+)c(1\))g Fl(\000)g Fp(16tr)2761 4695 y Fm(H)2824 4683 y Fn(Q)o Fp(])2922 4509 y Ff(\023)2984 4523 y Fk(1)p Fm(=)p Fk(4)3102 4626 y Fn(:)195 b Fp(\(27\))515 4810 y Fj(Her)l(e)28 b Fn(a)756 4822 y Fk(0)821 4810 y Fj(and)h Fn(a)1025 4822 y Fk(1)1090 4810 y Fj(ar)l(e)g(some)f(absolute)h(c)l(onstants)e (and)i Fn(c)2316 4822 y Fm(E)2400 4810 y Fj(is)g(the)f(same)h(as)f(in)g (L)l(emma)h(3.6.)1950 5059 y Fp(17)p eop %%Page: 18 18 18 17 bop 515 523 a Fj(Pr)l(o)l(of.)52 b Fp(It)28 b(follo)n(ws)f(from)g (Lemma)g(3.6)g(that)647 740 y(\006)707 752 y Fm(k)771 740 y Fl(\024)890 684 y Fp(4)p 869 721 84 4 v 869 797 a Fn(\027)915 773 y Fk(2)962 740 y Fl(k)p Fn(f)9 b Fl(k)1096 706 y Fk(2)1096 761 y Fm(V)1148 744 y Fc(0)1193 740 y Fp(+)1307 684 y(8)p 1286 721 V 1286 797 a Fn(\027)1332 773 y Fk(2)1380 740 y Fi(E)1443 673 y Ff(\000)1487 740 y Fl(k)p Fn(z)t Fl(k)1614 706 y Fk(2)1614 761 y Fm(H)1675 740 y Fl(k)p Fn(z)t Fl(k)1802 706 y Fk(2)1802 761 y Fm(V)1858 673 y Ff(\001)1915 740 y Fp(+)18 b Fn(k)2044 706 y Fk(2)2081 740 y Fi(E)8 b Fl(k)p Fn(z)t Fl(k)2257 706 y Fk(2)2257 761 y Fm(V)2338 740 y Fp(+)2431 684 y Fn(c)2467 696 y Fm(E)p 2431 721 93 4 v 2454 797 a Fn(\027)2533 740 y(M)h Fi(E)f Fl(k)p Fn(z)t Fl(k)2799 706 y Fk(2)2799 763 y Fm(H)2857 746 y Fd(3)2917 740 y Fp(+)3032 684 y Fn(c)3068 696 y Fm(E)p 3010 721 136 4 v 3010 797 a Fn(\027)d(M)3156 740 y Fi(E)j Fn(R)3269 706 y Fk(4)3312 740 y Fn(:)515 982 y Fp(If)28 b(w)n(e)f(c)n(ho)r(ose)f Fn(M)32 b Fp(=)1184 915 y Ff(\000)1222 982 y Fi(E)8 b Fl(k)p Fn(z)t Fl(k)1398 952 y Fk(2)1398 1009 y Fm(H)1456 992 y Fd(3)1498 915 y Ff(\001)1536 932 y Fh(\000)p Fk(1)p Fm(=)p Fk(2)1710 982 y Fl(\001)1752 915 y Ff(\000)1790 982 y Fi(E)g Fn(R)1903 952 y Fk(4)1946 915 y Ff(\001)1984 932 y Fk(1)p Fm(=)p Fk(2)2089 982 y Fp(,)27 b(then)i(w)n(e)e(obtain)1097 1196 y(\006)1157 1208 y Fm(k)1286 1196 y Fl(\024)1402 1163 y Fk(4)p 1384 1177 70 4 v 1384 1225 a Fm(\027)1421 1208 y Fd(2)1463 1196 y Fl(k)p Fn(f)9 b Fl(k)1597 1166 y Fk(2)1597 1219 y Fm(V)1649 1202 y Fc(0)1694 1196 y Fp(+)1805 1163 y Fk(8)p 1787 1177 V 1787 1225 a Fm(\027)1824 1208 y Fd(2)1881 1129 y Ff(\000)1919 1196 y Fi(E)f Fl(k)p Fn(z)t Fl(k)2095 1166 y Fk(4)2095 1219 y Fm(H)2162 1129 y Ff(\001)2200 1146 y Fk(1)p Fm(=)p Fk(2)2323 1196 y Fl(\001)2365 1129 y Ff(\000)2403 1196 y Fi(E)g Fl(k)p Fn(z)t Fl(k)2579 1166 y Fk(4)2579 1219 y Fm(V)2641 1129 y Ff(\001)2679 1146 y Fk(1)p Fm(=)p Fk(2)1280 1352 y Fp(+)p Fn(k)1391 1322 y Fk(2)1428 1352 y Fi(E)g Fl(k)p Fn(z)t Fl(k)1604 1322 y Fk(2)1604 1375 y Fm(V)1685 1352 y Fp(+)1778 1318 y Fk(2)p Fm(c)1841 1326 y Fe(E)p 1778 1333 111 4 v 1815 1381 a Fm(\027)1913 1285 y Ff(\000)1951 1352 y Fi(E)g Fl(k)p Fn(z)t Fl(k)2127 1322 y Fk(2)2127 1379 y Fm(H)2185 1362 y Fd(3)2227 1285 y Ff(\001)2265 1302 y Fk(1)p Fm(=)p Fk(2)2387 1352 y Fl(\001)2429 1285 y Ff(\000)2467 1352 y Fi(E)g Fn(R)2580 1322 y Fk(4)2623 1285 y Ff(\001)2661 1302 y Fk(1)p Fm(=)p Fk(2)2779 1352 y Fn(:)515 1535 y Fp(Using)23 b(the)h(de\014nition)g(of)g Fn(z)j Fp(it)d(is)g(easy)f(to)g(\014nd)h(that)g(for)f(an)n(y)g(p)r (ositiv)n(e)h Fn(\013)f Fl(2)h Fp([0)p Fn(;)14 b Fp(3)p Fn(=)p Fp(2])21 b(w)n(e)j(ha)n(v)n(e)1324 1756 y Fi(E)9 b Fl(k)p Fn(A)1478 1722 y Fm(\013)1531 1756 y Fn(z)t Fl(k)1616 1722 y Fk(2)1616 1777 y Fm(H)1701 1756 y Fp(=)1948 1700 y(1)p 1798 1737 342 4 v 1798 1813 a(4\()p Fn(k)21 b Fp(+)d(1\))p Fn(\027)2149 1756 y Fp(tr)2214 1768 y Fm(H)2277 1756 y Fp(\()p Fn(QA)2437 1722 y Fk(2)p Fm(\013)p Fh(\000)p Fk(1)2603 1756 y Fp(\))p Fn(:)662 b Fp(\(28\))515 1983 y(F)-7 b(urthermore)26 b(it)i(is)g(clear)e(that)1213 2166 y Fi(E)9 b Fl(k)p Fn(A)1367 2131 y Fm(\013)1420 2166 y Fn(z)t Fl(k)1505 2131 y Fk(2)p Fm(l)1505 2186 y(H)1590 2166 y Fl(\024)22 b Fn(c)1713 2178 y Fm(l)1753 2098 y Ff(\000)1791 2166 y Fi(E)8 b Fl(k)p Fn(A)1944 2131 y Fm(\013)1997 2166 y Fn(z)t Fl(k)2082 2131 y Fk(2)2082 2186 y Fm(H)2144 2098 y Ff(\001)2182 2115 y Fm(l)2221 2166 y Fn(;)97 b(l)25 b Fp(=)d(1)p Fn(;)14 b Fp(2)p Fn(;)g(:)g(:)g(:)f (;)551 b Fp(\(29\))515 2348 y(with)28 b(appropriate)e(constan)n(ts)g Fn(c)1555 2360 y Fm(l)1581 2348 y Fp(.)h(Therefore)g(w)n(e)g(ha)n(v)n (e)660 2586 y(\006)720 2598 y Fm(k)784 2586 y Fl(\024)903 2530 y Fp(4)p 882 2567 84 4 v 882 2643 a Fn(\027)928 2619 y Fk(2)975 2586 y Fl(k)p Fn(f)9 b Fl(k)1109 2551 y Fk(2)1109 2606 y Fm(V)1161 2590 y Fc(0)1206 2586 y Fp(+)1354 2530 y Fn(c)1390 2542 y Fk(0)1427 2530 y Fp([tr)1515 2542 y