Content-Type: multipart/mixed; boundary="-------------0704241152621" This is a multi-part message in MIME format. ---------------0704241152621 Content-Type: text/plain; name="07-101.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-101.comments" MSC classification: 47D08, 60H30, 81T10 ---------------0704241152621 Content-Type: text/plain; name="07-101.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-101.keywords" Brownian motion, double Ito integral, stochastic currents, rough paths, Gibbs measure, cluster expansion ---------------0704241152621 Content-Type: application/postscript; name="BrownCurrent.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BrownCurrent.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: BrownCurrent.dvi %%Pages: 51 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMBX12 CMR10 CMTI10 CMTT10 CMBX10 CMMI10 CMSY10 CMSY8 %%+ CMMI8 CMEX10 MSBM7 CMMI6 CMR8 MSBM10 CMR6 CMSY6 CMCSC10 MSAM10 CMTI8 %%+ TeX-cmex8 CMBX8 CMBXTI10 CMMI5 CMSY5 CMBX6 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips BrownCurrent %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2007.04.24:1813 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 69 /E put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-29 -250 1274 754}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 120 /x put readonly def /FontBBox{-59 -250 1235 750}readonly def currentdict end currentfile eexec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0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 80 /P put dup 82 /R put dup 99 /c put dup 100 /d put readonly def /FontBBox {-29 -2957 1554 772} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 73 /I put dup 94 /circumflex put dup 111 /o put dup 116 /t put readonly def /FontBBox{-35 -250 1190 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 80 /P put dup 102 /f put dup 111 /o put dup 114 /r put readonly def /FontBBox{14 -250 1077 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 71 /G put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 59 /emptyset put dup 67 /C put dup 75 /K put dup 85 /U put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 19 /acute put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 90 /Z put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 4 /Xi put dup 9 /Psi put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 70 /F put dup 73 /I put dup 80 /P put dup 83 /S put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 100 /d put dup 105 /i put dup 108 /l put dup 109 /m put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t 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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 28 /tau put dup 33 /omega put dup 37 /rho1 put dup 58 /period put dup 59 /comma put dup 65 /A put dup 67 /C put dup 71 /G put dup 73 /I put dup 74 /J put dup 75 /K put dup 78 /N put dup 83 /S put dup 84 /T put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 93 /sharp put dup 98 /b put dup 99 /c put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 56 /bracelefttp put dup 58 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 93 /sharp put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DDF753642CD526A96ACEDA4120 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 10 /circlemultiply put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 59 /emptyset put dup 68 /D put dup 69 /E put dup 71 /G put dup 74 /J put dup 75 /K put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 34 /arrowup put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 0ABF1D42C5DDD0C384C7E22F8F8047BE1D4C1CC8E33368FB1AC82B4E96146730 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 8 /Phi put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 600 600 (BrownCurrent.dvi) @start /Fa 177[36 78[{}1 49.8132 /MSBM7 rf /Fb 135[36 120[{}1 49.8132 /CMBX6 rf /Fc 252[31 3[{}1 41.511 /CMSY5 rf /Fd 150[22 105[{}1 41.511 /CMMI5 rf /Fe 134[51 2[48 56 35 44 46 1[54 54 59 86 4[54 48 1[48 54 1[48 54 27[69 2[74 66[{}18 90.9091 /CMBXTI10 rf /Ff 135[43 120[{}1 66.4176 /CMBX8 rf /Fg 155[102 71 16[33 1[74 80[{}4 66.4176 /TeX-cmex8 rf /Fh 139[24 4[36 16[36 20[27 73[{}4 66.4176 /CMTI8 rf /Fi 252[71 3[{}1 90.9091 /MSAM10 rf /Fj 141[55 2[58 8[49 21[67 33[29 46[{}5 90.9091 /CMCSC10 rf /Fk 134[55 55 2[58 41 41 43 1[58 52 58 87 29 55 1[29 58 2[48 1[46 58 51 12[73 1[78 1[71 1[82 99 63 4[82 2[80 76 9[52 52 52 52 52 52 52 52 52 52 1[29 4[41 41 27[58 12[{}41 90.9091 /CMBX10 rf /Fl 134[59 59 81 59 62 44 44 46 59 62 56 62 93 31 2[31 62 56 34 51 62 50 62 54 2[56 6[116 2[78 1[84 1[77 2[106 67 2[42 1[88 1[74 86 81 80 11[56 56 56 56 56 56 2[31 33[62 12[{}44 99.6264 /CMBX12 rf /Fm 149[19 2[32 32 9[42 42 5[39 9[47 7[34 7[32 4[0 2[42 42 61 18 20[48 48 4[48 48 13[48 2[32 2[48{}21 49.8132 /CMSY6 rf /Fn 138[34 24 1[24 2[30 34 50 4[34 2[27 1[27 1[30 2[30 18 1[18 29[47 2[18 2[30 2[30 30 30 30 30 4[47 1[24 24 17[30 21[38{}26 49.8132 /CMR6 rf /Fo 128[46 4[37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 2[46 5[68 91 68 68 65 51 66 70 62 70 68 82 57 70 48 35 68 70 59 62 69 65 64 68 5[28 28 46 46 1[46 46 46 46 46 46 46 46 28 33 28 2[37 37 28 70 18[46 2[28 1[80 53 51 56 11[{}77 90.9091 /CMTI10 rf /Fp 165[61 66 66 5[66 3[66 8[61 69[{}6 90.9091 /MSBM10 rf /Fq 129[35 5[37 3[27 28 28 1[39 35 1[59 20 2[20 4[39 2[35 2[35 20 1[20 7[39 2[48 6[25 2[46 8[55 2[20 35 35 35 35 35 35 35 35 35 35 4[55 1[27 27 17[35 12[55 4[47 49 1[59 44{}40 66.4176 /CMR8 rf /Fr 134[32 35 2[37 24 29 29 3[38 54 20 33 25 22 4[32 28 27 4[25 2[42 36 50 57 2[36 38 4[48 2[51 34 27 1[47 3[44 1[46 5[19 19 20[32 3[39 4[29 14[32 35 40 11[{}37 49.8132 /CMMI6 rf /Fs 165[44 48 48 5[48 3[48 78[{}5 66.4176 /MSBM7 rf /Ft 142[91 91 61 61 4[43 43 3[131 91 51 7[51 116 131 5[43 86 96 12[80 80 80 80 1[81 1[81 1[81 1[81 2[61 61 61 61 80 80 7[73 4[53 53 72 72 5[68 4[48 48 67 67 54 54 3[30 8[38 38 42 42{}46 90.9091 /CMEX10 rf /Fu 134[35 40 51 34 41 25 33 32 32 36 1[43 62 21 37 29 24 2[34 1[36 31 30 37 3[27 3[41 58 67 41 48 41 43 4[56 1[48 60 39 31 58 55 1[52 58 50 53 53 2[55 35 55 20 20 20[36 59 42 33 44 1[44 2[31 40 36 1[31 35 42 41 2[33 35 31 1[31 36 40 45 11[{}66 66.4176 /CMMI8 rf /Fv 149[20 27 27 35 35 6[47 2[47 47 3[70 1[44 39 43 60 1[49 4[54 48 2[42 1[37 55 8[35 4[0 2[47 47 71 19 14[71 11[55 55 9[55 3[55 2[35 1[20 55{}33 66.4176 /CMSY8 rf /Fw 141[76 1[76 1[45 2[45 25 35 35 45 45 7[61 1[61 61 66 1[65 90 56 57 50 55 77 3[75 2[69 62 1[77 54 65 48 70 48 60 73 5[45 2[51 0 0 2[61 61 91 14[45 91 6[71 1[71 2[71 71 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y(b)s(ound)28 b(\(2.3\))k(are)f(then)f (satis\014ed.)141 3727 y(F)-8 b(or)25 b(our)f(purp)s(oses)f(b)s(elo)m (w)h(w)m(e)h(need)g(a)f(sligh)m(t)i(extension)f(since)g(w)m(e)g(w)m(an) m(t)g(to)g(b)s(e)f(able)h(to)g(in)m(tegrate)0 3840 y(functions)45 b(explicitly)i(dep)s(enden)m(t)d(on)h(the)h(time)g(parameter,)j(suc)m (h)c(as)2630 3767 y Ft(R)2691 3793 y Fu(T)2673 3872 y Fq(0)2761 3840 y Fx(')p Fy(\()p Fx(u;)15 b(X)3022 3854 y Fu(u)3068 3840 y Fy(\))p Fx(dX)3225 3854 y Fu(u)3271 3840 y Fy(.)86 b(One)0 3953 y(p)s(ossibilit)m(y)39 b(is)g(to)g (consider)g(the)g(couple)g(\()p Fx(u;)15 b(X)7 b Fy(\))40 b(as)f(a)g(new)f(path)h(and)f(construct)h(the)g(asso)s(ciated)0 4066 y(rough)c(path.)57 b(This)36 b(construction,)i(ho)m(w)m(ev)m(er,)h (has)c(the)h(disadv)-5 b(an)m(tage)38 b(of)e(requiring)f(rather)h(m)m (uc)m(h)0 4179 y(regularit)m(y)41 b(of)f(the)g(function)g Fx(')p Fy(.)70 b(A)40 b(more)g(e\013ectiv)m(e)j(approac)m(h)d(is)g(to)g (treat)i(the)e(line-in)m(tegral)i(as)0 4292 y(a)35 b(Y)-8 b(oung)35 b(in)m(tegral)i(with)d(resp)s(ect)h(to)g(the)g Fx(u)g Fy(dep)s(endence)e(of)i(the)g(in)m(tegrand)g(as)g(so)s(on)g(as)g (the)g(map)0 4405 y Fx(u)25 b Fw(7!)g Fx(')p Fy(\()p Fx(u;)15 b(\030)t Fy(\))32 b(has)e(H\177)-45 b(older)31 b(exp)s(onen)m(t)g(greater)g(than)f(1)21 b Fw(\000)f Fx(\015)35 b Fy(\(as)c(w)m(e)g(shall)g(see)g(b)s(elo)m(w\).)141 4518 y(F)-8 b(or)31 b(the)g(\(time-dep)s(enden)m(t\))g(test)g(v)m (ector)h(\014elds)e(in)g Fx(C)7 b Fy(\()p Fp(R)20 b Fw(\002)g Fp(R)2317 4485 y Fu(d)2357 4518 y Fy(;)15 b Fp(R)2463 4485 y Fu(d)2503 4518 y Fy(\))31 b(w)m(e)g(use)f(the)h(norm)111 4799 y Fw(k)p Fx(')p Fw(k)260 4813 y Fu(\032;)p Fq(2)p Fu(;s;t)479 4799 y Fy(=)25 b(sup)623 4871 y Fu(x)727 4644 y Ft(")814 4799 y Fy(sup)780 4882 y Fu(u)p Fv(2)p Fq([)p Fu(s;t)p Fq(])1000 4671 y Ft(\022)1101 4799 y Fy(max)1067 4860 y Fu(k)r Fq(=0)p Fu(;)p Fq(1)p Fu(;)p Fq(2)1320 4799 y Fw(jr)1421 4761 y Fu(k)1464 4799 y Fx(')p Fy(\()p Fx(u;)15 b(x)p Fy(\))p Fw(j)1762 4671 y Ft(\023)1851 4799 y Fy(+)82 b(sup)1942 4882 y Fu(u;v)r Fv(2)p Fq([)p Fu(s;t)p Fq(])2225 4799 y Fy(max)2218 4860 y Fu(k)r Fq(=0)p Fu(;)p Fq(1)2427 4737 y Fw(jr)2528 4704 y Fu(k)2570 4737 y Fx(')p Fy(\()p Fx(u;)15 b(x)p Fy(\))22 b Fw(\000)e(r)3032 4704 y Fu(k)3075 4737 y Fx(')p Fy(\()p Fx(v)s(;)15 b(x)p Fy(\))p Fw(j)p 2427 4778 943 4 v 2748 4861 a(j)p Fx(u)20 b Fw(\000)g Fx(v)s Fw(j)3008 4835 y Fu(\032)3380 4644 y Ft(#)0 5088 y Fy(with)34 b Fx(\032)f(>)f Fy(0)j(and)g(the)g(con)m(v)m (en)m(tion)h Fw(r)1351 5055 y Fq(0)1391 5088 y Fx(')p Fy(\()p Fx(t;)15 b(x)p Fy(\))34 b(=)e Fx(')p Fy(\()p Fx(t;)15 b(x)p Fy(\).)55 b(Then)34 b(our)g(basic)h(result)g(on)f(the)h (step-2)0 5201 y(rough)30 b(path)g(\()p Fx(X)r(;)15 b Fp(X)p Fy(\))31 b(is)1749 5549 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 0 431 a Fk(Theorem)35 b(2.3)46 b Fo(L)-5 b(et)43 b Fx(')g Fw(2)g Fx(C)7 b Fy(\()p Fp(R)26 b Fw(\002)h Fp(R)1359 398 y Fu(d)1400 431 y Fx(;)15 b Fp(R)1506 398 y Fu(d)1546 431 y Fy(\))43 b Fo(b)-5 b(e)42 b(such)g(that)i(it)e(is)h Fx(C)2437 398 y Fq(2)2518 431 y Fo(with)g(r)-5 b(esp)g(e)g(ct)44 b(to)f(its)g(se)-5 b(c)g(ond)0 544 y(variable)31 b(and)g(H\177)-46 b(older)31 b(c)-5 b(ontinuous)32 b(with)f(exp)-5 b(onent)32 b Fx(\032)e Fo(with)h(r)-5 b(esp)g(e)g(ct)32 b(to)f(its)f(\014rst)h(variable,)h (such)e(that)0 657 y Fx(\032)20 b Fy(+)g Fx(\015)30 b(>)25 b Fy(1)p Fo(.)43 b(Then)33 b(the)g(sums)470 787 y Ft(X)513 977 y Fu(\013)616 772 y Ft(\020)671 873 y Fx(')730 887 y Fu(i)758 873 y Fy(\()p Fx(\034)833 887 y Fu(\013)883 873 y Fx(;)15 b(X)998 887 y Fu(\034)1029 895 y Fr(\013)1078 873 y Fy(\))p Fx(X)1195 835 y Fu(i)1188 895 y(\034)1219 904 y Fr(\013)p Fn(+1)1342 895 y Fu(\034)1373 903 y Fr(\013)1442 873 y Fy(+)k Fw(r)1608 887 y Fu(j)1645 873 y Fx(')1704 887 y Fu(i)1732 873 y Fy(\()p Fx(\034)1807 887 y Fu(\013)1857 873 y Fx(;)c(X)1972 887 y Fu(\034)2003 895 y Fr(\013)2052 873 y Fy(\))p Fp(X)2153 835 y Fu(ij)2153 895 y(\034)2184 903 y Fr(\013)2229 895 y Fu(\034)2260 904 y Fr(\013)p 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Fx(')p Fw(k)1804 1555 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)2026 1541 y Fy(\(1)21 b(+)f Fw(k)p Fx(X)7 b Fw(k)2390 1555 y Fu(\015)2456 1541 y Fy(+)20 b Fw(k)p Fp(X)2658 1503 y Fq(2)2698 1541 y Fw(k)2743 1555 y Fq(2)p Fu(\015)2823 1541 y Fy(\))2858 1503 y Fq(3)2898 1541 y Fx(:)0 1798 y Fj(Pr)n(oof.)53 b Fy(W)-8 b(e)36 b(pro)m(v)m(e)g(con)m (v)m(ergence)h(o)m(v)m(er)f(the)f(dy)m(adic)g(partition)g(of)g([0)p Fx(;)15 b Fy(1],)38 b(then)d(con)m(v)m(ergence)i(of)e(a)0 1926 y(general)f(partition)g(follo)m(ws)g(then)f(b)m(y)g(the)g(argumen) m(ts)h(dev)m(elop)s(ed)f(in)g([23)q(].)49 b(Let)34 b Fx(\034)2907 1878 y Fq(\()p Fu(n)p Fq(\))2897 1938 y Fu(\013)3039 1926 y Fy(=)29 b Fx(T)13 b(\013=)p Fy(2)3353 1893 y Fu(n)3434 1926 y Fy(for)0 2039 y Fx(\013)26 b Fy(=)f(0)p Fx(;)15 b(:)g(:)g(:)i(;)e Fy(2)472 2006 y Fu(n)520 2039 y Fy(,)30 b(and)g(let)453 2319 y Fx(S)509 2333 y Fu(n)581 2319 y Fy(=)677 2205 y Fq(2)712 2181 y Fr(n)755 2205 y Fv(\000)p Fq(1)695 2232 y Ft(X)693 2426 y Fu(\013)p Fq(=0)860 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2783 y Fu(n)p Fv(\000)p Fq(1)1406 2769 y Fy(=)1502 2655 y Fq(2)1537 2631 y Fr(n)1579 2655 y Fv(\000)p Fq(1)1520 2682 y Ft(X)1518 2877 y Fu(\013)p Fq(=0)1670 2769 y Fy(\()p Fx(A)1773 2731 y Fq(1)1773 2791 y Fu(\013)1843 2769 y Fy(+)g Fx(A)2002 2731 y Fq(2)2002 2791 y Fu(\013)2072 2769 y Fy(+)g Fx(A)2231 2731 y Fq(3)2231 2791 y Fu(\013)2300 2769 y Fy(+)g Fx(A)2459 2731 y Fq(4)2459 2791 y Fu(\013)2509 2769 y Fy(\))p Fx(;)787 b Fy(\(2.4\))0 3008 y(where)166 3234 y Fx(A)234 3196 y Fq(1)234 3256 y Fu(\013)309 3234 y Fy(=)405 3106 y Ft(\024)453 3234 y Fx(')512 3248 y Fu(i)541 3234 y Fy(\()p Fx(\034)626 3186 y Fq(\()p Fu(n)p Fq(\))616 3260 y(2)p Fu(\013)p Fq(+1)791 3234 y Fx(;)15 b(X)906 3275 y Fu(\034)945 3240 y Fn(\()p Fr(n)p Fn(\))937 3297 y(2)p Fr(\013)p Fn(+2)1093 3234 y Fy(\))21 b Fw(\000)f Fx(')1299 3248 y Fu(i)1328 3234 y Fy(\()p Fx(\034)1413 3186 y Fq(\()p Fu(n)p Fq(\))1403 3260 y(2)p Fu(\013)p Fq(+1)1578 3234 y Fx(;)15 b(X)1693 3275 y Fu(\034)1732 3240 y Fn(\()p Fr(n)p Fn(\))1724 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y Fx(;)g(X)1018 3527 y Fu(\034)1057 3491 y Fn(\()p Fr(n)p Fn(\))1049 3548 y(2)p Fr(\013)p Fn(+2)1206 3485 y Fy(\))20 b Fw(\000)g(r)1428 3499 y Fu(j)1464 3485 y Fx(')1523 3499 y Fu(i)1552 3485 y Fy(\()p Fx(\034)1637 3437 y Fq(\()p Fu(n)p Fq(\))1627 3511 y(2)p Fu(\013)p Fq(+1)1802 3485 y Fx(;)15 b(X)1917 3527 y Fu(\034)1956 3491 y Fn(\()p Fr(n)p Fn(\))1948 3548 y(2)p Fr(\013)p Fn(+1)2105 3485 y Fy(\))2140 3357 y Ft(\025)2204 3485 y Fp(X)2270 3441 y Fu(ij)2270 3541 y(\034)2309 3506 y Fn(\()p Fr(n)p Fn(\))2301 3562 y(2)p Fr(\013)p Fn(+1)2452 3541 y Fu(\034)2491 3506 y Fn(\()p Fr(n)p Fn(\))2483 3562 y(2)p Fr(\013)2586 3485 y Fx(;)166 3737 y(A)234 3699 y Fq(3)234 3759 y Fu(\013)309 3737 y Fy(=)405 3609 y Ft(\024)453 3737 y Fx(')512 3751 y Fu(i)541 3737 y Fy(\()p Fx(\034)626 3689 y Fq(\()p Fu(n)p Fq(\))616 3763 y(2)p Fu(\013)p Fq(+2)791 3737 y Fx(;)g(X)906 3778 y Fu(\034)945 3743 y Fn(\()p Fr(n)p Fn(\))937 3799 y(2)p Fr(\013)p Fn(+2)1093 3737 y Fy(\))21 b Fw(\000)f Fx(')1299 3751 y Fu(i)1328 3737 y Fy(\()p Fx(\034)1413 3689 y Fq(\()p Fu(n)p Fq(\))1403 3763 y(2)p Fu(\013)p Fq(+1)1578 3737 y Fx(;)15 b(X)1693 3778 y Fu(\034)1732 3743 y Fn(\()p Fr(n)p Fn(\))1724 3799 y(2)p Fr(\013)p Fn(+2)1880 3737 y Fy(\))1915 3609 y Ft(\025)1979 3737 y Fx(X)2061 3699 y Fu(i)2054 3786 y(\034)2093 3751 y Fn(\()p Fr(n)p Fn(\))2085 3808 y(2)p Fr(\013)p Fn(+1)2237 3786 y Fu(\034)2276 3751 y Fn(\()p Fr(n)p Fn(\))2268 3808 y(2)p Fr(\013)2371 3737 y Fx(;)166 3954 y(A)234 3917 y Fq(4)234 3977 y Fu(\013)309 3954 y Fy(=)25 b(\()p Fw(r)516 3968 y Fu(j)553 3954 y Fx(')612 3968 y Fu(i)640 3954 y Fy(\()p Fx(\034)725 3906 y Fq(\()p Fu(n)p Fq(\))715 3980 y(2)p Fu(\013)p Fq(+2)890 3954 y Fx(;)15 b(X)1005 3995 y Fu(\034)1044 3960 y Fn(\()p Fr(n)p Fn(\))1036 4017 y(2)p Fr(\013)p Fn(+2)1193 3954 y Fy(\))21 b Fw(\000)f(r)1416 3968 y Fu(j)1452 3954 y Fx(')1511 3968 y Fu(i)1539 3954 y Fy(\()p Fx(\034)1624 3906 y Fq(\()p Fu(n)p Fq(\))1614 3980 y(2)p Fu(\013)p Fq(+1)1790 3954 y Fx(;)15 b(X)1905 3995 y Fu(\034)1944 3960 y Fn(\()p Fr(n)p Fn(\))1936 4017 y(2)p Fr(\013)p Fn(+2)2092 3954 y Fy(\)\))p Fp(X)2228 3910 y Fu(ij)2228 4010 y(\034)2267 3975 y Fn(\()p Fr(n)p Fn(\))2259 4031 y(2)p Fr(\013)p Fn(+1)2412 4010 y Fu(\034)2451 3975 y Fn(\()p Fr(n)p Fn(\))2443 4031 y(2)p Fr(\013)2545 3954 y Fx(:)0 4208 y Fy(By)34 b(using)f(the)g(fact)h(that)g Fx(')d Fw(2)f Fx(C)1181 4175 y Fq(2)1220 4208 y Fy(,)k(w)m(e)g(ha)m(v)m (e)g(the)g(b)s(ounds)d Fw(j)p Fx(A)2200 4175 y Fq(1)2200 4230 y Fu(\013)2250 4208 y Fw(j)g(\024)f Fx(T)2473 4175 y Fq(3)p Fu(\015)2552 4208 y Fw(k)p Fx(')p Fw(k)2701 4222 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)2923 4208 y Fw(k)p Fx(X)7 b Fw(k)3095 4175 y Fq(3)3095 4230 y Fu(\015)3141 4208 y Fy(2)3186 4175 y Fv(\000)p Fq(3)p Fu(\015)t(n)3397 4208 y Fy(and)0 4321 y Fw(j)p Fx(A)93 4288 y Fq(2)93 4343 y Fu(\013)143 4321 y Fw(j)38 b(\024)f Fx(T)380 4288 y Fq(3)p Fu(\015)460 4321 y Fw(k)p Fx(')p Fw(k)609 4335 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)831 4321 y Fw(k)p Fx(X)7 b Fw(k)1003 4335 y Fu(\015)1049 4321 y Fw(k)p Fp(X)p Fw(k)1205 4335 y Fq(2)p Fu(\015)1285 4321 y Fy(2)1330 4288 y Fv(\000)p Fq(3)p Fu(\015)t(n)1546 4321 y Fy(on)37 b(the)h(\014rst)g(t)m(w)m(o)h(terms.)63 b(F)-8 b(or)39 b(the)f(last)g(t)m(w)m(o)i(w)m(e)e(ha)m(v)m(e)0 4434 y Fw(j)p Fx(A)93 4401 y Fq(3)93 4456 y Fu(\013)143 4434 y Fw(j)25 b(\024)g Fx(T)355 4401 y Fu(\032)p Fq(+)p Fu(\015)491 4434 y Fw(k)p Fx(')p Fw(k)640 4448 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)862 4434 y Fw(k)p Fx(X)7 b Fw(k)1034 4448 y Fu(\015)1079 4434 y Fy(2)1124 4401 y Fv(\000)p Fu(n)p Fq(\()p Fu(\032)p Fq(+)p Fu(\015)t Fq(\))1443 4434 y Fy(and)30 b Fw(j)p Fx(A)1713 4401 y Fq(4)1713 4456 y Fu(\013)1763 4434 y Fw(j)25 b(\024)g Fx(T)1975 4401 y Fu(\032)p Fq(+2)p Fu(\015)2146 4434 y Fw(k)p Fx(')p Fw(k)2295 4448 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)2517 4434 y Fw(k)p Fp(X)p Fw(k)2673 4448 y Fq(2)p Fu(\015)2753 4434 y Fy(2)2798 4401 y Fv(\000)p Fu(n)p Fq(\()p Fu(\032)p Fq(+2)p Fu(\015)t Fq(\))3122 4434 y Fy(.)40 b(Th)m(us)540 4642 y Fw(j)p Fx(S)621 4656 y Fu(n)688 4642 y Fw(\000)20 b Fx(S)835 4656 y Fu(n)p Fv(\000)p Fq(1)972 4642 y Fw(j)51 b(\024)f Fx(C)7 b Fy(\()p Fx(T)1342 4604 y Fq(3)p Fu(\015)1441 4642 y Fy(+)20 b Fx(T)1598 4604 y Fu(\032)p Fq(+)p Fu(\015)1733 4642 y Fy(\))p Fw(k)p Fx(')p Fw(k)1917 4656 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)2140 4642 y Fy(\(2)2220 4604 y Fq(\(1)p Fv(\000)p Fq(3)p Fu(\015)t Fq(\))p Fu(n)2508 4642 y Fy(+)g(2)2644 4604 y Fq(\(1)p Fv(\000)p Fu(\015)t Fv(\000)p Fu(\032)p Fq(\))p Fu(n)2968 4642 y Fy(\))0 4840 y(where)40 b Fx(C)47 b Fy(is)40 b(a)h(function)f(dep)s(enden)m(t)f (only)i(on)f Fx(X)r(;)15 b Fp(X)p Fy(.)71 b(W)-8 b(rite)42 b Fx(S)2331 4854 y Fu(n)2418 4840 y Fy(as)e(the)h(telescopic)i(sum)c Fx(S)3384 4854 y Fu(n)3473 4840 y Fy(=)0 4953 y Fx(S)56 4967 y Fq(0)115 4953 y Fy(+)206 4885 y Ft(P)302 4911 y Fu(n)302 4980 y(k)r Fq(=1)435 4953 y Fy(\()p Fx(S)526 4968 y Fu(k)589 4953 y Fw(\000)20 b Fx(S)736 4968 y Fu(k)r Fv(\000)p Fq(1)869 4953 y Fy(\).)41 b(Then,)29 b(since)467 5112 y Fu(n)423 5139 y Ft(X)424 5337 y Fu(k)r Fq(=1)569 5226 y Fw(j)p Fx(S)650 5241 y Fu(k)713 5226 y Fw(\000)20 b Fx(S)860 5241 y Fu(k)r Fv(\000)p Fq(1)993 5226 y Fw(j)25 b(\024)g Fx(C)7 b Fy(\()p Fx(T)1312 5188 y Fq(3)p Fu(\015)1412 5226 y Fy(+)20 b Fx(T)1569 5188 y Fu(\032)p Fq(+)p Fu(\015)1704 5226 y Fy(\))p Fw(k)p Fx(')p Fw(k)1888 5240 y Fu(\032;)p Fq(2)p Fu(;)p Fq(0)p Fu(;T)2170 5112 y(n)2125 5139 y Ft(X)2127 5337 y Fu(k)r Fq(=1)2257 5226 y Fy(\(2)2337 5188 y Fq(\(1)p Fv(\000)p Fq(3)p Fu(\015)t Fq(\))p Fu(n)2625 5226 y Fy(+)g(2)2761 5188 y Fq(\(1)p Fv(\000)p Fu(\015)t Fv(\000)p Fu(\032)p Fq(\))p Fu(n)3085 5226 y Fy(\))1749 5549 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 431 a Fy(con)m(v)m(erges)31 b(geometrically)h(whenev)m(er)c Fx(\015)j(>)25 b Fy(1)p Fx(=)p Fy(3)30 b(and)e Fx(\015)23 b Fy(+)17 b Fx(\032)25 b(>)g Fy(1,)30 b(w)m(e)f(obtain)g(absolute)h(con)m(v)m(ergence)0 544 y(of)h(the)f(sequence)h Fx(S)693 558 y Fu(n)740 544 y Fy(.)40 b(The)30 b(b)s(ound)e(on)j(the)f(in)m(tegral)i(th)m(us)e (follo)m(ws.)1095 b Fi(\003)141 697 y Fy(As)36 b(this)g(theorem)g(sho)m (ws,)h(b)m(y)f(using)g(the)g(additional)g(information)h(pro)m(vided)e (b)m(y)h(the)g(path)g Fp(X)0 810 y Fy(w)m(e)31 b(are)g(able)g(to)g(mak) m(e)g(Riemann)f(sums)f(con)m(v)m(erge.)43 b(The)30 b(follo)m(wing)i (consequence)f(is)f(immediate.)0 973 y Fk(Corollary)35 b(2.4)46 b Fo(The)33 b(inte)-5 b(gr)g(al)34 b(de\014ne)-5 b(d)34 b(in)e(The)-5 b(or)g(em)35 b(2.3)e(has)h Fp(X)2332 929 y Fu(ij)2332 997 y(st)2419 973 y Fy(=)2515 900 y Ft(R)2575 926 y Fu(t)2558 1005 y(s)2620 973 y Fx(X)2702 940 y Fu(i)2695 996 y(su)2773 973 y Fx(dX)2902 929 y Fu(j)2895 985 y(u)2941 973 y Fo(.)0 1136 y Fj(Pr)n(oof.)40 b Fy(W)-8 b(e)31 b(ha)m(v)m(e)936 1159 y Ft(Z)1027 1186 y Fu(t)987 1365 y(s)1072 1283 y Fx(X)1154 1246 y Fu(i)1147 1306 y(us)1225 1283 y Fx(dX)1354 1246 y Fu(j)1347 1306 y(u)1418 1283 y Fy(=)1514 1159 y Ft(Z)1605 1186 y Fu(t)1564 1365 y(s)1649 1283 y Fx(X)1731 1246 y Fu(i)1724 1306 y(u)1770 1283 y Fx(dX)1899 1246 y Fu(j)1892 1306 y(u)1958 1283 y Fw(\000)20 b Fx(X)2131 1246 y Fu(i)2124 1306 y(s)2161 1283 y Fy(\()p Fx(X)2278 1246 y Fu(j)2271 1306 y(u)2337 1283 y Fw(\000)f Fx(X)2509 1246 y Fu(j)2502 1306 y(s)2546 1283 y Fy(\))p Fx(;)0 1476 y Fy(as)k(it)g(is)g(easily)h(seen)f(b)m(y)f (using)h(the)g(de\014nition.)37 b(That)23 b(is,)i(the)e(rough)f(in)m (tegral)i(has)f(the)g(same)g(linearit)m(y)0 1589 y(prop)s(ert)m(y)33 b(as)h(Riemann)f(in)m(tegral)i(and)e(b)s(eha)m(v)m(es)h(the)g(same)g(w) m(a)m(y)g(with)g(resp)s(ect)f(to)h(the)g(in)m(tegration)0 1702 y(of)j(constan)m(ts.)63 b(Moreo)m(v)m(er,)41 b(the)d(in)m(tegral)h (in)e(the)g(righ)m(t)h(hand)e(side)h(corresp)s(onds)f(to)i(the)g (function)0 1815 y Fx(')59 1830 y Fu(k)102 1815 y Fy(\()p Fx(\030)t Fy(\))26 b(=)f Fx(\030)382 1782 y Fu(i)410 1815 y Fx(\016)450 1830 y Fu(k)r(j)552 1815 y Fy(and)i(b)m(y)f(using)h (\(2.4\))h(w)m(e)g(ha)m(v)m(e)g Fx(S)1689 1829 y Fu(n)1761 1815 y Fy(=)c Fx(S)1912 1829 y Fu(n)p Fv(\000)p Fq(1)2076 1815 y Fy(on)j(the)g(dy)m(adic)g(partition)h(ab)s(o)m(v)m(e.)40 b(Hence)0 1928 y Fx(S)56 1942 y Fu(n)128 1928 y Fy(=)25 b Fx(S)280 1942 y Fq(0)349 1928 y Fy(and)361 2027 y Ft(Z)452 2053 y Fu(t)412 2233 y(s)497 2150 y Fx(X)579 2113 y Fu(i)572 2173 y(u)617 2150 y Fx(dX)746 2113 y Fu(j)739 2173 y(u)810 2150 y Fy(=)g Fx(S)962 2164 y Fq(0)1026 2150 y Fy(=)1122 2064 y Ft(X)1169 2262 y Fu(k)1269 2150 y Fx(')1328 2165 y Fu(k)1371 2150 y Fy(\()p Fx(X)1481 2164 y Fu(s)1518 2150 y Fy(\))p Fx(X)1635 2113 y Fu(k)1628 2173 y(st)1712 2150 y Fy(+)1803 2064 y Ft(X)1808 2262 y Fu(k)r(;m)1949 2150 y Fw(r)2025 2164 y Fu(m)2091 2150 y Fx(')2150 2165 y Fu(k)2193 2150 y Fy(\()p Fx(X)2303 2164 y Fu(s)2341 2150 y Fy(\))p Fp(X)2442 2113 y Fu(k)r(m)2442 2173 y(st)2572 2150 y Fy(=)g Fx(X)2750 2113 y Fu(i)2743 2173 y(s)2780 2150 y Fx(X)2862 2106 y Fu(j)2855 2174 y(st)2938 2150 y Fy(+)20 b Fp(X)3095 2106 y Fu(ij)3095 2174 y(st)3157 2150 y Fx(;)0 2417 y Fy(whic)m(h)30 b(completes)i(the)e(pro)s(of.)2404 b Fi(\003)0 2601 y Fk(Remark)35 b(2.5)46 b Fy(The)d(function)g Fp(X)g Fy(can)g(b)s(e)f(iden)m(ti\014ed)i(as)f(giving)h(the)f(v)-5 b(alue)44 b(of)f(a)h Fo(twic)-5 b(e)44 b(iter)-5 b(ate)g(d)0 2714 y(inte)g(gr)g(al)41 b Fy(o)m(v)m(er)32 b Fx(X)7 b Fy(,)1251 2878 y Fp(X)1317 2834 y Fu(ij)1317 2902 y(st)1404 2878 y Fy(=)1500 2754 y Ft(Z)1591 2780 y Fu(t)1551 2960 y(s)1636 2750 y Ft(\022)1703 2754 y(Z)1794 2780 y Fu(u)1753 2960 y(s)1854 2878 y Fx(dX)1983 2840 y Fu(i)1976 2900 y(v)2017 2750 y Ft(\023)2099 2878 y Fx(dX)2228 2840 y Fu(j)2221 2900 y(u)2267 2878 y Fx(:)1064 b Fy(\(2.5\))0 3075 y(Th)m(us)36 b(Theorem)g(2.3)i(can)f(b)s(e)f(alternativ)m(ely)k (in)m(terpreted)d(as)g(sa)m(ying)h(that)f(the)g(kno)m(wledge)h(of)f (the)0 3188 y(t)m(wice)27 b(iterated)h(in)m(tegral)f(\(in)f(addition)h (with)e(some)i(H\177)-45 b(older)26 b(con)m(tin)m(uit)m(y\))i(is)e (su\016cien)m(t)g(for)g(determin-)0 3301 y(ing)31 b(the)f(v)-5 b(alue)31 b(of)f(the)h(in)m(tegral)1134 3228 y Ft(R)1210 3301 y Fx(')p Fy(\()p Fx(X)1379 3315 y Fu(t)1410 3301 y Fy(\))p Fx(dX)1567 3315 y Fu(t)1628 3301 y Fy(for)f(arbitrary)g Fx(C)2226 3268 y Fq(2)2295 3301 y Fy(function.)141 3484 y(Pro)m(vided)36 b Fx(\015)k(>)35 b Fy(1)p Fx(=)p Fy(2)i(and)f(\(2.5\)) i(holds,)f(whenev)m(er)f(the)h(righ)m(t)f(hand)f(side)i(is)f(de\014ned) f(b)m(y)h(using)0 3597 y(the)27 b(Y)-8 b(oung)27 b(in)m(tegral,)j(the)c (in)m(tegral)j(de\014ned)c(in)i(Theorem)f(2.3)i(coincides)g(with)e(the) h(Y)-8 b(oung)27 b(in)m(tegral.)0 3710 y(This)j(is)g(easy)h(to)g(see)g (due)f(to)h(the)f(estimate)1084 3885 y Fw(j)p Fp(X)1175 3841 y Fu(ij)1175 3909 y(st)1237 3885 y Fw(j)c(\024)f Fx(C)7 b Fy(\()p Fw(k)p Fx(X)g Fw(k)1663 3899 y Fv(1)1759 3885 y Fy(+)20 b Fw(k)p Fx(X)7 b Fw(k)2022 3899 y Fu(\015)2068 3885 y Fy(\))2103 3848 y Fq(2)2142 3885 y Fw(j)p Fx(t)21 b Fw(\000)f Fx(s)p Fw(j)2380 3848 y Fq(2)p Fu(\015)0 4070 y Fy(for)29 b(the)h(Y)-8 b(oung)30 b(in)m(tegral.)42 b(Since)30 b(2)p Fx(\015)g(>)25 b Fy(1,)31 b(the)e(sums)1881 4002 y Ft(P)1977 4097 y Fu(\013)2041 4070 y Fw(r)2117 4084 y Fu(j)2154 4070 y Fx(')2213 4084 y Fu(i)2241 4070 y Fy(\()p Fx(X)2351 4084 y Fu(\034)2382 4092 y Fr(\013)2432 4070 y Fy(\))p Fp(X)2533 4026 y Fu(ij)2533 4082 y(\034)2564 4091 y Fr(\013)p Fn(+1)2685 4082 y Fu(\034)2716 4090 y Fr(\013)2794 4070 y Fy(v)-5 b(anish)29 b(as)h(the)g(mesh)0 4183 y(go)s(es)35 b(to)h(zero,)h(so)e(the)g(mo)s(di\014ed)e(and)i (Riemann)f(sums)g(con)m(v)m(erge)j(to)e(the)g(same)g(limit.)55 b(Moreo)m(v)m(er,)0 4296 y(the)38 b(rough)g(in)m(tegral)i(is)e(con)m (tin)m(uous)h(in)e(the)i(natural)f(top)s(ology)h(asso)s(ciated)h(to)f (the)f(step-2)h(rough)0 4409 y(path)30 b(\()p Fx(X)r(;)15 b Fp(X)p Fy(\),)32 b(i.e.,)g(w)m(e)e(ha)m(v)m(e)0 4572 y Fk(Corollary)35 b(2.6)46 b Fo(L)-5 b(et)33 b Fy(\()p Fx(X)918 4539 y Fu(n)966 4572 y Fx(;)15 b Fp(X)1072 4539 y Fu(n)1119 4572 y Fy(\))32 b Fo(b)-5 b(e)33 b(a)f(se)-5 b(quenc)g(e)33 b(of)f(step-2)h(r)-5 b(ough)34 b(p)-5 b(aths)34 b(such)f(that)g Fw(k)p Fx(X)3122 4539 y Fu(n)3190 4572 y Fw(\000)20 b Fx(X)7 b Fw(k)3408 4586 y Fu(\015)3473 4572 y Fy(+)0 4685 y Fw(k)p Fp(X)111 4652 y Fu(n)178 4685 y Fw(\000)20 b Fp(X)p Fw(k)380 4699 y Fq(2)p Fu(\015)485 4685 y Fw(!)25 b Fy(0)p Fx(:)34 b Fo(Then)942 4612 y Ft(R)1017 4685 y Fx(')p Fy(\()p Fx(X)1193 4652 y Fu(n)1242 4685 y Fy(\))p Fx(dX)1406 4652 y Fu(n)1479 4685 y Fw(!)1595 4612 y Ft(R)1671 4685 y Fx(')p Fy(\()p Fx(X)7 b Fy(\))p Fx(dX)r(;)35 b Fo(for)e(al)5 b(l)33 b Fx(')26 b Fw(2)f Fx(C)2584 4652 y Fq(2)2623 4685 y Fo(.)141 4849 y Fy(Supp)s(ose)42 b Fx(X)593 4816 y Fu(n)684 4849 y Fy(is)i(a)h(sequence)f(of)g(regular)g (\(sa)m(y)-8 b(,)49 b(piecewise)c(linear\))f(appro)m(ximations)h(of)f (the)0 4962 y(H\177)-45 b(older)26 b(con)m(tin)m(uous)g(path)f Fx(X)7 b Fy(.)40 b(Then)25 b(b)m(y)g(putting)g Fp(X)1833 4929 y Fu(n)1905 4962 y Fy(=)2001 4888 y Ft(R)33 b(R)2153 4962 y Fx(dX)2282 4929 y Fu(n)2340 4962 y Fw(\012)11 b Fx(dX)2551 4929 y Fu(n)2598 4962 y Fy(,)27 b(where)e(the)h(in)m (tegrals)h(are)0 5074 y(Riemann)i(in)m(tegrals,)i(a)f(su\016cien)m(t)g (condition)g(for)f(the)g(con)m(v)m(ergence)j(of)e(the)f(appro)m(ximate) i(in)m(tegrals)0 5114 y Ft(R)76 5187 y Fx(')p Fy(\()p Fx(X)252 5154 y Fu(n)300 5187 y Fy(\))p Fx(dX)464 5154 y Fu(n)537 5187 y Fy(to)642 5114 y Ft(R)718 5187 y Fx(')p Fy(\()p Fx(X)7 b Fy(\))p Fx(dX)33 b Fy(is)25 b(that)g(the)g(sequence)g Fp(X)1949 5154 y Fu(n)2020 5187 y Fy(con)m(v)m(erges)h(to)f Fp(X)f Fy(in)h(the)f(top)s(ology)i(induced)0 5300 y(b)m(y)k Fw(k)21 b(\001)f(k)282 5314 y Fq(2)p Fu(\015)363 5300 y Fy(.)1726 5549 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 0 431 a Fl(2.2)112 b(Bro)m(wnian)38 b(motion)g(as)g(rough)g(path)0 603 y Fy(Let)32 b(no)m(w)g Fx(X)39 b Fy(b)s(e)32 b(a)g(sample)g(path)g(of)g(Bro)m(wnian)g(motion.) 