Content-Type: multipart/mixed; boundary="-------------0008281031761" This is a multi-part message in MIME format. ---------------0008281031761 Content-Type: text/plain; name="00-324.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-324.comments" AMS-Code: 37H20, 60H10 (primary), 34E15, 93E03 (secondary). ---------------0008281031761 Content-Type: text/plain; name="00-324.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-324.keywords" Dynamic bifurcation, pitchfork bifurcation, additive noise, bifurcation delay, singular perturbations, stochastic differential equations, random dynamical systems, pathwise description, concentration of measure. ---------------0008281031761 Content-Type: application/postscript; name="preprint.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: preprint.dvi %%Pages: 46 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips preprint -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.28:1541 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup 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 /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore 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 /IE 256 array N 2 string 0 1 255{IE S dup 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (preprint.dvi) @start %DVIPSBitmapFont: Fa ectt1000 10 17 /Fa 17 123 df45 D<121FEA7FC0A2EAFFE0A5EA7FC0A2EA1F00 0B0B708A2C>I 64 D<3801FFF8000F13FF4814C0488081819038C01FFC1407391F8001FE6CC7FCC8FC15 7EA2903803FFFE133F48B5FC1207121F48EBC07E387FF80013C048C7FC5A5AA36C14FE6C 1301387F80079038F03FFF6CB612FEA27E6C143F0003EBFC1FC6EBE00727247CA32C>97 DI100 DI< 903907F807FC90393FFF1FFE90B7FC5A5A5A9039FE1FFE7F3A0FF807FC3ED9F003130048 486C7EEBC00049137EA56D13FEEBE0016C6C485AEBF8073907FE1FF890B5FC5D485C5D01 BF90C7FCEB87F80180C8FC7FA2EA07EC90B512F015FF4815C04815F04815F89038C0003F 48C7EA03FC007E140048157EA248153EA36C157E6C15FED87F80EB03FC01E0130FD83FFE EBFFF86CB612F06C15E06C15C0000115006C6C13FC010713C028377EA32C>103 D105 D<387FFFF0B5FCA47EEA0003B3B3A3007FB61280B712C0A46C15 8022337BB22C>108 D<397FF81FF000FFEBFFFC01F97F90B6FC827E0001EBF07F9138C0 1FC01480EC000F5B5BA25BB1267FFFE0B5FCB500F11480A46C01E0140029247FA32C> 110 D114 D<90387FFCF80003B5FC120F123FA25A38FFE00FEB00034813015AA27E6C6CC7FCEA7FF0 EBFFC06C13FE000FEBFFC0000314F0C66C13F8010313FC9038000FFE1401007CEB00FF00 FC147F153F7E6C147F7F9038C001FF9038F00FFE90B5FC15FC15F815F000F814C090383F FE0020247AA32C>I<131F5BA9007FB61280B7FCA5D8003FC8FCAFED03C0ED07E0A4EC80 0F151F9138C03FC090381FF07F91B512806D1400A26D13FC010113F09038007FC0232E7E AD2C>I<3A7FF803FFC000FF5BA4007F7F0001EB000FB3A2151F153F6D137F9038FF03FF 91B6FC6C1680A27F011F13CF902607FE0F130029247FA32C>I<267FFF80B5FCB51580A4 6C16003A07C00001F0A56C6C495AA3143FEC7F83A2D801F0EB87C0ECFFC7A214F713F1EC F3E700005D13F9A201FB13EFA214E1017B91C7FC017F13FFA214C0A2013F5B90381F807C 29247FA32C>119 D<003FB612E04815F0A5007EC7EA7FE0EDFFC04A13804A13004A5A4A 5AC7485A4A5A4A5AECFF804990C7FC495A495A495A495AEB7FE0495A4890388003F04813 