Fm(H)1578 2530 y Fn(Q)p Fp(])1667 2499 y Fk(2)p 1299 2567 460 4 v 1299 2643 a Fn(\027)1345 2619 y Fk(4)1383 2643 y Fp(\()p Fn(k)21 b Fp(+)d(1\))1636 2619 y Fk(2)1673 2643 y Fn(\025)1721 2655 y Fk(1)1788 2586 y Fp(+)1902 2530 y Fn(k)p 1881 2567 88 4 v 1881 2643 a Fp(2)p Fn(\027)1997 2586 y Fl(\001)g Fp(tr)2103 2598 y Fm(H)2166 2586 y Fn(Q)g Fp(+)2343 2530 y Fn(c)2379 2542 y Fm(E)2435 2530 y Fp([tr)2523 2542 y Fm(H)2586 2530 y Fn(QA)2714 2499 y Fk(2)2751 2530 y Fp(])2774 2499 y Fk(1)p Fm(=)p Fk(2)p 2343 2567 536 4 v 2357 2644 a Fn(\027)2403 2620 y Fk(3)p Fm(=)p Fk(2)2507 2644 y Fp(\()p Fn(k)k Fp(+)c(1\))2761 2620 y Fk(1)p Fm(=)p Fk(2)2907 2586 y Fl(\001)2948 2518 y Ff(\000)2986 2586 y Fi(E)9 b Fn(R)3100 2551 y Fk(4)3143 2518 y Ff(\001)3181 2535 y Fk(1)p Fm(=)p Fk(2)3299 2586 y Fn(:)515 2850 y Fp(with)33 b(some)f(absolute)g(constan)n(t)g Fn(c)1631 2862 y Fk(0)1668 2850 y Fp(.)h(No)n(w)f(w)n(e)g(estimate)2383 2783 y Ff(\000)2421 2850 y Fi(E)8 b Fn(R)2535 2820 y Fk(4)2577 2783 y Ff(\001)2615 2800 y Fk(1)p Fm(=)p Fk(2)2720 2850 y Fp(.)33 b(W)-7 b(e)33 b(use)f(the)h(idea)g(of)515 2949 y(the)28 b(pro)r(of)f(of)g(Lemma)h(3.3.)e(It)i(is)g(clear)e(that) 527 3143 y Ff(\000)565 3210 y Fi(E)8 b Fn(R)678 3176 y Fk(4)721 3143 y Ff(\001)759 3159 y Fk(1)p Fm(=)p Fk(2)887 3210 y Fl(\024)22 b Fp(\(1)c(+)g Fn(")p Fp(\))1234 3093 y Ff(\022)1371 3154 y Fp(3)p 1305 3191 174 4 v 1305 3267 a(2)p Fn(\027)5 b(\025)1441 3279 y Fk(1)1489 3093 y Ff(\023)1550 3110 y Fk(3)p Fm(=)p Fk(4)1668 3143 y Ff(\000)1706 3210 y Fi(E)k Fn(m)1829 3176 y Fk(4)1872 3143 y Ff(\001)1910 3159 y Fk(1)p Fm(=)p Fk(4)2028 3093 y Ff(\022)2089 3210 y Fi(E)2152 3097 y Ff(Z)2241 3117 y Fk(0)2204 3286 y Fh(\0001)2340 3210 y Fn(e)2379 3165 y Fk(2)p Fm(\027)t(\025)2488 3173 y Fd(1)2521 3165 y Fm(\034)e Fk(+4)p Fm(c)2683 3105 y Ff(R)2739 3125 y Fd(0)2723 3201 y Fe(\034)2783 3165 y Fh(k)p Fm(z)r Fk(\()p Fm(\022)2909 3173 y Fe(s)2940 3165 y Fm(!)r Fk(\))p Fh(k)3044 3140 y Fd(2)3044 3182 y Fe(V)3094 3165 y Fm(ds)3164 3210 y Fn(d\034)3252 3093 y Ff(\023)3314 3101 y Fk(1)p Fm(=)p Fk(4)3432 3210 y Fn(;)515 3446 y Fp(where)33 b Fn(m)p Fp(\()p Fn(!)s Fp(\))i(is)f(giv)n (en)f(b)n(y)h(\(18\))g(and)g Fn(c)g Fp(=)g(8)p Fn(=\027)5 b Fp(.)34 b(Using)g(Girsano)n(v's)e(tric)n(k)i(and)g(\(23\))g(and)515 3546 y(\(24\))27 b(with)h Fn(c)23 b Fp(:=)g Fn(c=)p Fp(2)j(w)n(e)i(ha)n (v)n(e)714 3778 y Fi(E)8 b Fn(e)802 3734 y Fk(4)p Fm(c)882 3673 y Ff(R)938 3693 y Fe(\034)921 3769 y Fd(0)986 3734 y Fh(k)p Fm(z)r Fh(k)1088 3708 y Fd(2)1088 3750 y Fe(V)1166 3778 y Fl(\024)22 b Fp(exp)1394 3661 y Ff(\032)1747 3722 y Fp(2)p Fn(c)p Fp(tr)1889 3734 y Fm(H)1952 3722 y Fn(Q)p 1466 3759 833 4 v 1466 3835 a(\025)1514 3847 y Fk(1)1552 3835 y Fp(\()p Fn(k)f Fp(+)d(1\))1805 3811 y Fk(2)1842 3835 y Fn(\027)1888 3811 y Fk(2)1944 3835 y Fl(\000)g Fp(8)p Fn(c)p Fp(tr)2170 3847 y Fm(H)2233 3835 y Fn(Q)2308 3661 y Ff(\033)2389 3778 y Fl(\001)g Fp(exp)2571 3661 y Ff(\032)2754 3722 y Fp(4)p Fn(c)p 2643 3759 300 4 v 2643 3835 a Fp(\()p Fn(k)k Fp(+)c(1\))p Fn(\027)2953 3778 y(\034)23 b Fp(tr)3077 3790 y Fm(H)3140 3778 y Fn(Q)3206 3661 y Ff(\033)515 4016 y Fp(under)i(conditions)g(\(22\).)h(Ho)n(w)n (ev)n(er)d(\(22\))j(implies)g(that)g Fn(\025)2352 4028 y Fk(1)2389 4016 y Fp(\()p Fn(k)18 b Fp(+)c(1\))2635 3986 y Fk(2)2672 4016 y Fn(\027)2718 3986 y Fk(2)2779 4016 y Fl(\025)22 b Fp(32)p Fn(c)p Fp(tr)3050 4028 y Fm(H)3113 4016 y Fn(Q)p Fp(.)j(There-)515 4116 y(fore)1200 4244 y Fi(E)9 b Fn(e)1289 4200 y Fk(4)p Fm(c)1369 4139 y Ff(R)1424 4159 y Fe(\034)1408 4235 y Fd(0)1473 4200 y Fh(k)p Fm(z)r Fh(k)1575 4175 y Fd(2)1575 4216 y Fe(V)1652 4244 y Fl(\024)23 b Fp(exp)1880 4127 y Ff(\032)1973 4188 y Fp(1)p 1952 4225 84 4 v 1952 4301 a(12)2064 4244 y(+)2268 4188 y(4)p Fn(c)p 2157 4225 300 4 v 2157 4301 a Fp(\()p Fn(k)e Fp(+)d(1\))p Fn(\027)2467 4244 y(\034)23 b Fp(tr)2591 4256 y Fm(H)2654 4244 y Fn(Q)2720 4127 y Ff(\033)3320 4244 y Fp(\(30\))515 4444 y(under)k(conditions)g(\(22\).)h(F)-7 b(rom)27 b(\(30\))g(w)n(e)g(ha)n(v)n(e)559 4632 y Ff(\000)597 4699 y Fi(E)9 b Fn(R)711 4665 y Fk(4)753 4632 y Ff(\001)792 4648 y Fk(1)p Fm(=)p Fk(2)919 4699 y Fl(\024)23 b Fp(\(1)18 b(+)g Fn(")p Fp(\))p Fn(e)1292 4665 y Fk(1)p Fm(=)p Fk(48)1443 4582 y Ff(\022)1580 4643 y Fp(3)p 1514 4680 174 4 v 1514 4756 a(2)p Fn(\027)5 b(\025)1650 4768 y Fk(1)1697 4582 y Ff(\023)1758 4599 y Fk(3)p Fm(=)p Fk(4)1881 4699 y Fl(\001)1923 4582 y Ff(\022)1984 