46 b(By)32 b(Kolmogoro)m(v's)i(Lemma)e(the)g(pro)s(cess)0 715 y Fx(t)d Fw(7!)f Fx(X)256 729 y Fu(t)318 715 y Fy(has)33 b(a)f(v)m(ersion)h(that)g(is)g(H\177)-45 b(older)33 b(con)m(tin)m(uous) g(with)f(exp)s(onen)m(t)g Fx(\015)i Fw(2)29 b Fy(\(1)p Fx(=)p Fy(3)p Fx(;)15 b Fy(1)p Fx(=)p Fy(2\))36 b(\(actually)e Fx(\015)0 828 y Fy(can)c(b)s(e)f(tak)m(en)h(arbitrarily)g(close)h(to)f (1)p Fx(=)p Fy(2\).)42 b(In)29 b(the)h(follo)m(wing)h(w)m(e)f(will)g (use)f(suc)m(h)g(a)h(v)m(ersion)g(without)0 941 y(eac)m(h)k(time)g(men) m(tioning)g(explicitly)-8 b(,)36 b(i.e.,)f(consider)e(a)g(subspace)g Fw(X)2365 955 y Fu(\015)2438 941 y Fw(\032)d Fx(C)7 b Fy(\()p Fp(R)p Fx(;)15 b Fp(R)2818 908 y Fu(d)2858 941 y Fy(\))33 b(suc)m(h)g(that)g(ev)m(ery)0 1054 y Fx(X)g Fw(2)25 b(X)259 1068 y Fu(\015)333 1054 y Fy(is)31 b(H\177)-45 b(older)30 b(con)m(tin)m(uous)h(with)f(exp)s(onen)m(t)h Fx(\015)5 b Fy(.)141 1167 y(T)-8 b(o)29 b(apply)f(the)g(ab)s(o)m(v)m(e) h(results)f(to)h Fx(X)36 b Fy(w)m(e)28 b(need)g(a)h(c)m(hoice)h(for)d Fp(X)p Fy(.)40 b(This)27 b(candidate)i(is)g(not)f(unique,)0 1280 y(di\013eren)m(t)g(c)m(hoices)i(will)e(lead)h(to)g(di\013eren)m(t) f(in)m(tegrals)h(o)m(v)m(er)h Fx(X)7 b Fy(.)40 b(F)-8 b(or)29 b(instance,)g(in)f(order)f(to)i(construct)0 1393 y(a)i(p)s(ossible)f Fp(X)g Fy(w)m(e)g(can)h(start)g(b)m(y)f(setting) 1244 1603 y(\()p Fp(X)1345 1618 y Fq(It^)-35 b(o)1437 1603 y Fy(\))1472 1558 y Fu(ij)1472 1627 y(st)1560 1603 y Fy(=)1656 1479 y Ft(Z)1747 1505 y Fu(t)1707 1685 y(s)1792 1479 y Ft(Z)1883 1505 y Fu(u)1842 1685 y(s)1943 1603 y Fx(dX)2072 1565 y Fu(i)2065 1625 y(v)2106 1603 y Fx(dX)2235 1565 y Fu(j)2228 1625 y(u)2274 1603 y Fx(;)0 1802 y Fy(where)42 b(the)g(double)g(in)m(tegral)h(is)f(understo)s(o)s(d)f(in)g(It^)-45 b(o)43 b(sense.)76 b(In)41 b(this)h(w)m(a)m(y)h(w)m(e)g(obtain)f(a)h (family)0 1915 y(of)37 b(random)f(v)-5 b(ariables)37 b Fw(f)p Fy(\()p Fp(X)979 1930 y Fq(It^)-35 b(o)1072 1915 y Fy(\))1107 1871 y Fu(ij)1107 1939 y(st)1205 1915 y Fy(:)36 b Fx(i;)15 b(j)42 b Fy(=)36 b(1)p Fx(;)15 b(:)g(:)g(:)i(;)e (d)p Fy(;)52 b Fx(t;)15 b(s)36 b Fw(2)f Fy([0)p Fx(;)15 b(T)e Fy(])p Fw(g)38 b Fy(satisfying)f(the)g(m)m(ultiplicativ)m(e)0 2028 y(prop)s(ert)m(y)1002 2141 y(\()p Fp(X)1103 2156 y Fq(It^)-35 b(o)1196 2141 y Fy(\))1231 2097 y Fu(ij)1231 2165 y(st)1313 2141 y Fw(\000)20 b Fy(\()p Fp(X)1505 2156 y Fq(It^)-35 b(o)1598 2141 y Fy(\))1633 2097 y Fu(ij)1633 2165 y(ut)1724 2141 y Fw(\000)20 b Fy(\()p Fp(X)1916 2156 y Fq(It^)-35 b(o)2008 2141 y Fy(\))2043 2103 y Fu(ij)2043 2163 y(su)2146 2141 y Fy(=)25 b Fx(X)2324 2103 y Fu(i)2317 2163 y(ut)2388 2141 y Fx(X)2470 2103 y Fu(j)2463 2163 y(su)0 2285 y Fy(almost)31 b(surely)f(for)g(an)m(y)g(\014xed)f Fx(t;)15 b(s;)g(u)26 b Fw(2)f Fy([0)p Fx(;)15 b(T)e Fy(].)41 b(The)30 b(next)g(step)g(is)g(to)h(sho)m(w)f(that)h(this)f(family)g (has)g(a)0 2398 y(v)m(ersion)h(for)f(whic)m(h)1390 2511 y Fw(k)p Fp(X)1501 2466 y Fu(ij)1501 2540 y Fq(It^)-35 b(o)1593 2511 y Fw(k)1638 2525 y Fq(2)p Fu(\015)1744 2511 y Fx(<)25 b Fw(1)91 b Fy(a.s.)1203 b(\(2.6\))0 2654 y(In)33 b(order)g(to)h(pro)m(v)m(e)h(\(2.6\),)h(w)m(e)e(use)g(the)g (follo)m(wing)h(lemma)f(obtained)g(as)g(an)f(extension)i(of)e(a)h (result)0 2767 y(of)d(Garsia-Ro)s(demic)m(h-Rumsey)g(in)f([23)q(].)0 2917 y Fk(Lemma)35 b(2.7)46 b Fo(F)-7 b(or)38 b(any)f Fx(\022)e(>)d Fy(0)37 b Fo(and)h Fx(p)33 b Fw(\025)f Fy(1)37 b Fo(ther)-5 b(e)38 b(exists)f(a)g(c)-5 b(onstant)39 b Fx(C)k Fo(such)37 b(that)h(for)f(any)g Fx(R)d Fw(2)0 3030 y Fx(C)7 b Fy(\([0)p Fx(;)15 b(T)e Fy(])308 2997 y Fq(2)348 3030 y Fx(;)i Fw(B)s Fy(\))p Fo(,)33 b(wher)-5 b(e)34 b Fy(\()p Fw(B)s Fx(;)15 b Fw(j)20 b(\001)h(j)p Fy(\))33 b Fo(is)g(a)g(Banach)g(sp)-5 b(ac)g(e,)34 b(we)f(have)1105 3188 y Fw(k)p Fx(R)q Fw(k)1265 3203 y Fu(\022)1330 3188 y Fw(\024)25 b Fx(C)1513 3114 y Ft(\000)1555 3188 y Fx(U)1617 3207 y Fu(\022)r Fq(+2)p Fu(=p;p)1872 3188 y Fy(\()p Fx(R)q Fy(\))20 b(+)g Fx(V)2176 3203 y Fu(\022)2215 3188 y Fy(\()p Fx(R)q Fy(\))2355 3114 y Ft(\001)2413 3188 y Fx(;)918 b Fy(\(2.7\))0 3347 y Fo(with)1002 3537 y Fx(U)1064 3552 y Fu(\022)r(;p)1159 3537 y Fy(\()p Fx(R)q Fy(\))25 b(=)1420 3382 y Ft(")1473 3414 y(Z)1524 3620 y Fq([0)p Fu(;T)10 b Fq(])1670 3601 y Fn(2)1723 3409 y Ft(\022)1847 3476 y Fw(j)p Fx(R)1941 3490 y Fu(ts)2003 3476 y Fw(j)p 1800 3517 277 4 v 1800 3600 a(j)p Fx(t)20 b Fw(\000)g Fx(s)p Fw(j)2037 3574 y Fu(\022)2086 3409 y Ft(\023)2153 3432 y Fu(p)2207 3537 y Fx(dtds)2377 3382 y Ft(#)2431 3405 y Fq(1)p Fu(=p)0 3754 y Fo(and)955 3892 y Fx(V)1008 3907 y Fu(\022)1047 3892 y Fy(\()p Fx(R)q Fy(\))26 b(=)99 b(inf)1309 3956 y Fu(\022)1342 3965 y Fn(1)1376 3956 y Fv(2)p Fq(\(0)p Fu(;\022)r Fq(\))1619 3892 y Fy(sup)1583 3970 y Fu(t)p Fv(6)p Fq(=)p Fu(u)p Fv(6)p Fq(=)p Fu(s)1840 3830 y Fw(j)p Fx(R)1934 3844 y Fu(st)2017 3830 y Fw(\000)20 b Fx(R)2177 3844 y Fu(ut)2267 3830 y Fw(\000)g Fx(R)2427 3844 y Fu(su)2505 3830 y Fw(j)p 1817 3871 736 4 v 1817 3954 a(j)p Fx(t)g Fw(\000)g Fx(u)p Fw(j)2063 3928 y Fu(\022)2096 3937 y Fn(1)2135 3954 y Fw(j)p Fx(u)h Fw(\000)f Fx(s)p Fw(j)2392 3928 y Fu(\022)r Fv(\000)p Fu(\022)2515 3937 y Fn(1)2563 3892 y Fx(:)0 4123 y Fk(Corollary)35 b(2.8)46 b Fo(We)32 b(have)h Fw(k)p Fp(X)1129 4138 y Fh(It^)-36 b(o)1221 4123 y Fw(k)1266 4137 y Fq(2)p Fu(\015)1372 4123 y Fx(<)25 b Fw(1)1591 4131 y(W)1688 4123 y Fo(-almost)34 b(sur)-5 b(ely.)0 4289 y Fj(Pr)n(oof.)40 b Fy(Consider)29 b Fx(V)786 4303 y Fq(2)p Fu(\015)866 4289 y Fy(\()p Fp(X)967 4304 y Fq(It^)-35 b(o)1059 4289 y Fy(\).)41 b(By)31 b(using)f(the)h(m)m(ultiplicativ)m(e) i(prop)s(ert)m(y)d(w)m(e)g(ha)m(v)m(e)747 4499 y Fx(V)800 4513 y Fq(2)p Fu(\015)880 4499 y Fy(\()p Fp(X)981 4514 y Fq(It^)-35 b(o)1073 4499 y Fy(\))84 b Fw(\024)177 b Fy(inf)1345 4564 y Fu(\022)1378 4573 y Fn(1)1413 4564 y Fv(2)p Fq(\(0)p Fu(;)p Fq(2)p Fu(\015)t Fq(\))1696 4499 y Fy(sup)1660 4578 y Fu(t)p Fv(6)p Fq(=)p Fu(u)p Fv(6)p Fq(=)p Fu(s)2083 4438 y Fw(j)p Fx(X)2183 4452 y Fu(tu)2254 4438 y Fw(jj)p Fx(X)2379 4452 y Fu(us)2457 4438 y Fw(j)p 1894 4479 777 4 v 1894 4562 a(j)p Fx(t)21 b Fw(\000)f Fx(u)p Fw(j)2141 4536 y Fu(\022)2174 4545 y Fn(1)2212 4562 y Fw(j)p Fx(u)h Fw(\000)f Fx(s)p Fw(j)2469 4536 y Fq(2)p Fu(\015)t Fv(\000)p Fu(\022)2632 4545 y Fn(1)1192 4763 y Fw(\024)177 b Fy(inf)1345 4828 y Fu(\022)1378 4837 y Fn(1)1413 4828 y Fv(2)p Fq(\(0)p Fu(;)p Fq(2)p Fu(\015)t Fq(\))1660 4763 y Fy(sup)1668 4842 y Fu(t)p Fv(6)p Fq(=)p Fu(u)1883 4702 y Fw(j)p Fx(X)1983 4716 y Fu(tu)2054 4702 y Fw(j)p 1822 4742 319 4 v 1822 4826 a(j)p Fx(t)21 b Fw(\000)f Fx(u)p Fw(j)2069 4799 y Fu(\022)2102 4808 y Fn(1)2166 4763 y Fy(sup)2170 4842 y Fu(u)p Fv(6)p Fq(=)p Fu(s)2455 4702 y Fw(j)p Fx(X)2555 4716 y Fu(us)2633 4702 y Fw(j)p 2328 4742 459 4 v 2328 4826 a(j)p Fx(u)g Fw(\000)g Fx(s)p Fw(j)2584 4799 y Fq(2)p Fu(\015)t Fv(\000)p Fu(\022)2747 4808 y Fn(1)1192 5073 y Fw(\024)1345 4918 y Ft( )1417 5073 y Fy(sup)1425 5152 y Fu(t)p Fv(6)p Fq(=)p Fu(u)1627 5012 y Fw(j)p Fx(X)1727 5026 y Fu(tu)1798 5012 y Fw(j)p 1579 5052 291 4 v 1579 5135 a(j)p Fx(t)h Fw(\000)f Fx(u)p Fw(j)1826 5109 y Fu(\015)1880 4918 y Ft(!)1952 4940 y Fq(2)1192 5300 y Fy(=)82 b Fw(k)p Fx(X)7 b Fw(k)1517 5263 y Fq(2)1517 5323 y Fu(\015)1563 5300 y Fx(:)1726 5549 y Fy(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 0 431 a Fy(Moreo)m(v)m(er,)782 673 y Fp(E)p Fy([)p Fx(U)930 687 y Fq(2)p Fu(\015)t(;p)1065 673 y Fy(\()p Fp(X)1166 688 y Fq(It^)-35 b(o)1258 673 y Fy(\))1293 635 y Fu(p)1333 673 y Fy(])83 b(=)g Fp(E)1671 517 y Ft(")1724 549 y(Z)1774 755 y Fq([0)p Fu(;T)10 b Fq(])1920 736 y Fn(2)1973 545 y Ft(\022)2050 611 y Fw(j)p Fy(\()p Fp(X)2176 626 y Fq(It^)-35 b(o)2269 611 y Fy(\))2304 625 y Fu(ts)2366 611 y Fw(j)p 2050 652 342 4 v 2062 735 a(j)p Fx(t)21 b Fw(\000)f Fx(s)p Fw(j)2300 709 y Fq(2)p Fu(\015)2401 545 y Ft(\023)2468 567 y Fu(p)2523 673 y Fx(dtds)2693 517 y Ft(#)1441 949 y Fy(=)1595 826 y Ft(Z)1645 1032 y Fq([0)p Fu(;T)10 b Fq(])1791 1013 y Fn(2)1854 888 y Fp(E)p Fy([)p Fw(j)p Fy(\()p Fp(X)2066 903 y Fq(It^)-35 b(o)2159 888 y Fy(\))2194 902 y Fu(ts)2256 888 y Fw(j)2281 855 y Fu(p)2321 888 y Fy(])p 1854 929 493 4 v 1924 1012 a Fw(j)p Fx(t)21 b Fw(\000)e Fx(s)p Fw(j)2161 986 y Fq(2)p Fu(\015)t(p)2356 949 y Fx(dtds;)0 1177 y Fy(with)42 b(exp)s(ectation)i (with)e(resp)s(ect)g(to)h(Wiener)g(measure.)77 b(An)42 b(application)i(of)e(the)h(Burkholder-)0 1290 y(Da)m(vis-Gundy)f (inequalit)m(y)h([44)q(])f(allo)m(ws)h(to)g(estimate)g(the)f Fx(p)p Fy(-momen)m(t)g(of)g(the)g(double)f(sto)s(c)m(hastic)0 1403 y(in)m(tegral)32 b(as)821 1647 y Fp(E)p Fy([)p Fw(j)p Fy(\()p Fp(X)1033 1662 y Fq(It^)-35 b(o)1126 1647 y Fy(\))1161 1661 y Fu(ts)1223 1647 y Fw(j)1248 1610 y Fu(p)1288 1647 y Fy(])83 b Fw(\024)g Fx(c)1589 1661 y Fu(p)1629 1647 y Fp(E)1705 1519 y Ft(\024)1753 1524 y(Z)1844 1550 y Fu(t)1803 1730 y(s)1889 1647 y Fw(j)p Fx(X)1989 1661 y Fu(us)2067 1647 y Fw(j)2092 1610 y Fq(2)2131 1647 y Fx(ds)2221 1519 y Ft(\025)2269 1535 y Fu(p=)p Fq(2)1396 1905 y Fw(\024)g Fx(c)1589 1919 y Fu(p)1629 1905 y Fw(j)p Fx(t)20 b Fw(\000)g Fx(s)p Fw(j)1866 1868 y Fu(p=)p Fq(2)p Fv(\000)p Fq(1)2067 1905 y Fp(E)2143 1777 y Ft(\024)2190 1781 y(Z)2281 1808 y Fu(t)2241 1988 y(s)2326 1905 y Fw(j)p Fx(X)2426 1919 y Fu(us)2504 1905 y Fw(j)2529 1868 y 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3431 y(inte)g(gr)g(al)41 b Fy(o)m(v)m(er)32 b(\()p Fx(X)r(;)15 b Fp(X)744 3446 y Fq(It^)-35 b(o)837 3431 y Fy(\).)141 3544 y(The)30 b(relationship)h(b)s(et)m(w)m(een)g (the)f(rough)g(in)m(tegral)i(and)e(the)h(It^)-45 b(o)30 b(in)m(tegral)j(is)d(made)g(clear)i(b)m(y)0 3722 y Fk(Lemma)j(2.9)46 b Fo(The)37 b(r)-5 b(ough)39 b(inte)-5 b(gr)g(al)39 b(over)e(the)h(c)-5 b(ouple)38 b Fy(\()p Fx(X)r(;)15 b Fp(X)2184 3737 y Fh(It^)-36 b(o)2277 3722 y Fy(\))38 b Fo(c)-5 b(oincides)38 b(with)g(the)g(It^)-46 b(o)38 b(inte)-5 b(gr)g(al)0 3835 y(for)33 b(any)g Fx(')26 b Fw(2)f Fx(C)562 3802 y Fq(2)601 3835 y Fo(.)0 4013 y Fj(Pr)n(oof.)61 b Fy(By)37 b(It^)-45 b(o)38 b(theory)g(the)f(sums) 1366 3945 y Ft(P)1462 4040 y Fu(\013)1527 4013 y Fx(')1586 4027 y Fu(i)1614 4013 y Fy(\()p Fx(X)1724 4027 y Fu(\034)1755 4035 y Fr(\013)1805 4013 y Fy(\))p Fx(X)1922 3980 y Fu(i)1915 4036 y(\034)1946 4045 y Fr(\013)p Fn(+1)2068 4036 y Fu(\034)2099 4044 y Fr(\013)2185 4013 y Fy(con)m(v)m(erge)i(in)e(probabilit)m(y)h 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b(v)m(ersion)h(for)f(the)g(double)f(in)m(tegral)j Fp(X)d Fy(w)m(as)i(arbitrary)-8 b(.)0 770 y(Alternativ)m(ely)39 b(w)m(e)d(could)g(ha)m(v)m(e)i(considered)e(other)g(de\014nitions,)i (e.g.)59 b(Stratono)m(vic)m(h)38 b(in)m(tegral)g(and)0 883 y(let)1170 1029 y(\()p Fp(X)1271 1043 y Fq(Stra)1405 1029 y Fy(\))1440 985 y Fu(ij)1440 1053 y(st)1528 1029 y Fy(=)1624 905 y Ft(Z)1715 932 y Fu(t)1674 1112 y(s)1759 905 y Ft(Z)1850 932 y Fu(u)1810 1112 y(s)1911 1029 y Fw(\016)p Fx(dX)2085 992 y Fu(i)2078 1052 y(v)2140 1029 y Fw(\016)20 b Fx(dX)2334 992 y Fu(j)2327 1052 y(u)0 1238 y Fy(where)36 b Fw(\016)p Fx(dX)44 b Fy(stands)36 b(for)g(Stratono)m(vic)m(h)i(\(or)f(symmetric\))g(in)m(tegration.)60 b(By)37 b(the)g(same)f(pro)s(cedure)0 1351 y(w)m(e)e(\014nd)f(a)h (regular)g(v)m(ersion)g(of)g Fp(X)1199 1365 y Fq(Stra)1366 1351 y Fy(suc)m(h)g(that)g Fw(k)p Fp(X)1886 1365 y Fq(Stra)2020 1351 y Fw(k)2065 1365 y Fq(2)p Fu(\015)2176 1351 y Fx(<)d Fw(1)i Fy(and)g(can)i(construct)f(the)g(rough)0 1464 y(in)m(tegral)d(o)m(v)m(er)f(the)f(couple)g(\()p Fx(X)r(;)15 b Fp(X)1177 1478 y Fq(Stra)1312 1464 y Fy(\))29 b(whic)m(h)g(w)m(e)g (denote)g(again)h Fw(\016)p Fx(dX)7 b Fy(.)42 b(It)29 b(is)g(not)g(di\016cult)g(to)g(pro)m(v)m(e)0 1577 y(that)i(for)f Fx(')c Fw(2)f Fx(C)579 1544 y Fq(2)648 1577 y Fy(it)31 b(coincides)g(with)f(the)h(familiar)g(Stratono)m(vic)m(h)h(in)m (tegral.)141 1690 y(The)h(relationship)g(b)s(et)m(w)m(een)g(the)h (rough)e(in)m(tegrals)j(based)d(on)h(the)g(It^)-45 b(o)34 b(and)e(Stratono)m(vic)m(h)j(con-)0 1803 y(structions)30 b(follo)m(ws)i(from)e(the)g(iden)m(tit)m(y)1132 2032 y(\()p Fp(X)1233 2046 y Fq(Stra)1367 2032 y Fy(\))1402 1988 y Fu(ij)1402 2056 y(st)1490 2032 y Fy(=)25 b(\()p Fp(X)1687 2047 y Fq(It^)-35 b(o)1779 2032 y Fy(\))1814 1988 y Fu(ij)1814 2056 y(st)1897 2032 y Fy(+)1998 1971 y(1)p 1998 2011 46 4 v 1998 2095 a(2)2053 2032 y Fx(\016)2093 2046 y Fu(ij)2154 2032 y Fy(\()p Fx(t)20 b Fw(\000)g Fx(s)p Fy(\))0 2249 y(b)s(et)m(w)m(een)30 b(It^)-45 b(o)29 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b(it)f(is)f(not)h(di\016cult)f(to)i(see)f(the)f(follo)m(wing)i (regularit)m(y)g(result.)0 3763 y Fk(Corollary)j(2.11)46 b Fo(The)33 b(map)h Fx(')26 b Fw(7!)1283 3690 y Ft(R)1344 3716 y Fq(1)1326 3795 y(0)1398 3763 y Fx(')p Fy(\()p Fx(X)1567 3777 y Fu(u)1613 3763 y Fy(\))p Fx(dX)1770 3777 y Fu(u)1849 3763 y Fo(is)32 b(c)-5 b(ontinuous)34 b(fr)-5 b(om)34 b Fx(C)2691 3730 y Fq(2)2730 3763 y Fy(\()p Fp(R)2831 3730 y Fu(d)2871 3763 y Fx(;)15 b Fp(R)2977 3730 y Fu(d)3018 3763 y Fy(\))33 b Fo(to)g Fp(R)p Fo(.)0 4049 y FD(3)135 b(Sto)t(c)l(hastic)45 b(curren)l(ts)0 4255 y Fl(3.1)112 b(Lifting)38 b(Wiener)g(measure)h(to)e(the)g(space)h (of)g(curren)m(ts)0 4426 y Fy(Let)f Fw(X)50 b Fy(=)36 b Fx(C)7 b Fy(\()p Fp(R)p Fx(;)15 b Fp(R)670 4393 y Fu(d)710 4426 y Fy(\))37 b(b)s(e)f(path)h(space,)i(i.e.,)h(the)d(space)h(of)f (con)m(tin)m(uous)h(functions)e(from)h Fp(R)f Fy(to)i Fp(R)3478 4393 y Fu(d)3518 4426 y Fy(.)0 4539 y(The)33 b Fx(\033)s Fy(-\014eld)476 4547 y Fw(F)583 4539 y Fy(is)h(generated)g 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Fr(n)2063 5263 y Fv(\000)p Fu(t)2143 5272 y Fr(n)p Fm(\000)p Fn(1)2264 5263 y Fq(\))p Fu(H)2349 5272 y Fn(0)2389 5300 y Fx(f)2434 5314 y Fu(n)2480 5300 y Fy(\))2515 5322 y Fu(L)2563 5304 y Fn(2)2598 5322 y Fq(\()p Fs(R)2673 5304 y Fr(d)2710 5322 y Fu(;dx)p Fq(\))3356 5300 y Fy(\(3.1\))1726 5549 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 431 a Fy(where)31 b Fx(F)42 b Fy(=)27 b Fx(f)507 445 y Fq(1)546 431 y Fy(\()p Fx(X)656 445 y Fu(t)681 454 y Fn(1)721 431 y Fy(\))15 b Fw(\001)g(\001)g(\001)i Fx(f)938 445 y Fu(n)984 431 y Fy(\()p Fx(X)1094 445 y Fu(t)1119 453 y Fr(n)1167 431 y Fy(\).)46 b(Here)32 b(it)h(is)f (understo)s(o)s(d)e(that)j Fx(f)2396 445 y Fq(2)2435 431 y Fx(;)15 b(:)g(:)g(:)h(;)f(f)2681 445 y Fu(n)p Fv(\000)p Fq(1)2850 431 y Fy(act)33 b(as)g(m)m(ultiplica-)0 544 y(tion)d(op)s(erators)g(for)g(whic)m(h)f(w)m(e)i(use)e(the)h(same)h (sym)m(b)s(ol)e(as)h(for)g(the)g(corresp)s(onding)f(functions.)40 b(The)0 657 y(op)s(erator)31 b Fx(e)409 624 y Fv(\000)p Fu(tH)547 633 y Fn(0)616 657 y Fy(has)g(the)f(in)m(tegral)i(k)m(ernel) 728 894 y(\005)796 908 y Fu(t)826 894 y Fy(\()p Fx(x;)15 b(y)s Fy(\))26 b(=)1303 832 y(1)p 1168 873 316 4 v 1168 960 a(\(2)p Fx(\031)s(t)p Fy(\))1371 934 y Fu(d=)p Fq(2)1508 894 y Fy(exp)1662 766 y Ft(\022)1729 894 y Fw(\000)1826 832 y Fy(1)p 1810 873 79 4 v 1810 956 a(2)p Fx(t)1898 894 y Fy(\()p Fx(x)21 b Fw(\000)f Fx(y)s Fy(\))2180 856 y Fq(2)2219 766 y Ft(\023)2301 894 y Fx(;)107 b(x;)15 b(y)28 b Fw(2)d Fp(R)2750 856 y Fu(d)2790 894 y Fx(:)541 b Fy(\(3.2\))0 1149 y(W)-8 b(e)25 b(denote)f(b)m(y)556 1157 y Fw(W)646 1163 y Fu(I)709 1149 y Fy(the)g(measure)1207 1157 y Fw(W)1328 1149 y Fy(restricted)g(to)g Fw(X)1895 1163 y Fu(I)1960 1149 y Fy(=)h Fx(C)7 b Fy(\()p Fx(I)g(;)15 b Fp(R)2316 1116 y Fu(d)2357 1149 y Fy(\).)39 b(Similarly)-8 b(,)26 b(with)d(giv)m(en)i Fx(T)38 b(>)25 b Fy(0,)0 1262 y Fx(\030)t(;)15 b(\021)29 b Fw(2)c Fp(R)310 1229 y Fu(d)380 1262 y Fy(write)62 1480 y Fw(W)159 1428 y Fu(\030)s(;\021)152 1501 y(I)255 1472 y Fy(\()p Fx(F)13 b Fy(\))26 b(=)518 1371 y Ft(\020)572 1472 y Fx(e)614 1435 y Fv(\000)p Fq(\()p Fv(\000)p Fu(T)10 b Fv(\000)p Fu(t)882 1444 y Fn(1)917 1435 y Fq(\))p Fu(H)1002 1444 y Fn(0)1041 1472 y Fx(f)1086 1486 y Fq(1)1125 1472 y Fx(e)1167 1435 y Fv(\000)p Fq(\()p Fu(t)1274 1444 y Fn(2)1310 1435 y Fv(\000)p Fu(t)1390 1444 y Fn(1)1425 1435 y Fq(\))p Fu(H)1510 1444 y Fn(0)1549 1472 y Fx(f)1594 1486 y Fq(2)1648 1472 y Fx(:)15 b(:)g(:)i(e)1812 1435 y Fv(\000)p Fq(\()p Fu(t)1919 1443 y Fr(n)1962 1435 y Fv(\000)p Fu(t)2042 1444 y Fr(n)p Fm(\000)p Fn(1)2163 1435 y Fq(\))p Fu(H)2248 1444 y Fn(0)2303 1371 y Ft(\020)2357 1472 y Fx(f)2402 1486 y Fu(n)2449 1472 y Fx(e)2491 1435 y Fv(\000)p Fq(\()p Fu(T)10 b Fv(\000)p Fu(t)2704 1443 y Fr(n)2747 1435 y Fq(\))p Fu(H)2832 1444 y Fn(0)2871 1472 y Fy(\()p Fw(\001)p Fx(;)15 b(\021)s Fy(\))3054 1371 y Ft(\021)r(\021)3180 1472 y Fy(\()p Fx(\030)t Fy(\))62 b(\(3.3\))0 1730 y(with)23 b Fx(I)32 b Fy(=)25 b([)p Fw(\000)p Fx(T)8 b(;)15 b(T)e Fy(])24 b(and)f(let)985 1707 y Ft(c)974 1730 y Fw(W)1071 1665 y Fu(\030)s(;\021)1071 1752 y(I)1166 1730 y Fy(\()p Fx(F)13 b Fy(\))26 b(=)1429 1738 y Fw(W)1527 1685 y Fu(\030)s(;\021)1519 1758 y(I)1622 1730 y Fy(\()p Fx(F)13 b Fy(\))p Fx(=)p Fy(\005)1876 1744 y Fq(2)p Fu(T)1968 1730 y Fy(\()p Fx(\030)t(;)i(\021)s Fy(\))24 b(b)s(e)f(the)h(Bro)m(wnian)f(bridge)g(starting)i(in)0 1860 y Fx(\030)k Fy(at)c Fw(\000)p Fx(T)37 b Fy(and)24 b(ending)g(in)h Fx(\021)j Fy(at)d Fx(T)13 b Fy(.)39 b(Under)1482 1837 y Ft(c)1472 1860 y Fw(W)1569 1795 y Fu(x;y)1569 1883 y Fq([)p Fv(\000)p Fu(T)6 b(;T)k Fq(])1809 1860 y Fy(the)25 b(pro)s(cess)f Fx(X)2346 1874 y Fu(t)2401 1860 y Fy(is)h(a)g(Gaussian)g(semimartingale)0 1973 y(\(Bro)m(wnian)31 b(bridge\))f(satisfying)h(the)g(SDE)1256 2207 y Fx(dX)1378 2221 y Fu(u)1449 2207 y Fy(=)25 b Fw(\000)1626 2145 y Fx(y)e Fw(\000)d Fx(X)1860 2159 y Fu(u)p 1626 2186 280 4 v 1651 2269 a Fx(T)33 b Fw(\000)20 b Fx(u)1915 2207 y(du)h Fy(+)f Fx(dB)2242 2221 y Fu(u)0 2458 y Fy(where)30 b(\()p Fx(B)367 2472 y Fu(u)412 2458 y Fy(\))447 2476 y Fu(u)p Fv(2)p Fq([)p Fv(\000)p Fu(T)6 b(;T)k Fq(])781 2458 y Fy(is)31 b(a)959 2435 y Ft(c)949 2458 y Fw(W)1046 2393 y Fu(x;y)1046 2480 y Fq([)p Fv(\000)p Fu(T)6 b(;T)k Fq(])1262 2458 y Fy(-Bro)m(wian)31 b(motion.)141 2630 y(Next,)37 b(let)f Fw(D)h Fy(b)s(e)d(a)h(Banac)m(h)h(space)f(of)g (functions)f(from)g Fp(R)23 b Fw(\002)g Fp(R)2393 2597 y Fu(d)2467 2630 y Fy(to)36 b Fp(R)2649 2597 y Fu(d)2724 2630 y Fy(con)m(taining)g(the)f(space)0 2743 y(of)g(smo)s(oth)f (functions)g Fx(C)901 2710 y Fv(1)976 2743 y Fy(\()p Fp(R)23 b Fw(\002)f Fp(R)1259 2710 y Fu(d)1299 2743 y Fy(;)15 b Fp(R)1405 2710 y Fu(d)1446 2743 y Fy(\).)54 b(Denote)36 b(with)e Fw(k)23 b(\001)h(k)2249 2757 y Fv(D)2344 2743 y Fy(the)35 b(Banac)m(h)h(norm.)52 b(Let)36 b Fw(D)3360 2710 y Fv(0)3417 2743 y Fy(the)0 2856 y(top)s(ological)d(dual)d(of)h Fw(D)s Fy(,)f(\001)972 2870 y Fu(I)1037 2856 y Fy(=)25 b Fw(f)p Fy(\()p Fx(s;)15 b(t)p Fy(\))26 b Fw(2)f Fx(I)1523 2823 y Fq(2)1562 2856 y Fw(j)p Fx(s)h Fw(\024)f Fx(t)p Fw(g)p Fy(,)30 b(and)g(write)h(\001)25 b(=)g(\001)2568 2870 y Fs(R)2620 2856 y Fy(.)0 3033 y Fk(De\014nition)35 b(3.1)46 b Fx(C)32 b Fw(2)25 b Fx(C)7 b Fy(\(\001;)15 b Fw(D)1146 3000 y Fv(0)1169 3033 y Fy(\))24 b Fo(is)g(a)32 b Fy(sto)s(c)m(hastic)23 b(curren)m(t)g Fo(if)h(it)g(satis\014es)h(the) g(fol)5 b(lowing)25 b(pr)-5 b(op)g(erties:)61 3235 y(\(1\))46 b Fx(C)292 3249 y Fu(tt)347 3235 y Fy(\()p Fx(')p Fy(\))27 b(=)e(0)p Fx(;)66 b(C)800 3249 y Fu(su)878 3235 y Fy(\()p Fx(')p Fy(\))21 b(+)f Fx(C)1184 3249 y Fu(ut)1254 3235 y Fy(\()p Fx(')p Fy(\))27 b(=)e Fx(C)1571 3249 y Fu(st)1633 3235 y Fy(\()p Fx(')p Fy(\))p Fx(;)85 b Fo(for)33 b(any)g Fx(s)25 b Fw(\024)g Fx(u)g Fw(\024)g Fx(t)32 b Fo(and)i(any)f Fx(')26 b Fw(2)f(D)s Fo(;)61 3442 y(\(2\))46 b Fy(lo)s(calit)m(y)33 b(prop)s(ert)m(y:)134 b Fx(C)1114 3456 y Fu(st)1177 3442 y Fy(\()p Fx(')p Fy(\))26 b(=)f(0)58 b Fo(whenever)h Fx(')p Fy(\()p Fx(u;)15 b(x)p Fy(\))26 b(=)f(0)33 b Fo(for)h(al)5 b(l)33 b Fx(u)25 b Fw(2)g Fy([)p Fx(s;)15 b(t)p Fy(])p Fo(,)33 b Fx(x)25 b Fw(2)g Fp(R)3321 3409 y Fu(d)3361 3442 y Fo(.)0 3619 y(We)32 b(denote)i Fw(C)c(\032)25 b Fx(C)7 b Fy(\(\001;)15 b Fw(D)926 3586 y Fv(0)949 3619 y Fy(\))33 b Fo(the)g(sp)-5 b(ac)g(e)34 b(of)f(sto)-5 b(chastic)34 b(curr)-5 b(ents.)0 3797 y Fy(Set)32 b(\004)27 b(=)g Fw(X)34 b(\002)20 b(C)37 b Fy(endo)m(w)m(ed)32 b(with)f(the)h(pro)s(duct)e(top)s(ology)j(and)e(with)h(the)f(Borel)i Fx(\033)s Fy(-\014eld)f(\(on)f(whose)0 3910 y(comp)s(onen)m(t)h (generated)i(b)m(y)e Fw(C)37 b Fy(w)m(e)c(consider)f(the)g(top)s(ology) i(of)e(uniform)f(con)m(v)m(ergence)j(on)e(b)s(ounded)0 4023 y(in)m(terv)-5 b(als\).)65 b(\004)38 b(pla)m(ys)g(the)g(role)h(of) f(join)m(t)h(path-curren)m(t)f(con\014guration)g(space.)65 b(As)38 b(a)g(measurable)0 4136 y(space,)32 b(it)f(is)g(endo)m(w)m(ed)h (with)e(a)i(family)f(of)g Fx(\033)s Fy(-algebras)i Fw(fA)2047 4150 y Fu(st)2109 4136 y Fw(g)2154 4150 y Fu(t>s)2302 4136 y Fy(suc)m(h)e(that)h Fw(A)2779 4150 y Fu(st)2867 4136 y Fy(=)26 b Fx(\033)s Fy(\()p Fx(X)3129 4150 y Fu(u)3175 4136 y Fx(;)15 b(C)3280 4150 y Fu(uv)3362 4136 y Fy(\()p Fx(')p Fy(\))27 b(:)0 4249 y Fx(u;)15 b(v)42 b Fw(2)d Fy([)p Fx(t;)15 b(s)p Fy(])p Fx(;)g(')40 b Fw(2)e(D)s Fy(\).)65 b(Similarly)-8 b(,)41 b(w)m(e)e(can)g(de\014ne)f(the)g(forw)m (ard)g(\014ltration)h Fw(fA)2893 4210 y Fq(+)2893 4273 y Fu(t)2952 4249 y Fw(g)2997 4263 y Fu(t)3066 4249 y Fy(=)f Fw(fA)3293 4263 y Fq(+)p Fv(1)p Fu(;t)3468 4249 y Fw(g)3513 4263 y Fu(t)0 4361 y Fy(and)f(the)h(bac)m(kw)m(ard)g (\014ltration)g Fw(fA)1266 4323 y Fv(\000)1266 4386 y Fu(t)1325 4361 y Fw(g)1370 4375 y Fu(t)1437 4361 y Fy(=)f Fw(fA)1663 4375 y Fu(t;)p Fv(\0001)1838 4361 y Fw(g)1883 4375 y Fu(t)1913 4361 y Fy(.)62 b(The)37 b(ab)s(o)m(v)m(e)i (de\014nitions)e(mak)m(e)i(sense)e(also)0 4474 y(in)f(the)h(case)g (that)g(the)g(parameter)f Fx(t)g Fy(is)h(restricted)g(to)g(a)g(b)s (ounded)d(in)m(terv)-5 b(al)37 b Fx(I)43 b Fw(\032)35 b Fp(R)p 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5014 y Fy(\))p Fo(-semimartingale)34 b(and)991 5263 y Fx(C)1063 5226 y Fu(X)1056 5286 y(st)1130 5263 y Fy(\()p Fx(')p Fy(\))26 b(=)1381 5140 y Ft(Z)1472 5166 y Fu(t)1432 5346 y(s)1517 5263 y Fx(')p Fy(\()p Fx(u;)15 b(X)1778 5277 y Fu(u)1824 5263 y Fy(\))p Fx(dX)1981 5277 y Fu(u)2027 5263 y Fx(;)202 b(\021)23 b Fw(\000)d Fo(a.s.)804 b Fy(\(3.4\))1726 5549 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 431 a Fo(for)35 b(any)g Fy(\()p Fx(s;)15 b(t)p Fy(\))29 b Fw(2)g Fy(\001)34 b Fo(\(or)h Fy(\001)970 445 y Fu(I)1010 431 y Fo(\))g(and)g(any)g(adapte)-5 b(d)37 b Fx(')29 b Fw(2)g(D)37 b Fo(wher)-5 b(e)35 b(on)g(the)g(right)g (hand)h(side)f(we)g(have)0 544 y(the)d(standar)-5 b(d)35 b(It^)-46 b(o)32 b(inte)-5 b(gr)g(al)34 b(on)e(the)g(semimartingale)i Fx(X)7 b Fo(.)42 b(When)32 b Fx(X)39 b Fo(is)32 b(Br)-5 b(ownian)33 b(motion,)h(we)e(c)-5 b(al)5 b(l)0 657 y Fx(\021)36 b Fy(\(forw)m(ard\))30 b(Bro)m(wnian)h(curren)m(t)p Fo(.)0 862 y Fy(In)40 b(order)g(not)h(to)h(m)m(ultiply)f(terminology)-8 b(,)45 b(unless)40 b(confusion)h(ma)m(y)g(arise)g(w)m(e)h(will)f(use)f (the)h(term)0 975 y Fo(curr)-5 b(ent)43 b Fy(also)34 b(for)g(the)f(elemen)m(ts)i(of)f Fw(D)1370 942 y Fv(0)1393 975 y Fy(,)g(of)g(\004,)g(and)f(for)g(the)h(la)m(ws)g(on)f(\004)h (without)f(making)h(explicit)0 1088 y(distinction.)40 b(F)-8 b(or)26 b(unsp)s(eci\014ed)d(b)s(ounded)g(in)m(terv)-5 b(als)26 b Fx(I)32 b Fy(w)m(e)26 b(use)f(the)g(notation)i Fx(C)2764 1055 y Fu(X)2757 1115 y(I)2856 1088 y Fy(for)d(the)i(asso)s (ciated)0 1201 y(curren)m(t)k(with)g(in)m(tegrator)i Fx(X)7 b Fy(.)141 1314 y(Next)41 b(w)m(e)g(w)m(an)m(t)h(to)f(construct) f(a)h(\(non-trivial\))h(measure)f(on)f(\004)g(for)g(the)h(Bro)m(wnian)f (curren)m(t.)0 1427 y(Th)m(us)22 b(w)m(e)i(start)g(from)f(Wiener)g (measure)1425 1435 y Fw(W)1545 1427 y Fy(\(similarly)h(w)m(e)g(could)f (ha)m(v)m(e)i(w)m(ork)m(ed)f(with)f(the)g(Bro)m(wnian)0 1540 y(bridge)278 1548 y Fw(W)375 1496 y Fu(x;y)368 1569 y(I)476 1540 y Fy(\))31 b(and)e(pro)m(v)m(e)j(that)f(there)f(exists)h (a)g(map)f