00485A485A485A485A485AB7FCA57E24247DA32C>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb ectt1095 10.95 20 /Fb 20 120 df<121FEA7FC0A2EAFFE0A5EA7FC0A2EA1F000B0B6E8A2F>46 DI<121FEA7FC0A2EAFFE0A5EA7FC0A2EA1F00C7FCB1121F EA7FC0A2EAFFE0A5EA7FC0A2EA1F000B276EA62F>58 D98 D<903803FFF8010F13FF017F 148090B612C05A1207EC803F380FFC004848EB1F804848EB0F004990C7FC5B485AA248C9 FCA25AA67EA36C7EA26C6CEB0FE07F6D131F6C7ED80FFE133F3A07FFC0FFC091B5128012 016C15006D13FC010F13F00103138023277AA62F>I<913807FFE0A6EC000FABEB07FC90 383FFF8F4913EF48B6FC5A5A481307381FFC019038F0007F4848133F5B007F141F49130F 90C7FC5AA25AA77EA26C141F7F153F6C6C137F7F6C6C13FFEBF803380FFE0F6CB712C07E 6C14EF6C14CF013F130FEB0FFC2A387DB72F>II<4AB4FC020F13C0023F13 E04A13F091B5FC5B49130F14FE903907FC07E09138F803C04AC7FCA7007FB612C0B7FCA5 260007F0C7FCB3A9007FB6FCA624387DB72F>IIII<387FFFFCB5FCA47EEA0001B3B3A8007FB612F0B712F8A46C15 F025387BB72F>108 D<3A7FE1FC01FC3AFFE7FF07FF90B5009F138015BF92B512C07E00 07011FEB1FE09039FE0FFE0F01FC13FC9039F807F80701F013F0A301E013E0B3A23B7FFE 1FFE1FFEB5393FFF3FFFA43B7FFE1FFE1FFE302781A62F>I<39FFFC07FCEC3FFF91B512 C090B67EA2820001EBFC1F9138F00FF8ECC007EC80031400A25BA25BB3B539F87FFFF0A6 2C277FA62F>I<39FFFC0FFCEC7FFF01FDB512C090B67E828200019038FC1FFC9138E007 FEECC0034A6C7E91C7FC49EC7F805B163F17C0A2161FA7163FA26D1580167F6D14FF1700 6E5A6E485AECE00F9138F83FFC91B55A5E5E01FD14809026FC7FFEC7FCEC0FF891C9FCAE B512F8A62A3B7FA62F>112 D<3A7FFFC01FF8B5EBFFFE02C37F02C7148014DF6C90B6FC D8001FEBF87F15E09238803F00ED001E4A90C7FC5C5C5CA25CA35CAF007FB512FEB6FCA4 7E29277EA62F>114 D<90383FFF1F48B6FC1207121F5A5AEBF00738FF800148C7FC4880 A27E7E01C090C7FCEA7FFCEBFFE06CEBFF806C14F0000714FC000180D8003F7F01001480 020313C0EC007F007EEC1FE000FE140F15077EA26D130F6D131F01F0EB3FC09038FC03FF 90B6128016005D00FD5CD8FC7F13F0D8F80F138023277AA62F>II<3AFFFC01FFF8A60001EB0003B3A415 07150F151F6D133F9038FF80FF6C90B612F0A27F7F6D13F3010313832C277FA62F>I<3B 7FFFC03FFFE0B56C4813F0A46C496C13E0D807E0C7EA7E00A46C6C5CA6EC1F803A01F83F C1F8EC7FE1A3ECFFF1A2000001F95BA201FC13F301FD13FBA214F0017D5CA390387FE07F A46D486C5AA26D486C5A2C277FA62F>119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmr12 12 2 /Fc 2 42 df<141E143E147C14FCEB01F8EB03F0EB07E014C0EB0F80131FEB3F00133E13 7E137C13FC5B12015B12035B1207A2485AA3485AA390C7FC5AA4127EA75AB3A2127EA77E A47E7FA36C7EA36C7EA212037F12017F12007F137C137E133E133FEB1F80130FEB07C014 E0EB03F0EB01F8EB00FC147C143E141E176477CA26>40 D<12F07E127C127E7E6C7E6C7E 12076C7E7F6C7E12007F137C137E133E133F7F1480130F14C0A2EB07E0A3EB03F0A31301 14F8A4EB00FCA7147EB3A214FCA7EB01F8A414F01303A3EB07E0A3EB0FC0A21480131F14 005B133E137E137C13FC5B1201485A5B485A120F485A48C7FC127E127C5A5A17647BCA26 >I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsy10 12 1 /Fd 1 113 df<1B0E1B1E1B3EA21B7CA21BF8A2F201F0A2F203E0A2F207C0A2F20F80A2 F21F00A21A3EA262A262A24F5AA2621903A24F5AA24F5AA24FC7FCA2193EA261A261A24E 5AA24E5AA24E5AA24E5AA2010E4CC8FC131E017F163E5B486D5D1207485F487F007F4C5A 38FC7FE000F84C5A486C7E00004C5A6D7E4D5A6D7E4DC9FC6D7E173E13036E5CA26D6D5B