4699 y Fp(2)p Fn(\027)g(\025)2120 4711 y Fk(1)2176 4699 y Fl(\000)2396 4643 y Fp(32)p 2269 4680 337 4 v 2269 4756 a(\()p Fn(k)21 b Fp(+)d(1\))p Fn(\027)2568 4732 y Fk(2)2616 4699 y Fp(tr)2680 4711 y Fm(H)2743 4699 y Fn(Q)2809 4582 y Ff(\023)2870 4599 y Fh(\000)p Fk(1)p Fm(=)p Fk(4)3040 4632 y Ff(\000)3078 4699 y Fi(E)9 b Fn(m)3201 4665 y Fk(4)3244 4632 y Ff(\001)3282 4648 y Fk(1)p Fm(=)p Fk(4)3400 4699 y Fn(:)1950 5059 y Fp(18)p eop %%Page: 19 19 19 18 bop 515 534 a Fp(No)n(w)27 b(w)n(e)g(estimate)1159 466 y Ff(\000)1197 534 y Fi(E)8 b Fn(m)1319 504 y Fk(4)1362 466 y Ff(\001)1400 484 y Fk(1)p Fm(=)p Fk(4)1505 534 y Fp(.)27 b(It)h(is)g(clear)e(from)i(\(18\))f(that)740 689 y Ff(\000)778 756 y Fi(E)8 b Fn(m)900 721 y Fk(4)943 689 y Ff(\001)982 705 y Fk(1)p Fm(=)p Fk(4)1109 756 y Fl(\024)1209 700 y Fp(4)p 1207 737 47 4 v 1207 813 a Fn(\027)1277 639 y Ff(\032)1371 700 y Fp(2)p 1349 737 86 4 v 1349 813 a Fn(\025)1397 825 y Fk(1)1458 689 y Ff(\000)1496 756 y Fi(E)h Fl(k)p Fn(z)t Fp(\()p Fn(!)s Fp(\))p Fl(k)1792 721 y Fk(16)1792 776 y Fm(V)1867 689 y Ff(\001)1905 705 y Fk(1)p Fm(=)p Fk(4)2027 756 y Fp(+)18 b Fn(k)2156 721 y Fk(2)2194 756 y Fn(\027)2240 721 y Fk(2)2291 689 y Ff(\000)2329 756 y Fi(E)8 b Fl(k)p Fn(z)t Fp(\()p Fn(!)s Fp(\))p Fl(k)2624 721 y Fk(8)2624 776 y Fm(V)2687 689 y Ff(\001)2725 705 y Fk(1)p Fm(=)p Fk(4)2847 756 y Fp(+)18 b Fl(k)p Fn(f)9 b Fl(k)3064 721 y Fk(2)3064 776 y Fm(V)3116 760 y Fc(0)3143 639 y Ff(\033)3219 756 y Fn(:)515 970 y Fp(Therefore)26 b(using)h(\(29\))h(w)n(e)f(obtain)985 1120 y Ff(\000)1023 1187 y Fi(E)8 b Fn(m)1145 1153 y Fk(4)1189 1120 y Ff(\001)1227 1136 y Fk(1)p Fm(=)p Fk(4)1354 1187 y Fl(\024)1454 1131 y Fp(4)p 1452 1168 47 4 v 1452 1244 a Fn(\027)1522 1070 y Ff(\032)1584 1187 y Fl(k)p Fn(f)h Fl(k)1718 1153 y Fk(2)1718 1207 y Fm(V)1770 1191 y Fc(0)1815 1187 y Fp(+)1963 1131 y Fn(c)1999 1143 y Fk(1)2036 1131 y Fp([tr)2124 1143 y Fm(H)2187 1131 y Fn(Q)p Fp(])2276 1100 y Fk(2)p 1908 1168 460 4 v 1908 1244 a Fn(\027)1954 1220 y Fk(2)1992 1244 y Fp(\()p Fn(k)21 b Fp(+)d(1\))2245 1220 y Fk(2)2282 1244 y Fn(\025)2330 1256 y Fk(1)2396 1187 y Fp(+)g Fn(c)2515 1199 y Fk(2)2553 1187 y Fn(k)s(\027)23 b Fl(\001)c Fp(tr)2769 1199 y Fm(H)2832 1187 y Fn(Q)2898 1070 y Ff(\033)2974 1187 y Fn(;)515 1396 y Fp(where)k Fn(c)787 1408 y Fk(1)849 1396 y Fp(and)h Fn(c)1043 1408 y Fk(2)1104 1396 y Fp(are)g(absolute)f(constan)n(ts.)g (Put)i(all)f(these)g(estimates)g(together)f(w)n(e)h(obtain)515 1496 y(the)k(upp)r(er)f(b)r(ound)h(\(25\))g(for)f(\006)1514 1508 y Fm(k)1555 1496 y Fp(.)1827 b Fa(2)704 1695 y Fp(W)-7 b(e)36 b(ha)n(v)n(e)f(seen)g(that)i Fn(B)t Fp(,)f(de\014ned)g(in)g (\(21\),)g(is)g(b)r(ounded)g(in)g Fn(V)19 b Fp(,)36 b(and)g(hence)g(it) g(is)g(a)515 1795 y(compact)24 b(set)g(in)g Fn(H)32 b Fp(whic)n(h)24 b(is)g(temp)r(ered)h(with)g(resp)r(ect)f(the)g Fn(H)31 b Fp(norm.)24 b(W)-7 b(e)25 b(no)n(w)f(pro)n(v)n(e)e(that)515 1894 y Fn(B)32 b Fp(is)27 b(also)g(temp)r(ered)g(and)h(lo)r(cally)f(in) n(tegrable)f(in)i Fn(V)19 b Fp(.)515 2045 y Fg(Lemma)29 b(3.8)41 b Fj(The)31 b(r)l(andom)f(variable)i Fp(sup)1925 2066 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))2174 2045 y Fl(k)p Fn(x)p Fl(k)2305 2015 y Fk(2)2305 2068 y Fm(V)2392 2045 y Fj(is)e(temp)l(er)l(e)l(d)g(and)g(the)g(mapping)515 2156 y Fn(t)23 b Fl(!)g Fp(sup)799 2176 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(\022)993 2184 y Fe(t)1019 2176 y Fm(!)r Fk(\))1107 2156 y Fl(k)p Fn(x)p Fl(k)1238 2126 y Fk(2)1238 2179 y Fm(V)1325 2156 y Fj(is)30 b(lo)l(c)l(al)t(ly)i(inte)l(gr)l (able.)515 2307 y(Pr)l(o)l(of.)50 b Fp(T)-7 b(o)25 b(obtain)g(an)g (estimate)g(in)h Fn(V)44 b Fp(w)n(e)25 b(use)g(the)h(standard)f(metho)r (d)g(whic)n(h)h(is)f(based)g(on)515 2407 y(the)j(form)n(ula)1334 2483 y Fn(d)p 1319 2520 74 4 v 1319 2596 a(dt)1402 2539 y Fp(\()p Fn(t)p Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)1690 2505 y Fk(2)1690 2559 y Fm(V)1747 2539 y Fp(\))23 b(=)g Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)2116 2505 y Fk(2)2116 2559 y Fm(V)2191 2539 y Fp(+)18 b Fn(t)2314 2483 y(d)p Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)2583 2452 y Fk(2)2583 2505 y Fm(V)p 2314 2520 326 4 v 2441 2596 a Fn(dt)2650 2539 y(:)515 2705 y Fp(for)27 b Fn(t)c Fp(=)f(1)28 b(b)n(y)f(T)-7 b(emam)27 b([23])g(Lemma)h(I)r(I)r(I.3.8)f (and)1099 2896 y Fl(j)p Fp(\(\()p Fn(u)18 b Fl(\001)h(r)p Fp(\))p Fn(v)s(;)14 b(w)r