Fx(F)38 b Fy(:)26 b Fw(X)38 b(3)25 b Fx(!)j Fw(7!)d Fx(F)13 b Fy(\()p Fx(!)s Fy(\))26 b Fw(2)f(C)36 b Fy(suc)m(h)30 b(that)940 1800 y Fx(F)13 b Fy(\()p Fw(\001)p Fy(\))1106 1814 y Fu(st)1169 1800 y Fy(\()p Fx(')p Fy(\))26 b(=)1420 1676 y Ft(Z)1511 1702 y Fu(t)1471 1882 y(s)1556 1800 y Fx(')p Fy(\()p Fx(u;)15 b(X)1817 1814 y Fu(u)1864 1800 y Fy(\))p Fx(dX)2021 1814 y Fu(u)2067 1800 y Fx(;)2289 1808 y Fw(W)2401 1800 y(\000)p Fy(a.s.)753 b(\(3.5\))0 2038 y(for)38 b(an)m(y)g(adapted)f Fx(')h Fw(2)g(D)s Fy(,)h(with)e(the)h(standard)f(It^)-45 b(o)39 b(in)m(tegral)g(at)f(the) g(righ)m(t)h(hand)d(side.)63 b(Then)37 b(a)0 2151 y(measure)349 2159 y Fw(W)446 2118 y Fu(])502 2151 y Fy(on)25 b(\004)g(can)g(b)s(e)f (de\014ned)g(as)h(the)g(la)m(w)h(of)f(the)g(couple)g(\()p Fx(X)r(;)15 b(F)e Fy(\))27 b(under)c(the)i(measure)3275 2159 y Fw(W)3397 2151 y Fy(and)0 2264 y(it)30 b(will)g(b)s(e)e(a)i (forw)m(ard)f(curren)m(t.)40 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b(e)31 b(do)g(this)f(next.)141 3167 y(F)-8 b(or)30 b(an)m(y)f Fx(\013)c(>)g Fy(1,)30 b(let)f Fw(D)951 3181 y Fu(\013)1030 3167 y Fy(b)s(e)f(the)h(completion)g(of)g(the)g(space)g(of)g(smo)s(oth) g(test)g(v)m(ector)h(\014elds)f(with)0 3280 y(resp)s(ect)h(to)h(the)g (norm)1089 3393 y Fw(k)p Fx(')p Fw(k)1238 3407 y Fv(D)1293 3415 y Fr(\013)1367 3393 y Fy(=)25 b(sup)1467 3472 y Fu(k)r Fv(2)p Fs(Z)1600 3393 y Fy(\(1)c(+)f Fw(j)p Fx(k)s Fw(j)p Fy(\))1927 3355 y Fu(\013)1977 3393 y Fw(k)p Fx(')p Fw(k)2126 3408 y Fu(\032;)p Fq(2)p Fu(;k)r(;k)r Fq(+1)2429 3393 y Fx(:)0 3607 y Fy(In)30 b(the)g(follo)m(wing)i(w)m(e)f(will)g (\014x)e Fx(\013)d(>)f Fy(1)31 b(but)f(otherwise)g(arbitrarily)h(small) g(and)f(write)g Fw(D)3070 3621 y Fu(\013)3145 3607 y Fy(=)25 b Fw(D)s Fy(.)0 3789 y Fk(Lemma)35 b(3.3)46 b Fo(F)-7 b(or)38 b(any)f Fx(\015)h Fw(\025)32 b Fy(1)p Fx(=)p Fy(3)p Fo(,)39 b Fx(\032)33 b(>)f Fy(1)24 b Fw(\000)e Fx(\015)5 b Fo(,)38 b Fx(t)33 b(>)f(s)k Fo(and)i Fx(x;)15 b(y)36 b Fw(2)c Fp(R)2516 3756 y Fu(d)2556 3789 y Fo(,)38 b(ther)-5 b(e)37 b(exists)g(a)g(family)h(of)0 3902 y(r)-5 b(andom)35 b(variables)f Fx(F)39 b Fw(2)26 b(C)38 b Fo(such)33 b(that)h(\(3.4\))g(holds)h(with)e(r)-5 b(esp)g(e)g(ct)35 b(to)2437 3910 y Fw(W)2534 3858 y Fu(x;y)2526 3935 y Fq([)p Fu(s;t)p Fq(])2647 3902 y Fo(,)e(and)h(which)g(satisfy)g(the)0 4015 y(p)-5 b(athwise)35 b(b)-5 b(ound)751 4212 y Fw(j)p Fx(F)834 4226 y Fu(uv)916 4212 y Fy(\()p Fx(')p Fy(\))p Fw(j)27 b(\024)e Fx(C)1258 4231 y Fu(\015)t(;\032;)p Fv(j)p Fu(t)p Fv(\000)p Fu(s)p Fv(j)1530 4212 y Fw(j)p Fx(u)c Fw(\000)e Fx(v)s Fw(j)1790 4175 y Fu(\015)1835 4212 y Fy(\(1)i(+)f Fx(N)2100 4231 y Fq([)p Fu(s;t)p Fq(])2221 4212 y Fy(\()p Fx(X)7 b Fy(\)\))2408 4175 y Fq(3)2449 4212 y Fw(k)p Fx(')p Fw(k)2598 4226 y Fu(\032;)p Fq(2)p Fu(;s;t)3356 4212 y Fy(\(3.6\))0 4410 y Fo(for)45 b(any)g Fx(')i Fw(2)g(D)s Fo(,)f Fy(\()p Fx(u;)15 b(v)s Fy(\))49 b Fw(2)d Fy(\001)1145 4428 y Fq([)p Fu(s;t)p Fq(])1266 4410 y Fo(,)h(wher)-5 b(e)45 b Fx(N)1682 4428 y Fq([)p Fu(s;t)p Fq(])1804 4410 y Fy(\()p Fx(X)7 b Fy(\))47 b(=)g(\()p Fw(k)p Fx(X)7 b Fw(k)2328 4428 y Fu(\015)t(;)p Fq([)p Fu(s;t)p Fq(])2539 4410 y Fy(+)29 b Fw(k)p Fp(X)2750 4377 y Fq(2)2789 4410 y Fw(k)2834 4428 y Fq(2)p Fu(\015)t(;)p Fq([)p Fu(s;t)p Fq(])3051 4410 y Fy(\))45 b Fo(and)g(wher)-5 b(e)0 4523 y Fx(C)65 4541 y Fu(\015)t(;\032;)p Fv(j)p Fu(t)p Fv(\000)p Fu(s)p Fv(j)369 4523 y Fo(dep)g(ends)35 b(only)e(on)g Fx(\015)5 b(;)15 b(\032;)g Fw(j)p Fx(t)21 b Fw(\000)f Fx(s)p Fw(j)p Fo(.)141 4704 y Fj(Pr)n(oof.)40 b Fy(De\014ne)666 4902 y Fx(F)724 4916 y Fu(uv)806 4902 y Fy(\()p Fx(')p Fy(\))27 b(=)69 b(lim)1058 4963 y Fu(\016)r(\034)1122 4971 y Fr(\013)1167 4963 y Fv(!)p Fq(0)1288 4815 y Ft(X)1331 5006 y Fu(\013)1435 4801 y Ft(\020)1489 4902 y Fx(')p Fy(\()p Fx(\034)1623 4916 y Fu(\013)1673 4902 y Fx(;)15 b(X)1788 4916 y Fu(\034)1819 4924 y Fr(\013)1868 4902 y Fy(\))21 b(+)f Fw(r)p Fx(')p Fy(\()p Fx(\034)2225 4916 y Fu(\013)2275 4902 y Fx(;)15 b(X)2390 4916 y Fu(\034)2421 4924 y 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Fo(we)g(have)g Fw(N)1046 2420 y Fu(\013;)p Fq(3)1150 2406 y Fy(\()p Fx(X)7 b Fy(\))30 b Fx(<)f Fw(1)34 b Fo(a.s.,)h(the)h(b) -5 b(oundary)36 b(curr)-5 b(ents)36 b(ar)-5 b(e)35 b(wel)5 b(l)35 b(de\014ne)-5 b(d)36 b(under)0 2527 y Fw(W)97 2486 y Fu(])129 2519 y Fo(.)0 2731 y Fj(Pr)n(oof.)43 b Fy(The)31 b(existence)i(of)f(the)g(lifted)g(measure)f(for)g(Bro)m (wnian)h(bridge)2618 2739 y Fw(W)2715 2687 y Fu(x;y)2708 2760 y(I)2848 2731 y Fy(is)f(essen)m(tially)j(con-)0 2844 y(tained)h(in)g(Lemma)g(3.3.)55 b(Its)35 b(uniqueness)f(is)h(a)g (direct)g(consequence)h(of)f(the)g(prop)s(ert)m(y)g(\(3.4\).)56 b(The)0 2957 y(pro)s(of)35 b(in)h(the)g(case)h(of)825 2965 y Fw(W)958 2957 y Fy(is)f(similar)g(and)g(w)m(e)g(are)h(left)f(to) h(pro)m(v)m(e)g(that)g(under)2837 2965 y Fw(W)2934 2924 y Fu(])3002 2957 y Fy(the)f(b)s(oundary)0 3070 y(curren)m(ts)24 b(are)h(w)m(ell)g(de\014ned.)37 b(The)24 b(in)m(tegrals)i(in)e(ev)m (ery)h(in)m(terv)-5 b(al)25 b([)p Fx(i;)15 b(i)8 b Fy(+)g(1])26 b(are)f(w)m(ell)g(de\014ned,)g(moreo)m(v)m(er)664 3202 y Ft(\014)664 3256 y(\014)664 3311 y(\014)664 3366 y(\014)694 3210 y(Z)785 3236 y Fu(i)p Fq(+1)745 3416 y Fu(i)919 3334 y Fx(')p Fy(\()p Fx(u;)15 b(X)1180 3348 y Fu(u)1226 3334 y Fy(\))p Fx(dX)1383 3348 y Fu(u)1429 3202 y Ft(\014)1429 3256 y(\014)1429 3311 y(\014)1429 3366 y(\014)1485 3334 y Fw(\024)25 b Fy(\(1)c(+)f Fw(j)p Fx(i)p Fw(j)p Fy(\))1889 3296 y Fv(\000)p Fu(\013)1994 3334 y Fw(k)p Fx(')p Fw(k)2143 3348 y Fv(D)2205 3334 y Fy([1)h(+)f Fx(N)2460 3352 y Fq([)p Fu(i;i)p Fq(+1])2661 3334 y Fy(\()p Fx(X)7 b Fy(\)])2838 3296 y Fq(3)0 3611 y Fy(so)31 b(that)g(the)g(series)710 3537 y Ft(R)771 3564 y Fq(+)p Fv(1)753 3643 y Fu(t)916 3611 y Fx(')p Fy(\()p Fx(u;)15 b(X)1177 3625 y Fu(u)1223 3611 y Fy(\))p Fx(dX)1380 3625 y Fu(u)1452 3611 y Fy(=)1549 3542 y Ft(P)1645 3637 y Fu(i>t)p Fv(\000)p Fq(1)1859 3537 y Ft(R)1919 3564 y Fu(i)p Fq(+1)1902 3643 y Fu(i)p Fv(_)p Fu(t)2053 3611 y Fx(')p Fy(\()p Fx(u;)g(X)2314 3625 y Fu(u)2361 3611 y Fy(\))p Fx(dX)2518 3625 y Fu(u)2594 3611 y Fy(is)31 b(absolutely)h(con)m(v)m(ergen)m(t)0 3723 y(if)27 b Fw(N)155 3737 y Fu(\013;)p Fq(3)259 3723 y Fy(\()p Fx(X)7 b Fy(\))26 b Fx(<)f Fw(1)p Fy(.)40 b(Under)961 3731 y Fw(W)1085 3723 y Fy(w)m(e)28 b(ha)m(v)m(e)g Fp(E)1483 3743 y Fv(W)1562 3723 y Fy([\(1)14 b(+)g Fx(N)1839 3742 y Fq([)p Fu(i;i)p Fq(+1])2040 3723 y Fy(\()p Fx(X)7 b Fy(\)\))2227 3690 y Fq(3)2268 3723 y Fw(j)p Fx(X)2368 3737 y Fq(0)2434 3723 y Fy(=)25 b Fx(x)p Fy(])g Fw(\024)g Fx(C)33 b Fy(uniformly)27 b(in)f Fx(i)g Fw(2)f Fp(Z)0 3846 y Fy(and)30 b Fx(x)25 b Fw(2)g Fp(R)406 3813 y Fu(d)476 3846 y Fy(th)m(us)53 4050 y Fp(E)114 4070 y Fv(W)193 4050 y Fy([)p Fw(N)293 4064 y Fu(\013;)p Fq(3)397 4050 y Fy(\()p Fx(X)7 b Fy(\))p Fw(j)p Fx(X)649 4064 y Fq(0)716 4050 y Fy(=)25 b Fx(x)p Fy(])g(=)1010 3964 y Ft(X)1018 4161 y Fu(i)p Fv(2)p Fs(Z)1141 4050 y Fy(\(1)c(+)f Fw(j)p Fx(i)p Fw(j)p Fy(\))1449 4013 y Fv(\000)p Fu(\013)1555 4050 y Fp(E)1616 4070 y Fv(W)1695 4050 y Fy([)p Fx(N)1793 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4946 y(st)1583 4923 y Fy(\()p Fx(')p Fy(\))d(+)1825 4800 y Ft(Z)1916 4826 y Fu(t)1875 5006 y(s)1960 4923 y Fy(div)17 b Fx(')p Fy(\()p Fx(t;)e(X)2343 4937 y Fu(t)2374 4923 y Fy(\))p Fx(dt)0 5174 y Fy(and)31 b(obtain)h(a)g(di\013eren)m(t)g(lifted)g(measure)g(\(whic)m(h)f(is)h (no)f(longer)h(a)g(forw)m(ard)f(curren)m(t\).)45 b(It)32 b(w)m(ould)f(b)s(e)0 5286 y(in)m(teresting)g(to)g(explore)g(whether)e (di\013eren)m(t)h(lifts)h(ma)m(y)f(ha)m(v)m(e)i(a)e(di\013eren)m(t)g (ph)m(ysical)h(meaning)f(in)g(the)1726 5549 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 0 431 a Fy(mo)s(dels.)51 b(F)-8 b(or)34 b(instance,)i(non-relativistic)g(particles)f(with)e(spin)g(can) h(b)s(e)f(\(partially\))j(describ)s(ed)c(b)m(y)0 544 y(a)f(curren)m(t)f Fx(C)464 511 y Fu(\033)541 544 y Fy(de\014ned)f(as) 914 782 y Fx(C)986 744 y Fu(X)979 804 y(st)1053 782 y Fy(\()p Fx(')p Fy(\)[)p Fx(\033)s Fy(])e(=)e Fx(C)1482 744 y Fu(X)1475 804 y(st)1549 782 y Fy(\()p Fx(')p Fy(\))d(+)1791 658 y Ft(Z)1881 684 y 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y Fy(\()p Fx(x;)g(y)s Fy(\))1831 3293 y Fp(E)1900 3312 y Fg(c)1892 3329 y Fv(W)1967 3283 y Fr(x;y)1967 3346 y Fn([)p Fm(\000)p Fr(T)5 b(;T)j Fn(])2178 3192 y Ft(h)2221 3293 y Fy(1)2266 3307 y Fu(A)2323 3293 y Fy(\()p Fx(X)f Fy(\))p Fx(e)2517 3255 y Fv(\000)p Fu(V)2613 3269 y Fn([)p Fm(\000)p Fr(T)e(;T)j Fn(])2807 3255 y Fq(\()p Fu(X)d Fq(\))2929 3192 y Ft(i)2987 3293 y Fx(;)299 b Fy(\(3.12\))0 3546 y(with)27 b(\005)272 3560 y Fu(t)330 3546 y Fy(the)h(Bro)m(wnian)g (transition)g(k)m(ernel)g(\(3.2\).)42 b(This)27 b(form)m(ula)h(can)g(b) s(e)f(c)m(hec)m(k)m(ed)j(b)m(y)d(noting)i(that)430 3749 y(^)423 3773 y Fp(E)484 3787 y Fu(\027)519 3798 y Fr(T)586 3672 y Ft(h)629 3773 y Fx(f)10 b Fy(\()p Fx(x)p Fy(\))p Fx(g)s Fy(\()p Fx(y)s Fy(\))p Fp(E)1031 3795 y Fu(\027)1070 3763 y Fr(x;y)1066 3818 y(T)1166 3773 y Fy([)p Fx(Q)p Fy(])1288 3672 y Ft(i)539 3973 y Fy(=)25 b Fx(Z)704 3935 y Fv(\000)p Fq(1)697 4002 y Fu(T)805 3950 y Fy(^)798 3973 y Fp(E)859 3987 y Fv(W)929 4001 y Fn([)p Fm(\000)p Fr(T)5 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Fv(\000)p Fu(tH)1929 3961 y Fy(,)h Fx(t)25 b Fw(\025)g Fy(0,)30 b(to)f(hold.)40 b(This)28 b(is)h(generated)g(b)m(y)g(the)0 4074 y(Sc)m(hr\177)-45 b(odinger)34 b(op)s(erator)h Fx(H)k Fy(=)32 b Fx(H)1167 4088 y Fq(0)1229 4074 y Fy(+)22 b Fx(V)55 b Fy(de\014ned)33 b(on)h Fx(L)1939 4041 y Fq(2)1979 4074 y Fy(\()p Fp(R)2080 4041 y Fu(d)2120 4074 y Fx(;)15 b(dx)p Fy(\))36 b(as)e(a)h(form)f(sum)g(\()p Fx(V)55 b Fy(regarded)34 b(as)0 4187 y(a)i(m)m(ultiplication)h(op)s(erator\).) 56 b(F)-8 b(or)36 b(Kato-class)h(p)s(oten)m(tials)g Fx(H)42 b Fy(is)35 b(essen)m(tially)i(self-adjoin)m(t)g(on)e(the)0 4299 y(form)30 b(core)h Fx(C)479 4266 y Fv(1)472 4324 y Fq(0)553 4299 y Fy(\()p Fp(R)654 4266 y Fu(d)695 4299 y Fy(\).)41 b(In)29 b(addition,)i(w)m(e)g(will)g(require)f(of)g Fx(V)51 b Fy(to)31 b(b)s(e)f(suc)m(h)g(that)66 4481 y(\(1\))46 b Fx(H)39 b Fy(has)32 b(a)g(unique)f(strictly)i(p)s(ositiv)m(e)g (eigenfunction)g(\(ground)e(state\))j(\011)d(at)i Fx(E)g Fy(=)28 b(inf)22 b(Sp)s(ec)14 b Fx(H)7 b 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b(ac)g(ontr)g(active)46 b Fy(when)35 b Fx(e)2068 398 y Fv(\000)p Fu(tH)2206 406 y Fr(!)2291 431 y Fy(is)h Fo(ultr)-5 b(ac)g(ontr)g(active)p Fy(,)40 b(i.e.,)e(it)f(maps)0 544 y Fx(L)62 511 y Fq(2)101 544 y Fy(\()p Fp(R)202 511 y Fu(d)243 544 y Fx(;)15 b(d!)s Fy(\))27 b(in)m(to)g Fx(L)694 511 y Fv(1)769 544 y Fy(\()p Fp(R)870 511 y Fu(d)910 544 y Fx(;)15 b(d!)s Fy(\))28 b(con)m(tin)m(uously)-8 b(.)40 b(Equiv)-5 b(alen)m(tly)d(,)29 b(this)e(means)f(that)h Fw(k)p Fx(e)2943 511 y Fv(\000)p Fu(tH)3081 519 y Fr(!)3131 544 y Fw(k)3176 558 y Fq(2)p Fu(;)p Fv(1)3331 544 y Fx(<)e Fw(1)p Fy(,)0 657 y Fw(8)p Fx(t)30 b(>)h Fy(0,)k(and)e(it)i(is)e(a)h(monotonically)i(decreasing)f (function)e(in)h Fx(t)p Fy(.)50 b(Moreo)m(v)m(er,)37 b(the)d(in)m(tegral)i(k)m(ernel)0 770 y(\(3.10\))d(of)d Fx(e)408 737 y Fv(\000)p Fu(tH)546 745 y Fr(!)626 770 y Fy(satis\014es)h(0)26 b Fw(\024)f Fx(\031)1180 784 y Fu(t)1209 770 y Fy(\()p Fx(x;)15 b(y)s Fy(\))26 b Fw(\024)f(k)p Fx(e)1628 737 y Fv(\000)p Fq(\()p Fu(t=)p 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y Fw(\024)25 b Fx(V)1708 1297 y Fq(+)1767 1334 y Fy(\()p Fx(x)p Fy(\))h Fw(\024)f Fx(C)2076 1348 y Fq(2)2116 1334 y Fw(j)p Fx(x)p Fw(j)2218 1297 y Fu(b)2273 1334 y Fy(+)20 b Fx(C)2429 1348 y Fq(4)2468 1334 y Fx(:)863 b Fy(\(4.3\))141 1601 y(A)30 b Fo(p)-5 b(air)33 b(inter)-5 b(action)33 b(p)-5 b(otential)41 b Fy(is)30 b(a)g(Leb)s(esgue)f(measurable)h(function)f Fx(W)38 b Fy(:)25 b Fp(R)2891 1568 y Fu(d)2950 1601 y Fw(\002)18 b Fp(R)25 b Fw(!)g Fp(R)p Fy(,)30 b(ev)m(en)0 1714 y(in)g(b)s(oth)g(of)g(its)h(v)-5 b(ariables,)31 b(whic)m(h)f(w)m(e)h(require)f(to)66 1901 y(\(1\))46 b(ha)m(v)m(e)32 b(p)s(ositiv)m(e)f(F)-8 b(ourier)31 b(transform;)66 2089 y(\(2\))46 b(satisfy)31 b(the)g(regularit)m(y)g(condition)g(that)g (there)g(exists)g Fx(M)2250 2104 y Fu(I)5 b(;\014)2378 2089 y Fw(2)25 b Fp(R)30 b Fy(suc)m(h)g(that)1398 2293 y(sup)1317 2379 y Fu(x)p Fv(2)p Fs(R)1452 2361 y Fr(d)1488 2379 y Fu(;t)p Fv(2)p Fu(I)1631 2293 y Fw(k)p Fx(W)13 b Fy(\()p Fx(x;)i(t)p Fy(\))p Fw(k)2015 2307 y Fv(D)2070 2319 y Fr(\014)2142 2293 y Fw(\024)25 b Fx(M)2326 2308 y Fu(I)5 b(;\014)2428 2293 y Fx(;)903 b Fy(\(4.4\))227 2569 y(for)30 b Fx(\014)h(>)25 b Fy(max)p Fw(f)p Fx(\013;)15 b Fy(3)p Fw(g)33 b Fy(and)c(ev)m(ery)j(b)s(ounded)c Fx(I)k Fw(\032)25 b Fp(R)p Fy(.)0 2757 y(The)j(requiremen)m(t)h Fx(\014)i(>)25 b(\013)k Fy(is)g(needed)f(for)h(ha)m(ving)g(a)h(w)m(ell) f(de\014ned)f(in)m(teraction)j(energy)e(\(actually)i(in)0 2870 y(our)i(applications)h(for)g(an)m(y)f Fx(\014)j(>)30 b Fy(1)k(there)f(is)h(a)g(suitable)g Fx(\013)f Fy(so)h(that)g(this)f (holds\),)i(while)e Fx(\014)j(>)30 b Fy(3)j(is)h(a)0 2983 y(deca)m(y)27 b(condition)f(su\016cien)m(t)g(for)f(ensuring)g(the) g(con)m(v)m(ergence)j(of)e(the)f(cluster)h(expansion)g(in)f(Chapter)0 3096 y(5)31 b(b)s(elo)m(w.)141 3209 y(An)f(example)h(satisfying)g (these)g(conditions)g(is)f Fx(W)1909 3176 y Fu(\032)1979 3209 y Fy(seen)h(in)f(Section)h(1.)141 3421 y(Finally)c(w)m(e)e(write)h (do)m(wn)f(the)g(energies)i(app)s(earing)e(in)g(the)g(de\014nition)g (of)h(the)g(densities)f(of)h(Gibbs)0 3534 y(measures)f(w)m(e)h(are)f (going)i(to)f(study)-8 b(.)38 b(With)26 b(giv)m(en)g(pair)f(p)s(oten)m (tial)i Fx(W)13 b Fy(,)26 b(for)f(all)h Fx(a;)15 b(b)26 b Fw(2)f(D)3020 3501 y Fv(0)3068 3534 y Fy(consider)g(the)0 3647 y(function)30 b Fx( )415 3662 y Fu(k)r(;$)538 3647 y Fy(\()p Fx(x;)15 b(t)p Fy(\))26 b(=)f Fx(e)897 3614 y Fu(ik)r Fv(\001)p Fu(x)p Fq(+)p Fu(i$)r(t)1214 3647 y Fw(2)g(D)33 b Fy(and)c(de\014ne)h(the)h(\(p)s(ossibly)e(un)m(b)s (ounded\))f(quadratic)j(form)917 3898 y Fw(h)p Fx(a;)15 b(b)p Fw(i)1114 3912 y Fu(W)1221 3898 y Fy(:=)1342 3774 y Ft(Z)1452 3875 y(c)1448 3898 y Fx(W)e Fy(\()p Fx(k)s(;)i($)s Fy(\))p Fx(a)p Fy(\()p Fx( )1927 3913 y Fu(k)r(;$)2051 3898 y Fy(\))p Fx(b)p Fy(\()p 2160 3824 183 4 v Fx( )2219 3913 y Fu(k)r(;$)2343 3898 y Fy(\))p Fx(dk)s(d$)s(:)0 4154 y Fy(By)21 b(using)g(the)g(quadratic)g(form,)i(for)d(all)i(b)s (ounded)d Fx(I)32 b Fw(\032)25 b Fp(R)c Fy(and)f(ev)m(ery)i Fp(X)j Fy(=)g(\()p Fx(X)r(;)15 b(C)2811 4121 y Fu(X)2879 4154 y Fy(\))p Fx(;)30 b Fp(Y)25 b Fy(=)g(\()p Fx(Y)5 b(;)15 b(C)3361 4121 y Fu(Y)3422 4154 y Fy(\))26 b Fw(2)0 4267 y Fy(\004)k(de\014ne)g(the)g Fo(internal)k(ener)-5 b(gy)33 b(functional)1123 4509 y Fx(H)1199 4523 y Fu(I)1238 4509 y Fy(\()p Fp(X)p Fy(\))26 b(=)f Fx(V)1549 4523 y Fu(I)1589 4509 y Fy(\()p Fx(X)7 b Fy(\))21 b(+)1863 4448 y Fx(\025)p 1863 4488 54 4 v 1867 4571 a Fy(2)1926 4509 y Fw(h)p Fx(C)2033 4472 y Fu(X)2026 4532 y(I)2100 4509 y Fx(;)15 b(C)2212 4472 y Fu(X)2205 4532 y(I)2279 4509 y Fw(i)2314 4523 y Fu(W)2395 4509 y Fx(;)936 b Fy(\(4.5\))0 4734 y(and)30 b Fo(inter)-5 b(action)34 b(functional)761 4976 y Fx(H)837 4990 y Fu(I)877 4976 y Fy(\()p Fp(X)p Fw(j)p Fp(Y)p Fy(\))26 b(=)e Fx(V)1278 4990 y Fu(I)1318 4976 y Fy(\()p Fx(X)7 b Fy(\))22 b(+)1593 4915 y Fx(\025)p 1593 4955 V 1597 5039 a Fy(2)1656 4976 y Fw(h)p Fx(C)1763 4939 y Fu(X)1756 4999 y(I)1830 4976 y Fx(;)15 b(C)1942 4939 y Fu(X)1935 4999 y(I)2009 4976 y Fw(i)2044 4990 y Fu(W)2145 4976 y Fy(+)20 b Fx(\025)p Fw(h)p Fx(C)2396 4939 y Fu(X)2389 4999 y(I)2463 4976 y Fx(;)15 b(C)2575 4939 y Fu(Y)2568 4999 y(I)2604 4980 y Fr(c)2641 4976 y Fw(i)2676 4990 y Fu(W)2757 4976 y Fx(;)574 b Fy(\(4.6\))0 5211 y(with)30 b(parameter)h Fx(\025)25 b Fw(2)g Fp(R)p Fy(,)30 b(where)g(w)m(e)h(wrote)g Fx(V)1630 5225 y Fu(I)1670 5211 y Fy(\()p Fx(X)7 b Fy(\))26 b(=)1944 5138 y Ft(R)1987 5243 y Fu(I)2042 5211 y Fx(V)20 b Fy(\()p Fx(X)2225 5225 y Fu(t)2256 5211 y Fy(\))p Fx(dt)p Fy(.)1726 5549 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 0 431 a Fl(4.2)112 b(Gibbs)39 b(sp)s(eci\014cations)0 603 y Fy(On)28 b(\004)g(with)g(its)h(asso)s (ciated)h Fx(\033)s Fy(-Borel)g(\014eld)e Fw(A)g Fy(w)m(e)i(tak)m(e)g (no)m(w)2136 611 y Fw(W)2234 570 y Fu(])2294 603 y Fy(as)e(reference)i (measure)e(and)g(de\014ne)0 715 y(a)j(Gibbs)f(sp)s(eci\014cation.)0 890 y Fk(De\014nition)35 b(4.1)h(\(Gibbs)f(sp)s(eci\014cation\))46 b Fo(T)-7 b(ake)35 b(the)h(r)-5 b(e)g(gular)37 b(version)2650 898 y Fw(W)2747 846 y Fu(])2740 919 y(I)2780 890 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))e Fo(of)h(the)g(me)-5 b(a-)0 1003 y(sur)g(e)193 1011 y Fw(W)290 970 y Fu(])354 1003 y Fo(c)g(onditional)34 b(on)e Fp(Y)f Fo(in)h Fw(A)1230 1017 y Fu(I)1266 998 y Fr(c)1301 1003 y Fo(,)g(and)h Fx(H)1613 1017 y Fu(I)1652 1003 y Fy(\()p Fp(X)p Fy(\))f Fo(given)f(by)h(\(4.5\).)42 b(We)32 b(c)-5 b(al)5 b(l)32 b(the)g(family)h(of)f(pr)-5 b(ob)g(a-)0 1124 y(bility)33 b(kernels)g Fw(f)p Fx(\026)636 1080 y Fu(])636 1153 y(I)676 1124 y Fw(g)721 1138 y Fu(I)794 1124 y Fo(on)g Fy(\004)f Fo(indexe)-5 b(d)33 b(by)g(the)g(b)-5 b(ounde)g(d)34 b(intervals)g Fx(I)e Fw(\032)25 b Fp(R)p Fo(,)1249 1381 y Fx(\026)1304 1337 y Fu(])1304 1410 y(I)1344 1381 y Fy(\()p Fx(d)p Fp(X)p Fy(\))h(=)1658 1320 y Fx(e)1700 1287 y Fv(\000)p Fu(H)1813 1298 y Fr(I)1849 1287 y Fq(\()p Fs(X)p Fq(\))p 1658 1360 298 4 v 1756 1443 a Fx(Z)1818 1457 y Fu(I)1981 1389 y Fw(W)2078 1337 y Fu(])2071 1410 y(I)2111 1381 y Fy(\()p Fx(d)p Fp(X)p Fy(\))1062 b(\(4.7\))0 1610 y Fo(a)39 b Fy(Gibbs)30 b(sp)s(eci\014cation)g(on)g(Bro)m(wnian)g (curren)m(ts)f(with)h(free)f(b)s(oundary)f(condition)p Fo(.)42 b(T)-7 b(ake)33 b Fx(H)3277 1624 y Fu(I)3316 1610 y Fy(\()p Fp(X)p Fw(j)p Fp(Y)p Fy(\))0 1731 y Fo(given)f(by)g (\(4.6\).)43 b(We)32 b(c)-5 b(al)5 b(l)34 b(the)f(family)g Fw(f)p Fx(\032)1472 1687 y Fu(])1472 1760 y(I)1513 1731 y Fw(g)1558 1745 y Fu(I)1630 1731 y Fo(on)h Fy(\004)p Fo(,)1128 1988 y Fx(\032)1175 1943 y Fu(])1175 2016 y(I)1215 1988 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))26 b(=)1621 1926 y Fx(e)1663 1893 y Fv(\000)p Fu(H)1776 1904 y Fr(I)1811 1893 y Fq(\()p Fs(X)p Fv(j)p Fs(Y)p Fq(\))p 1621 1967 366 4 v 1684 2050 a Fx(Z)1746 2064 y Fu(I)1786 2050 y Fy(\()p Fp(Y)p Fy(\))2011 1996 y Fw(W)2108 1943 y Fu(])2101 2016 y(I)2141 1988 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))941 b(\(4.8\))0 2220 y Fo(a)40 b Fy(Gibbs)30 b(sp)s (eci\014cation)h(on)f(Bro)m(wnian)h(curren)m(ts)f(with)g(b)s(oundary)e (condition)j Fp(Y)p Fo(.)0 2417 y Fk(De\014nition)k(4.2)h(\(Gibbs)f (measure\))45 b Fo(A)h(pr)-5 b(ob)g(ability)48 b(me)-5 b(asur)g(e)48 b Fx(\026)e Fo(on)h Fy(\(\004)p Fx(;)15 b Fw(A)p Fx(;)2888 2425 y Fw(W)2986 2384 y Fu(])3017 2417 y Fy(\))47 b Fo(is)f(a)54 b Fy(Gibbs)0 2538 y(measure)38 b Fo(for)g(the)h(p)-5 b(otentials)40 b Fx(V)58 b Fo(and)39 b Fx(W)50 b Fo(if)37 b(it)h(is)g(c)-5 b(onsistent)39 b(with)g(the)f(sp)-5 b(e)g(ci\014c)g(ation)40 b Fw(f)p Fx(\032)3197 2494 y Fu(])3197 2567 y(I)3238 2538 y Fw(g)3283 2552 y Fu(I)3323 2538 y Fo(,)f(i.e.,)0 2651 y(ther)-5 b(e)27 b(exists)f(a)h(version)f(of)h(its)f(c)-5 b(onditional)28 b(pr)-5 b(ob)g(abilities)28 b(with)f(r)-5 b(esp)g(e)g(ct)28 b(to)f(the)f(family)h Fw(fA)3137 2665 y Fu(I)3173 2646 y Fr(c)3209 2651 y Fw(g)3254 2665 y Fu(I)3320 2651 y Fo(which)0 2772 y(agr)-5 b(e)g(es)34 b(with)f Fw(f)p Fx(\032)560 2728 y Fu(])560 2801 y(I)601 2772 y Fw(g)646 2786 y Fu(I)718 2772 y Fo(for)h(al)5 b(l)33 b(b)-5 b(ounde)g(d)34 b Fx(I)e Fw(\032)25 b Fp(R)p Fo(.)0 2969 y Fy(In)37 b(the)g(follo)m (wing)i(c)m(hapter)f(our)e(main)i(concern)f(will)h(b)s(e)e(to)i(pro)m (v)m(e)g(the)g(existence)g(of)g(suc)m(h)f(Gibbs)0 3082 y(measures.)141 3195 y(On)30 b(Gibbs)f(sp)s(eci\014cations)i(here)g(is) f(a)h(\014rst)e(result.)0 3369 y Fk(Lemma)35 b(4.3)46 b Fo(The)j(family)g Fw(f)p Fx(\032)1136 3325 y Fu(])1136 3398 y(I)1176 3369 y Fw(g)1221 3383 y Fu(I)1310 3369 y Fo(is)g(c)-5 b(onsistent,)53 b(i.e.,)f(for)d(every)f(p)-5 b(air)50 b(of)f(b)-5 b(ounde)g(d)50 b(intervals)0 3497 y Fx(I)k Fw(\032)46 b Fx(J)56 b Fw(\032)47 b Fp(R)d Fo(we)g(have)912 3423 y Ft(R)33 b(R)1064 3497 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))p Fx(\032)1318 3452 y Fu(])1318 3525 y(I)1358 3497 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))p Fx(\032)1679 3452 y Fu(])1679 3525 y(J)1729 3497 y Fy(\()p Fx(d)p Fp(Y)p Fw(j)p Fp(Z)p Fy(\))47 b(=)2163 3423 y Ft(R)2238 3497 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))p Fx(\032)2492 3452 y Fu(])2492 3525 y(J)2542 3497 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Z)p Fy(\))p Fo(,)47 b(for)f(any)f(b)-5 b(ounde)g(d)0 3610 y(me)g(asur)g(able)34 b Fx(F)39 b Fy(:)25 b(\004)g Fw(!)g Fp(R)p Fo(.)0 3784 y Fj(Pr)n(oof.)40 b Fy(The)30 b(family)g Fw(f)863 3792 y(W)961 3740 y Fu(])954 3813 y(I)994 3784 y Fw(g)1039 3798 y Fu(I)1109 3784 y Fy(is)h(consisten)m(t)g(b)m(y)g(its)g(de\014nition.)40 b(Hence)410 3897 y Ft(Z)55 b(Z)607 4020 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))p Fx(\032)861 3976 y Fu(])861 4049 y(I)901 4020 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))p Fx(\032)1222 3976 y Fu(])1222 4049 y(J)1272 4020 y Fy(\()p Fx(d)p Fp(Y)p Fw(j)p Fp(Z)p Fy(\))632 4285 y(=)728 4161 y Ft(Z)55 b(Z)940 4285 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))1157 4223 y Fx(e)1199 4189 y Fv(\000)p Fu(V)1295 4200 y Fr(I)1331 4189 y Fq(\()p Fu(X)5 b Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1750 4166 y Fr(X)1745 4212 y(I)1811 4189 y Fu(;C)1886 4166 y Fr(X)1881 4212 y(I)1944 4189 y Fv(i)1971 4200 y Fr(W)2041 4189 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)2219 4166 y Fr(X)2214 4212 y(I)2277 4189 y Fu(;C)2352 4166 y Fr(Y)2347 4213 y(I)2378 4199 y(c)2414 4189 y Fv(i)2441 4200 y Fr(W)p 1158 4264 1358 4 v 1718 4347 a Fx(Z)1780 4361 y Fu(I)1820 4347 y Fy(\()p Fp(Y)p Fy(\))809 4556 y Fw(\002)910 4495 y Fx(e)952 4461 y Fv(\000)p Fu(V)1048 4472 y Fr(J)1091 4461 y Fq(\()p Fu(Y)16 b Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1504 4437 y Fr(Y)1499 4483 y(J)1560 4461 y Fu(;C)1635 4437 y Fr(Y)1630 4483 y(J)1688 4461 y Fv(i)1715 4472 y Fr(W)1785 4461 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1963 4437 y Fr(Y)1958 4483 y(J)2016 4461 y Fu(;C)2091 4437 y Fr(Z)2086 4485 y(J)2125 4471 y(c)2161 4461 y Fv(i)2188 4472 y Fr(W)p 910 4535 1353 4 v 1465 4618 a Fx(Z)1527 4632 y Fu(J)1576 4618 y Fy(\()p Fp(Z)p Fy(\))2287 4564 y Fw(W)2385 4512 y Fu(])2377 4585 y(I)2417 4556 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))2706 4564 y Fw(W)2804 4512 y Fu(])2797 4585 y(J)2846 4556 y Fy(\()p Fx(d)p Fp(Y)p Fw(j)p Fp(Z)p Fy(\))632 4827 y(=)728 4704 y Ft(Z)55 b(Z)940 4827 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))1157 4766 y Fx(e)1199 4732 y Fv(\000)p Fu(V)1295 4743 y Fr(I)1331 4732 y Fq(\()p Fu(X)5 b Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1750 4708 y Fr(X)1745 4754 y(I)1811 4732 y Fu(;C)1886 4708 y Fr(X)1881 4754 y(I)1944 4732 y Fv(i)1971 4743 y Fr(W)2041 4732 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)2219 4708 y Fr(X)2214 4754 y(I)2277 4732 y Fu(;C)2352 4708 y Fr(Y)2347 4756 y(I)2378 4742 y(c)2414 4732 y Fv(i)2441 4743 y Fr(W)p 1158 4807 1358 4 v 1718 4890 a Fx(Z)1780 4904 y Fu(I)1820 4890 y Fy(\()p Fp(Y)p Fy(\))809 5041 y Fw(\002)20 b Fx(e)942 5003 y Fv(\000)p Fu(V)1038 5014 y Fr(I)1074 5003 y Fq(\()p Fu(Y)c Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1487 4980 y Fr(Y)1482 5026 y(I)1542 5003 y Fu(;C)1617 4980 y Fr(Y)1612 5026 y(I)1670 5003 y Fv(i)1697 5014 y Fr(W)1767 5003 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1945 4980 y Fr(Y)1940 5026 y(I)1999 5003 y Fu(;C)2074 4980 y Fr(Z)2069 5027 y(J)2108 5013 y(c)2144 5003 y Fv(i)2171 5014 y Fr(W)2241 5003 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)2419 4980 y Fr(Y)2414 5026 y(I)2472 5003 y Fu(;C)2547 4980 y Fr(Y)2542 5026 y(K)2601 5003 y Fv(i)2628 5014 y Fr(W)809 5261 y Fw(\002)910 5199 y Fx(e)952 5165 y Fv(\000)p Fu(V)1048 5176 y Fr(K)1107 5165 y Fq(\()p Fu(Y)g Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1520 5142 y Fr(Y)1515 5188 y(K)1576 5165 y Fu(;C)1651 5142 y Fr(Y)1646 5188 y(K)1705 5165 y Fv(i)1732 5176 y Fr(W)1802 5165 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1980 5142 y Fr(Y)1975 5188 y(K)2034 5165 y Fu(;C)2109 5142 y Fr(Z)2104 5190 y(J)2143 5176 y(c)2179 5165 y Fv(i)2206 5176 y Fr(W)p 910 5240 1371 4 v 1474 5323 a Fx(Z)1536 5337 y Fu(J)1585 5323 y Fy(\()p Fp(Z)p Fy(\))2305 5269 y Fw(W)2403 5217 y Fu(])2395 5289 y(I)2435 5261 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))2724 5269 y Fw(W)2822 5217 y Fu(])2815 5289 y(J)2864 5261 y Fy(\()p Fx(d)p Fp(Y)p Fw(j)p Fp(Z)p Fy(\))1726 5549 y(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 0 431 a Fy(where)30 b(w)m(e)h(split)f(o\013)h Fx(C)800 398 y Fu(Y)793 458 y(J)885 431 y Fy(=)25 b Fx(C)1053 398 y Fu(Y)1046 458 y(I)1134 431 y Fy(+)20 b Fx(C)1297 398 y Fu(Y)1290 458 y(K)1388 431 y Fy(with)30 b Fx(K)i Fy(=)25 b Fx(J)9 b Fw(n)p Fx(I)e Fy(.)42 b(This)29 b(giv)m(es)j(for)e (the)h(righ)m(t)g(hand)e(side)352 587 y Ft(Z)55 b(Z)565 711 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))782 649 y Fx(e)824 615 y Fv(\000)p Fu(V)920 626 y Fr(I)956 615 y Fq(\()p Fu(X)5 b Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1375 592 y Fr(X)1370 638 y(I)1435 615 y Fu(;C)1510 592 y Fr(X)1505 638 y(I)1568 615 y Fv(i)1595 626 y Fr(W)1665 615 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1843 592 y Fr(X)1838 638 y(I)1901 615 y Fu(;C)1976 592 y Fr(Y)1971 639 y(I)2002 625 y(c)2038 615 y Fv(i)2065 626 y Fr(W)p 782 690 1358 4 v 1322 773 a Fx(Z)1384 787 y Fu(I)1424 773 y Fy(\()p Fp(Y)1525 787 y Fq(2)1564 773 y Fy(\))539 961 y Fw(\002)630 832 y Ft(\022)697 837 y(Z)803 961 y Fx(e)845 923 y Fv(\000)p Fu(V)941 934 y Fr(I)977 923 y Fq(\()p Fu(Y)1045 932 y Fn(1)1080 923 y Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1409 885 y Fr(Y)1445 900 y Fn(1)1404 946 y Fr(I)1486 923 y Fu(;C)1561 885 y Fr(Y)1597 900 y Fn(1)1556 946 y Fr(I)1636 923 y Fv(i)1663 934 y Fr(W)1733 923 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1911 885 y Fr(Y)1947 900 y Fn(1)1906 946 y Fr(I)1986 923 y Fu(;C)2061 900 y Fr(Z)2056 947 y(J)2095 933 y(c)2131 923 y Fv(i)2158 934 y Fr(W)2228 923 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)2406 885 y Fr(Y)2442 900 y Fn(1)2401 946 y Fr(I)2481 923 y Fu(;C)2556 885 y Fr(Y)2592 900 y Fn(2)2551 946 y Fr(K)2631 923 y Fv(i)2658 934 y Fr(W)2747 969 y Fw(W)2844 916 y Fu(])2837 989 y(I)2877 961 y Fy(\()p Fx(d)p Fp(Y)3025 975 y Fq(1)3064 961 y Fw(j)p Fp(Y)3155 975 y Fq(2)3195 961 y Fy(\))3230 832 y Ft(\023)539 1248 y Fw(\002)640 1186 y Fx(e)682 1152 y Fv(\000)p Fu(V)778 1163 y Fr(K)837 1152 y Fq(\()p Fu(Y)905 1161 y Fn(2)940 1152 y Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1269 1114 y Fr(Y)1305 1129 y Fn(2)1264 1175 y Fr(K)1346 1152 y Fu(;C)1421 1114 y Fr(Y)1457 1129 y Fn(2)1416 1175 y Fr(K)1496 1152 y Fv(i)1523 1163 y Fr(W)1593 1152 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1771 1114 y Fr(Y)1807 1129 y Fn(2)1766 1175 y Fr(K)1846 1152 y Fu(;C)1921 1129 y Fr(Z)1916 1176 y(J)1955 1162 y(c)1991 1152 y Fv(i)2018 1163 y Fr(W)p 640 1227 1453 4 v 1245 1310 a Fx(Z)1307 1324 y Fu(J)1356 1310 y Fy(\()p Fp(Z)p Fy(\))2117 1256 y Fw(W)2215 1203 y Fu(])2207 1276 y(I)2247 1248 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)2486 1262 y Fq(2)2526 1248 y Fy(\))2576 1256 y Fw(W)2673 1203 y Fu(])2666 1276 y(K)2734 1248 y Fy(\()p Fx(d)p Fp(Y)2882 1262 y Fq(2)2922 1248 y Fw(j)p Fp(Z)p Fy(\))p Fx(;)0 1523 y Fy(where)43 b(w)m(e)h(used)f(the)g(fact)i(that) 1216 1449 y Ft(R)1291 1531 y Fw(W)1389 1478 y Fu(])1381 1551 y(I)1421 1523 y Fy(\()p Fx(d)p Fp(Y)1569 1537 y Fq(1)1609 1523 y Fw(j)p Fp(Y)1700 1537 y Fq(2)1739 1523 y Fy(\))1789 1531 y Fw(W)1887 1478 y Fu(])1879 1551 y(K)1948 1523 y Fy(\()p Fx(d)p Fp(Y)2096 1537 y Fq(2)2136 1523 y Fw(j)p Fp(Z)p Fy(\))i(=)2422 1531 y Fw(W)2519 1478 y Fu(])2511 1551 y(J)2560 1523 y Fy(\()p Fx(d)p Fp(Y)2708 1537 y Fq(1)2748 1523 y Fw(j)p Fp(Z)p Fy(\).)80 b(Note)45 b(that)f(the)0 1636 y(expression)30 b(b)s(et)m(w)m(een)h(the)g(brac)m (k)m(ets)h(equals)e Fx(Z)1640 1650 y Fu(I)1680 1636 y Fy(\()p Fp(Y)1781 1650 y Fq(2)1821 1636 y Fy(\),)h(th)m(us)f(w)m(e)g (further)g(obtain)304 1758 y Ft(Z)55 b(Z)517 1882 y Fx(F)13 b Fy(\()p Fp(X)p Fy(\))p Fx(e)766 1844 y Fv(\000)p Fu(V)862 1855 y Fr(I)898 1844 y Fq(\()p Fu(X)5 b Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1317 1821 y Fr(X)1312 1867 y(I)1377 1844 y Fu(;C)1452 1821 y Fr(X)1447 1867 y(I)1510 1844 y Fv(i)1537 1855 y Fr(W)1607 1844 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1785 1821 y Fr(X)1780 1867 y(I)1843 1844 y Fu(;C)1918 1821 y Fr(Y)1913 1868 y(I)1944 1854 y(c)1981 1844 y Fv(i)2008 1855 y Fr(W)491 2161 y Fw(\002)592 2099 y Fx(e)634 2065 y Fv(\000)p Fu(V)730 2076 y Fr(K)789 2065 y Fq(\()p Fu(Y)857 2074 y Fn(2)892 2065 y Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1221 2027 y Fr(Y)1257 2042 y Fn(2)1216 2088 y Fr(K)1298 2065 y Fu(;C)1373 2027 y Fr(Y)1409 2042 y Fn(2)1368 2088 y Fr(K)1448 2065 y Fv(i)1475 2076 y Fr(W)1545 2065 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1723 2027 y Fr(Y)1759 2042 y Fn(2)1718 2088 y Fr(K)1798 2065 y Fu(;C)1873 2042 y Fr(Z)1868 2089 y(J)1907 2075 y(c)1943 2065 y Fv(i)1970 2076 y Fr(W)p 592 2140 V 1197 2223 a Fx(Z)1259 2237 y Fu(J)1308 2223 y Fy(\()p Fp(Z)p Fy(\))2069 2169 y Fw(W)2167 2116 y Fu(])2159 2189 y(I)2199 2161 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)2438 2175 y Fq(2)2478 2161 y Fy(\))2528 2169 y Fw(W)2625 2116 y Fu(])2618 2189 y(K)2686 2161 y Fy(\()p Fx(d)p Fp(Y)2834 2175 y Fq(2)2874 2161 y Fw(j)p Fp(Z)p Fy(\))314 2432 y(=)410 2308 y Ft(Z)517 2432 y Fx(F)13 b Fy(\()p Fp(Y)p Fy(\))734 2370 y Fx(e)776 2336 y Fv(\000)p Fu(V)872 2347 y Fr(J)915 2336 y Fq(\()p Fu(Y)j Fq(\))p Fv(\000)p Fq(\()p Fu(\025=)p Fq(2\))p Fv(h)p Fu(C)1328 2313 y Fr(Y)1323 2359 y(J)1384 2336 y Fu(;C)1459 2313 y Fr(Y)1454 2359 y(J)1512 2336 y Fv(i)1539 2347 y Fr(W)1609 2336 y Fv(\000)p Fu(\025)p Fv(h)p Fu(C)1787 2313 y Fr(Y)1782 2359 y(J)1840 2336 y Fu(;C)1915 2313 y Fr(Z)1910 2361 y(J)1949 2347 y(c)1985 2336 y Fv(i)2012 2347 y Fr(W)p 734 2411 1353 4 v 1289 2494 a Fx(Z)1351 2508 y Fu(J)1400 2494 y Fy(\()p Fp(Z)p Fy(\))2111 2440 y Fw(W)2209 2388 y Fu(])2201 2461 y(J)2250 2432 y Fy(\()p Fx(d)p Fp(Y)p Fw(j)p Fp(Z)p Fy(\))26 b(=)2641 2308 y Ft(Z)2747 2432 y Fx(F)13 b Fy(\()p Fp(Y)p Fy(\))p Fx(\032)3001 2446 y Fu(J)3050 2432 y Fy(\()p Fx(d)p Fp(Y)p Fw(j)p Fp(Z)p Fy(\))p Fx(:)3473 2678 y Fi(\003)141 2791 y Fy(The)40 b(forw)m(ard)g(curren)m(t)1008 2799 y Fw(W)1105 2758 y Fu(])1177 2791 y Fy(has)g(the)g(k)m(ey)h(prop)s (ert)m(y)e(that)i Fx(C)2349 2758 y Fu(X)2342 2813 y(st)2416 2791 y Fy(\()p Fx(')p Fy(\))i(=)2700 2718 y Ft(R)2761 2744 y Fu(t)2743 2823 y(s)2806 2791 y Fx(')p Fy(\()p Fx(u;)15 b(X)3067 2805 y Fu(u)3113 2791 y Fy(\))p Fx(dX)3270 2805 y Fu(u)3316 2791 y Fy(,)3384 2799 y Fw(W)3481 2758 y Fu(])3513 2791 y Fy(-)0 2904 y(a.s.)49 b(for)33 b(all)h(\()p Fx(s;)15 b(t)p Fy(\))30 b Fw(2)f Fy(\001)k(and)g(all)g(adapted)h Fx(')c Fw(2)f(D)s Fy(.)48 b(This)33 b(will)g(enable)g(us)g(to)h(sho)m (w)e(that)i(the)f(\014nite)0 3025 y(v)m(olume)25 b(measures)e Fx(\026)745 3039 y Fu(I)809 3025 y Fy(coincide)h(with)g(the)g (marginals)g(of)g(the)g(measures)g Fx(\026)2595 2981 y Fu(])2595 3054 y(I)2658 3025 y Fy(on)g(the)g(\014rst)f(comp)s(onen)m (t)0 3152 y(of)38 b(the)f(pro)s(duct)g(\004.)61 b(The)37 b(sp)s(eci\014cation)i Fw(f)p Fx(\032)1582 3108 y Fu(])1582 3181 y(I)1622 3152 y Fw(g)f Fy(can)g(then)f(b)s(e)g(considered)g(as)h (a)g(suitable)g(rigorous)0 3265 y(replacemen)m(t)h(for)f(the)g(DLR)g (description)g(of)g(the)g(in\014nite-v)m(olume)g(limit.)64 b(A)38 b(Gibbs)g(measure)f(on)0 3378 y Fw(X)48 b Fy(will)35 b(then)f(b)s(e)g(a)h(measure)g(for)f(whic)m(h)h(there)g(exists)g(a)g (unique)f(lift)h(to)g(the)g(space)h(\004)e(of)h(curren)m(ts)0 3491 y(satisfying)c(the)g(relation)62 b(\(3.4\))32 b(ensuring)e(the)h (iden)m(ti\014cation)h(of)e(the)h(curren)m(t)f(with)h(the)f(sto)s(c)m (hastic)0 3613 y(in)m(tegral)i(and)e(whic)m(h)g(satisfy)h(the)f(DLR)h (conditions)g(with)f(resp)s(ect)g(to)h(the)g(sp)s(eci\014cation)g Fw(f)p Fx(\032)3263 3568 y Fu(])3263 3641 y(I)3303 3613 y Fw(g)p Fy(.)141 3725 y(T)-8 b(o)35 b(sho)m(w)f(that)h(the)f(sp)s (eci\014cation)h(is)g(w)m(ell)g(de\014ned)e(w)m(e)i(rewrite)f(the)h(v) -5 b(arious)34 b(terms)g(using)g(the)0 3838 y(fact)k(that,)h(under)c (the)i(measure)1204 3846 y Fw(W)1301 3794 y Fu(])1294 3867 y(I)1334 3838 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))g(w)m(e)g(ha)m(v)m(e)h(path)m(wise)g(equalit)m(y)g(b)s(et)m(w)m (een)f(the)g(curren)m(t)0 3951 y Fx(C)72 3918 y Fu(X)169 3951 y Fy(and)29 b(the)i(sto)s(c)m(hastic)h(in)m(tegral)f(with)f(resp)s (ect)g(to)h Fx(X)38 b Fy(for)30 b(adapted)g(in)m(tegrands)g(b)s (elonging)h(to)f Fw(D)s Fy(.)0 4064 y(Then)511 4288 y Fw(h)p Fx(C)618 4250 y Fu(X)611 4310 y(I)685 4288 y Fx(;)15 b(C)797 4250 y Fu(X)790 4310 y(I)864 4288 y Fw(i)899 4302 y Fu(W)1005 4288 y Fy(=)1101 4164 y Ft(Z)1211 4265 y(c)1207 4288 y Fx(W)e Fy(\()p Fx(k)s(;)i($)s Fy(\))p Fx(C)1616 4250 y Fu(X)1609 4310 y(I)1684 4288 y Fy(\()p Fx( )1778 4303 y Fu(k)r(;$)1901 4288 y Fy(\))p Fx(C)2008 4250 y Fu(X)2001 4310 y(I)2076 4288 y Fy(\()p 2111 4214 183 4 v Fx( )2170 4303 y Fu(k)r(;$)2293 4288 y Fy(\))p Fx(dk)s(d$)1005 4523 y Fy(=)1101 4399 y Ft(Z)1211 4500 y(c)1207 4523 y Fx(W)e Fy(\()p Fx(k)s(;)i($)s Fy(\))p Fw(j)p Fx(C)1641 4485 y Fu(X)1634 4545 y(I)1709 4523 y Fy(\()p Fx( )1803 4538 y Fu(k)r(;$)1927 4523 y Fy(\))p Fw(j)1987 4485 y Fq(2)2027 4523 y Fx(dk)s(d$)1005 4783 y Fy(=)1101 4659 y Ft(Z)1211 4760 y(c)1207 4783 y Fx(W)e Fy(\()p Fx(k)s(;)i($)s Fy(\))1559 4651 y Ft(\014)1559 4705 y(\014)1559 4760 y(\014)1559 4814 y(\014)1591 4659 y(Z)1641 4865 y Fu(I)1696 4783 y Fx( )1755 4798 y Fu(k)r(;$)1878 4783 y Fy(\()p Fx(t;)g(X)2061 4797 y Fu(t)2092 4783 y Fy(\))p Fx(dX)2249 4797 y Fu(t)2279 4651 y Ft(\014)2279 4705 y(\014)2279 4760 y(\014)2279 4814 y(\014)2310 4677 y Fq(2)2364 4783 y Fx(dk)s(d$)29 b Fy(=)c(2)p Fx(W)2839 4797 y Fu(I)2879 4783 y Fy(\()p Fx(X)7 b Fy(\))1726 5549 y(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 0 431 a Fy(and)588 651 y Fw(h)p Fx(C)695 613 y Fu(X)688 673 y(I)763 651 y Fx(;)15 b(C)875 613 y Fu(Y)868 673 y(I)904 654 y Fr(c)940 651 y Fw(i)975 665 y Fu(W)1081 651 y Fy(=)1177 527 y Ft(Z)1287 628 y(c)1283 651 y Fx(W)e Fy(\()p Fx(k)s(;)i($)s Fy(\))p Fx(C)1692 613 y Fu(X)1685 673 y(I)1759 651 y Fy(\()p Fx( )1853 666 y Fu(k)r(;$)1977 651 y Fy(\))p Fx(C)2084 613 y Fu(Y)2077 673 y(I)2113 654 y Fr(c)2149 651 y Fy(\()p 2184 577 183 4 v Fx( )2243 666 y Fu(k)r(;$)2367 651 y Fy(\))p Fx(dk)s(d$)1081 894 y Fy(=)25 b Fx(C)1249 856 y Fu(X)1242 916 y(I)1331 766 y Ft(\022)1398 770 y(Z)1508 871 y(c)1504 894 y Fx(W)12 b Fy(\()p Fx(k)s(;)j($)s Fy(\))p Fx( )1899 909 y Fu(k)r(;$)2024 894 y Fx(C)2096 856 y Fu(Y)2089 916 y(I)2125 897 y Fr(c)2161 894 y Fy(\()p 2196 820 V Fx( )2255 909 y Fu(k)r(;$)2378 894 y Fy(\))p Fx(dk)s(d$)2635 766 y Ft(\023)1081 1156 y Fy(=)25 b Fx(C)1249 1119 y Fu(X)1242 1179 y(I)1331 1055 y Ft(\020)1385 1156 y Fx(w)1452 1119 y Fu(C)1507 1095 y Fr(Y)1502 1143 y(I)1533 1129 y(c)1575 1055 y Ft(\021)1654 1156 y Fy(=)1750 1033 y Ft(Z)1841 1059 y Fu(T)1801 1239 y Fv(\000)p Fu(T)1926 1156 y Fx(C)1998 1119 y Fu(Y)1991 1179 y(I)2027 1160 y Fr(c)2063 1156 y Fy(\()p Fx(W)13 b Fy(\()p Fw(\001)20 b(\000)g Fx(X)2443 1170 y Fu(s)2480 1156 y Fx(;)15 b Fw(\001)21 b(\000)f Fx(s)p Fy(\)\))p Fx(dX)2892 1170 y Fu(s)2930 1156 y Fx(;)0 1405 y Fy(with)438 1537 y Fx(w)505 1499 y Fu(C)564 1537 y Fy(\()p Fx(x;)15 b(t)p Fy(\))26 b(=)881 1413 y Ft(Z)991 1514 y(c)988 1537 y Fx(W)12 b Fy(\()p Fx(k)s(;)j($)s Fy(\))p Fx( )1383 1552 y Fu(k)r(;$)1507 1537 y Fy(\()p Fx(t;)g(x)p Fy(\))p Fx(C)7 b Fy(\()p 1809 1463 V Fx( )1868 1552 y Fu(k)r(;$)1992 1537 y Fy(\))p Fx(dk)s(d$)29 b Fy(=)c Fx(C)7 b Fy(\()p Fx(W)13 b Fy(\()p Fx(x)20 b Fw(\000)g(\001)p Fx(;)15 b(t)21 b Fw(\000)e(\001)p Fy(\)\))p Fx(:)0 1744 y Fy(By)31 b(using)f(these)g(equalities)i(it)f(is)g(seen)f(that)h(the)g(sp)s (eci\014cation)g(\(4.8\))h(tak)m(es)g(the)e(form)652 2044 y Fx(\032)699 2000 y Fu(])699 2073 y(I)739 2044 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))c(=)1145 1983 y Fx(e)1187 1949 y Fv(\000)p Fu(V)1283 1960 y Fr(I)1318 1949 y Fq(\()p Fu(X)5 b Fq(\))p Fv(\000)p Fu(\025W)1598 1960 y Fr(I)1634 1949 y Fq(\()p Fu(X)g Fq(\))p Fv(\000)p Fu(\025)1860 1895 y Fg(R)1893 1973 y Fr(I)1940 1949 y Fu(w)1993 1917 y Fr(C)2040 1896 y(Y)2037 1942 y(I)2068 1928 y(c)2108 1949 y Fq(\()p Fu(u;X)2254 1957 y Fr(u)2295 1949 y Fq(\))p Fu(dX)2416 1957 y Fr(u)p 1145 2024 1318 4 v 1684 2107 a Fx(Z)1746 2121 y Fu(I)1786 2107 y Fy(\()p Fp(Y)p Fy(\))2487 2052 y Fw(W)2584 2000 y Fu(])2577 2073 y(I)2617 2044 y Fy(\()p Fx(d)p Fp(X)p Fw(j)p Fp(Y)p Fy(\))465 b(\(4.9\))0 2289 y(and)41 b(it)i(is)f(w)m(ell)h(de\014ned)e(as)h(so)s (on)g(as)g(the)g(exp)s(onen)m(tial)h(w)m(eigh)m(t)g(is)f(in)m(tegrable) i(and)d(the)i(in)m(tegral)0 2401 y(is)37 b(di\013eren)m(t)h(from)f (zero.)61 b(The)37 b(conditions)h(on)f Fx(V)57 b Fy(and)37 b Fx(W)49 b Fy(mak)m(e)39 b(sure)d(this)h(is)g(true.)61 b(Indeed,)38 b(for)0 2514 y(Kato-class)d(p)s(oten)m(tials)g(exp)s(onen) m(tial)f(in)m(tegrabilit)m(y)i(is)d(a)g(consequence)h(of)g (Khasminskii's)e(Lemma)0 2627 y([47)q(].)59 b(Moreo)m(v)m(er,)40 b(since)d(the)f(F)-8 b(ourier)37 b(transform)f(of)g Fx(W)49 b Fy(is)36 b(p)s(ositiv)m(e)i(b)m(y)e(assumption)g(and)f Fx(\025)h(>)f Fy(0,)0 2740 y(the)26 b(in)m(ternal)h(energy)f(term)g(is) g(negativ)m(e)i(and)e(th)m(us)f(exp)s(onen)m(tially)j(in)m(tegrable)f (without)f(an)m(y)h(further)0 2853 y(restriction.)42 b(F)-8 b(or)31 b(the)f(in)m(teraction)j(with)d(the)g(b)s(oundary)f (curren)m(t)h(w)m(e)h(ha)m(v)m(e)937 3054 y Fw(j)p Fx(w)1029 3016 y Fu(C)1084 2993 y Fr(Y)1079 3040 y(I)1110 3026 y(c)1151 3054 y Fy(\()p Fx(x;)15 b(t)p Fy(\))p Fw(j)27 b Fy(=)e Fw(j)p Fx(C)1591 3016 y Fu(Y)1584 3076 y(I)1620 3057 y Fr(c)1656 3054 y Fy(\()p Fx(W)13 b Fy(\()p Fx(x;)i(t)p Fy(\)\))p Fw(j)26 b(\024)f Fx(M)2255 3068 y Fu(I)2296 3054 y Fw(k)p Fx(C)2413 3016 y Fu(Y)2406 3076 y(I)2442 3057 y Fr(c)2478 3054 y Fw(k)2523 3069 y Fv(D)2580 3050 y Fm(0)0 3254 y Fy(with)41 b Fx(M)306 3268 y Fu(I)391 3254 y Fy(=)j(sup)642 3276 y Fu(x)p Fv(2)p Fs(R)777 3257 y Fr(d)813 3276 y Fu(;t)p Fv(2)p Fu(I)961 3254 y Fw(k)p Fx(W)13 b Fy(\()p Fx(x;)i(t)p Fy(\))p Fw(k)1345 3268 y Fv(D)1406 3254 y Fy(,)45 b(whic)m(h)d(b)m(y)f(condition)i(\(2\))f(on) g Fx(W)54 b Fy(is)42 b(\014nite.)75 b(Hence)42 b(the)0 3367 y(sto)s(c)m(hastic)g(in)m(tegral)f(in)f(the)g(exp)s(onen)m(t)g (has)g(a)g(b)s(ounded)e(and)h(adapted)h(in)m(tegrand)g(and)g(th)m(us)f (b)m(y)0 3480 y(standard)30 b(tec)m(hniques)g(it)h(follo)m(ws)h(that)f (it)g(is)f(exp)s(onen)m(tially)i(in)m(tegrable)g(for)e(an)m(y)h(v)-5 b(alue)30 b(of)h Fx(\025)p Fy(.)141 3641 y(By)e(making)f(use)g(of)h (the)f(It^)-45 b(o)29 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b(b=)p Fy(2)27 b Fx(>)d Fy(0)33 b Fo(we)g(have)1101 3073 y Fx(Z)1163 3087 y Fu(T)1244 3073 y Fy(=)25 b(1)20 b(+)1497 2987 y Ft(X)1496 3183 y Fu(n)p Fv(\025)p Fq(1)1817 2987 y Ft(X)1654 3176 y Fm(f)p Fn(\000)1724 3191 y(1)1758 3176 y Fr(;:::;)p Fn(\000)1891 3184 y Fr(n)1933 3176 y Fm(g2)2007 3181 y(K)2053 3193 y Fr(N)1691 3242 y Fn(\000)1729 3225 y Fm(\003)1729 3268 y Fr(i)1765 3242 y Fm(\\)p Fn(\000)1845 3225 y Fm(\003)1845 3268 y Fr(j)1880 3242 y Fn(=)p Fm(;)p Fr(;i)p Fm(6)p Fn(=)p Fr(j)2172 2960 y Fu(n)2136 2987 y Ft(Y)2138 3185 y Fu(l)q Fq(=1)2267 3073 y Fx(K)2344 3087 y Fq(\000)2388 3099 y Fr(l)2417 3073 y Fx(:)914 b Fy(\(5.9\))0 3449 y Fo(If)32 b(the)h(activities)g Fx(K)709 3463 y Fq(\000)790 3449 y Fo(satisfy)h(the)f(b)-5 b(ound)1445 3576 y Ft(X)1419 3761 y Fn(\000)p Fm(2)1499 3766 y(K)1544 3778 y Fr(N)1348 3828 y Fn(\000)1386 3814 y Fm(\003)1421 3828 y(3)p Fn(0)p Fr(;)p Fm(j)1535 3815 y Fn(\026)1531 3828 y(\000)p Fm(j)p Fn(=)p Fr(n)1698 3662 y Fw(j)p Fx(K)1800 3676 y Fq(\000)1848 3662 y Fw(j)51 b(\024)f Fx(c)26 b(\021)2158 3624 y Fu(n)3311 3662 y Fy(\(5.10\))0 4013 y Fo(for)33 b Fx(\021)c(>)c Fy(0)33 b Fo(smal)5 b(l)33 b(enough,)g(then)g(the)g (series)g(ab)-5 b(ove)33 b(and)h(at)f(the)g(right)g(hand)h(side)f(of) 850 4281 y Fy(log)18 b Fx(Z)1045 4295 y Fu(T)1125 4281 y Fy(=)1222 4195 y Ft(X)1221 4391 y Fu(n)p Fv(\025)p Fq(1)1542 4195 y Ft(X)1379 4384 y Fm(f)p Fn(\000)1449 4399 y(1)1483 4384 y Fr(;:::;)p Fn(\000)1616 4392 y Fr(n)1658 4384 y Fm(g2)1732 4389 y(K)1778 4401 y Fr(N)1535 4450 y Fn(0)p Fm(2)p Fn(\000)1645 4433 y Fm(\003)1645 4475 y Fn(1)1861 4281 y Fx(\036)1915 4244 y Fu(T)1970 4281 y Fy(\(\000)2062 4295 y Fq(1)2101 4281 y Fx(;)d(:::;)g Fy(\000)2313 4295 y Fu(n)2362 4281 y Fy(\))2449 4168 y Fu(n)2412 4195 y Ft(Y)2414 4393 y Fu(l)q Fq(=1)2544 4281 y Fx(K)2621 4295 y Fq(\000)2665 4307 y Fr(l)3311 4281 y Fy(\(5.11\))0 4646 y Fo(ar)-5 b(e)39 b(absolutely)h(c)-5 b(onver)g(gent,)40 b(uniformly)f(in)f Fx(N)10 b Fo(,)40 b(and)f(the)g(latter)g(one)f(gives)g(the)h(lo)-5 b(garithm)40 b(of)f(the)0 4759 y(p)-5 b(artition)35 b(function)e(for)g(the)g (interval)g Fy([)p Fw(\000)p Fx(T)8 b(;)15 b(T)e Fy(])p Fo(.)0 4947 y Fy(The)33 b(expression)h(of)g(the)g(logarithm)h(and)f (the)g(absolute)g(con)m(v)m(ergence)j(of)d(the)g(sums)f(are)h(a)g (general)0 5060 y(result)c(of)h(cluster)g(expansion)f(tec)m(hniques,)h (for)f(details)h(of)g(pro)s(of)f(see)h([40)q(].)1726 5549 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 0 431 a Fl(5.2)112 b(Con)m(v)m(ergence)38 b(of)g(cluster)f(expansion)0 603 y Fk(Prop)s(osition)f(5.3)46 b Fo(Supp)-5 b(ose)39 b(that)f(ther)-5 b(e)37 b(exist)g(a)g(function)g Fx(D)f Fy(:)d Fp(N)23 b Fw(\002)g Fp(N)32 b Fw(!)h Fy(\(0)p Fx(;)15 b Fw(1)p Fy(\))38 b Fo(and)g(numb)-5 b(ers)0 712 y Fx(")42 715 y(;)15 b(C)54 b(>)46 b Fy(0)f Fo(with)h Fy(sup)755 737 y Fu(i)p Fv(2)p Fs(N)893 647 y Ft(P)989 742 y Fu(j)t Fv(2)p Fs(N)1136 715 y Fx(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))48 b Fw(\024)f Fx(C)j 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b(onstant)37 b Fx(c)30 b(>)f Fy(0)36 b Fo(and)g(a)f(function)g Fy(0)30 b Fx(<)f(\021)s Fy(\()p Fx(")p Fy(\))j Fx(<)d Fy(1)35 b Fo(with)h Fx(\021)d Fw(!)c Fy(0)36 b Fo(as)f Fx(")30 b Fw(!)g Fy(0)0 1562 y Fo(such)j(that)1432 1588 y Ft(X)1406 1774 y Fn(\000)p Fm(2)1486 1779 y(K)1532 1791 y Fr(N)1335 1841 y Fn(\000)1373 1827 y Fm(\003)1409 1841 y(3)p Fn(0)p Fr(;)p Fm(j)1523 1828 y Fn(\026)1519 1841 y(\000)o Fm(j)p Fn(=)p Fr(n)1685 1675 y Fw(j)p Fx(K)1787 1689 y Fq(\000)1836 1675 y Fw(j)51 b(\024)f Fx(c)25 b(\021)2145 1637 y Fu(n)2193 1675 y Fx(:)1093 b Fy(\(5.13\))0 2039 y(The)30 b(function)g Fx(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))32 b(will)f(b)s(e)f(sp)s(eci\014ed)f(in)h(Prop)s(osition)h(5.5)g (b)s(elo)m(w.)141 2201 y Fj(Pr)n(oof.)44 b Fy(W)-8 b(e)33 b(put)e(for)g(a)i(shorthand)d Fw(D)s Fy(\()p Fx(\015)5 b Fy(\))28 b(=)1795 2133 y Ft(Q)1881 2228 y Fq(\()p Fu(\034)1939 2238 y Fr(i)1966 2228 y Fu(;\034)2017 2238 y Fr(j)2050 2228 y Fq(\))p Fv(2)p Fu(\015)2184 2198 y Fx(")2241 2201 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\).)47 b(Consider)31 b(the)h(function)f(of)0 2314 y(complex)g(v)-5 b(ariable)31 b Fx(z)149 2527 y(H)7 b Fy(\()p Fx(z)t Fy(;)15 b Fx(")p Fy(\))27 b(=)680 2441 y Ft(X)553 2638 y Fq(\000)p Fv(2)644 2644 y(K)698 2649 y Fr(N)756 2638 y Fq(:\000)820 2619 y Fm(\003)855 2638 y Fv(3)p Fq(0)953 2527 y Fx(K)1030 2541 y Fq(\000)1078 2527 y Fx(z)1124 2490 y Fv(j)1148 2473 y Fq(\026)1144 2490 y(\000)p Fv(j)1237 2527 y Fy(=)1486 2441 y Ft(X)1378 2630 y Fn(\000)p Fm(2)1458 2635 y(K)1504 2647 y Fr(N)1562 2630 y Fn(:\000)1618 2616 y Fm(\003)1653 2630 y(3)p Fn(0)1343 2690 y(\000)p Fm(\033)17 b Fn(one)k(con)n(tour) 1785 2527 y Fx(K)1862 2541 y Fq(\000)1911 2527 y Fx(z)1957 2490 y Fv(j)1981 2473 y Fq(\026)1977 2490 y(\000)p Fv(j)2065 2527 y Fy(+)2456 2441 y Ft(X)2348 2630 y Fn(\000)p Fm(2)2428 2635 y(K)2474 2647 y Fr(N)2532 2630 y Fn(:\000)2588 2616 y Fm(\003)2623 2630 y(3)p Fn(0)2166 2690 y(\000)p Fm(\033)c Fn(more)k(than)f(one)i(con)n(tour)2902 2527 y Fx(K)2979 2541 y Fq(\000)3028 2527 y Fx(z)3074 2490 y Fv(j)3098 2473 y Fq(\026)3094 2490 y(\000)p Fv(j)3311 2527 y Fy(\(5.14\))0 2866 y(W)-8 b(e)38 b(sho)m(w)e(that)h(for)f(su\016cien)m(tly)h(small) 1447 2863 y Fx(")1525 2866 y(>)e Fy(0)i(this)f(is)h(an)f(analytic)i (function)e(of)h Fx(z)j Fy(in)d(a)f(circle)i(of)0 2979 y(radius)29 b Fx(R)q Fy(\()378 2976 y Fx(")420 2979 y Fy(\))h(whic)m(h)f(div)m(erges)h(as)1203 2976 y Fx(")1271 2979 y Fw(!)25 b Fy(0.)41 b(Moreo)m(v)m(er,)32 b(w)m(e)e(sho)m(w)f (that)h(within)f(this)g(circle)i Fx(H)7 b Fy(\()p Fx(z)t Fy(;)3374 2976 y Fx(")3417 2979 y Fy(\))30 b(is)0 3092 y(uniformly)f(b)s(ounded)g(in)892 3089 y Fx(")934 3092 y Fy(.)41 b(This)29 b(will)i(then)f(imply)g(\(5.13\))j(b)m(y)d(c)m(ho)s (osing)h Fx(\021)s Fy(\()2682 3089 y Fx(")2725 3092 y Fy(\))26 b(=)f(1)p Fx(=R)q Fy(\()3077 3089 y Fx(")3120 3092 y Fy(\).)141 3205 y(W)-8 b(e)45 b(start)f(b)m(y)f(estimating)j (the)d(second)h(sum;)49 b(the)44 b(\014rst)f(is)g(simpler)h(as)f(it)h (in)m(v)m(olv)m(es)i(clusters)0 3318 y(ha)m(ving)40 b(a)f(single)h(con) m(tour.)67 b(Our)38 b(strategy)i(is)f(\014rst)f(to)i(b)s(ound)d(it)i(b) m(y)g(a)h(sum)e(tak)m(en)i(o)m(v)m(er)g(graphs)0 3431 y(whose)35 b(v)m(ertices)h(are)g(the)f(con)m(tours)g(of)h(the)f(same)g (cluster.)55 b(The)34 b(sums)g(o)m(v)m(er)j(graphs)d(will)h(then)g(b)s (e)0 3544 y(b)s(ounded)28 b(b)m(y)i(sums)g(tak)m(en)h(o)m(v)m(er)h (trees.)0 3707 y Fe(Bounds)41 b(by)g(sums)g(over)g(gr)-5 b(aphs)130 b Fy(F)-8 b(or)34 b(eac)m(h)g Fx(r)e Fw(\025)d Fy(2)k(consider)g(in)2545 3715 y Fw(K)2614 3721 y Fu(N)2714 3707 y Fy(those)h(clusters)f(\000)g(that)0 3819 y(ha)m(v)m(e)27 b Fx(r)h Fy(con)m(tours.)40 b(F)-8 b(or)27 b(giv)m(en)g(\000)e Fw(2)1235 3827 y(K)1304 3833 y Fu(N)1397 3819 y Fy(let)i Fw(V)1580 3833 y Fu(r)1643 3819 y Fy(=)e Fw(f)p Fx(\015)1831 3833 y Fq(1)1870 3819 y Fx(;)15 b(:::;)g(\015)2072 3833 y Fu(r)2112 3819 y Fw(g)26 b Fy(b)s(e)g(the)g(collection)i(of)e(these)g (con)m(tours.)0 3932 y(W)-8 b(e)26 b(construct)g(connected)f(graphs)g Fx(G)g Fy(b)m(y)g(dra)m(wing)g(edges)g(b)s(et)m(w)m(een)h(the)f(elemen) m(ts)i(of)e Fw(V)3066 3946 y Fu(r)3128 3932 y Fy(considered)0 4045 y(as)38 b(v)m(ertex)h(set.)63 b(Connected)38 b(graphs)f(are)h (those)g(for)f(whic)m(h)h(either)g Fx(\015)2494 4012 y Fv(\003)2489 4071 y Fu(i)2558 4045 y Fw(\\)25 b Fx(\015)2696 4012 y Fv(\003)2691 4071 y Fu(j)2773 4045 y Fw(6)p Fy(=)37 b Fw(;)h Fy(or)g(there)g(exists)0 4158 y Fx(\045)47 4173 y Fu(l)98 4158 y Fw(2)25 b Fy(\000)30 b(suc)m(h)g(that)h Fx(\045)720 4125 y Fv(\003)720 4186 y Fu(l)780 4158 y Fw(\\)20 b Fx(\015)913 4125 y Fv(\003)908 4184 y Fu(i)978 4158 y Fw(6)p Fy(=)k Fw(;)i(6)p Fy(=)f Fx(\045)1287 4125 y Fv(\003)1287 4186 y Fu(l)1347 4158 y Fw(\\)20 b Fx(\015)1480 4125 y Fv(\003)1475 4184 y Fu(j)1519 4158 y Fy(.)41 b(Let)31 b Fw(G)1802 4172 y Fu(r)1870 4158 y Fy(denote)g(the)g(set)f(of)h(all)g (p)s(ossible)f(suc)m(h)g(graphs.)141 4271 y(Consider)g(the)g (collection)j(of)e Fo(r)-5 b(e)g(duc)g(e)g(d)32 b Fy(c)m(hains)f Fw(f)8 b Fy(^)-53 b Fx(\045)1870 4285 y Fq(1)1910 4271 y Fx(;)15 b(:::;)25 b Fy(^)-55 b Fx(\045)2112 4285 y Fu(s)2150 4271 y Fw(g)31 b Fy(with)f(the)g(prop)s(erties:)66 4459 y(\(1\))46 b(for)35 b(ev)m(ery)g(pair)f Fw(f)p Fx(\015)898 4473 y Fu(i)927 4459 y Fx(;)15 b(\015)1014 4473 y Fu(j)1051 4459 y Fw(g)33 b(2)e Fx(G)i Fw(2)f(G)1472 4473 y Fu(r)1510 4459 y Fy(,)k Fx(\015)1623 4426 y Fv(\003)1618 4484 y Fu(i)1685 4459 y Fw(\\)23 b Fx(\015)1821 4426 y Fv(\003)1816 4484 y Fu(j)1893 4459 y Fy(=)32 b Fw(;)p Fy(,)k(there)f(is)f(at)h (least)h(one)f(c)m(hain)43 b(^)-53 b Fx(\045)3228 4474 y Fu(l)3289 4459 y Fy(of)34 b(this)227 4587 y(collection)e(connecting)e Fx(\015)1130 4601 y Fu(i)1186 4587 y Fy(and)e Fx(\015)1408 4601 y Fu(j)1474 4587 y Fy(\(i.e.)p 1665 4513 74 4 v 49 w(^)-53 b Fx(\045)1712 4602 y Fu(l)1755 4587 y Fw(\\)20 b Fy(\026)-49 b Fx(\015)1879 4601 y Fu(i)1933 4587 y Fy(=)25 b Fw(;)h Fy(=)p 2196 4513 V 33 w(^)-53 b Fx(\045)2243 4602 y Fu(l)2285 4587 y Fw(\\)20 b Fy(\026)-48 b Fx(\015)2410 4601 y Fu(j)2475 4587 y Fy(and)37 b(^)-54 b Fx(\045)2697 4554 y Fv(\003)2697 4614 y Fu(l)2753 4587 y Fw(\\)17 b Fx(\015)2883 4554 y Fv(\003)2878 4612 y Fu(i)2948 4587 y Fw(6)p Fy(=)25 b Fw(;)g(6)p Fy(=)34 b(^)-54 b Fx(\045)3257 4554 y Fv(\003)3257 4614 y Fu(l)3314 4587 y Fw(\\)16 b Fx(\015)3443 4554 y Fv(\003)3438 4612 y Fu(j)3483 4587 y Fy(\),)227 4700 y(and)30 b(for)g(an)m(y)h(pair)f Fw(f)p Fx(\015)994 4714 y Fu(i)1023 4700 y Fx(;)15 b(\015)1110 4714 y Fu(j)1147 4700 y Fw(g)25 b(62)g Fx(G)31 b Fy(suc)m(h)f(a)h(c)m (hain)g(do)s(es)f(not)g(o)s(ccur;)66 4887 y(\(2\))46 b Fw([)288 4901 y Fu(i)319 4887 y Fy(\026)-48 b Fx(\015)363 4901 y Fu(i)412 4887 y Fw(\\)19 b([)553 4901 y Fu(j)p 589 4813 48 4 v 597 4887 a Fy(^)-53 b Fx(\045)636 4909 y Fu(j)698 4887 y Fy(=)25 b Fw(;)31 b Fy(and)f Fw([)1108 4901 y Fu(j)1144 4887 y Fw(f)p Fx(@)1242 4854 y Fv(\000)1310 4887 y Fy(^)-53 b Fx(\045)1349 4901 y Fu(j)1385 4887 y Fx(;)15 b(@)1478 4854 y Fq(+)1546 4887 y Fy(^)-53 b Fx(\045)1585 4901 y Fu(j)1621 4887 y Fw(g)26 b(\032)f([)1849 4901 y Fu(i)1877 4887 y Fx(\015)1929 4854 y Fv(\003)1924 4913 y Fu(i)1968 4887 y Fy(;)66 5075 y(\(3\))46 b(0)26 b Fw(2)f Fy(\()p Fw([)480 5089 y Fu(i)508 5075 y Fx(\015)560 5042 y Fv(\003)555 5100 y Fu(i)600 5075 y Fy(\))20 b Fw([)g Fy(\()p Fw([)832 5089 y Fu(j)877 5075 y Fy(^)-54 b Fx(\045)915 5042 y Fv(\003)915 5100 y Fu(j)955 5075 y Fy(\);)66 5263 y(\(4\))46 b(eac)m(h)32 b(c)m(hain)39 b(^)-53 b Fx(\045)719 5278 y Fu(k)792 5263 y Fy(connects)31 b(a)g(pair)f Fw(f)p Fx(\015)1516 5277 y Fu(i)1544 5263 y Fx(;)15 b(\015)1631 5277 y Fu(j)1668 5263 y Fw(g)26 b(2)f Fx(G)h Fw(2)f(G)2062 5277 y Fu(r)2130 5263 y Fy(or)30 b(\014lls)g(a)h(gap)g(within)f(a)h(con)m(tour)g Fx(\015)3393 5277 y Fu(i)3421 5263 y Fy(.)1726 5549 y(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 0 431 a Fy(W)-8 b(e)26 b(call)h(a)e(collection) j(of)d(reduced)g(c)m(hains)g(compatible)h(with)f(graph)g Fx(G)g Fy(if)g(it)h(satis\014es)f(the)h(conditions)0 544 y(ab)s(o)m(v)m(e)32 b(and)e(denote)i(it)f Fw(f)8 b Fy(^)-53 b Fx(\045)912 558 y Fq(1)952 544 y Fx(;)15 b(:::;)25 b Fy(^)-55 b Fx(\045)1154 558 y Fu(s)1193 544 y Fw(g)1238 558 y Fu(G)1297 544 y Fy(.)42 b(Note)33 b(that)e(eac)m(h)41 b(^)-54 b Fx(\045)31 b