Fp(\))1568 2908 y Fm(V)1627 2896 y Fl(j)23 b(\024)g Fn(c)p Fl(k)p Fn(u)p Fl(k)1939 2825 y Fd(1)p 1937 2834 29 4 v 1937 2867 a(2)1929 2920 y Fm(H)1990 2896 y Fl(k)p Fn(u)p Fl(k)2132 2825 y Fd(1)p 2131 2834 V 2131 2867 a(2)2122 2920 y Fm(V)2178 2896 y Fl(k)p Fn(v)s Fl(k)2315 2825 y Fd(1)p 2315 2834 V 2315 2867 a(2)2305 2920 y Fm(V)2362 2896 y Fl(k)p Fn(Av)s Fl(k)2561 2825 y Fd(1)p 2561 2834 V 2561 2867 a(2)2551 2920 y Fm(H)2614 2896 y Fl(k)p Fn(Aw)r Fl(k)2821 2908 y Fm(H)515 3060 y Fp(for)k(su\016cien)n(tly)g(regular)f Fn(u;)h(v)s(;)h(w)j Fp(and)c Fn(c)c(>)g Fp(0)k(whic)n(h)h(allo)n(ws)e (us)h(to)h(write)f(b)n(y)g(\(15\))667 3211 y Fn(d)p 652 3248 74 4 v 652 3324 a(dt)735 3267 y Fp(\()p Fn(t)p Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)1023 3233 y Fk(2)1023 3288 y Fm(V)1080 3267 y Fp(\))83 b Fl(\024)g Fn(K)6 b Fp(\()p Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)1678 3233 y Fk(2)1678 3288 y Fm(H)1739 3267 y Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)1965 3233 y Fk(2)1965 3288 y Fm(V)2041 3267 y Fp(+)18 b Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)2350 3233 y Fk(2)2350 3288 y Fm(H)2430 3267 y Fp(+)g Fl(k)p Fn(z)t Fp(\()p Fn(\022)2669 3279 y Fm(t)2697 3267 y Fn(!)s Fp(\))p Fl(k)2826 3233 y Fk(2)2826 3288 y Fm(H)2889 3267 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)3045 3279 y Fm(t)3073 3267 y Fn(!)s Fp(\))p Fl(k)3202 3233 y Fk(2)3202 3288 y Fm(V)3259 3267 y Fp(\))1195 3429 y Fl(\002)83 b Fp(\()p Fn(t)p Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)1631 3395 y Fk(2)1631 3450 y Fm(V)1688 3429 y Fp(\))1195 3554 y(+)g Fn(p)p Fp(\()p Fl(k)p Fn(f)9 b Fl(k)1551 3566 y Fm(V)1603 3550 y Fc(0)1629 3554 y Fn(;)14 b Fl(k)p Fn(z)t Fp(\()p Fn(\022)1822 3566 y Fm(t)1851 3554 y Fn(!)s Fp(\))p Fl(k)1980 3566 y Fm(H)2042 3554 y Fn(;)g Fl(k)p Fn(z)t Fp(\()p Fn(\022)2235 3566 y Fm(t)2263 3554 y Fn(!)s Fp(\))p Fl(k)2392 3566 y Fm(V)2449 3554 y Fn(;)g Fl(k)p Fn(Az)t Fp(\()p Fn(\022)2704 3566 y Fm(t)2733 3554 y Fn(!)s Fp(\))p Fl(k)2862 3566 y Fm(H)2924 3554 y Fp(\))19 b(+)f Fl(k)p Fn(u)p Fp(\()p Fn(t)p Fp(\))p Fl(k)3284 3519 y Fk(2)3284 3574 y Fm(V)515 3718 y Fp(where)36 b Fn(p)h Fp(is)f(an)h(appropriate)e(p)r(olynomial)h(and)h Fn(K)42 b Fp(an)37 b(appropriate)e(p)r(ositiv)n(e)h(constan)n(t.)515 3817 y(Consequen)n(tly)-7 b(,)27 b(b)n(y)g(the)h(Gron)n(w)n(all)e (lemma)719 4036 y(sup)714 4106 y Fm(x)p Fh(2)p Fm(B)863 4036 y Fl(k)p Fn(')p Fp(\(1)p Fn(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)1283 4002 y Fk(2)1283 4057 y Fm(V)1423 4036 y Fl(\024)83 b Fp(exp)1711 3919 y Ff(\022)1772 4036 y Fn(K)25 b Fp(sup)1863 4106 y Fm(x)p Fh(2)p Fm(B)2012 3923 y Ff(Z)2095 3944 y Fk(1)2058 4112 y(0)2146 4036 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)2565 4002 y Fk(2)2565 4057 y Fm(H)2628 4036 y Fn(d\034)2716 3919 y Ff(\023)1423 4282 y Fl(\002)83 b Fp(exp)1711 4165 y Ff(\022)1772 4282 y Fn(K)25 b Fp(sup)1863 4351 y Fm(x)p Fh(2)p Fm(B)2012 4169 y Ff(Z)2095 4189 y Fk(1)2058 4357 y(0)2146 4282 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)2565 4247 y Fk(2)2565 4302 y Fm(H)2628 4282 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)3047 4247 y Fk(2)3047 4302 y Fm(V)3104 4282 y Fn(d\034)3192 4165 y Ff(\023)1423 4527 y Fl(\002)83 b Fp(exp)1711 4410 y Ff(\022)1772 4527 y Fn(K)1863 4414 y Ff(Z)1946 4435 y Fk(1)1909 4603 y(0)1997 4527 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)2153 4539 y Fm(\034)2193 4527 y Fn(!)s Fp(\))p Fl(k)2322 4493 y Fk(2)2322 4548 y Fm(V)2379 4527 y Fl(k)p Fn(z)t Fp(\()p Fn(\022)2535 4539 y Fm(\034)2576 4527 y Fn(!)s Fp(\))p Fl(k)2705 4493 y Fk(2)2705 4548 y Fm(H)2767 4527 y Fn(d\034)2855 4410 y Ff(\023)1423 4769 y Fl(\002)1571 4652 y Ff(\022)1632 4656 y(Z)1715 4677 y Fk(1)1678 4845 y(0)1766 4769 y Fn(p)p Fp(\()p Fn(\034)9 b Fp(\))p Fn(d\034)29 b Fp(+)24 b(sup)2108 4839 y Fm(x)p Fh(2)p Fm(B)2258 4656 y Ff(Z)2341 4677 y Fk(1)2304 4845 y(0)2392 4769 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)2811 4735 y Fk(2)2811 4790 y Fm(V)2868 4769 y Fn(d\034)2956 4652 y Ff(\023)3032 4769 y Fn(:)1950 5059 y Fp(19)p eop %%Page: 20 20 20 19 bop 515 523 a Fp(Note)31 b(that)h(a)f(pro)r(duct)h(of)f(random)g (v)-5 b(ariables)30 b(is)h(temp)r(ered)h(if)g(eac)n(h)f(factor)g(is)g (temp)r(ered.)515 623 y(T)-7 b(o)26 b(see)h(that)g(the)g(\014rst)g (factor)f(of)g(the)i(righ)n(t)e(hand)h(side)f(is)h(temp)r(ered)g(w)n(e) g(use)f(the)i(estimate)798 