Fy(can)g(join)g(only)g(one)g(pair)g(of)g(con)m (tours.)43 b(A)0 657 y(collection)33 b(of)e(reduced)e(c)m(hains)i(is)f (then)g(constructed)h(through)f(the)g(follo)m(wing)i(steps:)66 835 y(\(1\))46 b(\014rst)30 b(remo)m(v)m(e)i(all)f(c)m(hains)g Fx(\045)1175 850 y Fu(l)1226 835 y Fw(2)25 b Fy(\000)30 b(for)g(whic)m(h)39 b(\026)-54 b Fx(\045)1845 850 y Fu(l)1897 835 y Fw(\032)24 b([)2053 802 y Fu(r)2053 860 y(i)p Fq(=1)2175 835 y Fy(\026)-49 b Fx(\015)2218 849 y Fu(i)2247 835 y Fy(;)66 1019 y(\(2\))46 b(for)f(all)g(remaining)g(c)m(hains)g Fx(\045)1297 1034 y Fu(k)1389 1019 y Fw(2)k Fy(\000,)57 b(\026)-54 b Fx(\045)1676 1034 y Fu(k)1768 1019 y Fw(6\032)49 b([)1949 986 y Fu(r)1949 1044 y(i)p Fq(=1)2070 1019 y Fy(\026)-48 b Fx(\015)2114 1033 y Fu(i)2187 1019 y Fy(remo)m(v)m(e)46 b(all)g(in)m(terv)-5 b(als)46 b(from)e(the)h(set)236 1132 y(\026)-54 b Fx(\045)274 1147 y Fu(k)337 1132 y Fw(\\)20 b Fy(\()p Fw([)514 1099 y Fu(r)514 1157 y(i)p Fq(=1)636 1132 y Fy(\026)-49 b Fx(\015)679 1146 y Fu(i)708 1132 y Fy(\);)66 1316 y(\(3\))46 b(of)31 b(the)f(remaining)g(in)m(terv) -5 b(als)31 b(form)f(all)h(p)s(ossible)e(collections)k(of)d(non-empt)m (y)h(c)m(hains)f(denoted)227 1429 y(b)m(y)h Fw(f)8 b Fy(^)-53 b Fx(\045)446 1443 y Fq(1)485 1429 y Fx(;)15 b(:::;)25 b Fy(^)-55 b Fx(\045)687 1443 y Fu(s)726 1429 y Fw(g)p Fy(.)0 1607 y(Then)29 b(b)m(y)i(Prop)s(osition)f(5.32)i(w)m(e) f(write)311 1731 y Ft(\014)311 1786 y(\014)657 1722 y(X)622 1911 y Fn(\000:\000)716 1897 y Fm(\003)751 1911 y(3)p Fn(0)367 1967 y(\000)p Fm(\033)17 b Fn(more)k(than)g(one)g(con)n(tour) 1104 1808 y Fx(K)1181 1822 y Fq(\000)1229 1808 y Fx(z)1275 1771 y Fv(j)1299 1754 y Fq(\026)1295 1771 y(\000)p Fv(j)1363 1731 y Ft(\014)1363 1786 y(\014)1444 1808 y Fw(\024)1796 b Fy(\(5.15\))477 2079 y Ft(X)481 2275 y Fu(r)r Fv(\025)p Fq(2)713 2079 y Ft(X)624 2280 y Fv(f)p Fu(\015)695 2289 y Fn(1)730 2280 y Fu(;:::;\015)866 2288 y Fr(r)899 2280 y Fv(g)973 2079 y Ft(X)950 2276 y Fu(G)p Fv(2G)1094 2284 y Fr(r)1143 2079 y Ft(X)1148 2275 y Fu(s)p Fv(\025)p Fq(0)1331 2052 y Fu(r)1290 2079 y Ft(Y)1291 2274 y Fu(i)p Fq(=1)1406 2165 y Fy(\(\()p Fw(j)p Fx(z)t Fw(j)p Fy(\(1)22 b(+)1765 2162 y Fx(")1823 2165 y Fw(j)p Fx(z)t Fw(j)p Fy(\)\))1989 2128 y Fv(j)r Fq(\026)-37 b Fu(\015)2045 2138 y Fr(i)2073 2128 y Fv(j)2096 2165 y Fw(D)s Fy(\()p Fx(\015)2251 2179 y Fu(i)2280 2165 y Fy(\))2465 2079 y Ft(X)2360 2268 y Fm(f)6 b Fn(^)-36 b Fr(\045)2424 2283 y Fn(1)2459 2268 y Fr(;:::;)6 b Fn(^)-36 b Fr(\045)2586 2276 y(s)2618 2268 y Fm(g)2650 2285 y Fr(G)2340 2334 y Fn(0)18 b Fm(2)f([)2489 2345 y Fd(i)2515 2334 y Fr(\015)2550 2313 y Fc(\003)2547 2351 y Fd(i)2600 2334 y Fm([)i Fn(^)-36 b Fr(\045)2687 2313 y Fc(\003)2687 2351 y Fd(i)2792 2052 y Fu(s)2750 2079 y Ft(Y)2747 2274 y Fu(j)t Fq(=1)2870 2165 y Fy(\()2905 2162 y Fx(")2963 2165 y Fw(j)p Fx(z)t Fw(j)p Fy(\))3094 2128 y Fv(j)p 3114 2073 69 3 v 6 w Fq(^)-41 b Fu(\045)3150 2138 y Fr(j)3183 2128 y Fv(j)3207 2165 y Fx(:)0 2516 y Fy(Note)31 b(that)f(for)f(\014xed)g Fw(f)p Fx(\015)864 2530 y Fq(1)904 2516 y Fx(;)15 b(:::;)g(\015)1106 2530 y Fu(r)1146 2516 y Fw(g)30 b Fy(the)f(collection)j Fw(f)8 b Fy(^)-53 b Fx(\045)1871 2530 y Fq(1)1911 2516 y Fx(;)15 b(:::;)25 b Fy(^)-55 b Fx(\045)2113 2530 y Fu(m)2181 2516 y Fw(g)30 b Fy(can)g(b)s(e)f(obtained)h(from)f(man)m(y)h (p)s(os-)0 2629 y(sible)h(collections)j(of)d(c)m(hains)g Fw(f)p Fx(\045)1121 2643 y Fq(1)1161 2629 y Fx(;)15 b(:::;)g(\045)1363 2643 y Fu(s)1401 2629 y Fw(g)p Fy(.)43 b(This)30 b(giv)m(es)j(the)e (factor)h(\(1)21 b(+)2556 2626 y Fx(")2614 2629 y Fw(j)p Fx(z)t Fw(j)p Fy(\))2745 2596 y Fv(j)r Fq(\026)-37 b Fu(\015)2801 2606 y Fr(i)2828 2596 y Fv(j)2883 2629 y Fy(app)s(earing)30 b(at)i(the)0 2742 y(righ)m(t)g(hand)e(side)h(of)h (\(5.15\).)45 b(F)-8 b(rom)32 b(no)m(w)f(on)g(w)m(e)h(assume)f(that) 2242 2739 y Fx(")2299 2742 y Fw(j)p Fx(z)t Fw(j)d(\024)e Fy(1)32 b(so)f(that)h(w)m(e)g(can)f(estimate)0 2855 y(this)f(factor)i (b)m(y)e(2)609 2822 y Fv(j)r Fq(\026)-37 b Fu(\015)665 2832 y Fr(i)692 2822 y Fv(j)715 2855 y Fy(.)141 2968 y(No)m(w)35 b(consider)e(the)h(last)h(sum)e(ab)s(o)m(v)m(e)i(in)m(v)m (olving)g(the)f(reduced)f(c)m(hains.)51 b(In)33 b(this)h(sum,)g(either) g(0)0 3081 y(b)s(elongs)28 b(to)h(a)f(con)m(tour)h(or)f(some)h(c)m (hains.)40 b(In)27 b(the)i(second)f(case)h(there)f(is)h(a)f(factor)h (of)f(\()3014 3078 y Fx(")3072 3081 y Fw(j)p Fx(z)t Fw(j)p Fy(\))3203 3048 y Fq(dist)r(\(0)p Fu(;)p Fv(f)p Fu(\015)t Fv(g)p Fq(\))0 3194 y Fy(app)s(earing)i(in)g(the)h(sum,)e(so)i(w)m(e)g (can)f(estimate)j(the)d(sum)g(b)m(y)90 3451 y(\()125 3448 y Fx(")182 3451 y Fw(j)p Fx(z)t Fw(j)p Fy(\))313 3413 y Fq(dist)r(\(0)p Fu(;\015)t Fq(\))p Fu(=)p Fq(2)804 3365 y Ft(X)699 3553 y Fm(f)6 b Fn(^)-36 b Fr(\045)763 3568 y Fn(1)797 3553 y Fr(;:::;)6 b Fn(^)-36 b Fr(\045)924 3561 y(s)957 3553 y Fm(g)989 3570 y Fr(G)679 3620 y Fn(0)18 b Fm(2)f([)828 3631 y Fd(i)854 3620 y Fr(\015)889 3599 y Fc(\003)886 3636 y Fd(i)938 3620 y Fm([)i Fn(^)-35 b Fr(\045)1026 3599 y Fc(\003)1026 3636 y Fd(i)1130 3337 y Fu(s)1089 3365 y Ft(Y)1086 3560 y Fu(j)t Fq(=1)1208 3451 y Fy(\()1243 3448 y Fx(")1301 3451 y Fw(j)p Fx(z)t Fw(j)p Fy(\))1432 3413 y Fv(j)p 1452 3358 V 6 w Fq(^)-41 b Fu(\045)1488 3423 y Fr(j)1521 3413 y Fv(j)p Fu(=)p Fq(2)1641 3451 y Fw(\024)1737 3365 y Ft(X)1783 3562 y Fu(k)1868 3451 y Fy(\()1903 3448 y Fx(")1961 3451 y Fw(j)p Fx(z)t Fw(j)p Fy(\))2092 3413 y Fq(dist)q(\(0)p Fu(;\015)2325 3425 y Fr(k)2364 3413 y Fq(\))p Fu(=)p Fq(2)2617 3365 y Ft(X)2512 3553 y Fm(f)6 b Fn(^)-36 b Fr(\045)2576 3568 y Fn(1)2610 3553 y Fr(;:::;)6 b Fn(^)-36 b Fr(\045)2737 3561 y(s)2770 3553 y Fm(g)2802 3570 y Fr(G)2491 3620 y Fn(0)19 b Fm(2)e([)2641 3631 y Fd(i)2667 3620 y Fr(\015)2702 3599 y Fc(\003)2699 3636 y Fd(i)2751 3620 y Fm([)i Fn(^)-35 b Fr(\045)2839 3599 y Fc(\003)2839 3636 y Fd(i)2943 3337 y Fu(s)2902 3365 y Ft(Y)2898 3560 y Fu(j)t Fq(=1)3021 3451 y Fy(\()3056 3448 y Fx(")3114 3451 y Fw(j)p Fx(z)t Fw(j)p Fy(\))3245 3413 y Fv(j)p 3265 3358 V 6 w Fq(^)-41 b Fu(\045)3301 3423 y Fr(j)3334 3413 y Fv(j)p Fu(=)p Fq(2)3429 3451 y Fx(:)0 3797 y Fy(F)-8 b(or)33 b(eac)m(h)h(con)m(tour)f Fx(\015)752 3811 y Fu(i)813 3797 y Fy(the)g(sum)e(o)m(v)m(er)j(all)f (reduced)f(c)m(hains)h(b)s(elonging)g(to)g(this)g(con)m(tour)g(cannot)g (b)s(e)0 3910 y(larger)e(than)e(2)514 3877 y Fv(j)p 534 3839 63 3 v Fu(\015)570 3887 y Fr(i)597 3877 y Fv(j)651 3910 y Fy(since)h(the)g(n)m(um)m(b)s(er)e(of)j(suc)m(h)e(c)m(hains)h (is)g(b)s(ounded)e(b)m(y)p 2530 3860 76 4 v 30 w Fx(\015)2577 3924 y Fu(i)2635 3910 y Fy(\(when)h(ev)m(ery)i(c)m(hain)f(sep-)0 4022 y(arates)g(eac)m(h)g(t)m(w)m(o)h(successiv)m(e)f(in)m(terv)-5 b(als)30 b(in)f(the)g(con)m(tour\).)41 b(Moreo)m(v)m(er)31 b(for)e(eac)m(h)h(couple)g(of)f(con)m(tours)0 4135 y(\()p Fx(\015)82 4149 y Fu(i)111 4135 y Fx(;)15 b(\015)198 4149 y Fu(j)235 4135 y Fy(\))30 b(the)h(con)m(tribution)g(to)g(the)f (sum)g(of)g(the)h(c)m(hains)g(connecting)g(them)g(is)f(giv)m(en)h(b)m (y)1304 4338 y(2)1349 4300 y Fv(j)p 1369 4263 63 3 v Fu(\015)1405 4310 y Fr(i)1432 4300 y Fv(j)p Fq(+)p Fv(j)p 1527 4263 69 3 v Fu(\015)1563 4310 y Fr(j)1595 4300 y Fv(j)1619 4338 y Fy(\()1654 4335 y Fx(")1712 4338 y Fw(j)p Fx(z)t Fw(j)p Fy(\))1843 4300 y Fq(dist)q(\()p Fu(\015)2021 4310 y Fr(i)2048 4300 y Fu(;\015)2104 4310 y Fr(j)2137 4300 y Fq(\))p Fu(=)p Fq(2)0 4531 y Fy(since)23 b(there)g(is)g(at)h (least)g(one)f(c)m(hain)h(longer)f(than)g(dist\()p Fx(\015)1924 4545 y Fu(i)1952 4531 y Fx(;)15 b(\015)2039 4545 y Fu(j)2076 4531 y Fy(\))24 b(and)e(the)h(rest)g(of)g(the)g(c)m(hains)g(con)m (tribute)0 4644 y(in)m(to)31 b(the)g(com)m(binatorial)h(prefactor.)42 b(This)29 b(giv)m(es)165 4768 y Ft(\014)165 4822 y(\014)510 4759 y(X)476 4948 y Fn(\000:\000)570 4934 y Fm(\003)604 4948 y(3)p Fn(0)220 5004 y(\000)p Fm(\033)17 b Fn(more)k(than)g(one)g (con)n(tour)957 4845 y Fx(K)1034 4859 y Fq(\000)1082 4845 y Fx(z)1128 4807 y Fv(j)1152 4791 y Fq(\026)1148 4807 y(\000)p Fv(j)1216 4768 y Ft(\014)1216 4822 y(\014)1297 4845 y Fw(\024)1943 b Fy(\(5.16\))331 5116 y Ft(X)334 5312 y Fu(r)r Fv(\025)p Fq(2)567 5116 y Ft(X)477 5317 y Fv(f)p Fu(\015)548 5326 y Fn(1)583 5317 y Fu(;:::;\015)719 5325 y Fr(r)752 5317 y Fv(g)803 5116 y Ft(X)849 5313 y Fu(k)990 5088 y(r)949 5116 y Ft(Y)950 5311 y Fu(i)p Fq(=1)1065 5202 y Fy(\(8)p Fw(j)p Fx(z)t Fw(j)p Fy(\))1276 5165 y Fv(j)r Fq(\026)-37 b Fu(\015)1332 5175 y Fr(i)1361 5165 y Fv(j)1384 5202 y Fw(D)s Fy(\()p Fx(\015)1539 5216 y Fu(i)1568 5202 y Fy(\)\()1638 5199 y Fx(")1696 5202 y Fw(j)p Fx(z)t Fw(j)p Fy(\))1827 5165 y Fq(dist)q(\(0)p Fu(;\015)2060 5177 y Fr(k)2099 5165 y Fq(\))p Fu(=)p Fq(2)2240 5116 y Ft(X)2216 5313 y Fu(G)p Fv(2G)2360 5321 y Fr(r)2514 5116 y Ft(Y)2410 5317 y Fv(f)p Fu(\015)2481 5327 y Fr(i)2508 5317 y Fu(;\015)2564 5327 y Fr(j)2596 5317 y Fv(g2)p Fu(G)2734 5202 y Fy(\()2769 5199 y Fx(")2827 5202 y Fw(j)p Fx(z)t Fw(j)p Fy(\))2958 5165 y Fq(dist)q(\()p Fu(\015)3136 5175 y Fr(i)3163 5165 y Fu(;\015)3219 5175 y Fr(j)3251 5165 y Fq(\))p Fu(=)p Fq(2)3354 5202 y Fx(:)1726 5549 y Fy(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 0 431 a Fe(Bounds)47 b(by)f(sums)h(over)f(tr)-5 b(e)g(es)137 b Fy(W)-8 b(e)39 b(use)g(the)f(tree-graph)h(b)s(ound)d (\(cf.)65 b(Lemma)39 b(8,)i(Ch.)64 b(2,)0 544 y(Sect.)41 b(4)31 b(of)g([40)q(]\))g(to)g(get)89 731 y Ft(X)65 928 y Fu(G)p Fv(2G)209 936 y Fr(r)363 731 y Ft(Y)259 932 y Fv(f)p Fu(\015)330 942 y Fr(i)357 932 y Fu(;\015)413 942 y Fr(j)445 932 y Fv(g2)p Fu(G)582 818 y Fy(\()617 815 y Fx(")675 818 y Fw(j)p Fx(z)t Fw(j)p Fy(\))806 780 y Fq(dist)r(\()p Fu(\015)985 790 y Fr(i)1012 780 y Fu(;\015)1068 790 y Fr(j)1100 780 y Fq(\))p Fu(=)p Fq(2)1228 818 y Fw(\024)1390 704 y Fu(r)1349 731 y Ft(Y)1324 927 y Fu(i;j)t Fq(=1)1490 818 y Fy(\(1)21 b(+)f(\()1717 815 y Fx(")1775 818 y Fw(j)p Fx(z)t Fw(j)p Fy(\))1906 780 y 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4615 y Fv(f)p Fu(\034)2013 4625 y Fr(i)2039 4615 y Fu(;\034)2090 4625 y Fr(j)2123 4615 y Fv(g2)p Fu(G)2275 4497 y Fx(")2333 4500 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))717 b(\(5.28\))0 4821 y(Here)233 4798 y(^)216 4821 y Fw(G)37 b Fy(denotes)c(the)f(set)g (of)g(connected)g(graphs)g(with)f(v)m(ertices)i Fx(\034)2343 4835 y Fq(1)2383 4821 y Fx(;)15 b(:::;)g(\034)2578 4836 y Fu(k)2622 4821 y Fy(.)44 b(Note)34 b(that)e(for)f(\014xed)h Fx(i)3504 4835 y Fq(0)0 4933 y Fy(w)m(e)f(ha)m(v)m(e)343 4865 y Ft(P)439 4960 y Fu(\034)470 4970 y Fr(j)503 4960 y Fq(:)p Fu(j)t Fv(6)p Fq(=)p Fu(i)635 4969 y Fn(0)688 4930 y Fx(")746 4933 y(D)s Fy(\()p Fx(i)890 4947 y Fq(0)929 4933 y Fx(;)15 b(j)5 b Fy(\))27 b Fw(\024)e Fx(C)1256 4930 y(")1328 4933 y Fy(with)30 b(some)h Fx(C)h(>)25 b Fy(0.)41 b(Th)m(us)29 b(b)m(y)i(using)e(\(5.17\))k(w)m(e)e(\014nd)806 5089 y Ft(X)798 5303 y Fu(G)p Fv(2)913 5287 y Fq(^)900 5303 y Fv(G)1060 5089 y Ft(Y)961 5290 y Fv(f)p Fu(\034)1027 5300 y Fr(i)1054 5290 y Fu(;\034)1105 5300 y Fr(j)1137 5290 y Fv(g2)p Fu(G)1290 5172 y Fx(")1347 5175 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))27 b Fw(\024)e Fx(e)1773 5138 y Fu(k)r(C)1879 5136 y(")1942 5089 y Ft(X)1931 5303 y Fu(T)10 b Fv(2)2041 5287 y Fq(^)2029 5303 y Fv(T)2197 5089 y Ft(Y)2100 5290 y Fv(f)p Fu(\034)2166 5300 y Fr(i)2193 5290 y Fu(;\034)2244 5300 y Fr(j)2277 5290 y Fv(g2)p Fu(T)2425 5172 y Fx(")2482 5175 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))568 b(\(5.29\))1726 5549 y(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 0 431 a Fy(where)282 408 y(^)266 431 y Fw(T)57 b Fy(are)34 b(trees)g(for)g(the)g(same)g(v)m(ertex)h(set) f(as)g(for)1982 408 y(^)1965 431 y Fw(G)5 b Fy(.)51 b(Next)35 b(order)e(the)h(collection)i Fw(f)p Fx(\034)3219 445 y Fq(1)3259 431 y Fx(;)15 b(:::;)g(\034)3454 446 y Fu(k)3498 431 y Fw(g)0 544 y Fy(further)29 b(to)i(get)669 679 y Ft(X)583 881 y Fv(f)p Fu(\034)649 890 y Fn(1)684 881 y Fu(;:::;\034)815 893 y Fr(k)851 881 y Fv(g)910 679 y Ft(X)902 894 y Fu(T)10 b Fv(2)1013 877 y Fq(^)1000 894 y Fu(T)1132 679 y Ft(Y)1066 892 y Fv(f)p Fu(i;j)t Fv(g2)1272 875 y Fq(~)1260 892 y Fv(T)1331 763 y Fx(")1388 766 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))27 b(=)1797 704 y(1)p 1782 745 76 4 v 1782 828 a Fx(k)s Fy(!)1961 679 y Ft(X)1883 881 y Fq(\()p Fu(\034)1941 890 y Fn(1)1976 881 y Fu(;:::;\034)2107 893 y Fr(k)2144 881 y Fq(\))2198 679 y Ft(X)2200 875 y Fq(~)2186 892 y Fu(T)10 b Fv(2)2296 875 y Fq(~)2284 892 y Fv(T)2420 679 y Ft(Y)2355 892 y Fv(f)p Fu(i;j)t Fv(g2)2563 875 y Fq(~)2549 892 y Fu(T)2615 763 y Fx(")2673 766 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))p Fx(;)352 b Fy(\(5.30\))0 1086 y(with)30 b(the)h(same)608 1063 y(~)591 1086 y Fw(T)53 b Fy(as)31 b(previously)-8 b(.)41 b(W)-8 b(e)31 b(then)f(obtain)h(inductiv)m(ely)1142 1197 y Ft(X)1063 1398 y Fq(\()p Fu(\034)1121 1407 y Fn(1)1156 1398 y Fu(;:::;\034)1287 1410 y Fr(k)1324 1398 y Fq(\))1458 1197 y Ft(Y)1366 1398 y Fq(\()p Fu(\034)1424 1408 y Fr(i)1452 1398 y Fu(;\034)1503 1408 y Fr(j)1535 1398 y Fq(\))p Fv(2T)1681 1280 y Fx(")1738 1283 y(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))27 b Fw(\024)e Fy(\()p Fx(C)2244 1280 y(")2287 1283 y Fy(\))2322 1246 y Fu(k)r Fv(\000)p Fq(1)2455 1283 y Fx(:)831 b Fy(\(5.31\))0 1590 y(Since)27 b(the)g(n)m(um)m(b)s(er)f (of)i(trees)f(ha)m(ving)h Fx(k)i Fy(v)m(ertices)f(is)e Fx(k)1861 1557 y Fu(k)r Fv(\000)p Fq(2)2021 1590 y Fy([40)q(],)i(b)m(y) e(using)f(Stirling's)i(form)m(ula,)g(\(5.29\))0 1703 y(and)i(\(5.30\))i(w)m(e)f(complete)h(the)f(pro)s(of)e(of)i(the)f (lemma.)1575 b Fi(\003)0 1944 y Fl(5.3)112 b(Cluster)38 b(estimates)0 2116 y Fy(Ha)m(ving)22 b(the)e(abstract)h(cluster)g (expansion)f(at)h(hand,)g(w)m(e)g(turn)e(no)m(w)i(to)g(establishing)f (the)h(b)s(ounds)d(\(5.12\))0 2229 y(on)30 b(the)h(cluster)g (activities.)0 2403 y Fk(Prop)s(osition)36 b(5.5)46 b Fo(Ther)-5 b(e)25 b(exists)f Fx(\016)30 b(>)25 b Fy(1)f Fo(and)h(a)f(function)2060 2400 y Fx(")2102 2403 y Fy(\()p Fx(\025)p Fy(\))i Fx(<)f Fw(1)f Fo(with)2651 2400 y Fx(")2693 2403 y Fy(\()p Fx(\025)p Fy(\))i Fw(!)f Fy(0)g Fo(as)f 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2751 y Fv(j)6 b Fq(\026)-41 b Fu(\045)2628 2759 y Fr(m)2687 2751 y Fv(j)3311 2789 y Fy(\(5.32\))0 3050 y Fo(holds.)0 3225 y Fy(The)28 b(\014rst)g(pro)s(duct)f(\(o)m(v)m(er)j(the)f(con)m (tours\))g(ab)s(o)m(v)m(e)h(is)f(our)f Fx(D)s Fy(\()p Fx(i;)15 b(j)5 b Fy(\))30 b(in)e(Prop)s(osition)h(5.3)g(ab)s(o)m(v)m (e,)i(and)d(it)0 3338 y(is)i(readily)h(seen)g(that)g(it)g(satis\014es)f (the)h(condition)g(giv)m(en)g(there.)141 3496 y Fj(Pr)n(oof.)40 b Fy(By)31 b(H\177)-45 b(older)30 b(inequalit)m(y)399 3753 y Fw(j)p Fx(K)501 3767 y Fq(\000)550 3753 y Fw(j)83 b(\024)853 3639 y Fu(r)812 3666 y Ft(Y)814 3864 y Fu(l)q Fq(=1)1037 3666 y Ft(Y)943 3868 y Fq(\()p Fu(\034)1001 3878 y Fr(i)1028 3868 y Fu(;\034)1079 3878 y Fr(j)1112 3868 y Fq(\))p Fv(2)p Fu(\015)1222 3880 y Fr(l)1262 3625 y Ft(\022)1329 3629 y(Z)1435 3753 y Fp(E)1496 3767 y Fu(\027)1535 3744 y Fb(x)1531 3790 y Fr(T)1598 3652 y Ft(h)1641 3753 y Fw(j)p Fx(e)1708 3709 y Fv(\000)p Fu(\025W)1871 3717 y Fr(\034)1900 3733 y(i)1926 3717 y(;\034)1974 3733 y(j)2035 3753 y Fw(\000)20 b Fy(1)p Fw(j)2196 3715 y Fu(n)2239 3725 y Fr(ij)2299 3652 y Ft(i)2357 3753 y Fx(d!)2464 3715 y Fv(\012)p Fu(N)2586 3753 y Fy(\()p Fk(x)p Fy(\))2711 3625 y Ft(\023)2779 3647 y Fq(1)p Fu(=n)2892 3657 y Fr(ij)2971 3753 y Fw(\002)269 b Fy(\(5.33\))883 4085 y Fw(\002)1028 3972 y Fu(s)987 3999 y Ft(Y)969 4193 y Fu(m)p Fq(=1)1213 3999 y Ft(Y)1136 4196 y Fu(k)r Fq(:)p Fu(\034)1226 4208 y Fr(k)1264 4196 y Fv(2)p Fu(\045)1347 4204 y Fr(m)1421 3957 y Ft(\022)1488 3961 y(Z)1594 4085 y Fw(j)p Fx(\031)1671 4100 y Fu(b)1705 4085 y Fy(\()p Fx(x)1792 4100 y Fu(k)r Fq(+1)1925 4085 y Fx(;)15 b(x)2017 4100 y Fu(k)2060 4085 y Fy(\))21 b Fw(\000)f Fy(1)p Fw(j)2277 4048 y Fu(\014)2325 4085 y Fx(d!)s Fy(\()p Fx(x)2519 4100 y Fu(k)2562 4085 y Fy(\))p Fx(d!)s Fy(\()p Fx(x)2791 4100 y Fu(k)r Fq(+1)2924 4085 y Fy(\))2959 3957 y Ft(\023)3027 3980 y Fq(1)p Fu(=\014)0 4381 y Fy(with)27 b(suitable)h(exp)s(onen)m(ts.)39 b(W)-8 b(e)29 b(c)m(ho)s(ose)f Fx(\014)j Fy(=)25 b(4,)j Fx(n)1771 4395 y Fu(ij)1857 4381 y Fy(=)d Fx(A)p Fw(j)p Fx(i)14 b Fw(\000)g Fx(j)20 b Fy(+)14 b(1)p Fw(j)2388 4348 y Fq(\001)2451 4381 y Fy(,)29 b(with)e(\001)e Fx(>)g Fy(1)i(to)h(b)s(e)f (sp)s(eci\014ed)0 4494 y(b)s(elo)m(w.)41 b(T)-8 b(ak)m(en)31 b(with)f(correct)i(m)m(ultiplicities,)g(w)m(e)f(pic)m(k)g Fx(A)g Fy(suc)m(h)f(that)1116 4686 y(2)p 1111 4726 57 4 v 1111 4809 a Fx(\014)1197 4747 y Fy(+)1288 4661 y Ft(X)1301 4847 y Fr(j)s Fm(2)p Fa(N)1305 4903 y Fr(j)s Fm(\025)p Fr(i)1480 4686 y Fy(2)p 1445 4726 116 4 v 1445 4809 a Fx(n)1500 4823 y Fu(ij)1595 4747 y Fy(=)1701 4686 y(1)p 1701 4726 46 4 v 1701 4809 a(2)1777 4747 y(+)1889 4686 y(2)p 1877 4726 69 4 v 1877 4809 a Fx(A)2001 4633 y Fv(1)1971 4661 y Ft(X)1972 4858 y Fu(k)r Fq(=1)2161 4686 y Fy(1)p 2127 4726 114 4 v 2127 4809 a Fx(k)2177 4783 y Fq(\001)2276 4747 y Fw(\024)25 b Fy(1)p Fx(:)0 5074 y Fy(The)c(\014rst)g(part)h(of)f(estimate)j(\(5.32\))g(follo)m(ws) e(b)m(y)g(Lemma)g(5.7,)i(the)e(second)g(b)m(y)f(Lemma)h(5.6)h(b)s(elo)m (w.)38 b(By)0 5187 y(c)m(ho)s(osing)30 b Fx(b)25 b Fy(=)g Fw(\000)15 b Fy(log)i Fw(j)p Fx(\025)p Fw(j)p Fx(=)p Fy(\(\003)i(+)d Fx(C)7 b Fy(\))29 b(with)f(suitable)h Fx(C)j(>)25 b Fy(0,)30 b(w)m(e)f(ha)m(v)m(e)h Fx(\025e)2504 5154 y Fu(C)5 b(b)2619 5187 y Fy(=)25 b Fx(e)2757 5154 y Fv(\000)p Fq(\003)p Fu(b)2921 5187 y Fy(=)f Fw(j)p Fx(\025)p Fw(j)3119 5154 y Fq(\003)p Fu(=)p Fq(\(\003+)p Fu(C)5 b Fq(\))3447 5187 y Fy(=:)0 5297 y Fx(")42 5300 y(;)31 b Fy(th)m(us)f(the)h(estimate)h(\(5.32\))g(is)f(\014nally)f (obtained.)1657 b Fi(\003)1726 5549 y Fy(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 0 431 a Fk(Lemma)35 b(5.6)46 b Fo(F)-7 b(or)33 b(lar)-5 b(ge)34 b(enough)f Fx(b)25 b(>)g Fy(0)33 b Fo(ther)-5 b(e)34 b(is)e(a)h(c)-5 b(onstant)35 b Fx(C)c(>)25 b Fy(0)33 b Fo(such)g(that)1305 635 y Fw(j)p Fx(\031)1382 650 y Fu(b)1417 635 y Fy(\()p Fx(x;)15 b(y)s Fy(\))21 b Fw(\000)f Fy(1)p Fw(j)41 b(\024)f Fx(C)7 b(e)2075 598 y Fv(\000)p Fq(\003)p Fu(b)2213 635 y Fx(;)1073 b Fy(\(5.34\))0 839 y Fo(uniformly)47 b(in)e Fx(x;)15 b(y)52 b Fw(2)c Fp(R)916 806 y Fu(d)957 839 y Fo(,)g(wher)-5 b(e)47 b Fy(\003)i Fx(>)f Fy(0)e Fo(is)g(the)g(sp)-5 b(e)g(ctr)g(al)48 b(gap)e(of)g(the)g(Schr\177)-46 b(odinger)47 b(op)-5 b(er)g(ator)0 952 y Fx(H)32 b Fy(=)25 b Fx(H)280 966 y Fq(0)339 952 y Fy(+)20 b Fx(V)h Fo(.)0 1165 y Fj(Pr)n(oof.)39 b Fy(By)29 b(assumption)f(the)h(p)s(oten)m(tial)h(is)f(c)m(hosen)g(so)g (that)g Fx(H)35 b Fy(is)29 b(in)m(trinsically)h(ultracon)m(tractiv)m (e,)0 1278 y(i.e.,)h(for)f(eac)m(h)i Fx(b)25 b(>)g Fy(0,)31 b Fx(C)841 1293 y Fu(b)900 1278 y Fy(=)25 b Fw(k)p Fx(\031)1093 1293 y Fu(b)1128 1278 y Fw(k)1173 1305 y Fu(L)1221 1286 y Fm(1)1286 1305 y Fq(\()p Fs(R)1361 1286 y Fn(2)p Fr(d)1428 1305 y Fq(\))1485 1278 y Fx(<)g Fw(1)p Fy(.)40 b(By)31 b(the)f(semigroup)g(prop)s(ert)m(y)f(of)h Fx(\031)2997 1293 y Fu(b)3062 1278 y Fy(and)f(the)h(fact)0 1403 y(that)197 1330 y Ft(R)273 1403 y Fx(\031)325 1418 y Fu(b)359 1403 y Fy(\()p Fx(x;)15 b(y)s Fy(\))p Fx(d!)s Fy(\()p Fx(y)s Fy(\))27 b(=)e(1)30 b(for)h(eac)m(h)g Fx(x)p Fy(,)g(for)f Fx(b)25 b(>)g Fy(2)31 b(w)m(e)g(ha)m(v)m(e)261 1654 y Fw(j)p Fx(\031)338 1669 y Fu(b)373 1654 y Fy(\()p Fx(x;)15 b(y)s Fy(\))21 b Fw(\000)f Fy(1)p Fw(j)26 b Fy(=)887 1549 y Ft(\014)887 1604 y(\014)887 1658 y(\014)932 1530 y(Z)1038 1654 y Fx(d\030)1145 1530 y Ft(Z)1251 1654 y Fx(d\021)18 b(\031)1413 1668 y Fq(1)1453 1654 y Fy(\()p Fx(x;)d(\030)t Fy(\)\011)1730 1616 y Fq(2)1770 1654 y Fy(\()p Fx(\030)t Fy(\)\()p Fx(\031)1971 1669 y Fu(b)p Fv(\000)p Fq(2)2096 1654 y Fy(\()p Fx(\030)t(;)g(\021)s Fy(\))22 b Fw(\000)e Fy(1\)\011)2562 1616 y Fq(2)2602 1654 y Fy(\()p Fx(\021)s Fy(\))p Fx(\031)2772 1668 y Fq(1)2812 1654 y Fy(\()p Fx(\021)s(;)15 b(y)s Fy(\))3018 1549 y Ft(\014)3018 1604 y(\014)3018 1658 y(\014)791 1889 y Fw(\024)25 b Fx(C)959 1852 y Fq(2)952 1912 y(1)1013 1765 y Ft(Z)1119 1889 y Fx(d\030)1225 1765 y Ft(Z)1331 1889 y Fx(d\021)19 b Fy(\011\()p Fx(\030)t Fy(\))p Fw(j)5 b Fy(^)-50 b Fx(\031)1704 1904 y Fu(b)p Fv(\000)p Fq(2)1829 1889 y Fy(\()p Fx(\030)t(;)15 b(\021)s Fy(\))22 b Fw(\000)e Fy(\011\()p Fx(\030)t Fy(\)\011\()p Fx(\021)s Fy(\))p Fw(j)p Fy(\011\()p Fx(\021)s Fy(\))791 2156 y Fw(\024)25 b Fx(C)959 2118 y Fq(2)952 2178 y(1)1013 2028 y Ft(\022)1080 2032 y(Z)1186 2156 y Fx(d\030)1292 2032 y Ft(Z)1398 2156 y Fx(d\021)s Fy(\()5 b(^)-50 b Fx(\031)1580 2171 y Fu(b)p Fv(\000)p Fq(2)1706 2156 y Fy(\()p Fx(\030)t(;)15 b(\021)s Fy(\))21 b Fw(\000)f Fy(\011\()p Fx(\030)t Fy(\)\011\()p Fx(\021)s Fy(\)\))2429 2118 y Fq(2)2470 2028 y Ft(\023)2537 2050 y Fq(1)p Fu(=)p Fq(2)791 2485 y Fy(=)25 b Fx(C)959 2448 y Fq(2)952 2508 y(1)997 2485 y Fx(e)1039 2448 y Fv(\000)p Fq(\()p Fu(b)p Fv(\000)p Fq(2\)\()p Fu(E)1347 2457 y Fn(1)1384 2448 y Fv(\000)p Fu(E)t Fq(\))1541 2303 y Ft(0)1541 2466 y(@)1620 2399 y(X)1622 2597 y Fu(k)r Fv(\025)p Fq(2)1767 2485 y Fx(e)1809 2448 y Fv(\000)p Fq(2\()p Fu(b)p Fv(\000)p Fq(2\)\()p Fu(E)2152 2460 y Fr(k)2192 2448 y Fv(\000)p Fu(E)2299 2457 y Fn(1)2333 2448 y Fq(\))2365 2303 y Ft(1)2365 2466 y(A)2444 2325 y Fq(1)p Fu(=)p Fq(2)2579 2485 y Fw(\024)g Fx(C)7 b(e)2789 2448 y Fv(\000)p Fq(\003)p Fu(b)2927 2485 y Fx(;)3311 2101 y Fy(\(5.35\))0 2794 y(where)33 b Fx(E)i Fy(=)29 b(inf)22 b(Sp)s(ec)14 b Fx(H)7 b Fy(.)49 b(The)32 b(last)i(but)f (\014rst)f(step)h(comes)h(ab)s(out)f(as)g(follo)m(ws.)50 b Fw(k)p Fx(\031)2956 2809 y Fu(b)2990 2794 y Fw(k)3036 2821 y Fu(L)3084 2802 y Fm(1)3149 2821 y Fq(\()p Fs(R)3224 2802 y Fn(2)p Fr(d)3291 2821 y Fq(\))3352 2794 y Fx(<)29 b Fw(1)0 2925 y Fy(implies)f(that)34 b(^)-50 b Fx(\031)553 2940 y Fu(b)613 2925 y Fw(2)24 b Fx(L)760 2892 y Fq(2)800 2925 y Fy(\()p Fp(R)901 2892 y Fq(2)p Fu(d)976 2925 y Fx(;)15 b(dx)p Fy(\))30 b(for)d(eac)m(h)j Fx(b)25 b(>)g Fy(0.)40 b(Th)m(us)27 b Fx(e)2059 2892 y Fv(\000)p Fu(bH)2240 2925 y Fy(is)h(a)h(Hilb)s(ert-Sc)m(hmidt)f(op)s(erator)g(for)0 3038 y(eac)m(h)37 b Fx(b)e(>)f Fy(0,)k(in)e(particular)g Fx(H)43 b Fy(has)36 b(a)g(purely)f(discrete)i(sp)s(ectrum)e(with)g (eigen)m(v)-5 b(alues)38 b Fx(E)i(<)34 b(E)3398 3052 y Fq(1)3473 3038 y Fw(\024)0 3151 y Fx(E)67 3165 y Fq(2)132 3151 y Fw(\024)25 b Fx(:)15 b(:)g(:)q Fy(.)39 b(With)27 b Fx(P)687 3165 y Fq(\011)747 3151 y Fy(,)g(the)g(pro)5 b(jection)27 b(on)m(to)h(the)e(subspace)g(of)h Fx(L)2271 3118 y Fq(2)2310 3151 y Fy(\()p Fp(R)2411 3118 y Fu(d)2452 3151 y Fx(;)15 b(dx)p Fy(\))27 b(spanned)e(b)m(y)i(\011,)g(the)f(last)0 3264 y(equalit)m(y)32 b(giv)m(es)g(the)e(Hilb)s(ert-Sc)m(hmidt)h(norm)e (of)i Fx(e)1778 3231 y Fv(\000)p Fq(\()p Fu(b)p Fv(\000)p Fq(2\))p Fu(H)2096 3264 y Fw(\000)20 b Fx(P)2245 3278 y Fq(\011)2304 3264 y Fy(.)1144 b Fi(\003)0 3526 y Fk(Lemma)35 b(5.7)46 b Fo(Assume)30 b(that)h Fx(\014)f(>)25 b Fy(3)p Fo(.)42 b(Then)30 b(ther)-5 b(e)31 b(exists)f Fx(\016)f(>)c Fy(1)31 b Fo(and)g(c)-5 b(onstants)32 b Fx(C)2943 3540 y Fq(8)2982 3526 y Fx(;)15 b(C)3087 3540 y Fq(9)3152 3526 y Fx(<)25 b Fw(1)30 b Fo(such)0 3639 y(that,)k(for)f Fx(\025)f Fo(smal)5 b(l)34 b(enough)285 3784 y Ft(\022)352 3788 y(Z)458 3912 y Fp(E)519 3926 y Fu(\027)558 3903 y Fb(x)554 3949 y Fr(T)621 3811 y Ft(h)664 3912 y Fw(j)p Fx(e)731 3868 y Fu(\025W)839 3876 y Fr(\034)868 3892 y(i)895 3876 y(;\034)943 3892 y(j)1004 3912 y Fw(\000)20 b Fy(1)p Fw(j)1165 3874 y Fu(n)1208 3884 y Fr(ij)1267 3811 y Ft(i)1325 3912 y Fx(d!)1432 3874 y Fv(\012)p Fu(N)1554 3912 y Fy(\()p Fk(x)p Fy(\))1679 3784 y Ft(\023)1747 3806 y Fq(1)p Fu(=n)1860 3816 y Fr(ij)1945 3912 y Fw(\024)25 b Fx(C)2106 3926 y Fq(8)2145 3912 y Fx(\025e)2240 3874 y Fu(C)5 b(b)2330 3912 y Fy(\(1)21 b(+)f Fx(b)p Fw(j)p Fx(i)g Fw(\000)g Fx(j)26 b Fw(\000)20 b Fy(1)p Fw(j)p Fy(\))2987 3874 y Fu(\016)3311 3900 y Fy(\(5.36\))0 4161 y Fo(for)33 b(al)5 b(l)33 b Fx(i;)15 b(j)39 b Fo(and)34 b Fx(b)25 b Fw(\025)g Fy(1)p Fo(.)0 4373 y Fj(Pr)n(oof.)83 b Fy(Note)46 b(that)g(the)f(conditional)h(exp)s(ectation)g Fp(E)2063 4387 y Fu(\027)2102 4364 y Fb(x)2098 4410 y Fr(T)2150 4373 y Fy([)p Fw(j)p Fx(e)2242 4330 y Fu(\025W)2350 4338 y Fr(\034)2379 4354 y(i)2406 4338 y(;\034)2454 4354 y(j)2525 4373 y Fw(\000)29 b Fy(1)p Fw(j)2695 4340 y Fu(n)2738 4350 y Fr(ij)2798 4373 y Fy(])44 b(dep)s(ends)f(only)i(on)0 4486 y Fx(x)52 4500 y Fu(i)80 4486 y Fx(;)15 b(x)172 4500 y Fu(i)p Fq(+1)291 4486 y Fx(;)g(x)383 4500 y Fu(j)420 4486 y Fx(;)g(x)512 4500 y Fu(j)t Fq(+1)639 4486 y Fy(.)40 b(Th)m(us)30 b(w)m(e)h(can)f(write)h(with)f(a)h(sligh)m(t)g(abuse)f(of) h(notation,)105 4612 y Ft(Z)211 4736 y Fp(E)272 4750 y Fu(\027)311 4727 y Fb(x)307 4773 y Fr(T)374 4635 y Ft(h)417 4736 y Fw(j)p Fx(e)484 4693 y Fu(\025W)592 4701 y Fr(\034)621 4717 y(i)648 4701 y(;\034)696 4717 y(j)757 4736 y Fw(\000)19 b Fy(1)p Fw(j)917 4699 y Fu(n)960 4709 y Fr(ij)1020 4635 y Ft(i)1078 4736 y Fx(d!)1185 4699 y Fv(\012)p Fu(N)1307 4736 y Fy(\()p Fk(x)p Fy(\))26 b(=)1554 4612 y Ft(Z)1660 4736 y Fp(E)1721 4750 y Fu(\027)1760 4727 y Fb(x)1756 4773 y Fr(T)1824 4635 y Ft(h)1867 4736 y Fw(j)p Fx(e)1934 4693 y Fu(\025W)2042 4701 y Fr(\034)2071 4717 y(i)2097 4701 y(;\034)2145 4717 y(j)2206 4736 y Fw(\000)20 