832 y Fi(E)131 b Fp(sup)946 909 y Fm(x)19 b Fh(2)f Fm(B)908 975 y(s)i Fh(2)e Fk([0)p Fm(;)11 b Fk(1])1212 719 y Ff(Z)1295 740 y Fk(1)1258 908 y(0)1346 832 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(\022)1591 844 y Fm(s)1627 832 y Fn(!)s(;)g(x)p Fp(\))p Fl(k)1840 798 y Fk(2)1840 853 y Fm(H)1903 832 y Fn(d\034)33 b Fl(\024)22 b Fi(E)2166 719 y Ff(Z)2255 740 y Fk(2)2218 908 y(0)2306 832 y Fn(R)2370 798 y Fk(2)2407 832 y Fp(\()p Fn(\022)2478 844 y Fm(s)2513 832 y Fn(!)s Fp(\))p Fn(ds)h Fp(=)g(2)p Fi(E)8 b Fn(R)2948 798 y Fk(2)3014 832 y Fn(<)23 b Fl(1)515 1129 y Fp(b)n(y)29 b(Lemma)g(3.3)g(and)g(the)h(forw)n(ard)d (in)n(v)-5 b(ariance)29 b(of)g Fn(B)t Fp(,)h(see)f(Arnold)g([1)o(])h (Prop)r(osition)e(4.1.3.)515 1229 y(Similarly)-7 b(,)27 b(w)n(e)g(get)h(for)f(the)h(next)f(factor)1022 1439 y Fi(E)131 b Fp(sup)1170 1515 y Fm(x)19 b Fh(2)f Fm(B)1132 1582 y(s)i Fh(2)e Fk([0)p Fm(;)11 b Fk(1])1436 1326 y Ff(Z)1519 1346 y Fk(1)1483 1514 y(0)1571 1439 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(\022)1816 1451 y Fm(s)1851 1439 y Fn(!)s(;)g(x)p Fp(\))p Fl(k)2064 1404 y Fk(2)2064 1459 y Fm(H)2127 1439 y Fl(k)p Fn(')p Fp(\()p Fn(\034)5 b(;)14 b(\022)2372 1451 y Fm(s)2407 1439 y Fn(!)s(;)g(x)p Fp(\))p Fl(k)2620 1404 y Fk(2)2620 1459 y Fm(V)2678 1439 y Fn(d\034)33 b(<)22 b Fl(1)515 1731 y Fp(whic)n(h)h(follo)n(ws)g(from) g(Lemma)h(3.6.)e(Ho)n(w)n(ev)n(er,)g(to)i(justify)g(this)g(estimate)g (w)n(e)f(also)g(need)g(that)515 1831 y Fi(E)f Fp(sup)709 1851 y Fm(s)p Fh(2)p Fk([0)p Fm(;)p Fk(1])926 1831 y Fn(R)990 1801 y Fk(4)1027 1831 y Fp(\()p Fn(\022)1098 1843 y Fm(s)1134 1831 y Fn(!)s Fp(\))h Fn(<)g Fl(1)p Fp(.)f(F)-7 b(or)22 b(this)g(expression)f(w)n(e)h(obtain)g(an)g (estimate)g(if)g(w)n(e)g(calculate)515 1941 y(in)30 b(\(20\))f Fn(R)855 1911 y Fk(4)854 1962 y(0)892 1941 y Fp(\()p Fn(\022)963 1953 y Fm(s)999 1941 y Fn(!)s Fp(\))e(=)f Fn(\032)1247 1911 y Fk(2)1285 1941 y Fp(\()p Fn(s)p Fp(\))k(b)n(y)g (the)g(c)n(hain)f(rule.)h(Then)g(w)n(e)f(can)h(estimate)g(this)g (suprem)n(um)515 2041 y(b)n(y)24 b Fn(R)691 2011 y Fk(4)727 2041 y Fp(\()p Fn(!)s Fp(\))h(and)f(some)f(in)n(tegrals)g(of)h(norms)f (from)h Fn(z)j Fp(whic)n(h)d(ha)n(v)n(e)f(a)h(\014nite)h(exp)r (ectation.)f(The)515 2141 y(temp)r(eredness)j(of)h(the)g(remaining)e (factors)h(follo)n(w)g(similarly)-7 b(.)515 2240 y(The)27 b(lo)r(cal)g(in)n(tegrabilit)n(y)f(follo)n(ws)h(b)n(y)g(the)h(con)n (tin)n(uit)n(y)f(of)g Fn(t)c Fl(!)g Fn(R)q Fp(\()p Fn(\022)2660 2252 y Fm(t)2689 2240 y Fn(!)s Fp(\))28 b(and)f(the)h(lo)r(cal)f(in)n (te-)515 2340 y(grabilit)n(y)f(of)i(the)g(norms)e(of)i Fn(z)t Fp(.)1910 b Fa(2)704 2539 y Fp(W)-7 b(e)28 b(are)e(no)n(w)h(in)h (a)f(p)r(osition)h(to)f(form)n(ulate)g(the)h(main)g(theorem)f(of)g (this)h(section.)515 2683 y Fg(Theorem)i(3.9)41 b Fj(L)l(et)29 b Fl(L)h Fj(b)l(e)f(a)h(set)f(of)i(line)l(ar)f(functionals)g(on)g Fn(V)48 b Fj(with)31 b(c)l(ompleteness)f(defe)l(ct)515 2783 y Fn(")554 2795 y Fh(L)604 2783 y Fj(.)j(Assume)g(that)g(for)i (some)f Fn(k)i Fj(satisfying)f(\(22\))f(the)f(c)l(ompleteness)h(defe)l (ct)h Fn(")3058 2795 y Fh(L)3141 2783 y Fj(p)l(ossesses)515 2882 y(the)30 b(pr)l(op)l(erty)597 3026 y Fp(4)p 595 3063 47 4 v 595 3139 a Fn(\027)670 3082 y Fl(\001)711 2965 y Ff(\022)803 3026 y Fp(4)p 782 3063 84 4 v 782 3139 a Fn(\027)828 3115 y Fk(2)876 3082 y Fl(k)p Fn(f)9 b Fl(k)1010 3048 y Fk(2)1010 3103 y Fm(V)1062 3086 y Fc(0)1107 3082 y Fp(+)18 b Fn(g)1230 3094 y Fm(k)1271 3082 y Fp(\()p Fn(\027)q(;)c(\025)1430 3094 y Fk(1)1467 3082 y Fn(;)g(Q)p Fp(\))1602 2965 y Ff(\023)1681 3082 y Fl(\001)19 b Fp(\(1)f(+)g Fn(h)1946 3094 y Fm(k)1987 3082 y Fp(\()p Fn(\027)q(;)c(A;)g(Q)p Fp(\)\))19 b(+)2568 3026 y(2)p 2439 3063 300 4 v 2439 3139 a(\()p Fn(k)i Fp(+)d(1\))p Fn(\027)2748 3082 y Fp(tr)2813 3094 y Fm(H)2876 3082 y Fn(Q)23 b(<)f(\027)5 b(")3137 3047 y Fh(\000)p Fk(2)3137 3107 y Fh(L)3226 3082 y Fn(;)71 b Fp(\(31\))515 3287 y Fj(wher)l(e)35 b Fn(g)794 3299 y Fm(k)835 3287 y Fp(\()p Fn(\027)q(;)14 b(\025)994 3299 y Fk(1)1031 3287 y Fn(;)g(Q)p Fp(\))35 b Fj(and)g Fn(h)1415 3299 y Fm(k)1455 3287 y Fp(\()p Fn(\027)q(;)14 b(A;)g(Q)p Fp(\))35 b Fj(ar)l(e)g(given)g(by)h(\(26\))f(and)g(\(27\).)h(Then)g Fl(L)e Fj(is)h(a)h(sys-)515 3387 y(tem)31 b(of)i(determining)g (functionals)g(in)f(pr)l(ob)l(ability)i(for)f(the)g(2D)f(sto)l(chastic) h(Navier-Stokes)515 3486 y(e)l(quation)d(\(13\).)515 3630 y(Pr)l(o)l(of.)56 b Fp(W)-7 