b Fy(1)p Fw(j)2367 4699 y Fu(n)2410 4709 y Fr(ij)2469 4635 y Ft(i)2528 4736 y Fx(d!)2635 4699 y Fv(\012)p Fq(4)2729 4736 y Fy(\()p Fx(x)2816 4750 y Fu(i)2845 4736 y Fx(;)15 b(x)2937 4750 y Fu(i)p Fq(+1)3055 4736 y Fx(;)g(x)3147 4750 y Fu(j)3184 4736 y Fx(;)g(x)3276 4750 y Fu(j)t Fq(+1)3403 4736 y Fy(\))1726 5549 y(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 0 431 a Fy(when)24 b Fw(j)p Fx(i)10 b Fw(\000)g Fx(j)5 b Fw(j)28 b Fx(>)c Fy(1,)k(otherwise)d(one)h(in)m (tegral)h(m)m(ust)f(b)s(e)e(ignored)i(in)f(this)g(expression.)39 b(Then)25 b(b)m(y)g(\(3.12\))0 553 y(the)31 b(problem)e(reduces)h(to)h (estimates)h(on)f(the)f(m)m(ultiple)h(Bro)m(wnian)g(bridge)2680 530 y Ft(d)2673 553 y Fw(W)2770 522 y Ff(x)2763 582 y Fu(T)2818 553 y Fy(:)2 805 y Fx(A)25 b Fy(=)191 681 y Ft(Z)297 805 y Fp(E)358 819 y Fu(\027)397 796 y Fb(x)393 842 y Fr(T)460 704 y Ft(h)503 805 y Fw(j)p Fx(e)570 762 y Fu(\025W)678 770 y Fr(\034)707 786 y(i)734 770 y(;\034)782 786 y(j)843 805 y Fw(\000)20 b Fy(1)p Fw(j)1004 768 y Fu(n)1047 778 y Fr(ij)1106 704 y Ft(i)1164 805 y Fx(d!)1271 768 y Fv(\012)p Fq(4)1366 805 y Fy(\()p Fx(x)1453 819 y Fu(i)1481 805 y Fx(;)15 b(x)1573 819 y Fu(i)p Fq(+1)1692 805 y Fx(;)g(x)1784 819 y Fu(j)1821 805 y Fx(;)g(x)1913 819 y Fu(j)t Fq(+1)2040 805 y Fy(\))27 1116 y Fw(\024)123 992 y Ft(Z)239 1021 y Fp(E)308 1040 y Fg(d)300 1056 y Fv(W)376 1034 y Fb(x)370 1080 y Fr(T)438 920 y Ft(h)481 1021 y Fw(j)p Fx(e)548 978 y Fu(\025W)656 986 y Fr(\034)685 1002 y(i)711 986 y(;\034)759 1002 y(j)820 1021 y Fw(\000)20 b Fy(1)p Fw(j)981 988 y Fu(n)1024 998 y Fr(ij)1083 1021 y Fx(e)1125 978 y Fv(\000)p Fu(V)1221 986 y Fr(\034)1250 1002 y(i)1281 978 y Fv(\000)p Fu(V)1377 986 y Fr(\034)1406 1002 y(j)1447 920 y Ft(i)p 239 1095 1251 4 v 326 1178 a Fx(Z)388 1193 y Fu(b)422 1178 y Fy(\011\()p Fx(x)580 1192 y Fu(i)608 1178 y Fy(\)\011\()p Fx(x)801 1192 y Fu(i)p Fq(+1)920 1178 y Fy(\))p Fx(\031)1007 1193 y Fu(b)1042 1178 y Fy(\()p Fx(x)1129 1192 y Fu(i)1157 1178 y Fx(;)15 b(x)1249 1192 y Fu(i)p Fq(+1)1368 1178 y Fy(\))1593 1054 y(\005)1661 1069 y Fu(b)1695 1054 y Fy(\()p Fx(x)1782 1068 y Fu(i)1811 1054 y Fx(;)g(x)1903 1068 y Fu(i)p Fq(+1)2022 1054 y Fy(\)\005)2125 1069 y Fu(b)2159 1054 y Fy(\()p Fx(x)2246 1068 y Fu(j)2283 1054 y Fx(;)g(x)2375 1068 y Fu(j)t Fq(+1)2502 1054 y Fy(\))p 1510 1095 1111 4 v 1510 1178 a Fx(Z)1572 1193 y Fu(b)1606 1178 y Fy(\011\()p Fx(x)1764 1192 y Fu(j)1801 1178 y Fy(\)\011\()p Fx(x)1994 1192 y Fu(j)t Fq(+1)2121 1178 y Fy(\))p Fx(\031)2208 1193 y Fu(b)2242 1178 y Fy(\()p Fx(x)2329 1192 y Fu(j)2366 1178 y Fx(;)g(x)2458 1192 y Fu(j)t Fq(+1)2585 1178 y Fy(\))2631 1116 y Fx(d!)2738 1078 y Fv(\012)p Fq(4)2832 1116 y Fy(\()p Fx(x)2919 1130 y Fu(i)2948 1116 y Fx(;)g(x)3040 1130 y Fu(i)p Fq(+1)3158 1116 y Fx(;)g(x)3250 1130 y Fu(j)3287 1116 y Fx(;)g(x)3379 1130 y Fu(j)t Fq(+1)3506 1116 y Fy(\))27 1362 y Fw(\024)25 b Fx(C)7 b(Z)264 1323 y Fv(\000)p Fq(2)257 1391 y Fu(b)372 1238 y Ft(Z)479 1362 y Fp(E)548 1380 y Fg(d)540 1397 y Fv(W)616 1374 y Fb(x)610 1420 y Fr(T)677 1261 y Ft(h)720 1362 y Fw(j)p Fx(e)787 1318 y Fu(\025W)895 1326 y Fr(\034)924 1342 y(i)951 1326 y(;\034)999 1342 y(j)1060 1362 y Fw(\000)20 b Fy(1)p Fw(j)1221 1324 y Fu(n)1264 1334 y Fr(ij)1323 1362 y Fx(e)1365 1318 y Fv(\000)p Fu(V)1461 1326 y Fr(\034)1490 1342 y(i)1521 1318 y Fv(\000)p Fu(V)1617 1326 y Fr(\034)1646 1342 y(j)1687 1261 y Ft(i)1745 1362 y Fy(\005)1813 1377 y Fu(b)1847 1362 y Fy(\()p Fx(x)1934 1376 y Fu(i)1963 1362 y Fx(;)15 b(x)2055 1376 y Fu(i)p Fq(+1)2173 1362 y Fy(\)\005)2276 1377 y Fu(b)2311 1362 y Fy(\()p Fx(x)2398 1376 y Fu(j)2435 1362 y Fx(;)g(x)2527 1376 y Fu(j)t Fq(+1)2654 1362 y Fy(\))p Fw(\002)204 1552 y(\002)20 b Fy(\011\()p Fx(x)453 1566 y Fu(i)481 1552 y Fy(\)\011\()p Fx(x)674 1566 y Fu(i)p Fq(+1)793 1552 y Fy(\)\011\()p Fx(x)986 1566 y Fu(j)1022 1552 y Fy(\)\011\()p Fx(x)1215 1566 y Fu(j)t Fq(+1)1343 1552 y Fy(\))p Fx(dx)1477 1566 y Fu(i)1505 1552 y Fx(dx)1604 1566 y Fu(i)p Fq(+1)1723 1552 y Fx(dx)1822 1566 y Fu(j)1859 1552 y Fx(dx)1958 1566 y Fu(j)t Fq(+1)2085 1552 y Fx(;)0 1765 y Fy(where)k(w)m(e)i(used)e (Lemma)h(5.6)h(and)e(c)m(hose)i Fx(b)e Fy(large)i(enough)f(so)g(that)g (sup)2476 1787 y Fu(x;y)2592 1765 y Fw(j)p Fx(\031)2669 1780 y Fu(b)2704 1765 y Fy(\()p Fx(x;)15 b(y)s Fy(\))9 b Fw(\000)g Fy(1)p Fw(j)26 b(\024)f Fx(C)7 b(e)3309 1732 y Fv(\000)p Fq(\003)p Fu(b)3473 1765 y Fw(\024)0 1878 y Fy(1)p Fx(=)p Fy(2.)44 b(F)-8 b(or)32 b Fx(i)21 b Fy(+)f(1)27 b Fx(<)f(j)37 b Fy(notice)32 b(that)g(\011\()p Fx(x)1378 1892 y Fu(i)p Fq(+1)1496 1878 y Fy(\)\011\()p Fx(x)1689 1892 y Fu(j)t Fq(+1)1816 1878 y Fy(\))27 b Fw(\024)f Fx(C)7 b Fy(.)43 b(Then)30 b(b)m(y)h(in)m(tegrating)i(with)d(resp)s (ect)h(to)0 1991 y Fx(x)52 2005 y Fu(i)p Fq(+1)170 1991 y Fx(;)15 b(x)262 2005 y Fu(j)t Fq(+1)420 1991 y Fy(w)m(e)30 b(remo)m(v)m(e)i(the)f(conditional)h(exp)s(ectation)g(and)d(obtain)558 2241 y Fx(A)c Fw(\024)g Fx(C)834 2117 y Ft(Z)945 2217 y(b)940 2240 y Fp(E)1001 2254 y Fu(x)1041 2264 y Fr(i)1067 2254 y Fu(;x)1127 2264 y Fr(j)1178 2140 y Ft(h)1221 2241 y Fw(j)p Fx(e)1288 2197 y Fu(\025W)1396 2205 y Fr(\034)1425 2221 y(i)1451 2205 y(;\034)1499 2221 y(j)1560 2241 y Fw(\000)20 b Fy(1)p Fw(j)1721 2203 y Fu(n)1764 2213 y Fr(ij)1824 2241 y Fx(e)1866 2197 y Fv(\000)p Fu(V)1962 2205 y Fr(\034)1991 2221 y(i)2022 2197 y Fv(\000)p Fu(V)2118 2205 y Fr(\034)2147 2221 y(j)2187 2140 y Ft(i)2245 2241 y Fy(\011\()p Fx(x)2403 2255 y Fu(i)2432 2241 y Fy(\))p Fx(dx)2566 2255 y Fu(i)2594 2241 y Fy(\011\()p Fx(x)2752 2255 y Fu(j)2789 2241 y Fy(\))p Fx(dx)2923 2255 y Fu(j)2960 2241 y Fx(;)326 b Fy(\(5.37\))0 2507 y(where)20 b(no)m(w)437 2484 y Ft(b)432 2507 y Fp(E)493 2521 y Fu(x)533 2531 y Fr(i)559 2521 y Fu(;x)619 2531 y Fr(j)675 2507 y Fy(denotes)h(exp)s (ectation)g(o)m(v)m(er)h(the)e(t)m(w)m(o)i(pieces)f Fp(X)2285 2521 y Fu(\034)2316 2531 y Fr(i)2367 2507 y Fy(and)e Fp(X)2599 2521 y Fu(\034)2630 2531 y Fr(j)2687 2507 y Fy(w)m(eigh)m(ted)j(b)m(y)3172 2515 y Fw(W)3269 2475 y Fu(x)3309 2485 y Fr(i)3262 2530 y Fu(\034)3293 2540 y Fr(i)3355 2507 y Fw(\012)3441 2515 y(W)3538 2456 y Fu(x)3578 2466 y Fr(j)3530 2519 y Fu(\034)3561 2529 y Fr(j)3614 2507 y Fy(,)0 2620 y(i.e.)42 b(t)m(w)m(o)31 b(indep)s(enden)m(t)e(Wiener)i(measures)f(starting)i(at)f Fx(x)2052 2634 y Fu(i)2110 2620 y Fy(and)f Fx(x)2339 2634 y Fu(j)2375 2620 y Fy(,)h(resp)s(ectiv)m(ely)-8 b(.)141 2733 y(Next)31 b(w)m(e)g(estimate)h(the)f(exp)s(ectation)h(in)e (\(5.37\),)94 2937 y Ft(b)89 2960 y Fp(E)150 2974 y Fu(x)190 2984 y Fr(i)216 2974 y Fu(;x)276 2984 y Fr(j)327 2859 y Ft(h)370 2960 y Fw(j)p Fx(e)437 2917 y Fu(\025W)545 2925 y Fr(\034)574 2941 y(i)600 2925 y(;\034)648 2941 y(j)709 2960 y Fw(\000)20 b Fy(1)p Fw(j)870 2923 y Fu(n)918 2960 y Fx(e)960 2917 y Fv(\000)p Fu(V)1056 2925 y Fr(\034)1085 2941 y(i)1116 2917 y Fv(\000)p Fu(V)1212 2925 y Fr(\034)1241 2941 y(j)1281 2859 y Ft(i)1350 2960 y Fw(\024)1451 2937 y Ft(b)1446 2960 y Fp(E)1507 2974 y Fu(x)1547 2984 y Fr(i)1572 2974 y Fu(;x)1632 2984 y Fr(j)1683 2859 y Ft(h)1726 2960 y Fw(j)p Fx(\025W)1890 2974 y Fu(\034)1921 2984 y Fr(i)1948 2974 y Fu(;\034)1999 2984 y Fr(j)2036 2960 y Fw(j)2061 2923 y Fu(n)2108 2960 y Fx(e)2150 2917 y Fu(n)p Fv(j)p Fu(\025W)2321 2925 y Fr(\034)2350 2941 y(i)2376 2925 y(;\034)2424 2941 y(j)2460 2917 y Fv(j)2484 2960 y Fx(e)2526 2917 y Fv(\000)p Fu(V)2622 2925 y Fr(\034)2651 2941 y(i)2682 2917 y Fv(\000)p Fu(V)2778 2925 y Fr(\034)2807 2941 y(j)2848 2859 y Ft(i)296 3181 y Fw(\024)25 b(j)p Fx(\025)p Fw(j)495 3143 y Fu(n)558 3080 y Ft(\020)617 3157 y(b)612 3181 y Fp(E)673 3195 y Fu(x)713 3205 y Fr(i)738 3195 y Fu(;x)798 3205 y Fr(j)850 3107 y Ft(\002)888 3181 y Fw(j)p Fx(W)999 3195 y Fu(\034)1030 3205 y Fr(i)1056 3195 y Fu(;\034)1107 3205 y Fr(j)1144 3181 y Fw(j)1169 3143 y Fq(2)p Fu(n)1251 3107 y Ft(\003)1289 3080 y(\021)1344 3102 y Fq(1)p Fu(=)p Fq(2)1469 3080 y Ft(\020)1528 3157 y(b)1523 3181 y Fp(E)1584 3195 y Fu(x)1624 3205 y Fr(i)1650 3195 y Fu(;x)1710 3205 y Fr(j)1761 3080 y Ft(h)1804 3181 y Fx(e)1846 3137 y Fq(4)p Fu(n)p Fv(j)p Fu(\025W)2052 3145 y Fr(\034)2081 3161 y(i)2107 3145 y(;\034)2155 3161 y(j)2191 3137 y Fv(j)2215 3080 y Ft(i\021)2312 3102 y Fq(1)p Fu(=)p Fq(4)2438 3080 y Ft(\020)2497 3157 y(b)2492 3181 y Fp(E)2553 3195 y Fu(x)2593 3205 y Fr(i)2619 3195 y Fu(;x)2679 3205 y Fr(j)2730 3080 y Ft(h)2773 3181 y Fx(e)2815 3137 y Fv(\000)p Fq(4)p Fu(V)2946 3145 y Fr(\034)2975 3161 y(i)3006 3137 y Fv(\000)p Fq(4)p Fu(V)3137 3145 y Fr(\034)3166 3161 y(j)3207 3080 y Ft(i\021)3304 3102 y Fq(1)p Fu(=)p Fq(4)3429 3181 y Fx(:)0 3402 y Fy(Since)30 b Fx(V)51 b Fy(is)30 b(of)h(Kato-class,)i(w)m(e)d(ha)m(v)m(e)i(the)f (uniform)e(b)s(ound)631 3547 y Ft(\020)690 3624 y(b)685 3648 y Fp(E)746 3662 y Fu(x)786 3672 y Fr(i)812 3662 y Fu(;x)872 3672 y Fr(j)923 3547 y Ft(h)966 3648 y Fx(e)1008 3604 y Fv(\000)p Fq(4)p Fu(V)1139 3612 y Fr(\034)1168 3628 y(i)1199 3604 y Fv(\000)p Fq(4)p Fu(V)1330 3612 y Fr(\034)1359 3628 y(j)1400 3547 y Ft(i\021)1497 3570 y Fq(1)p Fu(=)p Fq(4)1633 3648 y Fw(\024)c Fy(sup)1777 3720 y Fu(x)1881 3574 y Ft(\000)1922 3648 y Fp(E)1983 3662 y Fu(x)2042 3574 y Ft(\002)2080 3648 y Fx(e)2122 3610 y Fv(\000)p Fq(4)p Fu(V)2253 3624 y Fn([0)p Fr(;b)p Fn(])2373 3574 y Ft(\003\001)2453 3591 y Fq(1)p Fu(=)p Fq(2)2588 3648 y Fw(\024)g Fx(C)7 b(e)2798 3610 y Fu(C)e(b)2887 3648 y Fx(;)0 3902 y Fy(with)34 b(some)h Fx(C)k(>)32 b Fy(0.)53 b(F)-8 b(urthermore,)36 b(write)f Fx(W)1654 3916 y Fu(\034)1685 3926 y Fr(i)1711 3916 y Fu(;\034)1762 3926 y Fr(j)1831 3902 y Fy(=)1934 3829 y Ft(R)1977 3934 y Fu(\034)2008 3944 y Fr(i)2054 3902 y Fx(dX)2176 3916 y Fu(t)2221 3829 y Ft(R)2264 3934 y Fu(\034)2295 3944 y Fr(j)2347 3902 y Fx(dX)2469 3916 y Fu(s)2506 3902 y Fx(g)2549 3916 y Fu(ts)2647 3902 y Fy(\(see)g(\(5.41\))i(b)s(elo)m(w\)) e(and)0 4025 y(estimate)d(the)f(double)f(in)m(tegral)i(b)s(elo)m(w)e(b) m(y)h(using)e(the)i(Burkholder-Da)m(vis-Gundy)f(inequalit)m(y)-8 b(,)568 4308 y Ft(b)563 4331 y Fp(E)624 4345 y Fu(x)664 4355 y Fr(i)689 4345 y Fu(;x)749 4355 y Fr(j)801 4257 y Ft(\002)838 4331 y Fw(j)p Fx(W)949 4345 y Fu(\034)980 4355 y Fr(i)1007 4345 y Fu(;\034)1058 4355 y Fr(j)1095 4331 y Fw(j)1120 4294 y Fq(2)p Fu(n)1202 4257 y Ft(\003)1265 4331 y Fw(\024)25 b Fx(c)1400 4345 y Fu(n)1453 4308 y Ft(b)1448 4331 y Fp(E)1509 4345 y Fu(x)1549 4355 y Fr(i)1574 4345 y Fu(;x)1634 4355 y Fr(j)1686 4148 y Ft(2)1686 4312 y(4)1746 4145 y(\014)1746 4199 y(\014)1746 4254 y(\014)1746 4308 y(\014)1746 4363 y(\014)1746 4417 y(\014)1776 4207 y(Z)1827 4414 y Fu(\034)1858 4424 y Fr(i)1904 4331 y Fx(dt)1999 4172 y Ft(\014)1999 4227 y(\014)1999 4281 y(\014)1999 4336 y(\014)1999 4390 y(\014)2029 4207 y(Z)2080 4414 y Fu(\034)2111 4424 y Fr(j)2163 4331 y Fx(dX)2285 4345 y Fu(s)2323 4331 y Fx(g)2366 4345 y Fu(ts)2428 4172 y Ft(\014)2428 4227 y(\014)2428 4281 y(\014)2428 4336 y(\014)2428 4390 y(\014)2458 4198 y Fq(2)2498 4145 y Ft(\014)2498 4199 y(\014)2498 4254 y(\014)2498 4308 y(\014)2498 4363 y(\014)2498 4417 y(\014)2528 4171 y Fu(n)2575 4148 y Ft(3)2575 4312 y(5)1265 4692 y Fw(\024)g Fx(c)1400 4706 y Fu(n)1448 4692 y Fw(j)p Fx(\034)1513 4706 y Fu(i)1541 4692 y Fw(j)1566 4654 y Fu(n)p Fv(\000)p Fq(1)1718 4568 y Ft(Z)1769 4774 y Fu(\034)1800 4784 y Fr(i)1846 4692 y Fx(dt)1931 4668 y Ft(b)1926 4692 y Fp(E)1987 4706 y Fu(x)2027 4716 y Fr(i)2053 4706 y Fu(;x)2113 4716 y Fr(j)2164 4509 y Ft(2)2164 4673 y(4)2224 4533 y(\014)2224 4587 y(\014)2224 4642 y(\014)2224 4696 y(\014)2224 4751 y(\014)2255 4568 y(Z)2305 4774 y Fu(\034)2336 4784 y Fr(j)2388 4692 y Fx(dX)2510 4706 y Fu(s)2548 4692 y Fx(g)2591 4706 y Fu(ts)2653 4533 y Ft(\014)2653 4587 y(\014)2653 4642 y(\014)2653 4696 y(\014)2653 4751 y(\014)2684 4559 y Fq(2)p Fu(n)2766 4509 y Ft(3)2766 4673 y(5)1265 4989 y Fw(\024)g Fx(c)1400 4952 y Fq(2)1400 5012 y Fu(n)1448 4989 y Fw(j)p Fx(\034)1513 5003 y Fu(i)1541 4989 y Fw(j)1566 4952 y Fu(n)p Fv(\000)p Fq(1)1703 4989 y Fw(j)p Fx(\034)1768 5003 y Fu(j)1805 4989 y Fw(j)1830 4952 y Fu(n)p Fv(\000)p Fq(1)1982 4866 y Ft(Z)2033 5072 y Fu(\034)2064 5082 y Fr(i)2110 4989 y Fx(dt)2205 4866 y Ft(Z)2256 5072 y Fu(\034)2287 5082 y Fr(j)2339 4989 y Fx(ds)2434 4966 y Ft(b)2429 4989 y Fp(E)2490 5003 y Fu(x)2530 5013 y Fr(i)2555 5003 y Fu(;x)2615 5013 y Fr(j)2666 4916 y Ft(\002)2704 4989 y Fw(j)p Fx(g)2772 5003 y Fu(ts)2835 4989 y Fw(j)2860 4952 y Fq(2)p Fu(n)2943 4916 y Ft(\003)1265 5209 y Fw(\024)g Fx(c)1400 5171 y Fq(2)1400 5231 y Fu(n)1448 5209 y Fx(b)1487 5171 y Fq(2)p Fu(n)1569 5209 y Fx(C)1641 5171 y Fu(n)1687 5209 y Fy(\(1)c(+)f Fx(b)p Fw(j)p Fx(i)h Fw(\000)f Fx(j)26 b Fw(\000)20 b Fy(1)p Fw(j)p Fy(\))2345 5171 y Fv(\000)p Fq(2)p Fu(n\014)2526 5209 y Fx(:)1726 5549 y Fy(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 0 431 a Fy(No)m(w)31 b(w)m(e)g(estimate)h(also) f(the)g(exp)s(onen)m(tial)g(of)g(the)f(energy)-8 b(.)42 b(By)31 b Fx(e)2295 398 y Fv(j)p Fu(x)p Fv(j)2403 431 y Fw(\024)25 b Fy(2)15 b(cosh)h Fx(x)30 b Fy(w)m(e)h(get)549 619 y Ft(b)544 643 y Fp(E)605 657 y Fu(x)645 667 y Fr(i)671 657 y Fu(;x)731 667 y Fr(j)782 542 y Ft(h)825 643 y Fx(e)867 599 y Fq(4)p Fu(n)p Fv(j)p Fu(\025W)1073 607 y Fr(\034)1102 623 y(i)1128 607 y(;\034)1176 623 y(j)1213 599 y Fv(j)1237 542 y Ft(i)1305 643 y Fw(\024)1406 619 y Ft(b)1401 643 y Fp(E)1462 657 y Fu(x)1502 667 y Fr(i)1527 657 y Fu(;x)1587 667 y Fr(j)1639 542 y Ft(h)1681 643 y Fx(e)1723 599 y Fq(4)p Fu(n\025W)1909 607 y Fr(\034)1938 623 y(i)1965 607 y(;\034)2013 623 y(j)2054 542 y Ft(i)2117 643 y Fy(+)2213 619 y Ft(b)2208 643 y Fp(E)2269 657 y Fu(x)2309 667 y Fr(i)2334 657 y Fu(;x)2394 667 y Fr(j)2446 542 y Ft(h)2489 643 y Fx(e)2531 599 y Fv(\000)p Fq(4)p Fu(n\025W)2772 607 y Fr(\034)2801 623 y(i)2827 607 y(;\034)2875 623 y(j)2916 542 y Ft(i)2974 643 y Fx(:)0 854 y Fy(Eac)m(h)47 b(of)g(the)g(exp)s(ectations)h(in)f(the)f(righ)m(t)i(hand)d(side)i(can) g(b)s(e)f(similarly)h(estimated)h(b)m(y)f(using)0 967 y(Lemma)30 b(5.9)i(b)s(elo)m(w:)961 1155 y Ft(b)956 1179 y Fp(E)1017 1193 y Fu(x)1057 1203 y Fr(i)1082 1193 y Fu(;x)1142 1203 y Fr(j)1193 1078 y Ft(h)1236 1179 y Fx(e)1278 1135 y Fq(4)p Fu(n)p Fv(j)p Fu(\025W)1484 1143 y Fr(\034)1513 1159 y(i)1539 1143 y(;\034)1587 1159 y(j)1624 1135 y Fv(j)1648 1078 y Ft(i)1716 1179 y Fw(\024)25 b Fy(\(1)c Fw(\000)f Fx(n)2059 1141 y Fq(2)2098 1179 y Fx(a)2146 1141 y Fq(2)2185 1179 y Fy(\))2220 1141 y Fv(\000)p Fu(C)2325 1150 y Fn(14)2420 1179 y Fw(\024)25 b Fx(C)0 1411 y Fy(with)j Fx(a)d Fy(=)g Fx(C)439 1425 y Fq(13)514 1411 y Fx(\025b)p Fy(\(1)17 b(+)g Fx(b)p Fw(j)p Fx(i)g Fw(\000)f Fx(j)22 b Fw(\000)16 b Fy(1)p Fw(j)p Fy(\))1241 1378 y Fv(\000)p Fu(\014)1372 1411 y Fy(and)28 b Fx(na)d Fw(\024)g Fy(1)p Fx(=)p Fy(2.)42 b(Hence,)30 b(b)m(y)e(making)h(use)f(these)h (inequalities)0 1524 y(\(uniform)h(in)g Fx(x)535 1538 y Fu(i)563 1524 y Fx(;)15 b(x)655 1538 y Fu(j)692 1524 y Fy(\),)30 b(and)g(the)h(fact)g(that)g(\011)25 b Fw(2)g Fx(L)1736 1538 y Fq(1)1775 1524 y Fy(\()p Fp(R)1876 1491 y Fu(d)1917 1524 y Fx(;)15 b(dx)p Fy(\),)31 b(w)m(e)g(arriv)m(e)g(at) 563 1672 y Ft(\022)630 1677 y(Z)741 1777 y(b)736 1801 y Fp(E)797 1815 y Fu(x)837 1825 y Fr(i)863 1815 y Fu(;x)923 1825 y Fr(j)974 1700 y Ft(h)1017 1801 y Fw(j)p Fx(e)1084 1757 y Fu(\025W)1192 1765 y Fr(\034)1221 1781 y(i)1248 1765 y(;\034)1296 1781 y(j)1356 1801 y Fw(\000)20 b Fy(1)p Fw(j)1517 1763 y Fu(n)1560 1773 y Fr(ij)1620 1801 y Fx(e)1662 1757 y Fv(\000)p Fu(V)1758 1765 y Fr(\034)1787 1781 y(i)1818 1757 y Fv(\000)p Fu(V)1914 1765 y Fr(\034)1943 1781 y(j)1983 1700 y Ft(i)2041 1801 y Fy(\011\()p Fx(x)2199 1815 y Fu(i)2228 1801 y Fy(\))p Fx(dx)2362 1815 y Fu(i)2390 1801 y Fy(\011\()p Fx(x)2548 1815 y Fu(j)2585 1801 y Fy(\))p Fx(dx)2719 1815 y Fu(j)2756 1672 y Ft(\023)2823 1695 y Fq(1)p Fu(=n)2936 1705 y Fr(ij)809 2021 y Fw(\024)25 b Fx(C)7 b(c)1016 1970 y Fq(1)p Fu(=n)1129 1980 y Fr(ij)1016 2033 y Fu(n)1059 2043 y Fr(ij)1188 2021 y Fx(\025be)1322 1984 y Fu(C)e(b)1412 2021 y Fy(\(1)21 b(+)f Fx(b)p Fw(j)p Fx(i)h Fw(\000)f Fx(j)26 b Fw(\000)20 b Fy(1)p Fw(j)p Fy(\))2070 1984 y Fv(\000)p Fu(\014)2173 2021 y Fx(:)3311 1873 y Fy(\(5.38\))0 2229 y(By)39 b(using)g(the)h(estimate)h Fx(c)976 2244 y Fu(k)1058 2229 y Fw(\024)f Fy(\(2)p Fx(k)s Fy(\))1334 2196 y Fq(2)p Fu(k)1453 2229 y Fy(for)e(the)i(univ)m(ersal)f (constan)m(t)i(in)e(the)g(Burkholder-Da)m(vis-)0 2341 y(Gundy)29 b(inequalit)m(y)-8 b(,)33 b(w)m(e)d(furthermore)g(obtain)697 2555 y Fx(c)736 2503 y Fq(1)p Fu(=n)849 2513 y Fr(ij)736 2566 y Fu(n)779 2576 y Fr(ij)908 2555 y Fy(\(1)21 b(+)f Fx(b)p Fw(j)p Fx(i)h Fw(\000)f Fx(j)26 b Fw(\000)20 b Fy(1)p Fw(j)p Fy(\))1566 2517 y Fv(\000)p Fu(\014)1694 2555 y Fw(\024)25 b Fy(\(2)p Fx(n)1925 2569 y Fu(ij)1986 2555 y Fy(\))2021 2517 y Fq(2)2061 2555 y Fy(\(1)c(+)f Fx(b)p Fw(j)p Fx(i)g Fw(\000)g Fx(j)26 b Fw(\000)20 b Fy(1)p Fw(j)p Fy(\))2718 2517 y Fv(\000)p Fu(\014)2821 2555 y Fx(:)0 2754 y Fy(Recall)29 b(that)f Fx(n)520 2768 y Fu(ij)606 2754 y Fy(=)c Fx(A)p Fw(j)p Fx(i)14 b Fw(\000)g Fx(j)5 b Fw(j)991 2721 y Fq(\001)1084 2754 y Fy(with)27 b(\001)e Fx(>)g Fy(1)j(and)f(c)m(ho)s(ose)h(\001)d Fx(<)g(\014)33 b Fy(so)28 b(that)g Fx(an)2704 2768 y Fu(ij)2789 2754 y Fw(!)d Fy(0)j(as)g Fw(j)p Fx(i)14 b Fw(\000)g Fx(j)5 b Fw(j)27 b(!)e(1)0 2867 y Fy(and)30 b(condition)h Fx(an)679 2881 y Fu(ij)765 2867 y Fw(\024)25 b Fy(1)p Fx(=)p Fy(2)31 b(is)g(satis\014ed)f(uniformly)g(in)g Fx(i;)15 b(j)37 b Fy(for)30 b Fx(\025)h Fy(small)g(enough.)40 b(Th)m(us)30 b(there)g(is)h(a)0 2980 y(constan)m(t)h Fx(C)432 2994 y Fq(10)532 2980 y Fx(>)25 b Fy(0)30 b(suc)m(h)g(that)131 3171 y(r.h.s.)40 b(\(5.38\))j Fw(\024)d Fx(C)818 3185 y Fq(10)893 3171 y Fx(\025b)985 3134 y Fq(2)p Fv(\000)p Fq(2\001)1173 3171 y Fx(e)1215 3134 y Fu(C)5 b(b)1305 3171 y Fy(\(1)21 b(+)f Fx(b)p Fw(j)p Fx(i)h Fw(\000)f Fx(j)26 b Fw(\000)19 b Fy(1)p Fw(j)p Fy(\))1962 3134 y Fq(2\001)p Fv(\000)p Fu(\014)2200 3171 y Fw(\024)40 b Fx(C)2376 3185 y Fq(10)2451 3171 y Fx(\025e)2546 3134 y Fu(C)5 b(b)2636 3171 y Fy(\(1)21 b(+)e Fx(b)p Fw(j)p Fx(i)i Fw(\000)f Fx(j)26 b Fw(\000)20 b Fy(1)p Fw(j)p Fy(\))3293 3134 y Fv(\000)p Fu(\016)3387 3171 y Fx(;)3311 3293 y Fy(\(5.39\))0 3489 y(for)34 b(all)g Fw(j)p Fx(i)23 b Fw(\000)g Fx(j)5 b Fw(j)32 b Fx(>)e Fy(1)35 b(and)e Fx(b)e Fw(\025)g Fy(1,)k(where)e Fx(\016)i Fy(=)c Fx(\014)d Fw(\000)22 b Fy(2\001)31 b Fx(>)g Fy(1)j(\(this)h(is)e(p)s(ossible)h(b) m(y)g(c)m(ho)s(osing)g(\001)d Fx(>)g Fy(1)0 3602 y(but)f(small)h (enough\).)40 b(F)-8 b(or)32 b(the)e(cases)h Fw(j)p Fx(i)21 b Fw(\000)f Fx(j)5 b Fw(j)26 b Fy(=)f(1)31 b(w)m(e)g(can)f(follo)m(w)i (a)f(similar)g(strategy)g(to)g(estimate)749 3746 y Ft(\020)803 3847 y Fp(E)864 3861 y Fu(\027)903 3838 y Fb(x)899 3884 y Fr(T)967 3746 y Ft(h)1010 3847 y Fw(j)p Fx(e)1077 3804 y Fu(\025W)1185 3812 y Fr(\034)1214 3828 y(i)1240 3812 y(;\034)1288 3828 y(j)1349 3847 y Fw(\000)20 b Fy(1)p Fw(j)1510 3810 y Fq(2)p Fu(n)1588 3820 y Fr(ij)1648 3847 y Fx(e)1690 3804 y Fv(\000)p Fu(V)1786 3812 y Fr(\034)1815 3828 y(i)1846 3804 y Fv(\000)p Fu(V)1942 3812 y Fr(\034)1971 3828 y(j)2011 3746 y Ft(i\021)2109 3769 y Fq(1)p Fu(=)p Fq(2)p Fu(n)2257 3779 y Fr(ij)2341 3847 y Fw(\024)25 b Fx(C)2502 3861 y Fq(11)2577 3847 y Fx(e)2619 3810 y Fu(C)2669 3819 y Fn(12)2735 3810 y Fu(b)2769 3847 y Fx(;)0 4064 y Fy(where)30 b(the)g(constan)m(ts)i(do)e(not)h(dep)s(end)e(on)h (either)h Fx(b)f Fy(or)g Fx(n)2045 4078 y Fu(ij)2105 4064 y Fy(.)1343 b Fi(\003)0 4307 y Fl(5.4)112 b(Energy)38 b(estimates)0 4478 y Fy(Here)31 b(w)m(e)g(estimate)719 4455 y Ft(b)713 4478 y Fp(E)774 4492 y Fu(x;y)875 4478 y Fy([)p Fx(e)942 4435 y Fu(\025W)1050 4443 y Fr(\034)1079 4459 y(i)1106 4443 y(;\034)1154 4459 y(j)1194 4478 y Fy(].)42 b(Belo)m(w)32 b(w)m(e)f(will)g(pro)m(v)m(e)g(t)m(w)m(o)h (lemmas)f(that)g(giv)m(e)h(the)f(basic)g(esti-)0 4591 y(mates)26 b(b)m(y)g(making)g(use)f(of)h(the)g(follo)m(wing)h(result)f (to)g(con)m(trol)h(exp)s(onen)m(tial)g(in)m(tegrabilit)m(y)h(of)e(sto)s (c)m(has-)0 4704 y(tic)36 b(in)m(tegrals.)56 b(The)35 b(\014rst)f(lemma)i(will)f(b)s(e)g(often)g(used)g(for)f(con)m(trolling) j(exp)s(onen)m(tial)g(momen)m(ts)e(of)0 4817 y(sto)s(c)m(hastic)d(in)m (tegrals.)0 5000 y Fk(Lemma)j(5.8)46 b Fo(If)30 b Fx(X)38 b Fo(is)31 b(Br)-5 b(ownian)32 b(motion)g(and)g Fx(f)40 b Fo(is)30 b(an)2053 5008 y Fw(F)2128 5000 y Fo(-adapte)-5 b(d)32 b(pr)-5 b(o)g(c)g(ess,)34 b(we)c(have)h(the)g(b)-5 b(ound)1102 5241 y Fp(E)p Fy([)p Fx(e)1230 5150 y Fg(R)1277 5171 y Fr(b)1263 5228 y Fn(0)1320 5204 y Fu(f)1354 5212 y Fr(s)1387 5204 y Fu(dX)1481 5212 y Fr(s)1519 5241 y Fy(])25 b Fw(\024)1665 5140 y Ft(n)1726 5241 y Fp(E)p Fy([)p Fx(e)1854 5204 y Fq(2)1901 5150 y Fg(R)1948 5171 y Fr(b)1935 5228 y Fn(0)1991 5204 y Fv(j)p Fu(f)2045 5212 y Fr(s)2078 5204 y Fv(j)2098 5180 y Fn(2)2132 5204 y Fu(ds)2206 5241 y Fy(])2231 5140 y Ft(o)2291 5162 y Fq(1)p Fu(=)p Fq(2)2417 5241 y Fx(:)869 b Fy(\(5.40\))1726 5549 y(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 0 431 a Fj(Pr)n(oof.)40 b Fy(The)30 b(pro)s(of)f(is)i(a)g(com)m(bination)g(of)g(Cauc)m(h)m(y-Sc)m(h)m(w)m (artz)h(and)e(Girsano)m(v)h(Theorems:)527 625 y Fp(E)p Fy([)p Fx(e)655 533 y Fg(R)703 554 y Fr(b)689 611 y Fn(0)746 587 y Fu(f)780 595 y Fr(s)813 587 y Fu(dX)907 595 y Fr(s)945 625 y Fy(])25 b(=)g Fp(E)p Fy([)p Fx(e)1219 533 y Fg(R)1267 554 y Fr(b)1253 611 y Fn(0)1310 587 y Fu(f)1344 595 y Fr(s)1377 587 y Fu(dX)1471 595 y Fr(s)1505 587 y Fv(\000)1560 533 y Fg(R)1606 554 y Fr(b)1593 611 y Fn(0)1650 587 y Fv(j)p Fu(f)1704 595 y Fr(s)1736 587 y Fv(j)1756 564 y Fn(2)1791 587 y Fu(ds)p Fq(+)1915 533 y Fg(R)1962 554 y Fr(b)1948 611 y Fn(0)2005 587 y Fv(j)p Fu(f)2059 595 y Fr(s)2091 587 y Fv(j)2111 564 y Fn(2)2146 587 y Fu(ds)2219 625 y Fy(])995 820 y Fw(\024)1091 719 y Ft(n)1152 820 y Fp(E)p Fy([)p Fx(e)1280 783 y Fq(2)1327 729 y Fg(R)1374 750 y Fr(b)1360 807 y Fn(0)1417 783 y Fu(f)1451 791 y Fr(s)1485 783 y Fu(dX)1579 791 y Fr(s)1612 783 y Fv(\000)1677 756 y Fn(1)p 1677 768 31 3 v 1677 809 a(2)1729 729 y Fg(R)1776 750 y Fr(b)1763 807 y Fn(0)1819 783 y Fv(j)p Fq(2)p Fu(f)1908 791 y Fr(s)1942 783 y Fv(j)1962 759 y Fn(2)1996 783 y Fu(ds)2069 820 y Fy(])2094 719 y Ft(o)2155 741 y Fq(1)p Fu(=)p Fq(2)2280 719 y Ft(n)2340 820 y Fp(E)p Fy([)p Fx(e)2468 783 y Fq(2)2516 729 y Fg(R)2563 750 y Fr(b)2549 807 y Fn(0)2606 783 y Fv(j)p Fu(f)2660 791 y Fr(s)2693 783 y Fv(j)2713 759 y Fn(2)2747 783 y Fu(ds)2820 820 y Fy(])2845 719 y Ft(o)2906 741 y Fq(1)p Fu(=)p Fq(2)995 1041 y Fw(\024)1091 940 y Ft(n)1152 1041 y Fp(E)p Fy([)p Fx(e)1280 1004 y Fq(2)1327 950 y Fg(R)1374 971 y Fr(b)1360 1028 y Fn(0)1417 1004 y Fv(j)p Fu(f)1471 1012 y Fr(s)1504 1004 y Fv(j)1524 980 y Fn(2)1558 1004 y Fu(ds)1631 1041 y Fy(])1656 940 y Ft(o)1717 962 y Fq(1)p Fu(=)p Fq(2)1842 1041 y Fx(:)3473 1232 y Fi(\003)0 1386 y Fe(E)p Fo(stimates)35 b(for)g(sep)-5 b(ar)g(ate)g(d)37 b(intervals)132 b Fy(W)-8 b(e)33 b(turn)e(to)i(estimating)g(exp)s(onen)m(tials)g(of)g(energy)f (con)m(tri-)0 1498 y(butions)23 b(in)h(\(5.1\))i(b)m(y)e(starting)h (with)e(pairs)h(of)g(in)m(terv)-5 b(als)25 b(that)g(are)f(not)h (adjacen)m(t.)39 b(Let)25 b(th)m(us)f Fx(i)h(>)g(j)13 b Fy(+)8 b(1;)0 1611 y(in)30 b(this)g(case)i(the)e(exp)s(onen)m(t)h (has)f(the)g(form)g Fx(W)1643 1625 y Fu(\034)1674 1635 y Fr(i)1701 1625 y Fu(;\034)1752 1635 y Fr(j)1814 1611 y Fy(=)1910 1615 y Fw(J)1971 1628 y Fu(ij)2047 1611 y Fy(+)2133 1615 y Fw(J)2194 1628 y Fu(j)t(i)2285 1611 y Fy(with)549 1863 y Fw(J)611 1876 y Fu(i;j)716 1859 y Fy(=)25 b Fw(h)p Fx(C)919 1822 y Fu(X)912 1882 y(\034)943 1892 y Fr(i)986 1859 y Fx(;)15 b(C)1098 1822 y Fu(X)1091 1882 y(\034)1122 1892 y Fr(j)1165 1859 y Fw(i)1200 1873 y Fu(W)1306 1859 y Fy(=)1402 1735 y Ft(Z)1493 1762 y Fu(t)1518 1772 y Fr(i)1545 1762 y Fq(+)p Fu(b)1453 1942 y(t)1478 1952 y Fr(i)1650 1859 y Fx(dX)1772 1873 y Fu(t)1817 1735 y Ft(Z)1908 1762 y Fu(t)1933 1772 y Fr(j)1966 1762 y Fq(+)p Fu(b)1867 1942 y(t)1892 1952 y Fr(j)2070 1859 y Fx(dX)2192 1873 y Fu(s)2245 1859 y Fx(W)e Fy(\()p Fx(X)2454 1873 y Fu(t)2504 1859 y Fw(\000)20 b Fx(X)2670 1873 y Fu(s)2707 1859 y Fx(;)15 b(t)20 b Fw(\000)g Fx(s)p Fy(\))p Fx(:)317 b Fy(\(5.41\))0 2112 y(Note)37 b(that)f(under)f(the)h(measure) 1212 2120 y Fw(W)1310 2079 y Fu(x)1350 2089 y Fr(i)1302 2134 y Fu(\034)1333 2144 y Fr(i)1395 2112 y Fw(\012)1481 2120 y(W)1578 2061 y Fu(x)1618 2071 y Fr(j)1571 2124 y Fu(\034)1602 2134 y Fr(j)1690 2112 y Fy(the)g(t)m(w)m(o)h(curren)m (ts)f Fp(X)2453 2126 y Fu(\034)2484 2136 y Fr(i)2550 2112 y Fy(and)f Fp(X)2798 2126 y Fu(\034)2829 2136 y Fr(j)2901 2112 y Fy(are)h(indep)s(enden)m(t)0 2225 y(and)24 b(the)g(in)m(teraction)j(energy)d(can)h(b)s(e)f(written)g(as)h(a)g (double)f(sto)s(c)m(hastic)i(It^)-45 b(o)25 b(in)m(tegral)h(and)e (estimated)0 2338 y(b)m(y)30 b(using)g(to)s(ols)h(b)s(orro)m(w)m(ed)f (from)g(sto)s(c)m(hastic)i(analysis.)0 2501 y Fk(Lemma)j(5.9)46 b Fo(L)-5 b(et)33 b Fx(i)25 b(>)g(j)h Fy(+)20 b(1)p Fo(.)42 b(Ther)-5 b(e)33 b(exist)g(p)-5 b(ositive)34 b(c)-5 b(onstants)34 b Fx(C)2393 2515 y Fq(13)2468 2501 y Fx(;)15 b(C)2573 2515 y Fq(14)2681 2501 y Fo(such)32 b(that)i(whenever)1077 2677 y Fx(a)25 b Fy(:=)g Fx(\025C)1389 2691 y Fq(13)1464 2677 y Fx(b)p Fy(\(1)c(+)f Fw(j)p Fx(t)1753 2691 y Fu(i)1801 2677 y Fw(\000)g Fx(t)1925 2691 