b(e)30 b(are)e(going)f(to)i(apply)g (Theorem)f(2.2.)h(The)g(temp)r(eredness)g(and)g(lo)r(cal)f(in)n(tegra-) 515 3730 y(bilit)n(y)23 b(of)g(sup)942 3750 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(\022)1136 3758 y Fe(t)1162 3750 y Fm(!)1224 3730 y Fl(k)p Fn(x)p Fl(k)1355 3700 y Fk(2)1355 3753 y Fm(V)1435 3730 y Fp(follo)n(w)f(b)n(y)h(the)g(last)g(lemma.)g (Then)g(w)n(e)f(get)h(the)g(assertion)f(if)h(w)n(e)515 3829 y(c)n(ho)r(ose)h Fn(m)i Fp(su\016cien)n(tly)f(large.)g(Indeed,)h (in)f(the)h(case)f(of)h(large)e Fn(m)i Fp(w)n(e)f(can)g(reduce)g(the)h (in\015u-)515 3929 y(ence)e(of)f Fi(E)9 b Fn(R)900 3899 y Fk(2)943 3929 y Fp(.)24 b(By)f(Corollary)f(3.5)h(it)h(is)g (su\016cien)n(t)g(to)g(sho)n(w)f(that)h Fl(L)h Fp(is)e(a)h(set)g(of)g (determining)515 4029 y(functionals)k(for)g Fn(')g Fp(generated)g(b)n (y)g(\(15\).)g(The)g(prop)r(erties)g(of)g Fn(F)40 b Fp(allo)n(w)27 b(us)i(to)f(estimate)g(the)515 4128 y(Lipsc)n(hitz)f(constan)n(t)1350 4262 y Fn(l)r Fp(\()p Fn(x)1456 4274 y Fk(1)1493 4262 y Fn(;)14 b(x)1577 4274 y Fk(2)1615 4262 y Fn(;)g(!)s Fp(\))23 b(=)1862 4206 y(2)p 1859 4243 47 4 v 1859 4319 a Fn(\027)1916 4262 y Fp(\()p Fl(k)p Fn(x)2037 4274 y Fk(1)2074 4262 y Fl(k)2116 4228 y Fk(2)2116 4283 y Fm(V)2192 4262 y Fp(+)18 b Fl(k)p Fn(z)t Fp(\()p Fn(!)s Fp(\))p Fl(k)2521 4228 y Fk(2)2521 4283 y Fm(V)2577 4262 y Fp(\))p Fn(:)515 4423 y Fp(The)25 b(measurabilit)n(y)e(of)i Fn(l)h Fp(follo)n(ws)e(straigh)n(tforw)n(ardly)-7 b(.)21 b(W)-7 b(e)26 b(should)e(also)g(tak)n(e)g Fn(c)f Fp(=)3172 4390 y Fm(\027)p 3172 4404 38 4 v 3174 4452 a Fk(2)3244 4423 y Fp(in)i(\(3\).)515 4523 y(Therefore)h(w)n(e)h(can)h(apply)f(Theorem)g (2.2)f(if)1038 4692 y(4)p 999 4729 120 4 v 999 4805 a Fn(\027)5 b(m)1129 4748 y Fi(E)1192 4606 y Ff(\()1318 4748 y Fp(sup)1265 4821 y Fm(x)p Fh(2)p Fm(B)s Fk(\()p Fm(!)r Fk(\))1510 4635 y Ff(Z)1593 4655 y Fm(m)1556 4823 y Fk(0)1670 4748 y Fl(k)p Fn(')p Fp(\()p Fn(\034)g(;)14 b(!)s(;)g(x)p Fp(\))p Fl(k)2089 4714 y Fk(2)2089 4768 y Fm(V)2146 4748 y Fn(d\034)2234 4606 y Ff(\))2320 4748 y Fp(+)2416 4692 y(4)p 2413 4729 47 4 v 2413 4805 a Fn(\027)2470 4748 y Fi(E)8 b Fl(k)p Fn(z)t Fl(k)2646 4714 y Fk(2)2646 4768 y Fm(V)2731 4748 y Fn(<)23 b(\027)5 b(")2904 4712 y Fh(\000)p Fk(2)2904 4772 y Fh(L)3320 4748 y Fp(\(32\))1950 5059 y(20)p eop %%Page: 21 21 21 20 bop 515 523 a Fp(for)27 b(some)g Fn(m)p Fp(.)g(W)-7 b(e)28 b(can)g(\014nd)g Fn(m)f Fp(with)h(the)g(prop)r(ert)n(y)f (\(32\),)g(if)1563 689 y(4)p 1561 726 47 4 v 1561 802 a Fn(\027)1617 745 y Fp(\006)1677 757 y Fm(k)1736 745 y Fp(+)1832 689 y(4)p 1829 726 V 1829 802 a Fn(\027)1885 745 y Fi(E)9 b Fl(k)p Fn(z)t Fl(k)2062 711 y Fk(2)2062 765 y Fm(V)2147 745 y Fn(<)22 b(\027)5 b(")2319 709 y Fh(\000)p Fk(2)2319 769 y Fh(L)2409 745 y Fn(:)515 955 y Fp(The)27 b(last)h(relation)e(follo)n(ws)h(from)g(Lemma)h(3.7,)e(the) i(relation)f(\(28\))g(and)h(\(31\).)380 b Fa(2)515 1238 y Fg(Remark)31 b(3.10)40 b Fp(In)28 b(the)g(limit)g(tr)1599 1250 y Fm(H)1662 1238 y Fn(QA)1790 1207 y Fk(2)1850 1238 y Fl(!)23 b Fp(0)k(relation)g(\(31\))g(turns)h(in)f(the)h(inequalit)n (y)1763 1473 y Fn(")1802 1485 y Fh(L)1875 1473 y Fn(<)2017 1417 y Fp(4)p Fn(\027)2105 1387 y Fk(2)p 1973 1454 213 4 v 1973 1530 a Fl(k)p Fn(f)9 b Fl(k)2107 1542 y Fm(V)2159 1526 y Fc(0)2196 1473 y Fn(:)1101 b Fp(\(33\))515 1705 y(Th)n(us)32 b(if)g(the)h(estimate)f(\(33\))g(is)g(v)-5 b(alid,)32 b(then)h(there)f(exists)g(a)g(constan)n(t)f Fn(\016)2886 1717 y Fk(0)2954 1705 y Fn(>)g Fp(0)g(suc)n(h)h(that)515 1804 y(under)22 b(condition)h(tr)1170 1816 y Fm(H)1233 1804 y Fn(QA)1361 1774 y Fk(2)1421 1804 y Fn(<)g(\016)1546 1816 y Fk(0)1606 1804 y Fp(the)g(set)g Fl(L)g Fp(is)g(a)f(set)h(of)g (determining)g(functionals)f(in)i(prob-)515 1904 y(abilit)n(y)k(for)g (the)h(2D)f(sto)r(c)n(hastic)g(Na)n(vier-Stok)n(es)e(equations.)i(W)-7 b(e)29 b(also)e(note)i(that)g(estimate)515 2004 y(\(33\))g(is)g(the)h (same)f(order)f(as)h(the)h(b)r(est)g(kno)n(wn)f(estimate)g(for)g(the)h (completeness)f(defect)h(in)515 2103 y(the)24 b(case)f(of)g (deterministic)h(2)p Fn(D)h Fp(Na)n(vier)e(-)g(Stok)n(es)g(equations)g (with)h(the)g(p)r(erio)r(dic)f(b)r(oundary)515 2203 y(conditions)h (\(see)h(the)h(surv)n(ey)d([7])i(and)g(the)h(references)d(therein\).)j (Ho)n(w)n(ev)n(er)d(in)i(the)h(last)e(case)515 2302 y(relation)29 b(\(33\))g(in)n(v)n(olv)n(es)g(the)h(completeness)f(defect)i(with)g (resp)r(ect)e(to)h(the)h(pair)e Fn(D)r Fp(\()p Fn(A)p Fp(\))i(and)515 2402 y Fn(H)j Fp(and)28 b(it)g(leads)f(to)g(b)r(etter)h (estimates)f(for)g(the)h(n)n(um)n(b)r(er)g(of)f(determining)h (functionals.)515 2585 y Fg(Remark)j(3.11)40 b Fp(As)f(an)f (application)h(of)f(Theorem)h(2.2)f(and)g(2.3)g(w)n(e)h(can)f(consider) g(the)515 2684 y(equation)1460 2784 y Fn(@)1504 2796 y Fm(t)1533 2784 y Fn(u)23 b Fp(=)g(\001)p Fn(u)18 b Fl(\000)g Fn(f)9 b Fp(\()p Fn(u)p Fp(\))18 b(+)g Fn(@)2217 2796 y Fm(t)2246 2784 y Fn(W)12 b Fp(\()p Fn(t;)i(!)s Fp(\))515 2933 y(in)38 b(a)f(b)r(ounded)h(domain,)g(where)f Fn(f)9 b Fp(\()p Fn(u)p Fp(\))38 b(is)g(a)f(p)r(olynomial)g(of)h(o)r (dd)g(degree)f(with)h(p)r(ositiv)n(e)515 3033 y(leading)e(co)r (e\016cien)n(t.)h(W)-7 b(e)38 b(can)f(also)f(consider)g(2D)h(sto)r(c)n (hastic)g(Na)n(vier-Stok)n(es)d(equations)515 3133 y(with)c(m)n (ultiplicativ)n(e)g(white)h(noise)e Fn(u)14 b(dw)r Fp(,)31 b(where)e Fn(w)k Fp(is)d(a)g(scalar)e(Wiener)i(pro)r(cess.)f(In)h(this) 515 3232 y(case)f(w)n(e)i(ha)n(v)n(e)e(to)h(use)h(the)g(transformation) e Fn(T)12 b Fp(\()p Fn(!)s(;)i(x)p Fp(\))27 b(=)h Fn(x)14 b(e)2463 3202 y Fh(\000)p Fm(z)r Fk(\()p Fm(!)r Fk(\))2679 3232 y Fp(where)30 b Fn(z)k Fp(de\014nes)d(a)f(one)515 3332 y(dimensional)e(stationary)g(Ornstein-Uhlen)n(b)r(ec)n(k)g(pro)r (cess)g(generated)g(b)n(y)h Fn(dz)23 b Fp(+)c Fn(z)e(dt)26 b Fp(=)g Fn(dw)r Fp(.)515 3432 y(This)c(equation)g(has)g(b)r(een)h(in)n (v)n(estigated)e(for)h(instance)h(in)g(Sc)n(hmalfu\031)f([22)o(])h(but) g(with)g(a)f(little)515 3531 y(bit)28 b(di\013eren)n(t)g (transformation)e Fn(T)12 b Fp(.)704 3764 y Fg(Ac)m(kno)m(wledgmen)m (t.)23 b Fp(A)h(part)g(of)g(this)h(w)n(ork)e(w)n(as)g(done)h(at)g(the)g (Ob)r(erw)n(olfac)n(h)f(Math-)515 3863 y(ematical)k(Researc)n(h)g (Institute,)i(German)n(y)-7 b(,)28 b(while)g(J.)g(Duan)g(and)h(B.)f(Sc) n(hmalfu\031)g(w)n(ere)f(Re-)515 3963 y(searc)n(h)f(in)i(P)n(airs)d(F) -7 b(ello)n(ws,)27 b(supp)r(orted)h(b)n(y)f(the)h Fj(V)-6 b(olkswagen)31 b(Stiftung)p Fp(.)515 4237 y Fo(References)556 4419 y Fp([1])41 b(L.)28 b(Arnold.)36 b Fj(Random)31 b(Dynamic)l(al)f(Systems)p Fp(.)37 b(Springer,)26 b(Berlin,)i(1998.)556 4585 y([2])41 b(L.)22 b(Arnold)g(and)f(B.)h(Sc)n(hmalfu\031.)27 b(Ly)n(apuno)n(v)20 b(second)i(metho)r(d)g(for)f(random)g(dynamical)685 4685 y(systems.)30 b(T)-7 b(ec)n(hnical)23 b(Rep)r(ort)g(452,)f (Institut)j(f)r(\177)-44 b(ur)24 b(Dynamisc)n(he)f(Systeme,)g(Univ)n (ersit\177)-42 b(at)685 4785 y(Bremen,)28 b(1999.)34 b(T)-7 b(o)28 b(app)r(ear)e(in)i(Journal)e(of)i(Di\013eren)n(tial)g (Equations.)1950 5059 y(21)p eop %%Page: 22 22 22 21 bop 556 523 a Fp([3])41 b(A.)g(Bensoussan)e(and)h(R.)g(T)-7 b(emam.)75 b(Equations)39 b(sto)r(c)n(hastiques)g(du)i(t)n(yp)r(e)f(Na) n(vier{)685 623 y(Stok)n(es.)c Fj(Journal)30 b(of)h(F)-6 b(unctional)29 b(A)n(nalysis)p Fp(,)g(13:195{222,)23 b(1973.)556 789 y([4])41 b(L.)21 b(Berselli)f(and)h(F.)g(Flandoli.)k (Remarks)20 b(on)g(determining)h(pro)5 b(jections)19 b(for)h(sto)r(c)n(hastic)685 888 y(dissipativ)n(e)h(equations.)k Fj(Discr)l(ete)f(and)g(Continuous)g(Dynamic)l(al)h(Systems)p Fp(,)c(5\(8\):197{)685 988 y(214,)27 b(1999.)556 1154 y([5])41 b(C.)36 b(Castaing)e(and)h(M.)g(V)-7 b(aladier.)59 b Fj(Convex)37 b(A)n(nalysis)h(and)f(Me)l(asur)l(able)h(Multifunc-)685 1254 y(tions)p Fp(.)f(Lecture)28 b(Notes)f(in)h(Mathematics)f(580.)f (Springer,)h(New)h(Y)-7 b(ork,)27 b(1977.)556 1420 y([6])41 b(I.)32 b(Ch)n(uesho)n(v.)48 b(On)32 b(asymptotically)e(determining)i (functionals)f(for)h(dissipativ)n(e)f(sys-)685 1519 y(tems.)59 b(T)-7 b(ec)n(hnical)35 b(Rep)r(ort)g(414,)e(Institut)j(f)r(\177)-44 b(ur)36 b(Dynamisc)n(he)e(Systeme,)h(Univ)n(ersit\177)-42 b(at)685 1619 y(Bremen,)28 b(1997.)556 1785 y([7])41 b(I.)47 b(Ch)n(uesho)n(v.)90 b(Theory)45 b(of)h(functionals)g(that)g (uniquely)g(determine)g(asymptotic)685 1885 y(dynamics)41 b(of)g(in\014nite-dimensional)f(dissipativ)n(e)h(systems.)76 b Fj(Usp)l(ekhi)43 b(Mat.)g(Nauk)p Fp(,)685 1984 y(53\(4\):77{124,)24 b(1998.)36 b(\(in)28 b(Russian\).)g(English)f(translation)f(in)i Fj(R)n(ussian)h(Mathemati-)685 2084 y(c)l(al)i(Surveys)c Fp(53:731{776,)d(1998.)556 2250 y([8])41 b(I.)32 b(Ch)n(uesho)n(v.)46 b Fj(Intr)l(o)l(duction)33 b(to)g(the)g(The)l(ory)i(of)f (In\014nite-Dimensional)f(Dissip)l(ative)685 2350 y(Systems)p Fp(.)k(Acta,)28 b(Khark)n(o)n(v,)c(1999.)35 b(\(in)29 b(Russian\).)556 2516 y([9])41 b(I.)27 b(Ch)n(uesho)n(v.)33 