y Fu(j)1982 2677 y Fw(\000)g Fx(b)p Fw(j)p Fy(\))2172 2639 y Fv(\000)p Fu(\014)2300 2677 y Fx(<)25 b Fy(1)p Fx(;)845 b Fy(\(5.42\))0 2852 y Fo(we)33 b(have)1196 3027 y Ft(b)1191 3050 y Fp(E)1252 3064 y Fu(x;y)1352 3050 y Fy([)p Fx(e)1419 3006 y Fu(\025W)1527 3014 y Fr(\034)1556 3031 y(i)1583 3014 y(;\034)1631 3031 y(j)1672 3050 y Fy(])25 b Fw(\024)1818 2976 y Ft(\002)1856 3050 y Fy(1)c Fw(\000)e Fx(a)2060 3013 y Fq(2)2100 2976 y Ft(\003)2138 2996 y Fv(\000)p Fu(C)2243 3005 y Fn(14)2327 3050 y Fx(:)0 3228 y Fj(Pr)n(oof.)40 b Fy(W)-8 b(e)31 b(ha)m(v)m(e)759 3323 y Ft(\020)818 3401 y(b)813 3424 y Fp(E)874 3438 y Fu(x;y)975 3424 y Fy([)p Fx(e)1042 3380 y Fu(\025W)1150 3388 y Fr(\034)1179 3405 y(i)1205 3388 y(;\034)1253 3405 y(j)1294 3424 y Fy(])1319 3323 y Ft(\021)1374 3346 y Fq(4)1438 3424 y Fw(\024)1534 3323 y Ft(\020)1594 3401 y(b)1589 3424 y Fp(E)1650 3438 y Fu(x;y)1750 3424 y Fy([)p Fx(e)1817 3386 y Fq(2)p Fu(\025)1906 3389 y Fv(J)1954 3399 y Fr(ij)2013 3424 y Fy(])2038 3323 y Ft(\021)2092 3346 y Fq(2)2147 3323 y Ft(\020)2206 3401 y(b)2201 3424 y Fp(E)2262 3438 y Fu(x;y)2362 3424 y Fy([)p Fx(e)2429 3386 y Fq(2)p Fu(\025)2518 3389 y Fv(J)2566 3399 y Fr(j)s(i)2625 3424 y Fy(])2650 3323 y Ft(\021)2704 3346 y Fq(2)2759 3424 y Fx(:)0 3622 y Fy(By)g(using)f(\(5.40\))i(w)m(e) f(obtain)168 3801 y Ft(\020)227 3878 y(b)222 3902 y Fp(E)283 3916 y Fu(x;y)383 3902 y Fy([)p Fx(e)450 3864 y Fu(\025)504 3867 y Fv(J)552 3877 y Fr(ij)611 3902 y Fy(])636 3801 y Ft(\021)690 3824 y Fq(2)755 3902 y Fw(\024)856 3878 y Ft(b)851 3902 y Fp(E)912 3916 y Fu(x;y)1027 3719 y Ft(2)1027 3883 y(4)1088 3902 y Fy(exp)1242 3719 y Ft(0)1242 3883 y(@)1321 3902 y Fy(2)p Fx(\025)1419 3864 y Fq(2)1475 3778 y Ft(Z)1565 3805 y Fu(t)1590 3815 y Fr(i)1617 3805 y Fq(+)p Fu(b)1525 3984 y(t)1550 3994 y Fr(i)1722 3902 y Fx(dt)1817 3743 y Ft(\014)1817 3797 y(\014)1817 3852 y(\014)1817 3907 y(\014)1817 3961 y(\014)1847 3778 y(Z)1938 3805 y Fu(t)1963 3815 y Fr(j)1996 3805 y Fq(+)p Fu(b)1898 3984 y(t)1923 3994 y Fr(j)2101 3902 y Fx(dX)2223 3916 y Fu(s)2260 3902 y Fx(W)13 b Fy(\()p Fx(X)2469 3916 y Fu(t)2519 3902 y Fw(\000)20 b Fx(X)2685 3916 y Fu(s)2722 3902 y Fx(;)15 b(t)20 b Fw(\000)g Fx(s)p Fy(\))2984 3743 y Ft(\014)2984 3797 y(\014)2984 3852 y(\014)2984 3907 y(\014)2984 3961 y(\014)3015 3769 y Fq(2)3054 3719 y Ft(1)3054 3883 y(A)3134 3719 y(3)3134 3883 y(5)755 4262 y Fw(\024)851 4139 y Ft(Z)942 4165 y Fu(t)967 4175 y Fr(i)994 4165 y Fq(+)p Fu(b)901 4345 y(t)926 4355 y Fr(i)1108 4201 y Fx(dt)p 1108 4242 81 4 v 1128 4325 a(b)1203 4239 y Ft(b)1198 4262 y Fp(E)1259 4276 y Fu(x;y)1374 4080 y Ft(2)1374 4243 y(4)1435 4262 y Fy(exp)1589 4080 y Ft(0)1589 4243 y(@)1669 4262 y Fy(2)p Fx(b\025)1806 4225 y Fq(2)1861 4103 y Ft(\014)1861 4158 y(\014)1861 4212 y(\014)1861 4267 y(\014)1861 4322 y(\014)1891 4139 y(Z)1982 4165 y Fu(t)2007 4175 y Fr(j)2040 4165 y Fq(+)p Fu(b)1942 4345 y(t)1967 4355 y Fr(j)2144 4262 y Fx(dX)2266 4276 y Fu(s)2304 4262 y Fx(W)13 b Fy(\()p Fx(X)2513 4276 y Fu(t)2563 4262 y Fw(\000)20 b Fx(X)2729 4276 y Fu(s)2766 4262 y Fx(;)15 b(t)20 b Fw(\000)g Fx(s)p Fy(\))3028 4103 y Ft(\014)3028 4158 y(\014)3028 4212 y(\014)3028 4267 y(\014)3028 4322 y(\014)3058 4130 y Fq(2)3098 4080 y Ft(1)3098 4243 y(A)3177 4080 y(3)3177 4243 y(5)755 4596 y Fw(\024)851 4472 y Ft(Z)942 4498 y Fu(t)967 4508 y Fr(i)994 4498 y Fq(+)p Fu(b)901 4678 y(t)926 4688 y Fr(i)1108 4534 y Fx(dt)p 1108 4575 V 1128 4658 a(b)1203 4572 y Ft(b)1198 4596 y Fp(E)1259 4610 y Fu(x;y)1359 4596 y Fp(E)1420 4610 y Fu(G)1494 4440 y Ft(")1547 4596 y Fy(exp)1701 4440 y Ft( )1773 4596 y Fy(2)p Fx(\025G)1942 4514 y Fw(p)p 2019 4514 40 4 v 2019 4596 a Fx(b)2073 4472 y Ft(Z)2164 4498 y Fu(t)2189 4508 y Fr(j)2222 4498 y Fq(+)p Fu(b)2124 4678 y(t)2149 4688 y Fr(j)2326 4596 y Fx(dX)2448 4610 y Fu(s)2486 4596 y Fx(W)13 b Fy(\()p Fx(X)2695 4610 y Fu(t)2745 4596 y Fw(\000)20 b Fx(X)2911 4610 y Fu(s)2948 4596 y Fx(;)15 b(t)20 b Fw(\000)g Fx(s)p Fy(\))3210 4440 y Ft(!#)3350 4596 y Fx(;)0 4843 y Fy(with)31 b Fx(G)g Fy(a)g(normal)g(random)g(v)-5 b(ariable)31 b(de\014ned)f(on)h(a)g(new)g (probabilit)m(y)h(space)f(and)f Fp(E)3023 4857 y Fu(G)3113 4843 y Fy(the)h(related)0 4956 y(Gaussian)g(exp)s(ectation.)42 b(Then,)29 b(using)h(again)i(\(5.40\))g(yields)0 5131 y Ft(\020)59 5209 y(b)54 5232 y Fp(E)115 5246 y Fu(x;y)216 5233 y Fy([)p Fx(e)283 5189 y Fu(\025W)391 5197 y Fr(\034)420 5213 y(i)446 5197 y(;\034)494 5213 y(j)535 5233 y Fy(])560 5131 y Ft(\021)615 5154 y Fq(2)679 5233 y Fw(\024)775 5109 y Ft(Z)866 5135 y Fu(t)891 5145 y Fr(i)918 5135 y Fq(+)p Fu(b)826 5315 y(t)851 5325 y Fr(i)1032 5171 y Fx(dt)p 1032 5211 81 4 v 1053 5295 a(b)1138 5077 y Ft( )1215 5209 y(b)1210 5232 y Fp(E)1271 5246 y Fu(x;y)1371 5233 y Fp(E)1432 5247 y Fu(G)1506 5077 y Ft(")1559 5233 y Fy(exp)1713 5077 y Ft( )1785 5233 y Fy(8)p Fx(\025)1883 5195 y Fq(2)1923 5233 y Fx(G)1994 5195 y Fq(2)2034 5233 y Fx(b)2088 5109 y Ft(Z)2179 5135 y Fu(t)2204 5145 y Fr(j)2237 5135 y Fq(+)p Fu(b)2138 5315 y(t)2163 5325 y Fr(j)2341 5233 y Fx(ds)p Fw(j)p Fx(W)13 b Fy(\()p Fx(X)2665 5247 y Fu(t)2715 5233 y Fw(\000)20 b Fx(X)2881 5247 y Fu(s)2918 5233 y Fx(;)15 b(t)21 b Fw(\000)f Fx(s)p Fy(\))p Fw(j)3206 5195 y Fq(2)3245 5077 y Ft(!#!)3442 5099 y Fq(1)p Fu(=)p Fq(2)3567 5233 y Fx(:)1726 5549 y Fy(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 0 431 a Fy(The)30 b(assumptions)g(on)g Fx(W)43 b Fy(giv)m(e)142 554 y Ft(Z)233 580 y Fu(t)258 590 y Fr(j)291 580 y Fq(+)p Fu(b)192 760 y(t)217 770 y Fr(j)395 678 y Fx(ds)p Fw(j)p Fx(W)13 b Fy(\()p Fx(Y)697 692 y Fu(t)747 678 y Fw(\000)20 b Fx(Z)900 692 y Fu(s)937 678 y Fx(;)15 b(t)20 b Fw(\000)g Fx(s)p Fy(\))p Fw(j)1224 640 y Fq(2)1289 678 y Fw(\024)1385 554 y Ft(Z)1476 580 y Fu(t)1501 590 y Fr(j)1534 580 y Fq(+)p Fu(b)1435 760 y(t)1460 770 y Fr(j)1638 678 y Fx(ds)1976 616 y(C)p 1738 656 547 4 v 1738 740 a Fy(\(1)h(+)f Fw(j)p Fx(t)g Fw(\000)g Fx(s)p Fw(j)p Fy(\))2202 713 y Fq(2)p Fu(\014)2320 678 y Fw(\024)25 b Fx(C)2481 692 y Fq(15)2556 678 y Fx(b)p Fy(\(1)c(+)f Fw(j)p Fx(t)2845 692 y Fu(i)2893 678 y Fw(\000)g Fx(t)3017 692 y Fu(j)3074 678 y Fw(\000)f Fx(b)p Fw(j)p Fy(\))3263 640 y Fv(\000)p Fq(2)p Fu(\014)0 933 y Fy(for)30 b(some)h(constan)m(t)h Fx(C)799 947 y Fq(15)873 933 y Fy(,)f(where)f(w)m(e)h(used)e(that)i Fx(t)25 b Fw(2)g Fy([)p Fx(t)1932 947 y Fu(i)1960 933 y Fx(;)15 b(t)2033 947 y Fu(i)2082 933 y Fy(+)20 b Fx(b)p Fy(].)41 b(Hence)393 1068 y Ft(\020)452 1145 y(b)447 1169 y Fp(E)508 1183 y Fu(x;y)609 1170 y Fy([)p Fx(e)676 1131 y Fq(2)p Fu(\025)765 1134 y Fv(J)812 1144 y Fr(ij)871 1170 y Fy(])896 1068 y Ft(\021)951 1091 y Fq(2)1015 1170 y Fw(\024)1111 1068 y Ft(\020)1166 1170 y Fp(E)1227 1184 y Fu(G)1301 1068 y Ft(h)1344 1170 y Fy(exp)1498 1068 y Ft(h)1541 1170 y Fy(32)p Fx(\025)1684 1131 y Fq(2)1724 1170 y Fx(C)1789 1184 y Fq(15)1864 1170 y Fx(b)1903 1131 y Fq(2)1942 1170 y Fy(\(1)21 b(+)f Fw(j)p Fx(t)2192 1184 y Fu(i)2240 1170 y Fw(\000)g Fx(t)2364 1184 y Fu(j)2421 1170 y Fw(\000)g Fx(b)p Fw(j)p Fy(\))2611 1131 y Fv(\000)p Fq(2)p 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y Fw(\000)g Fx(t)1414 1845 y Fu(j)1470 1831 y Fw(\000)g Fx(b)p Fw(j)p Fy(\))1660 1798 y Fv(\000)p Fq(2)p Fu(\014)1823 1831 y Fx(>)25 b Fy(0)p Fx(:)31 b Fy(Th)m(us)f(the)g(claim)i(follo)m (ws.)531 b Fi(\003)0 1989 y Fe(E)p Fo(stimates)47 b(for)f(adjac)-5 b(ent)47 b(intervals)144 b Fy(The)45 b(in)m(teraction)h(energy)f (estimates)i(b)s(et)m(w)m(een)e(adjacen)m(t)0 2101 y(in)m(terv)-5 b(als)31 b(are)g(giv)m(en)h(b)m(y)0 2300 y Fk(Lemma)j(5.10)46 b Fo(F)-7 b(or)36 b(al)5 b(l)36 b Fx(i)31 b Fy(=)f(0)p Fx(;)15 b(:::;)g(N)34 b Fw(\000)22 b Fy(2)35 b Fo(we)h(have)1905 2276 y Ft(b)1900 2300 y Fp(E)1961 2314 y Fu(x;y)2061 2300 y Fy([)p Fx(e)2128 2259 y Fu(\025W)2236 2267 y Fr(\034)2265 2283 y(i)2292 2267 y(;\034)2340 2283 y(i)p Fn(+1)2451 2300 y Fy(])31 b Fw(\024)f Fx(C)2673 2314 y Fq(16)2747 2300 y Fx(e)2789 2267 y Fu(C)2839 2276 y Fn(17)2905 2267 y Fu(b)2970 2300 y Fx(<)g Fw(1)35 b Fo(for)h(su\016-)0 2412 y(ciently)d(smal)5 b(l)33 b Fx(\025)p Fo(.)0 2611 y Fj(Pr)n(oof.)52 b Fy(By)35 b(using)f(that)i Fx(W)13 b Fy(\()p Fx(x;)i(t)p Fy(\))34 b(is)h(b)s(ounded)e(and)h(argumen)m(ts)g (similar)h(to)h(those)f(of)f(the)h(previ-)0 2723 y(ous)k(lemma,)j(the)d (required)f(exp)s(onen)m(tial)i(in)m(tegrabilit)m(y)h(of)e Fx(W)2245 2737 y Fu(\034)2276 2747 y Fr(i)2303 2737 y Fu(;\034)2354 2747 y Fr(i)p Fn(+1)2500 2723 y Fy(easily)h(follo)m(ws,)j (at)c(least)h(for)0 2836 y(su\016cien)m(tly)31 b(small)g Fx(\025)p Fy(.)2702 b Fi(\003)0 3078 y Fl(5.5)112 b(Prop)s(erties)37 b(of)h(the)f(cluster)h(expansion)0 3249 y Fy(W)-8 b(e)37 b(\014nally)f(sho)m(w)f(ho)m(w)h(the)g(con)m(v)m(ergence)i(of)e(the)g (cluster)g(expansion)g(of)f Fx(Z)2706 3263 y Fu(T)2797 3249 y Fy(and)g(log)18 b Fx(Z)3174 3263 y Fu(T)3265 3249 y Fy(seen)36 b(in)0 3362 y(Prop)s(osition)30 b(5.2)i(imply)e(existence) i(of)e(a)h(limit)g(Gibbs)f(measure)g Fx(\026)p Fy(.)141 3475 y(F)-8 b(or)31 b(an)m(y)g(subset)f Fx(A)25 b Fw(\032)g Fp(R)30 b Fy(let)1127 3729 y Fx(Z)1196 3692 y Fu(A)1189 3752 y(T)1277 3729 y Fy(=)25 b(1)c(+)1531 3643 y Ft(X)1530 3839 y Fu(n)p Fv(\025)p Fq(1)1833 3643 y Ft(X)1712 3828 y Fn(\000)1750 3843 y(1)1784 3828 y Fr(;:::;)p Fn(\000)1917 3836 y Fr(r)1951 3828 y Fm(2C)2027 3845 y Fr(N)1698 3893 y Fn(\000)1736 3876 y Fm(\003)1736 3918 y Fr(i)1771 3893 y Fm(\\)p Fn(\000)1851 3876 y Fm(\003)1851 3918 y Fr(j)1886 3893 y Fn(=)p Fm(;)p Fr(;)c(i)p Fm(6)p Fn(=)p Fr(j)1721 3977 y(A)p Fm(\\)p Fn(\()p Fm([)1875 3993 y Fr(i)1900 3977 y Fn(\000)1938 3960 y Fm(\003)1938 4003 y Fr(i)1974 3977 y Fn(\)=)p Fm(;)2175 3616 y Fu(r)2134 3643 y Ft(Y)2135 3838 y Fu(i)p Fq(=1)2265 3729 y Fx(K)2342 3743 y Fq(\000)2386 3753 y Fr(i)0 4190 y Fy(and)30 b(write)g Fx(Z)478 4157 y Fq(\000)471 4217 y Fu(T)552 4190 y Fy(:=)25 b Fx(Z)p 742 4103 45 3 v 742 4157 a Fq(\000)p Fv([)p Fq(\000)877 4134 y Fm(\003)735 4217 y Fu(T)917 4190 y Fy(.)40 b(By)31 b(the)g(cluster)f(expansion)h(w)m(e)f(ha)m(v)m(e)795 4454 y(log)17 b Fx(Z)996 4416 y Fq(\000)989 4476 y Fu(T)1070 4454 y Fy(=)25 b(1)20 b(+)1323 4368 y Ft(X)1322 4563 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Fw(\000)1433 5060 y Ft(X)1432 5255 y Fu(n)p Fv(\025)p Fq(1)1716 5060 y Ft(X)1596 5245 y Fn(\000)1634 5260 y(1)1668 5245 y Fr(;:::;)p Fn(\000)1801 5253 y Fr(r)1835 5245 y Fm(2C)1911 5262 y Fr(N)1590 5311 y Fn(\000)1628 5297 y Fm(\003)1664 5311 y(\\)p Fn(\()p Fm([)1772 5327 y Fr(i)1797 5311 y Fn(\000)1835 5294 y Fm(\003)1835 5337 y Fr(i)1871 5311 y Fn(\))p Fm(6)p Fn(=)p Fm(;)1998 5146 y Fx(\036)2052 5108 y Fu(T)2107 5146 y Fy(\(\000)2199 5160 y Fq(1)2239 5146 y Fx(;)15 b(:::;)g Fy(\000)2451 5160 y Fu(n)2499 5146 y Fy(\))2591 5032 y Fu(r)2550 5060 y Ft(Y)2551 5255 y Fu(i)p Fq(=1)2681 5146 y Fx(K)2758 5160 y Fq(\000)2802 5170 y Fr(i)2833 4909 y Ft(1)2833 5069 y(C)2833 5123 y(C)2833 5181 y(A)3311 5146 y Fy(\(5.43\))1726 5549 y(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 0 431 a Fy(and)37 b(let)h Fx(f)377 398 y Fq(\000)461 431 y Fy(=)e(lim)694 445 y Fu(T)10 b Fv(!1)906 431 y Fx(f)961 398 y Fq(\000)951 458 y Fu(T)1045 431 y Fy(as)38 b(the)f(limit)h(exists)g(b)m(y)f(the)g(cluster)h (estimates)g(ab)s(o)m(v)m(e)h(and)d(general)0 544 y(argumen)m(ts)31 b(of)f(cluster)h(expansion)f([40)q(].)41 b(Moreo)m(v)m(er)33 b(w)m(e)e(ha)m(v)m(e)g(the)g(uniform)e(estimate)1568 740 y Fw(j)p Fx(f)1648 702 y Fq(\000)1638 762 y Fu(T)1696 740 y Fw(j)c(\024)g Fy(2)1887 702 y Fv(j)p 1907 648 45 3 v Fq(\000)p Fv(j)3311 740 y Fy(\(5.44\))0 920 y(for)30 b Fx(\025)f Fy(small)h(enough)g(\(the)g(constan)m(t)h(2)f(can)g (actually)i(b)s(e)d(replaced)h(with)g(an)m(y)g(n)m(um)m(b)s(er)e (larger)i(than)0 1033 y(1,)35 b(pro)m(vided)e Fx(\025)h Fy(is)g(c)m(hosen)g(corresp)s(ondingly)f(small\).)51 b(Then)33 b(existence)i(of)f(the)g(in\014nite)f(time)h(limit)0 1146 y(for)c(the)h(measures)f Fw(f)p Fx(\026)786 1160 y Fu(T)841 1146 y Fw(g)886 1160 y Fu(T)972 1146 y Fy(follo)m(ws)h (easily)h(and)e(w)m(e)h(ha)m(v)m(e)0 1313 y Fk(Prop)s(osition)36 b(5.11)47 b Fo(The)32 b(lo)-5 b(c)g(al)35 b(limit)e Fx(\026)25 b Fy(=)g(lim)1705 1327 y Fu(T)10 b Fv(!1)1916 1313 y Fx(\026)1971 1327 y Fu(T)2059 1313 y Fo(exists)33 b(and)g(satis\014es)h (the)f(e)-5 b(quality)688 1557 y Fp(E)749 1571 y Fu(\026)795 1557 y Fy([)p Fx(F)878 1571 y Fu(S)929 1557 y Fy(])26 b(=)f Fp(E)1137 1571 y Fu(\037)1184 1557 y Fy([)p Fx(F)1267 1571 y Fu(S)1319 1557 y Fy(])p Fx(f)1399 1520 y Fu(S)1470 1557 y Fy(+)1561 1471 y Ft(X)1560 1667 y Fu(n)p Fv(\025)p Fq(1)1863 1471 y Ft(X)1771 1657 y Fn(\000)1809 1672 y(1)1843 1657 y Fr(;:::;)p Fn(\000)1976 1665 y Fr(r)2010 1657 y Fm(2C)1728 1719 y Fn(\000)1766 1702 y Fm(\003)1766 1745 y Fr(i)1802 1719 y Fm(\\)p Fn(\000)1882 1702 y Fm(\003)1882 1745 y Fr(j)1917 1719 y Fn(=)p Fm(;)p Fr(;)17 b(i)p Fm(6)p Fn(=)p Fr(j)1792 1804 y(i)p Fn(:)p Fr(S)s Fm(\\)p Fn(\000)1953 1786 y Fm(\003)1953 1829 y Fr(i)1987 1804 y Fm(6)p Fn(=)p Fm(;)2164 1557 y Fp(E)2225 1571 y Fu(\037)2273 1557 y Fy([)p Fx(F)2356 1571 y Fu(S)2464 1444 y(r)2422 1471 y Ft(Y)2423 1666 y Fu(i)p Fq(=1)2554 1557 y Fx(\024)2606 1520 y Fq(\000)2650 1530 y Fr(i)2681 1557 y Fy(])p Fx(f)2761 1520 y Fv([)p 2808 1465 V Fq(\000)3311 1557 y Fy(\(5.45\))0 1977 y Fo(for)32 b(any)h(b)-5 b(ounde)g(d,)33 b Fw(F)756 1991 y Fu(S)807 1977 y Fo(-me)-5 b(asur)g(able)34 b(function)e Fx(F)1718 1991 y Fu(S)1769 1977 y Fo(,)g(wher)-5 b(e)32 b Fx(S)37 b Fo(is)32 b(a)g(\014nite)g(union)g(of)g(intervals)h(of)f (the)0 2090 y(p)-5 b(artition)38 b(c)-5 b(onsider)g(e)g(d)38 b(in)e(the)g(cluster)g(exp)-5 b(ansion.)54 b(Mor)-5 b(e)g(over,)38 b(the)e(me)-5 b(asur)g(e)37 b Fx(\026)f Fo(is)f(invariant)i(with)0 2202 y(r)-5 b(esp)g(e)g(ct)34 b(to)f(time)g(shift.)0 2370 y Fj(Pr)n(oof.)40 b Fy(W)-8 b(e)31 b(ha)m(v)m(e)660 2591 y Fp(E)721 2605 y Fu(\026)767 2591 y Fy([)p Fx(F)850 2605 y Fu(S)901 2591 y Fy(])26 b(=)58 b(lim)1048 2652 y Fu(T)10 b Fv(!1)1265 2455 y Ft(R)1341 2528 y Fx(e)1383 2495 y Fv(\000)p Fu(\025W)1546 2506 y Fr(T)1594 2495 y Fq(\()p Fu(X)5 b Fq(\))1717 2528 y Fx(F)13 b Fy(\()p Fx(X)7 b Fy(\))p Fx(d\027)2032 2542 y Fu(T)2088 2528 y Fy(\()p Fx(X)g Fy(\))p 1265 2570 977 4 v 1377 2584 a Ft(R)1453 2657 y Fx(e)1495 2631 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Fy(])p Fx(f)1537 4473 y Fu(S)1607 4511 y Fy(+)1852 4425 y Ft(X)1698 4631 y Fq(\000)1742 4640 y Fn(0)1777 4631 y Fq(:)1801 4614 y(\026)1797 4631 y(\000)1841 4640 y Fn(0)1875 4631 y Fv(\\)p Fq([0)p Fu(;b)p Fq(])p Fv(6)p Fq(=)p Fv(;)2152 4511 y Fp(E)2213 4525 y Fu(\037)2260 4511 y Fy([)p Fx(F)2343 4525 y Fu(S)2395 4511 y Fx(\024)2447 4473 y Fq(\000)2491 4482 y Fn(0)2530 4511 y Fy(])p Fx(f)2610 4473 y Fq(\000)2654 4482 y Fn(0)2692 4511 y Fx(;)0 4795 y Fy(where)31 b(in)g(the)h(second)f(term)g(the)h(sum)f(is)g(o)m(v)m(er) i(the)e(only)h(cluster)f(whic)m(h)h(can)f(o)m(v)m(erlap)i(with)e Fx(S)5 b Fy(.)44 b(W)-8 b(e)0 4908 y(ha)m(v)m(e)29 b Fw(j)p Fx(f)287 4875 y Fu(S)338 4908 y Fw(j)c(\024)g Fy(2,)k Fw(j)p Fx(f)663 4875 y Fq(\000)707 4884 y Fn(0)746 4908 y Fw(j)c(\024)g Fy(2)937 4875 y Fv(j)p Fq(\000)1001 4852 y Fm(\003)1001 4896 y Fn(0)1037 4875 y Fv(j)1061 4908 y Fy(.)40 b(On)27 b(the)i(other)f(hand,)g(b)m(y)g(using)g(Lemma)g (5.6)h(and)f(c)m(ho)s(osing)h Fx(b)f Fy(large)0 5021 y(enough)i(to)h(ensure)f(that)h(sup)1042 5043 y Fu(x;y)1158 5021 y Fw(j)p Fx(\031)1235 5036 y Fu(b)1269 5021 y Fy(\()p Fx(x;)15 b(y)s Fy(\))21 b Fw(\000)f Fy(1)p Fw(j)26 b(\024)f Fy(1)p Fx(=)p Fy(2,)32 b(w)m(e)f(ha)m(v)m(e)433 5261 y Fp(E)494 5275 y Fu(\037)542 5261 y Fy([)p Fx(F)625 5275 y Fu(S)676 5261 y Fy(])26 b(=)823 5137 y Ft(Z)929 5261 y Fx(F)987 5275 y Fu(S)1038 5261 y Fy(\()p Fx(X)7 b Fy(\))1200 5199 y Fx(d\027)1298 5166 y Fu(x)1338 5175 y Fn(0)1373 5166 y Fu(;x)1433 5175 y Fn(1)1471 5199 y Fy(\()p Fx(X)g Fy(\))p 1200 5240 424 4 v 1223 5323 a Fx(\031)1275 5338 y Fu(b)1309 5323 y Fy(\()p Fx(x)1396 5337 y Fq(0)1436 5323 y Fx(;)15 b(x)1528 5337 y Fq(1)1567 5323 y Fy(\))1634 5261 y Fx(\031)1686 5276 y Fu(b)1721 5261 y Fy(\()p Fx(x)1808 5275 y Fq(0)1847 5261 y Fx(;)g(x)1939 5275 y Fq(1)1979 5261 y Fy(\))p Fx(d!)s Fy(\()p Fx(x)2208 5275 y Fq(0)2248 5261 y Fy(\))p Fx(d!)s Fy(\()p Fx(x)2477 5275 y Fq(1)2518 5261 y Fy(\))51 b Fw(\024)f Fx(C)32 b Fp(E)2883 5275 y Fu(\027)2925 5261 y Fy([)p Fx(F)3008 5275 y Fu(S)3060 5261 y Fy(])p Fx(:)1726 5549 y Fy(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 0 442 a Fy(F)-8 b(urthermore,)29 b Fp(E)612 463 y Fu(\037)656 444 y Fr(N)717 442 y Fy([)p Fx(\024)794 409 y Fq(\000)838 418 y Fn(0)794 464 y Fu(a)877 442 y Fy(])d Fw(\024)1024 368 y Ft(\000)1066 442 y Fp(E)1127 463 y Fu(\037)1171 444 y Fr(N)1232 442 y Fy([)p Fx(\030)1297 456 y Fu(a)1339 442 y Fy(])1364 368 y Ft(\001)1405 391 y Fq(1)p Fu(=)p Fq(2)1556 368 y Ft(\000)1598 442 y Fp(E)1659 463 y Fu(\037)1703 444 y Fr(N)1764 442 y Fy([\()p Fx(\024)1876 409 y Fq(\000)1920 418 y Fn(0)1960 442 y Fy(\))1995 409 y Fq(2)2034 442 y Fy(])2059 368 y Ft(\001)2101 389 y Fq(1)p Fu(=)p Fq(2)2239 442 y Fy(and)i(b)m(y)g(the)g(same)h(argumen)m (ts)f(as)h(in)0 554 y(Prop)s(osition)h(5.3)i(ab)s(o)m(v)m(e)f(w)m(e)g (obtain)g(the)g(b)s(ound)1163 675 y Ft(X)1054 881 y Fq(\000)1098 890 y Fn(0)1133 881 y Fq(:[0)p Fu(;b)p Fq(])p Fv(2)1329 865 y Fq(\026)1325 881 y(\000)1369 890 y Fn(0)1403 761 y Fy([)p Fp(E)1489 775 y Fu(\037)1536 761 y Fy(\()p Fx(\024)1623 724 y Fq(\000)1667 733 y Fn(0)1707 761 y Fy(\))1742 724 y Fq(2)1782 761 y Fy(])1807 724 y Fq(1)p Fu(=)p Fq(2)1942 761 y Fy(2)1987 724 y Fv(j)2011 707 y Fq(\026)2007 724 y(\000)2051 733 y Fn(0)2086 724 y Fv(j)2160 761 y Fw(\024)50 b Fy(const)0 1084 y(with)31 b(some)g(constan)m(t.)44 b(Hence)31 b(w)m(e)h(get)g(that)f Fp(E)1655 1098 y Fu(\026)1702 1084 y Fy([)p Fx(F)1785 1098 y Fu(S)1836 1084 y Fy(])26 b Fw(\024)g Fx(C)7 b Fp(E)2117 1098 y Fu(\027)2160 1084 y Fy([)p Fx(F)2243 1098 y Fu(S)2294 1084 y Fy(])21 b(+)f Fx(C)7 b Fy(\()p Fp(E)2599 1098 y Fu(\027)2642 1084 y Fy([)p Fx(F)2738 1051 y Fq(2)2725 1111 y Fu(S)2778 1084 y Fy(]\))2838 1051 y Fq(1)p Fu(=)p Fq(2)2948 1084 y Fy(,)31 b(whic)m(h)g(implies)0 1196 y(the)g(claim.)3080 b Fi(\003)0 1403 y Fk(Theorem)35 b(5.13)46 b Fo(Ther)-5 b(e)37 b(exists)e(a)h (unique)f(forwar)-5 b(d)38 b(curr)-5 b(ent)36 b Fx(\026)2280 1370 y Fu(])2347 1403 y Fo(on)g Fy(\004)f Fo(such)h(that)g(its)g 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Fw(F)2438 5088 y Fu(S)2489 5074 y Fy(\).)51 b(It)33 b(su\016ces)h(to)g(pro)m(v)m(e)h (that)0 5187 y(for)30 b(an)m(y)h(increasing)h(sequence)f Fw(f)p Fx(T)1212 5201 y Fu(n)1259 5187 y Fw(g)p Fy(,)h Fx(T)1414 5201 y Fu(n)1486 5187 y Fw(!)26 b(1)p Fy(,)31 b(and)f(an)m(y)h(corresp)s(onding)e(sequence)i(of)g(b)s(oundary)0 5300 y(conditions)36 b(\()p Fp(Y)541 5314 y Fu(n)588 5300 y Fy(\))623 5314 y Fu(n)704 5300 y Fw(\032)d Fy(\004)869 5314 y Fu(a)946 5300 y Fy(\(6.2\))k(holds)e(for)g(arbitrary)g Fx(F)2005 5314 y Fu(S)2092 5300 y Fy(of)g(the)h(ab)s(o)m(v)m(e)h (class.)56 b(T)-8 b(o)36 b(sho)m(w)f(this)h(w)m(e)1726 5549 y(40)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 0 431 a Fy(express)20 b(the)h(conditional)h (exp)s(ectations)g(app)s(earing)e(ab)s(o)m(v)m(e)i(in)e(terms)g(of)h (the)g(cluster)g(represen)m(tation.)0 544 y(W)-8 b(e)36 b(supp)s(ose)d(without)h(loss)h(that)g Fx(S)40 b Fy(consists)35 b(of)f(a)h(\014nite)g(union)e(of)i(in)m(terv)-5 b(als)36 b(of)e(the)h(partition)g(of)0 657 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1817 y Fu(a)2560 1803 y Fy(\))2595 1822 y Fu(L)2643 1803 y Fn(2)2677 1822 y Fq(\()p Fu(d!)r Fq(\))2819 1803 y Fx(:)0 2029 y Fy(Moreo)m(v)m(er,)33 b(w)m(e)e(ha)m(v)m(e)600 2247 y Fw(jj)p Fx(f)695 2261 y Fu(a)737 2247 y Fw(jj)787 2210 y Fq(2)787 2275 y Fu(L)835 2256 y Fn(2)870 2275 y Fq(\()p Fu(d!)r Fq(\))1037 2247 y Fy(=)1133 2123 y Ft(Z)1239 2247 y Fx(f)1294 2210 y Fq(2)1284 2270 y Fu(a)1332 2247 y Fy(\()p Fx(x)p Fy(\))p Fx(d!)s Fy(\()p Fx(x)p Fy(\))c(=)1806 2123 y Ft(Z)1857 2330 y Fu(D)1915 2339 y Fn(1)1968 2247 y Fx(d!)s Fy(\()p Fx(x)p Fy(\))22 b(+)2309 2123 y Ft(Z)2360 2330 y Fu(D)2418 2339 y Fn(3)2491 2223 y Fy(~)2472 2247 y Fx(f)2527 2210 y Fq(2)2517 2270 y Fu(a)2565 2247 y Fy(\()p Fx(x)p Fy(\))p Fx(d!)s Fy(\()p Fx(x)p Fy(\))p Fx(:)0 2504 y Fy(Under)30 b(the)g(h)m(yp)s(othesis)h(on)f Fx(V)51 b Fy(the)30 b(standard)g (estimate)i(\011\()p Fx(x)p Fy(\))26 b Fw(\024)f Fx(C)7 b(e)2440 2471 y Fv(\000)p Fu(\022)r Fv(j)p Fu(x)p Fv(j)2610 2448 y Fr(s)p Fn(+1)2754 2504 y Fy(holds)30 b(b)m(y)g(Carmona's)0 2617 y(results)g([7)q(])g(for)h(the)f(ground)f(state)j(\011,)e(with)h (some)f Fx(C)q(;)15 b(\022)29 b(>)c Fy(0.)41 b(This)29 b(b)s(ound)g(further)g(leads)h(to)1315 2717 y Ft(Z)1366 2923 y Fu(D)1424 2932 y Fn(1)1478 2840 y Fx(d!)s Fy(\()p Fx(x)p Fy(\))51 b Fw(\024)f Fx(ce)1960 2803 y Fv(\000)p Fu(\022)r(a)2087 2779 y Fr(s)p Fn(+1)2203 2840 y Fx(;)1083 b Fy(\(6.24\))0 3072 y(where)34 b Fx(c;)15 b(\022)35 b(>)d Fy(0)j(are)g(indep)s(enden)m(t)e(of)i Fx(a)p Fy(.)54 b(A)34 b(similarly)h(estimate)i(is)d(v)-5 b(alid)35 b(for)g Fx(D)2902 3086 y Fq(3)2941 3072 y Fy(.)54 b(On)33 b(the)i(other)0 3185 y(hand,)30 b(since)495 3161 y(~)475 3185 y Fx(f)520 3199 y Fu(a)592 3185 y Fy(is)g(smo)s(oth)g(enough)g(and)g(max)1663 3199 y Fu(D)1721 3208 y Fn(3)1760 3185 y Fw(fjr)1926 3161 y Fy(~)1906 3185 y Fx(f)1951 3199 y Fu(a)1993 3185 y Fw(j)p Fx(;)15 b Fw(j)p Fy(\001)2179 3161 y(~)2159 3185 y Fx(f)2204 3199 y Fu(a)2245 3185 y Fw(j)p Fx(;)g Fy(\001)2406 3161 y(~)2386 3185 y Fx(f)2441 3152 y Fq(2)2431 3207 y Fu(a)2481 3185 y Fw(g)25 b(\024)g Fx(m)g(<)g Fw(1)p Fy(,)31 b(w)m(e)g(get)1352 3378 y(\()1407 3354 y(~)1387 3378 y Fx(f)1432 3392 y Fu(a)1474 3378 y Fx(;)15 b(L)1596 3354 y Fy(~)1576 3378 y Fx(f)1621 3392 y Fu(a)1662 3378 y Fy(\))26 b Fw(\024)f Fx(c)1858 3341 y Fv(0)1882 3378 y Fx(e)1924 3341 y Fv(\000)p Fu(\022)r(a)2051 3317 y Fr(s)p Fn(+1)2166 3378 y Fx(;)0 3560 y Fy(with)33 b(suitable)g Fx(c)590 3527 y Fv(0)643 3560 y Fx(>)d Fy(0.)49 b(A)33 b(similar)g(estimate)i(is)e(obtained)g(also)h(for)f(the)g(remaining)g (t)m(w)m(o)h(domains.)0 3673 y Fi(\003)0 3912 y Fl(6.3)112 b(Mixing)39 b(prop)s(erties)0 4084 y Fy(Since)h Fx(\026)h Fy(is)f(constructed)h(in)f(a)h(w)m(a)m(y)g(that)h(o\013ers)e(no)h (immediate)g(access)h(to)f(computations)h(with)0 4197 y(this)28 b(measure,)i(it)f(is)f(imp)s(ortan)m(t)h(to)g(deriv)m(e)h (further)d(basic)i(information)g(on)f Fx(\026)h Fy(b)m(y)f(using)g(the) h(cluster)0 4310 y(expansion.)41 b(W)-8 b(e)31 b(giv)m(e)h(here)e(one)h (last)g(result)f(of)h(this)f(pap)s(er.)0 4478 y Fk(Theorem)35 b(6.6)46 b Fo(L)-5 b(et)30 b Fx(F)s(;)15 b(G)31 b Fo(b)-5 b(e)30 b(two)h(b)-5 b(ounde)g(d)31 b(functions,)g(the)f(\014rst)h(me)-5 b(asur)g(able)31 b(with)g(r)-5 b(esp)g(e)g(ct)32 b(to)3410 4486 y Fw(F)3475 4492 y Fu(I)3515 4478 y Fo(,)0 4591 y(the)h(se)-5 b(c)g(ond)35 b(with)e(r)-5 b(esp)g(e)g(ct)35 b(to)1043 4599 y Fw(F)1109 4605 y Fu(J)1158 4591 y Fo(,)e(wher)-5 b(e)34 b Fx(I)7 b(;)15 b(J)42 b Fo(ar)-5 b(e)34 b(distinct)f(intervals) h(of)f(the)h(p)-5 b(artition)35 b(c)-5 b(onsider)g(e)g(d)0 4704 y(in)33 b(the)g(cluster)f(exp)-5 b(ansion)35 b(ab)-5 b(ove.)42 b(Then)33 b(the)g(estimate)h(on)f(the)g(c)-5 b(ovarianc)g(e)1099 4886 y Fy(co)m(v)1232 4900 y Fu(\026)1278 4886 y Fy(\()p Fx(F)13 b Fy(;)i Fx(G)p Fy(\))27 b(=)e Fp(E)1714 4900 y Fu(\026)1760 4886 y Fy([)p Fx(F)13 b(G)p Fy(])22 b Fw(\000)e Fp(E)2126 4900 y Fu(\026)2172 4886 y Fy([)p Fx(F)13 b Fy(])26 b Fp(E)2380 4900 y Fu(\026)2426 4886 y Fy([)p Fx(G)p Fy(])1099 5075 y Fw(j)15 b Fy(co)m(v)1272 5089 y Fu(\026)1344 5075 y Fy(\()p Fx(F)e Fy(;)i Fx(G)p Fy(\))p Fw(j)52 b(\024)f Fy(const)2037 5014 y(sup)14 b Fw(j)p Fx(F)f Fw(j)j Fy(sup)e Fw(j)p Fx(G)p Fw(j)p 2037 5054 563 4 v 2099 5138 a(j)p Fx(t)20 b Fw(\000)g Fx(s)p Fw(j)2336 5111 y Fu(#)2402 5138 y Fy(+)g(1)0 5300 y Fo(holds,)34 b(wher)-5 b(e)34 b Fx(#)25 b(>)g Fy(0)33 b Fo(and)g(the)g(c)-5 b(onstant)35 b(pr)-5 b(efactor)35 b(is)d(indep)-5 b(endent)34 b(of)f Fx(F)s(;)15 b(G)p Fo(.)1726 5549 y Fy(45)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 141 431 a Fj(Pr)n(oof.)40 b Fy(First)31 b(recall)g(form)m(ula)g(\(5.45\))i(whic)m(h)d(applied)g(to)h Fx(F)43 b Fy(\(and)30 b(similarly)h(to)h Fx(G)p Fy(\))f(giv)m(es)962 645 y Fp(E)1023 659 y Fu(\026)1069 645 y Fy([)p Fx(F)13 b Fy(])26 b(=)f Fp(E)1373 659 y Fu(\027)1416 645 y Fy([)p Fx(F)13 b Fy(])p Fx(f)1592 607 y Fu(I)1652 645 y Fy(+)1853 558 y Ft(X)1743 760 y Fq(\000)1787 769 y Fn(0)1821 760 y Fq(:)p Fu(I)5 b Fv(\\)p Fq(\000)1968 737 y Fm(\003)1968 781 y Fn(0)2004 760 y Fv(6)p Fq(=)p Fv(;)2109 645 y Fx(K)2193 607 y Fq(\000)2237 616 y Fn(0)2276 645 y Fy(\()p Fx(F)13 b Fy(\))p Fx(f)2472 607 y Fq(\000)2516 584 y Fm(\003)2516 628 y Fn(0)2556 645 y Fx(;)730 b Fy(\(6.25\))0 964 y(where)41 b(w)m(e)h(let)h Fx(K)639 978 y Fq(\000)683 987 y Fn(0)722 964 y Fy(\()p Fx(F)13 b Fy(\))45 b(=)f Fp(E)1084 978 y Fu(\037)1132 964 y Fy([)p Fx(F)13 b(\024)1280 978 y Fq(\000)1324 987 y Fn(0)1363 964 y Fy(].)75 b(F)-8 b(urthermore,)45 b(consider)d Fx(f)2476 931 y Fu(A)2574 964 y Fy(estimated)h(as)f(b)s (efore)f(lik)m(e)0 1077 y Fw(j)p Fx(f)80 1044 y Fu(A)137 1077 y Fw(j)25 b(\024)g Fy(2)328 1044 y Fv(j)p Fu(A)p Fv(j)425 1077 y Fy(.)40 b(F)-8 b(or)31 b Fx(A)721 1091 y Fq(1)781 1077 y Fw(\\)20 b Fx(A)930 1091 