b(On)26 b(determining)g(functionals)g(for)f(sto)r(c)n(hastic)g(Na)n (vier)g(-)h(Stok)n(es)f(equa-)685 2615 y(tions.)37 b Fj(Sto)l(chastics)30 b(and)h(Sto)l(chastics)f(R)l(ep)l(orts)p Fp(,)d(68:45{64,)e(1999.)515 2781 y([10])40 b(H.)35 b(Crauel,)e(A.)i (Debussc)n(he,)f(and)g(F.)g(Flandoli.)57 b(Random)33 b(attractors.)55 b Fj(Journal)36 b(of)685 2881 y(Dynamics)31 b(and)f(Di\013er)l(ential)g(Equations.)p Fp(,)f(9:307{341,)24 b(1997.)515 3047 y([11])40 b(H.)29 b(Crauel)f(and)g(F.)g(Flandoli.)39 b(A)n(ttractors)27 b(for)h(random)f(dynamical)h(systems.)38 b Fj(Pr)l(ob.)685 3147 y(The)l(ory)32 b(R)l(elat.)e(Fields)p Fp(,)f(100:365{393,)23 b(1994.)515 3313 y([12])40 b(F.)31 b(Flandoli)f(and)g(J.)g(A.)g(Langa.)43 b(On)30 b(determining)g(mo)r (des)g(for)g(dissipativ)n(e)g(random)685 3412 y(dynamical)d(systems.)37 b Fj(Sto)l(chastics)30 b(and)g(Sto)l(chastics)h(R)l(ep)l(orts)p Fp(,)c(66:1{25,)e(1999.)515 3578 y([13])40 b(F.)g(Flandoli)f(and)g(B.)g (Sc)n(hmalfu\031.)71 b(W)-7 b(eak)39 b(solutions)g(and)g(attractors)e (for)i(the)h(3D)685 3678 y(Na)n(vier{Stok)n(es)22 b(equation)i(with)i (nonregular)c(force.)31 b Fj(Journal)c(of)h(Dynamics)g(and)f(Dif-)685 3778 y(fer)l(ential)k(Equations)p Fp(,)d(11:355{397,)c(1999.)515 3944 y([14])40 b(C.)27 b(F)-7 b(oias)25 b(and)h(G.)h(Pro)r(di.)33 b(Sur)26 b(le)g(comp)r(ortemen)n(t)g(global)f(des)h(solutions)g (nonstation-)685 4043 y(naires)k(des)f(\023)-39 b(equations)29 b(de)i(Na)n(vier-Stok)n(es)d(en)j(dimension)g(deux.)47 b Fj(R)l(end.)33 b(Sem.)g(Mat.)685 4143 y(Univ.)e(Padova)p Fp(,)f(39:1{34,)24 b(1967.)515 4309 y([15])40 b(A.)32 b(W.)h(F)-7 b(ursik)n(o)n(v)30 b(and)h(M.)h(I.)g(Vishik.)49 b Fj(Mathematic)l(al)35 b(Pr)l(oblems)g(of)f(Statistic)l(al)g(Hy-)685 4408 y(dr)l(o)l(dynamics)p Fp(.)39 b(Klu)n(w)n(er)26 b(Academic)i(Publisher,)f(Dordrec)n(h)n(t,)f(1988.)515 4575 y([16])40 b(D.)g(A.)f(Jones)f(and)h(E.S.)g(Titi.)71 b(Upp)r(er)40 b(b)r(ounds)f(on)g(the)g(n)n(um)n(b)r(er)g(of)g (determining)685 4674 y(mo)r(des,)28 b(no)r(des)g(and)g(v)n(olume)g (elemen)n(ts)g(for)f(the)i(Na)n(vier)e(Stok)n(es)g(equations.)37 b Fj(Indiana)685 4774 y(Univ.)31 b(Math.)g(J.)p Fp(,)c(42:875{887,)d (1993.)1950 5059 y(22)p eop %%Page: 23 23 23 22 bop 515 523 a Fp([17])40 b(H.H.Kuo.)d Fj(Gaussian)31 b(Me)l(asur)l(es)e(in)h(Banach)i(Sp)l(ac)l(es)p Fp(.)37 b(Springer,)26 b(New)i(Y)-7 b(ork,)27 b(1972.)515 689 y([18])40 b(I.I.Gihman)24 b(and)e(A.)h(Sk)n(oroho)r(d.)28 b Fj(The)e(The)l(ory)h(of)f(Sto)l(chastic)g(Pr)l(o)l(c)l(esses)p Fp(,)e(v)n(olume)e(I)r(I)r(I.)685 789 y(Springer,)27 b(New)h(Y)-7 b(ork,)27 b(1979.)515 955 y([19])40 b(S.)20 b(Kozlo)n(v.)j(Some)c(problems)g(concerning)g(sto)r(c)n(hastic)f (equations)h(with)i(partial)e(deriv)-5 b(a-)685 1054 y(tiv)n(es.)37 b Fj(T)-6 b(rudy)30 b(Semin.)g(im.)g(I.G.)h(Petr)l (ovsko)l(go)p Fp(,)e(4:147{172,)24 b(1978.)35 b(\(in)28 b(Russian\).)515 1220 y([20])40 b(O.)20 b(Ladyzhensk)-5 b(a)n(y)n(a.)23 b(A)d(dynamical)g(system)f(generated)g(b)n(y)h(the)h (Na)n(vier{Stok)n(es)c(equa-)685 1320 y(tions.)37 b Fj(Journal)30 b(of)g(Soviet)g(Mathematics)p Fp(,)g(3:458{479,)24 b(1975.)515 1486 y([21])40 b(G.)e(D.)h(Prato)d(and)h(J.)h(Zab)r(czyk.)66 b Fj(Sto)l(chastic)40 b(Equations)g(in)f(In\014nite)f(Dimension)p Fp(.)685 1586 y(Univ)n(ersit)n(y)27 b(Press,)f(Cam)n(bridge,)h(1992.) 515 1752 y([22])40 b(B.)21 b(Sc)n(hmalfu\031.)26 b(Measure)20 b(attractors)f(and)i(sto)r(c)n(hastic)g(attractors)e(for)h(sto)r(c)n (hastic)h(par-)685 1851 y(tial)g(di\013eren)n(tial)g(equations.)k Fj(Sto)l(chastic)f(A)n(nalalysis)h(and)f(Applic)l(ations)p Fp(,)f(17\(6\):1075{)685 1951 y(1101,)j(1999.)515 2117 y([23])40 b(R.)26 b(T)-7 b(emam.)34 b Fj(Navier{Stokes)29 b(Equation{The)l(ory)h(and)f(Numeric)l(al)f(A)n(nalysis)p Fp(.)34 b(North-)685 2217 y(Holland,)28 b(Amsterdam,)f(1979.)515 2383 y([24])40 b(R.)49 b(T)-7 b(emam.)100 b Fj(In\014nite{Dimensional) 51 b(Dynamic)l(al)f(Systems)f(in)g(Me)l(chanics)i(and)685 2482 y(Physics)p Fp(.)39 b(Springer,)26 b(New)i(Y)-7 b(ork,)27 b(second)g(edition,)h(1997.)515 2648 y([25])40 b(M.)22 b(Viot.)27 b(Solutions)21 b(faible)g(d')n(\023)-39 b(equations)20 b(aux)h(d)n(\023)-39 b(eriv)n(\023)g(ees)18 b(partielles)j(sto)r(c)n(hastiques)f(non)685 2748 y(lin)n(\023)-39 b(eaires.)35 b(These)27 b(le)h(grade)e(do)r(cteur)f(\023)-39 b(es)27 b(sciences,)g(1976.)1950 5059 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0105281329634--