y Fq(2)995 1077 y Fy(=)25 b Fw(;)31 b Fy(w)m(e)g(ha)m(v)m(e)923 1340 y Fw(j)p Fx(f)1003 1302 y Fu(A)1056 1311 y Fn(1)1090 1302 y Fv([)p Fu(A)1190 1311 y Fn(2)1249 1340 y Fw(\000)20 b Fx(f)1395 1302 y Fu(A)1448 1311 y Fn(1)1486 1340 y Fx(f)1541 1302 y Fu(A)1594 1311 y Fn(2)1632 1340 y Fw(j)51 b(\024)f Fy(const)2149 1278 y(2)2194 1245 y Fv(j)p Fu(A)2267 1254 y Fn(1)2302 1245 y Fv(j)p Fq(+)p Fv(j)p Fu(A)2450 1254 y Fn(2)2483 1245 y Fv(j)p 2072 1319 514 4 v 2072 1402 a Fy(dist\()p Fx(A)2322 1416 y Fq(1)2362 1402 y Fx(;)15 b(A)2470 1416 y Fq(2)2510 1402 y Fy(\))2545 1376 y Fu(\020)2595 1340 y Fx(;)691 b Fy(\(6.26\))0 1588 y(with)28 b(some)g Fx(\020)k(>)25 b Fy(0.)40 b(This)27 b(estimate)j(can)f(easily) g(b)s(e)e(obtained)i(b)m(y)e(the)i(general)g(results)f(in)f([40)q(].)41 b(No)m(w)0 1701 y(w)m(e)31 b(write)62 1880 y Fp(E)123 1894 y Fu(\026)169 1880 y Fy([)p Fx(F)13 b(G)p Fy(])26 b(=)f Fp(E)544 1894 y Fu(\027)587 1880 y Fy([)p Fx(F)13 b Fy(])26 b Fp(E)795 1894 y Fu(\027)838 1880 y Fy([)p Fx(G)p Fy(])p Fx(f)1014 1842 y Fu(I)5 b Fv([)p Fu(J)564 2042 y Fy(+)837 1956 y Ft(X)867 2142 y Fn(\000)905 2157 y(1)665 2206 y(\000)703 2189 y Fm(\003)703 2231 y Fn(1)739 2206 y Fm(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)948 2189 y Fm(\003)948 2231 y Fn(1)982 2206 y Fm(\\)p Fr(J)g Fn(=)p Fm(;)1166 2042 y Fp(E)1227 2056 y Fu(\037)1274 2042 y Fy([)p Fx(F)13 b(\024)1422 2056 y Fq(\000)1466 2065 y Fn(1)1506 2042 y Fy(])25 b Fp(E)1617 2056 y Fu(\037)1665 2042 y Fy([)p Fx(G)p Fy(])p Fx(f)1841 2005 y Fu(J)6 b Fv([)p Fq(\000)1977 1981 y Fm(\003)1977 2026 y Fn(1)2037 2042 y Fy(+)2310 1956 y Ft(X)2340 2142 y Fn(\000)2378 2157 y(2)2138 2206 y(\000)2176 2189 y Fm(\003)2176 2231 y Fn(2)2212 2206 y Fm(\\)p Fr(J)f Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)2429 2189 y Fm(\003)2429 2231 y Fn(2)2463 2206 y Fm(\\)p Fr(I)t Fn(=)p Fm(;)2639 2042 y Fp(E)2700 2056 y Fu(\037)2748 2042 y Fy([)p Fx(G\024)2896 2056 y Fq(\000)2940 2065 y Fn(2)2979 2042 y Fy(])26 b Fp(E)3091 2056 y Fu(\037)3139 2042 y Fy([)p Fx(F)13 b Fy(])p Fx(f)3315 2005 y Fu(I)5 b Fv([)p Fq(\000)3442 1981 y Fm(\003)3442 2026 y Fn(2)564 2368 y Fy(+)837 2282 y Ft(X)679 2474 y Fn(\000)717 2489 y(1)751 2474 y Fr(;)p Fn(\000)808 2489 y(2)842 2474 y(:\000)898 2457 y Fm(\003)898 2498 y Fn(1)933 2474 y Fm(\\)p Fn(\000)1013 2457 y Fm(\003)1013 2498 y Fn(2)1048 2474 y(=)p Fm(;)665 2548 y Fn(\000)703 2531 y Fm(\003)703 2572 y Fn(2)739 2548 y Fm(\\)p Fr(J)g Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)956 2531 y Fm(\003)956 2572 y Fn(2)990 2548 y Fm(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)1166 2368 y Fp(E)1227 2382 y Fu(\037)1274 2368 y Fy([)p Fx(F)13 b(\024)1422 2382 y Fq(\000)1466 2391 y Fn(1)1506 2368 y Fy(])25 b Fp(E)1617 2382 y Fu(\037)1665 2368 y Fy([)p Fx(G\024)1813 2382 y Fq(\000)1857 2391 y Fn(2)1897 2368 y Fy(])p Fx(f)1977 2331 y Fq(\000)2021 2307 y Fm(\003)2021 2351 y Fn(1)2056 2331 y Fv([)p Fq(\000)2147 2307 y Fm(\003)2147 2351 y Fn(2)2385 2282 y Ft(X)2432 2467 y Fn(\000)2213 2524 y(\000)2251 2510 y Fm(\003)2286 2524 y(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)2503 2510 y Fm(\003)2537 2524 y(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)2713 2368 y Fp(E)2774 2382 y Fu(\037)2822 2368 y Fy([)p Fx(F)13 b(G\024)3041 2382 y Fq(\000)3090 2368 y Fy(])p Fx(f)3170 2331 y Fq(\000)3214 2307 y Fm(\003)3254 2368 y Fx(:)3311 2669 y Fy(\(6.27\))0 2866 y(F)-8 b(rom)31 b(here)f(and)g(\(6.25\))j(w)m(e)d(obtain)297 3068 y(co)m(v)429 3082 y Fu(\026)476 3068 y Fy(\()p Fx(F)13 b Fy(;)i Fx(G)p Fy(\))27 b(=)e Fp(E)912 3082 y Fu(\037)960 3068 y Fy([)p Fx(F)13 b Fy(])25 b Fp(E)1167 3082 y Fu(\037)1215 3068 y Fy([)p Fx(G)p Fy(]\()p Fx(f)1426 3030 y Fu(I)5 b Fv([)p Fu(J)1579 3068 y Fw(\000)19 b Fx(f)1724 3030 y Fu(I)1764 3068 y Fx(f)1819 3030 y Fu(J)1867 3068 y Fy(\))932 3230 y(+)1205 3144 y Ft(X)1234 3329 y Fn(\000)1272 3344 y(1)1033 3394 y(\000)1071 3376 y Fm(\003)1071 3418 y Fn(1)1106 3394 y Fm(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1315 3376 y Fm(\003)1315 3418 y Fn(1)1350 3394 y Fm(\\)p Fr(J)5 b Fn(=)p Fm(;)1533 3230 y Fp(E)1594 3244 y Fu(\037)1642 3230 y Fy([)p Fx(F)13 b(\024)1790 3244 y Fq(\000)1834 3253 y Fn(1)1873 3230 y Fy(])26 b Fp(E)1985 3244 y Fu(\037)2033 3230 y Fy([)p Fx(G)p Fy(]\()p Fx(f)2244 3193 y Fu(J)6 b Fv([)p Fq(\000)2380 3169 y Fm(\003)2380 3213 y Fn(1)2440 3230 y Fw(\000)20 b Fx(f)2586 3193 y Fu(J)2635 3230 y Fx(f)2690 3193 y Fq(\000)2734 3169 y Fm(\003)2734 3213 y Fn(1)2773 3230 y Fy(\))932 3556 y(+)1205 3470 y Ft(X)1234 3655 y Fn(\000)1272 3670 y(2)1033 3719 y(\000)1071 3702 y Fm(\003)1071 3744 y Fn(2)1106 3719 y Fm(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1323 3702 y Fm(\003)1323 3744 y Fn(2)1357 3719 y Fm(\\)p Fr(I)t Fn(=)p Fm(;)1533 3556 y Fp(E)1594 3570 y Fu(\037)1642 3556 y Fy([)p Fx(G\024)1790 3570 y Fq(\000)1834 3579 y Fn(2)1874 3556 y Fy(])25 b Fp(E)1985 3570 y Fu(\037)2033 3556 y Fy([)p Fx(F)13 b Fy(]\()p Fx(f)2244 3518 y Fu(I)5 b Fv([)p Fq(\000)2371 3495 y Fm(\003)2371 3539 y Fn(2)2431 3556 y Fw(\000)20 b Fx(f)2577 3518 y Fu(I)2617 3556 y Fx(f)2672 3518 y Fq(\000)2716 3495 y Fm(\003)2716 3539 y Fn(2)2755 3556 y Fy(\))932 3882 y(+)1205 3795 y Ft(X)1047 3987 y Fn(\000)1085 4002 y(1)1119 3987 y Fr(;)p Fn(\000)1176 4002 y(2)1210 3987 y(:\000)1266 3970 y Fm(\003)1266 4012 y Fn(1)1301 3987 y Fm(\\)p Fn(\000)1381 3970 y Fm(\003)1381 4012 y Fn(2)1416 3987 y(=)p Fm(;)1033 4061 y Fn(\000)1071 4044 y Fm(\003)1071 4086 y Fn(2)1106 4061 y Fm(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1323 4044 y Fm(\003)1323 4086 y Fn(2)1357 4061 y Fm(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)1533 3882 y Fp(E)1594 3896 y Fu(\037)1642 3882 y Fy([)p Fx(F)13 b(\024)1790 3896 y Fq(\000)1834 3905 y Fn(1)1873 3882 y Fy(])26 b Fp(E)1985 3896 y Fu(\037)2033 3882 y Fy([)p Fx(G\024)2181 3896 y Fq(\000)2225 3905 y Fn(2)2264 3882 y Fy(]\()p Fx(f)2379 3844 y Fq(\000)2423 3821 y Fm(\003)2423 3865 y Fn(1)2459 3844 y Fv([)p Fq(\000)2550 3821 y Fm(\003)2550 3865 y Fn(2)2611 3882 y Fw(\000)20 b Fx(f)2757 3844 y Fq(\000)2801 3821 y Fm(\003)2801 3865 y Fn(1)2840 3882 y Fx(f)2895 3844 y Fq(\000)2939 3821 y Fm(\003)2939 3865 y Fn(2)2979 3882 y Fy(\))932 4224 y(+)1205 4137 y Ft(X)1252 4323 y Fn(\000)1033 4379 y(\000)1071 4365 y Fm(\003)1106 4379 y(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1323 4365 y Fm(\003)1357 4379 y(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)1533 4224 y Fp(E)1594 4238 y Fu(\037)1642 4224 y Fy([)p Fx(F)13 b(G\024)1861 4238 y Fq(\000)1910 4224 y Fy(])p Fx(f)1990 4186 y Fq(\000)2034 4163 y Fm(\003)932 4527 y Fw(\000)1205 4440 y Ft(X)1252 4626 y Fn(\000)1033 4682 y(\000)1071 4668 y Fm(\003)1106 4682 y(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1323 4668 y Fm(\003)1357 4682 y(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)1533 4527 y Fp(E)1594 4541 y Fu(\037)1642 4527 y Fy([)p Fx(F)13 b Fy(])p Fp(E)1824 4541 y Fu(\037)1872 4527 y Fy([)p Fx(G\024)2020 4541 y Fq(\000)2069 4527 y Fy(])p Fx(f)2149 4489 y Fq(\000)2193 4466 y Fm(\003)2233 4527 y Fx(f)2288 4489 y Fu(I)2327 4527 y Fx(f)2382 4489 y Fq(\000)2426 4466 y Fm(\003)932 4830 y Fw(\000)1205 4743 y Ft(X)1252 4929 y Fn(\000)1033 4985 y(\000)1071 4971 y Fm(\003)1106 4985 y(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1323 4971 y Fm(\003)1357 4985 y(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)1533 4830 y Fp(E)1594 4844 y Fu(\037)1642 4830 y Fy([)p Fx(F)13 b(\024)1790 4844 y Fq(\000)1839 4830 y Fy(])p Fp(E)1925 4844 y Fu(\037)1973 4830 y Fy([)p Fx(G)p Fy(])p Fx(f)2149 4792 y Fq(\000)2193 4769 y Fm(\003)2233 4830 y Fx(f)2288 4792 y Fu(J)2336 4830 y Fx(f)2391 4792 y Fq(\000)2435 4769 y Fm(\003)932 5133 y Fw(\000)1205 5046 y Ft(X)1047 5238 y Fn(\000)1085 5253 y(1)1119 5238 y Fr(;)p Fn(\000)1176 5253 y(2)1210 5238 y(:\000)1266 5221 y Fm(\003)1266 5263 y Fn(1)1301 5238 y Fm(\\)p Fn(\000)1381 5221 y Fm(\003)1381 5263 y Fn(2)1416 5238 y Fm(6)p Fn(=)p Fm(;)1033 5312 y Fn(\000)1071 5295 y Fm(\003)1071 5337 y Fn(2)1106 5312 y Fm(\\)p Fr(J)5 b Fm(6)p Fn(=)p Fm(;)p Fr(;)p Fn(\000)1323 5295 y Fm(\003)1323 5337 y Fn(2)1357 5312 y Fm(\\)p Fr(I)t Fm(6)p Fn(=)p Fm(;)1533 5133 y Fp(E)1594 5147 y Fu(\037)1642 5133 y Fy([)p Fx(F)13 b(\024)1790 5147 y Fq(\000)1834 5156 y Fn(1)1873 5133 y Fy(])26 b Fp(E)1985 5147 y Fu(\037)2033 5133 y Fy([)p Fx(G\024)2181 5147 y Fq(\000)2225 5156 y Fn(2)2264 5133 y Fy(])p Fx(f)2344 5095 y Fq(\000)2388 5072 y Fm(\003)2388 5116 y Fn(1)2428 5133 y Fx(f)2483 5095 y Fq(\000)2527 5072 y Fm(\003)2527 5116 y Fn(2)3311 4189 y Fy(\(6.28\))1726 5549 y(46)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 0 431 a Fy(F)-8 b(or)32 b(estimating)i(the)e (\014rst)e(four)h(terms)h(at)g(the)g(righ)m(t)g(hand)f(side)h(ab)s(o)m (v)m(e)g(w)m(e)h(use)e(\(6.26\))j(along)f(with)0 544 y(the)e(b)s(ound)1001 682 y Fw(j)p Fp(E)1087 702 y Fv(P)1146 682 y Fy([)p Fx(F)13 b(\024)1294 696 y Fq(\000)1343 682 y Fy(])p Fw(j)51 b(\024)1737 620 y Fy(sup)14 b Fw(j)p Fx(F)f Fw(j)p 1575 661 598 4 v 1575 745 a Fy(\(diam)j(\000)1880 718 y Fv(\003)1919 745 y Fy(\))1954 718 y Fu(\020)1990 700 y Fm(0)2037 745 y Fy(+)j(1)2183 682 y Fx(E)2250 696 y Fq(\000)2298 682 y Fy(\()p Fx(\016)2376 644 y Fv(0)2401 682 y Fx(;)c(")2483 644 y Fv(0)2507 682 y Fy(\))769 b(\(6.29\))0 896 y Fx(i)26 b Fy(=)f(1)p Fx(;)15 b Fy(2,)27 b(where)e Fx(E)660 910 y Fq(\000)708 896 y Fy(\()p Fx(\016)786 863 y Fv(0)811 896 y Fx(;)15 b(")893 863 y Fv(0)917 896 y Fy(\))25 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Fw(6)p Fy(=)25 b Fw(;)p Fy(,)31 b Fx(J)e Fw(\\)20 b Fy(\000)823 2051 y Fv(\003)823 2108 y Fq(2)887 2084 y Fw(6)p Fy(=)25 b Fw(;)p Fy(,)31 b(and)f Fx(#)25 b Fy(=)g(min)p Fw(f)p Fx(\020)7 b(;)15 b(\020)1767 2051 y Fv(0)1790 2084 y Fw(g)25 b Fx(>)g Fy(0.)141 2197 y(Next,)36 b(in)e(the)h(\014fth)e(term) i(ab)s(o)m(v)m(e)g(w)m(e)g(use)f(that)h(diam)15 b(\000)2114 2164 y Fv(\003)2185 2197 y Fw(\025)31 b Fy(dist\()p Fx(I)7 b(;)15 b(J)9 b Fy(\))36 b(whenev)m(er)e Fx(I)c Fw(\\)22 b Fy(\000)3299 2164 y Fv(\003)3370 2197 y Fw(6)p Fy(=)32 b Fw(;)p Fy(,)0 2310 y Fx(J)d Fw(\\)20 b Fy(\000)217 2277 y Fv(\003)282 2310 y Fw(6)p Fy(=)25 b Fw(;)p Fy(,)31 b(and)e(that)857 2551 y Fw(j)p Fp(E)943 2565 y Fu(\037)990 2551 y Fy([)p Fx(F)13 b(G\024)1209 2565 y Fq(\000)1259 2551 y Fy(])p Fw(j)51 b(\024)f Fy(sup)14 b Fw(j)p Fx(F)f Fw(j)i Fy(sup)g Fw(j)p Fx(G)p Fw(j)2173 2490 y Fx(E)2240 2504 y Fq(\000)2288 2490 y Fy(\()p Fx(\016)2366 2457 y Fv(0)2391 2490 y Fx(;)g(")2473 2457 y Fv(0)2497 2490 y Fy(\))p 2053 2530 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Fy(\000)193 4004 y Fq(2)263 3990 y Fy(or)31 b(\000,)f(concluding)g(the)h(pro)s(of.)2142 b Fi(\003)0 4275 y FD(References)45 4478 y Fy([1])47 b(Bessaih,)28 b(H.,)f(Gubinelli,)h(M.)f(and)e(Russo,)i(F.:)46 b(The)26 b(ev)m(olution)i(of)e(a)h(random)e(v)m(ortex)i(\014lamen)m(t,) 187 4591 y Fo(A)n(nn.)32 b(Pr)-5 b(ob)g(ab.)31 b Fk(33)g Fy(\(2005\),)i(1825{1855)45 4776 y([2])47 b(Betz,)40 b(V.:)54 b(Existence)38 b(of)f(Gibbs)g(measures)f(relativ)m(e)j(to)f (Bro)m(wnian)f(motion,)i Fo(Markov)g(Pr)-5 b(o)g(c.)187 4889 y(R)g(el.)33 b(Fields)e Fk(9)g Fy(\(2003\),)i(85-102)45 5074 y([3])47 b(Betz,)e(V.,)g(Hiroshima,)f(F.,)g(L})-45 b(orinczi,)45 b(J.,)f(Minlos,)h(R.A.)c(and)g(Sp)s(ohn,)h(H.:)62 b(Ground)40 b(state)187 5187 y(prop)s(erties)21 b(of)g(the)h(Nelson)g (Hamiltonian)h(|)e(A)g(Gibbs)g(measure-based)h(approac)m(h,)h Fo(R)-5 b(ev.)25 b(Math.)187 5300 y(Phys.)30 b Fk(14)h Fy(\(2002\),)i(173-198)1726 5549 y(47)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 45 431 a Fy([4])47 b(Betz,)29 b(V.)e(and)g(L})-45 b(orinczi,)28 b(J.:)39 b(Uniqueness)26 b(of)h(Gibbs)g(measure)g(relativ)m(e)i(to)e(Bro)m(wnian)g(motion,)187 544 y Fo(A)n(nn.)32 b(IHP,)f(Pr)-5 b(ob)g(ab.)34 b(&)e(Stat.)f Fk(39)g Fy(\(2003\),)i(877-889)45 731 y([5])47 b(Bolthausen,)h(E.:)106 b(On)44 b(the)g(construction)h(of)f(the)h(three-dimensional)f(p)s (olymer)g(measure,)187 844 y Fo(Pr)-5 b(ob)g(ab.)34 b(The)-5 b(ory)34 b(R)-5 b(elate)g(d)34 b(Fields)d Fk(97)g Fy(\(1993\),)i (81{101)45 1032 y([6])47 b(Bo)m(vier,)29 b(A.,)f(F)-8 b(elder,)28 b(G.)f(and)f(F)-8 b(r\177)-45 b(ohlic)m(h,)29 b(J.:)46 b(On)25 b(the)i(critical)h(prop)s(erties)e(of)h(the)g(Edw)m (ards)e(and)187 1145 y(the)i(self-a)m(v)m(oiding)j(w)m(alk)e(mo)s(del)f (of)g(p)s(olymer)g(c)m(hains,)37 b Fo(Nucle)-5 b(ar)30 b(Phys.)g(B)p Fk(230)e Fy(\(1984\),)i(119{147)45 1333 y([7])47 b(Carmona,)31 b(R.:)53 b(P)m(oin)m(t)m(wise)33 b(b)s(ounds)c(for)i(Sc)m(hr\177)-45 b(odinger)30 b(eigenstates,)45 b Fo(Commun.)34 b(Math.)g(Phys.)187 1445 y Fk(62)d Fy(\(1978\),)i (97{106)45 1633 y([8])47 b(Coutin,)40 b(L.)f(and)f(Leja)m(y)-8 b(,)42 b(A.:)84 b(Semi-martingales)40 b(and)e(rough)g(paths)g(theory)-8 b(,)67 b Fo(Ele)-5 b(ctr)g(on.)42 b(J.)187 1746 y(Pr)-5 b(ob)g(ab.)31 b Fk(10)g Fy(\(2005\),)i(761{785)45 1934 y([9])47 b(Coutin,)d(L.)e(and)f(Qian,)k(Z.:)96 b(Sto)s(c)m(hastic)43 b(analysis,)j(rough)41 b(path)g(analysis)i(and)e(fractional)187 2047 y(Bro)m(wnian)30 b(motions,)42 b Fo(Pr)-5 b(ob)g(ab.)33 b(The)-5 b(ory)34 b(R)-5 b(el.)34 b(Fields)d Fk(122)g Fy(\(2002\),)i(108{140)0 2234 y([10])47 b(De)35 b(Angelis,)i(G.F.,)g (Jona-Lasinio,)g(G.)e(and)f(Sirugue,)h(M.:)50 b(Probabilistic)35 b(solution)g(of)g(Pauli)187 2347 y(t)m(yp)s(e)30 b(equations,)i Fo(J.)g(Phys.)h(A)d Fk(16)g Fy(\(1983\),)k(2433{2444)0 2535 y([11])47 b(De)26 b(Angelis,)h(G.F.)f(and)e(Jona-Lasinio,)j(G.:)39 b(A)25 b(sto)s(c)m(hastic)i(description)e(of)g(a)g(spin-1)p Fx(=)p Fy(2)h(particle)187 2648 y(in)k(a)h(magnetic)h(\014eld,)e Fo(J.)i(Phys.)h(A)p Fk(15)d Fy(\(1982\),)j(2053{2061)0 2835 y([12])47 b(Dobrushin,)g(R.L.:)69 b(Analyticit)m(y)46 b(of)f(correlation)h(functions)e(in)g(one-dimensional)h(classical)187 2948 y(systems)28 b(with)f(slo)m(wly)h(decreasing)h(p)s(oten)m(tials,)h Fo(Commun.)h(Math.)f(Phys.)e Fk(32)g Fy(\(1973\),)j(269-289)0 3136 y([13])47 b(Dobrushin,)42 b(R.L.:)62 b(Analyticit)m(y)43 b(of)e(correlation)i(functions)d(for)h(one-dimensional)g(classical)187 3249 y(systems)26 b(with)f(p)s(o)m(w)m(er-la)m(w)i(deca)m(y)g(of)f(the) g(p)s(oten)m(tial,)j Fo(Math.)f(USSR)h(Sb)-5 b(ornik)26 b Fk(23)h Fy(\(1973\),)i(13-44)0 3436 y([14])47 b(Donsk)m(er,)35 b(M.D.)f(and)f(V)-8 b(aradhan,)34 b(S.R.S.:)46 b(Asymptotics)35 b(for)e(the)g(p)s(olaron,)h Fo(Commun.)i(Pur)-5 b(e)187 3549 y(Appl.)33 b(Math.)e Fk(36)g Fy(\(1983\),)i(505-528)0 3737 y([15])47 b(F)-8 b(ey)m(el,)27 b(D.)e(and)e(de)g(La)h(Pradelle,)i (A.:)38 b(Curvilinear)23 b(in)m(tegrals)j(along)e(enric)m(hed)g(paths,) h Fo(Ele)-5 b(ctr)g(on.)187 3850 y(J.)32 b(Pr)-5 b(ob)g(ab.)31 b Fk(11)g Fy(\(2006\),)i(860{892)0 4037 y([16])47 b(Flandoli,)d(F.:)62 b(On)40 b(a)h(probabilistic)g(description)g(of)g(small)g(scale)h (structures)e(in)g(3D)h(\015uids,)187 4150 y Fo(A)n(nn.)32 b(IHP,)f(Pr)-5 b(ob)g(ab.)34 b(&)e(Stat.)f Fk(38)g Fy(\(2002\),)i (207-228)0 4338 y([17])47 b(Flandoli,)33 b(F.,)g(Giaquin)m(ta,)h(G.,)g (Gubinelli,)e(M.)h(and)e(T)-8 b(ortorelli,)34 b(V.M.:)45 b(Sto)s(c)m(hastic)33 b(curren)m(ts,)187 4451 y Fo(Sto)-5 b(ch.)34 b(Pr)-5 b(o)g(c.)33 b(Appl.)e Fk(155)g Fy(\(2005\),)i (1583-1601)0 4639 y([18])47 b(Flandoli,)42 b(F.)e(and)e(Gubinelli,)k (M.:)59 b(The)38 b(Gibbs)h(ensem)m(ble)g(of)g(a)h(v)m(ortex)g (\014lamen)m(t,)i Fo(Pr)-5 b(ob)g(ab.)187 4751 y(The)g(ory)34 b(R)-5 b(el.)33 b(Fields)39 b Fk(122)31 b Fy(\(2002\),)i(317-340)0 4939 y([19])47 b(Flandoli,)36 b(F.)f(and)f(Gubinelli,)h(M.:)49 b(Random)34 b(Curren)m(ts)f(and)g(Probabilistic)j(Mo)s(dels)e(of)g(V)-8 b(or-)187 5052 y(tex)29 b(Filamen)m(ts,)j Fo(Seminar)g(on)g(Sto)-5 b(chastic)33 b(A)n(nalysis,)e(R)-5 b(andom)34 b(Fields)e(and)h(Applic) -5 b(ations)33 b(IV,)187 5165 y(Asc)-5 b(ona)33 b(2002)p Fy(,)g(Progress)d(in)g(Probabilt)m(y)h(58,)h(Birkh\177)-45 b(auser)30 b(V)-8 b(erlag,)32 b(Basel,)g(2004)1726 5549 y(48)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 0 431 a Fy([20])47 b(F.)39 b(Flandoli)g(and)e (M.)i(Gubinelli.)64 b(Statistics)40 b(of)e(a)g(v)m(ortex)i(\014lamen)m (t)f(mo)s(del,)65 b Fo(Ele)-5 b(ctr.)40 b(Jour.)187 544 y(Pr)-5 b(ob.)31 b Fk(10)g Fy(\(2005\),)i(865{900)0 731 y([21])47 b(F.)39 b(Flandoli,)i(M.)e(Gubinelli,)i(and)d(F.)g(Russo.)64 b(On)38 b(the)g(regularit)m(y)i(of)e(sto)s(c)m(hastic)i(curren)m(ts,) 187 844 y(fractional)32 b(Bro)m(wnian)e(motion)h(and)f(applications)i (to)f(a)g(turbulence)e(mo)s(del,)41 b(preprin)m(t)29 b(\(2007\))0 1032 y([22])47 b(Georgii,)25 b(H.-O.:)36 b Fo(Gibbs)24 b(Me)-5 b(asur)g(es)24 b(and)h(Phase)g(T)-7 b(r)i(ansitions)p Fy(,)25 b(Berlin,)e(New)e(Y)-8 b(ork:)37 b(de)20 b(Gruyter,)187 1145 y(1988)0 1333 y([23])47 b(Gubinelli,)31 b(M.:)51 b(Con)m(trolling)31 b(rough)f(paths,)41 b Fo(J.)32 b(F)-7 b(unct.)32 b(A)n(nal.)f Fk(216)g Fy(\(2004\),)i(86{140)0 1520 y([24])47 b(Gubinelli,)36 b(M.)f(and)f(L})-45 b(orinczi,)37 b(J.:)50 b(Ultra)m(violet)37 b(renormalization)g(of)d(Nelson's)i (Hamiltonian)187 1633 y(through)30 b(functional)g(in)m(tegration,)j (preprin)m(t)c(\(2007\))0 1821 y([25])47 b(Hariy)m(a,)30 b(Y.)f(and)f(Osada,)h(H.:)40 b(Di\013usion)28 b(pro)s(cesses)g(on)g (path)h(spaces)f(with)g(in)m(teractions,)j Fo(R)-5 b(ev.)187 1934 y(Math.)33 b(Phys.)e Fk(13)f Fy(\(2001\),)j(199-220)0 2121 y([26])47 b(Hiroshima,)28 b(F.:)40 b(F)-8 b(unctional)29 b(in)m(tegral)g(represen)m(tations)f(of)f(quan)m(tum)g(electro)s (dynamics,)i Fo(R)-5 b(ev.)187 2234 y(Math.)33 b(Phys.)e Fk(9)f Fy(\(1997\),)j(489{530)0 2422 y([27])47 b(Hiroshima,)25 b(F.)g(and)e(L})-45 b(orinczi,)26 b(J.:)37 b(F)-8 b(unctional)25 b(in)m(tegral)g(represen)m(tation)g(of)f(a)g(non-relativistic)187 2535 y(quan)m(tum)30 b(particle)h(with)f(spin)g(1)p Fx(=)p Fy(2,)i(preprin)m(t)d(\(2007\))0 2722 y([28])47 b(Hiroshima,)39 b(F.)e(and)f(L})-45 b(orinczi,)39 b(J.:)53 b(Lo)s(calization)39 b(of)e(the)g(ground)e(state)j(in)e(the)h(P)m(auli-Fierz)187 2835 y(mo)s(del)30 b(at)h(w)m(eak)g(couplings,)g(preprin)m(t)f (\(2007\))0 3023 y([29])47 b(Kipnis,)35 b(C.)g(and)f(V)-8 b(aradhan,)36 b(S.R.S.:)69 b(Cen)m(tral)35 b(limit)h(theorem)f(for)g (additiv)m(e)h(functionals)f(of)187 3136 y(rev)m(ersible)f(Mark)m(o)m (v)h(pro)s(cesses)e(and)g(applications)i(to)f(simple)f(exclusions,)51 b Fo(Commun.)37 b(Math.)187 3249 y(Phys.)30 b Fk(104)i Fy(\(1986\),)h(1{19)0 3436 y([30])47 b(Le)32 b(Gall,)h(J.F.:)56 b(T)-8 b(emps)32 b(lo)s(caux)g(d'in)m(tersection)h(renormalis)m(\023) -43 b(es)33 b(et)f(d)m(\023)-43 b(ev)m(elopp)s(emen)m(t)34 b(asympto-)187 3549 y(tique)c(du)e(v)m(olume)i(de)f(la)h(saucisse)g(de) f(Wiener)h(plane,)39 b Fo(C.)31 b(R.)h(A)-5 b(c)g(ad.)32 b(Sci.)g(Paris)g(S)n(\023)-44 b(er.)31 b(I)h(Math.)187 3662 y Fk(304)f Fy(\(1987\),)i(339{342)0 3850 y([31])47 b(Le)21 b(Gall,)k(J.F.:)40 b(Exp)s(onen)m(tial)22 b(momen)m(ts)g(for)e (the)i(renormalized)f(self-in)m(tersection)j(lo)s(cal)e(time)g(of)187 3963 y(planar)f(Bro)m(wnian)h(motion,)29 b(In)21 b Fo(S)n(\023)-44 b(eminair)-5 b(e)26 b(de)e(Pr)-5 b(ob)g(abilit)n(\023)-44 b(es,)28 b(XXVIII)p Fy(,)21 b(LNM)h Fk(1583)p Fy(,)i(Springer,)187 4076 y(Berlin,)31 b(1994,)h(pp.)e(172{180)0 4263 y([32])47 b(Leja)m(y)-8 b(,)36 b(A.:)64 b(An)34 b(in)m(tro)s(duction)f(to)i (rough)e(paths,)51 b(in:)c Fo(S)n(\023)-44 b(eminair)-5 b(e)37 b(de)f(Pr)-5 b(ob)g(abilit)n(\023)-44 b(es)37 b(XXXVII)p Fy(,)187 4376 y Fo(L)-5 b(e)g(ctur)g(e)33 b(Notes)g(in)g(Math.)d Fk(1832)p Fy(,)i(Springer,)d(Berlin,)i(2003,)i (pp.)c(1-59)0 4564 y([33])47 b(Ly)m(ons,)31 b(T.:)52 b(Di\013eren)m(tial)33 b(equations)e(driv)m(en)f(b)m(y)h(rough)f (signals,)42 b Fo(R)-5 b(ev.)33 b(Mat.)g(Ib)-5 b(er)g(o)g(americ)g(ana) 187 4677 y Fk(14)31 b Fy(\(1998\),)i(215{310)0 4864 y([34])47 b(Ly)m(ons,)30 b(T.)g(and)g(Qian,)g(Z.:)50 b Fo(System)34 b(Contr)-5 b(ol)34 b(and)f(Rough)h(Paths)p Fy(,)41 b(Oxford)29 b(Univ)m(ersit)m(y)i(Press,)187 4977 y(2002)0 5165 y([35])47 b(L})-45 b(orinczi,)50 b(J.:)71 b(Gibbs)45 b(measures)g(on)h(Bro)m (wnian)f(paths,)50 b(in)45 b Fo(In)i(and)g(Out)f(of)h(Equilibrium:)187 5278 y(Physics)26 b(with)g(a)g(Pr)-5 b(ob)g(ability)28 b(Flavour)p Fy(,)e(V.)d(Sidora)m(vicius,)i(ed.,)g(Birkh\177)-45 b(auser,)24 b(2002,)i(pp.)c(349-362)1726 5549 y(49)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 0 431 a Fy([36])47 b(L})-45 b(orinczi,)32 b(J.)f(and)g(Mauser,)g(N.J.:)43 b(A)31 b(sto)s(c)m(hastic)i(approac)m (h)e(to)h(the)f(bip)s(olaron)g(mo)s(del,)g Fo(RIMS)187 544 y(S.)h(Kokyur)-5 b(oku)p Fy(,)32 b Fk(1482)f Fy(\(2006\),)i(1-10)0 731 y([37])47 b(L})-45 b(orinczi,)36 b(J.)e(and)g(Minlos,)i(R.A.:)50 b(Gibbs)33 b(measures)i(for)f(Bro)m(wnian)g(paths)g(under)f(the)i (e\013ect)187 844 y(of)30 b(an)h(external)g(and)f(a)g(small)h(pair)f(p) s(oten)m(tial,)j Fo(J.)f(Stat.)h(Phys.)e Fk(105)g Fy(\(2001\),)i (605-647)0 1032 y([38])47 b(L})-45 b(orinczi,)26 b(J.,)e(Minlos,)i (R.A.)e(and)f(Sp)s(ohn,)g(H.:)37 b(The)23 b(infrared)g(b)s(eha)m(viour) g(in)g(Nelson's)h(mo)s(del)f(of)187 1145 y(a)29 b(quan)m(tum)f (particle)i(coupled)e(to)i(a)f(massless)g(scalar)g(\014eld,)g Fo(A)n(nn.)i(Henri)f(Poinc)-5 b(ar)n(\023)-44 b(e)30 b Fk(3)f Fy(\(2002\),)187 1258 y(1-28)0 1445 y([39])47 b(L})-45 b(orinczi,)28 b(J.,)g(Minlos,)g(R.A.)f(and)f(Sp)s(ohn,)g(H.:) 39 b(Infrared)26 b(regular)g(represen)m(tation)i(of)f(the)g(three)187 1558 y(dimensional)j(massless)h(Nelson)g(mo)s(del,)f Fo(L)-5 b(ett.)33 b(Math.)g(Phys.)e Fk(59)g Fy(\(2002\),)i(189-198)0 1746 y([40])47 b(Malyshev,)31 b(V.A.)f(and)g(Minlos,)h(R.A.:)41 b Fo(Gibbs)32 b(R)-5 b(andom)34 b(Fields)p Fy(,)d(Klu)m(w)m(er)f (Academic)h(Publish-)187 1859 y(ers,)f(1991)0 2047 y([41])47 b(Nelson,)41 b(E.:)57 b(In)m(teraction)41 b(of)d(nonrelativistic)j (particles)e(with)f(a)h(quan)m(tized)h(scalar)f(\014eld,)i Fo(J.)187 2159 y(Math.)33 b(Phys.)e Fk(5)f Fy(\(1964\),)j(1990-1997)0 2347 y([42])47 b(Nualart,)d(D.,)f(Ro)m(vira,)h(C.)c(and)g(Tindel,)i (S.:)90 b(Probabilistic)41 b(mo)s(dels)f(for)g(v)m(ortex)i(\014lamen)m (ts)187 2460 y(based)32 b(on)g(fractional)h(Bro)m(wnian)g(motion,)47 b Fo(RA)n(CSAM)33 b(R)-5 b(ev.)34 b(R.)g(A)-5 b(c)g(ad.)35 b(Cienc.)f(Exactas)h(F)-9 b(\023)-37 b(\020s.)187 2573 y(Nat.)33 b(Ser.)f(A)g(Mat.)f Fk(95)f Fy(\(2001\),)k(213{218)0 2761 y([43])47 b(Osada,)25 b(H.)g(and)e(Sp)s(ohn,)h(H.:)38 b(Gibbs)23 b(measures)h(relativ)m(e)j(to)d(Bro)m(wnian)h(motion,)h Fo(A)n(nn.)g(Pr)-5 b(ob)g(ab.)187 2873 y Fk(27)31 b Fy(\(1999\),)i (1183-1207)0 3061 y([44])47 b(Revuz,)d(D.)e(and)e(Y)-8 b(or,)45 b(M.:)63 b Fo(Continuous)43 b(martingales)i(and)e(Br)-5 b(ownian)45 b(motion)p Fy(,)g(Springer,)187 3174 y(Berlin,)31 b(1999)0 3362 y([45])47 b(Sp)s(ohn,)28 b(H.:)40 b(The)29 b(p)s(olaron)g(at)h(large)g(total)h(momen)m(tum,)f Fo(J.)h(Phys.)h(A)d Fk(21)h Fy(\(1988\),)i(1199{1211.)0 3549 y([46])47 b(Simon,)39 b(B.:)55 b Fo(F)-7 b(unctional)41 b(Inte)-5 b(gr)g(ation)41 b(and)f(Quantum)f(Me)-5 b(chanics)p Fy(,)40 b(Academic)f(Press,)f(1975) 187 3662 y(\(2nd)30 b(edition,)h(2005\))0 3850 y([47])47 b(Simon,)30 b(B.:)41 b(Sc)m(hr\177)-45 b(odinger)30 b(op)s(erators,)h Fo(Bul)5 b(l.)32 b(AMS)e Fk(7)g Fy(\(1982\),)j(447-526)0 4037 y([48])47 b(Symanzik,)33 b(K.:)58 b(Euclidean)33 b(quan)m(tum)f(\014eld)g(theory)-8 b(,)47 b(in:)e Fo(L)-5 b(o)g(c)g(al)36 b(Quantum)f(The)-5 b(ory)p Fy(,)34 b(R.)f(Jost)187 4150 y(\(ed.\),)e(New)g(Y)-8 b(ork,)31 b(Academic)h(Press,)e(1969)0 4338 y([49])47 b(W)-8 b(est)m(w)m(ater,)30 b(J.:)45 b(On)25 b(Edw)m(ards')g(mo)s(del)g(for)h(p)s(olymer)f(c)m(hains,)34 b(in:)39 b Fo(T)-7 b(r)i(ends)29 b(and)g(Developments)187 4451 y(in)38 b(the)h(Eighties)f(\(Bielefeld,)i(1982/1983\))p Fy(,)i(W)-8 b(orld)38 b(Sci.)f(Publishing,)g(Singap)s(ore,)h(1985,)i (pp.)187 4564 y(384-404)0 4751 y([50])47 b(W)-8 b(est)m(w)m(ater,)32 b(J.:)48 b(On)27 b(Edw)m(ards')g(mo)s(del)h(for)g(p)s(olymer)g(c)m (hains.)h(I)s(I.)e(The)h(self-consisten)m(t)i(p)s(oten-)187 4864 y(tial,)42 b Fo(Comm.)33 b(Math.)g(Phys.)e Fk(79)g Fy(\(1981\),)i(53{73)0 5052 y([51])47 b(W)-8 b(est)m(w)m(ater,)27 b(J.:)42 b(On)22 b(Edw)m(ards')g(mo)s(del)g(for)h(p)s(olymer)f(c)m (hains)h(I)s(I)s(I.)e(Borel)j(summabilit)m(y)-8 b(,)30 b Fo(Comm.)187 5165 y(Math.)j(Phys.)e Fk(84)f Fy(\(1982\),)j(459{470) 1726 5549 y(50)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 0 431 a Fy([52])47 b(W)-8 b(est)m(w)m(ater,)32 b(J.:)49 b(On)28 b(Edw)m(ards')f(mo)s(del)i(for)f(long)i(p)s(olymer)e (c)m(hains,)38 b Fo(Comm.)32 b(Math.)g(Phys.)d Fk(72)187 544 y Fy(\(1980\),)k(131{174)0 731 y([53])47 b(Y)-8 b(or,)28 b(M.:)46 b(Pr)m(\023)-43 b(ecisions)28 b(sur)d(l'existence)j(et)f(la)f (con)m(tin)m(uit)m(\023)-43 b(e)29 b(des)d(temps)g(lo)s(caux)h(d'in)m (tersection)g(du)187 844 y(mouv)m(emen)m(t)35 b(bro)m(wnien)f(dans)g Fk(R)1369 811 y Fq(2)1409 844 y Fy(,)53 b(in:)c Fo(S)n(\023)-44 b(eminair)-5 b(e)37 b(de)g(Pr)-5 b(ob)g(abilit)n(\023)-44 b(es,)38 b(XX,)d(1984/85)p Fy(,)k Fk(LNM)187 957 y(1204)p Fy(,)32 b(Springer,)d(Berlin,)i(1986,)h(pp.)e(532{542)0 1145 y([54])47 b(Y)-8 b(oung,)29 b(L.C.:)47 b(An)27 b(inequalit)m(y)i (of)f(H\177)-45 b(older)28 b(t)m(yp)s(e)g(connected)g(with)g(Stieltjes) g(in)m(tegration,)39 b Fo(A)-5 b(cta)187 1258 y(Math.)31 b Fk(67)g Fy(\(1936\),)i(251{282)1726 5549 y(51)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0704241152621--