A26D6D5B1601EC7FE04C5AEC3FF04C5AEC1FF84C5A140F6F48CAFCA2913807FE3EA26E6C 5AA26E5BA26E5BA26F5AA25E153F5E151F93CBFC814F647A8353>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe msam10 12 1 /Fe 1 55 df<19F01803180F183FF0FFC005031300EF0FFCEF7FF0933801FFC0040790C7 FCEE1FFCEE7FE0923801FF80DB07FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF8 EC7FE0903803FF80D90FFECAFCEB3FF8EBFFE0000390CBFCEA0FFCEA3FF0EAFFC090CCFC A213C0EA3FF0EA0FFCEA03FFC613E0EB3FF8EB0FFE903803FF809038007FE0EC1FF8EC07 FE913801FF809138007FE0ED1FF8ED07FE923801FF809238007FE00070ED1FFC00FCED07 FFB4030113C001C09138007FF0D83FF0ED0FFCD80FFCED03FFD803FF030013C0C601E0ED 3FF0D93FF8150FD90FFE1503902603FF8014009026007FE01500EC1FF8EC07FE913801FF 809138007FE0ED1FF8ED07FE923801FF809238007FE0EE1FFCEE07FF040113C09338007F F0EF0FFCEF03FF050013C0F03FE0F00FF0180318003C4E78BE4D>54 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi12 12 3 /Ff 3 121 df34 D<147C14FEA21301A25CA21303A25CA21307A25CA2130FA25CA2B612FCA215F839001FC0 00133FA25CA2137FA291C7FCA25BA25BA21201A25BA21203A25BA21207A25BA2000F14F0 A29038E001E0A2001FEB03C0A29038C00780A2EC0F005C000F133EEBE0FC3807F3F8EBFF F06C13C0C690C7FC1E3F7EBD23>116 D120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmex8 8 1 /Fg 1 85 df84 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh eccc1095 10.95 14 /Fh 14 117 df<123FEA7F80EAFFC0A6EA7F80EA3F000A0A77891D>46 D<49B4FC010F13E0013F13F890387F01FC9038FC007E48487F4848EB1F804848EB0FC048 48EB07E0A24848EB03F0A3003F15F8A290C712014815FCA54815FEB36C15FCA46D130300 3F15F8A4001F15F06D1307000F15E06D130F000715C06C6CEB1F806C6CEB3F006C6C137E 90387F01FC6DB45A010F13E0010190C7FC273D7BBB32>48 D<14E013011303130F137FEA 07FFB5FC138FEAF80F1200B3B3AA497E497EB612FEA31F3C76BB32>I<150F5DA25D5DA2 5D5C5CA25C4A7E140E141E143C1438147814F014E01301EB03C0EB078014005B131E131C 133C5B137013F0485A5B1203485A48C7FC120E121E5A123812785AB81280A3C86CC7FCAC 913801FFC091B61280A3293C7CBB32>52 D<123FEA7F80EAFFC0A6EA7F80EA3F00C7FCB3 123FEA7F80EAFFC0A6EA7F80EA3F000A2777A61D>58 D80 D102 D105 D110 DI< B612FCEDFF8016E03A07FC000FF86C48EB03FCED00FE167FA2EE3F80A217C0A61780A2EE 7F00A216FEED03FCED0FF890B612E0168003FCC7FC01F8C9FCB1487EB512E0A32A2F7CAE 33>I114 D<90387FE01C3801FFFC0007EBFF3C390FF07FFC381F800F383F0003003E13 01481300A248147C153CA36C141CA27E6C14007FEA7FF013FF6C13F06C13FF15C06C14E0 000314F0C614F8011F13FC1300EC0FFE1403EC00FF157FA200E0143FA2151FA27EA26C14 3EA26C147E6C147C6C6C13FC9038E003F89038FC0FF000F3B512E000E0148090380FFE00 20317BAF2A>I<007FB712F8A39039801FF0073A7E000FE001007CED0078127800701638 A200F0163CA248161CA6C71500B3A84A7E011FB512F0A32E2F7CAE36>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi ecti0800 8 5 /Fi 5 117 df99 D<391FC03FE0486CB47E 393DF3F1F83979F7C0FC3978FF807C00701300EAF1FE5BEAE1F8A2D8E3F013FC1203495B 140112075DEBC003A2000FECE1C01407138015C3121F168001001387ED8F0015DFEC03FE 001EEB01F8221D7A9C28>110 D I115 D<133EA2137EA2137CA213FCA25BA21201A2B512E0 A33803F0005BA21207A25BA2120FA25BA2121FA290C7FCA2383F01C0A2EA3E03A2EB0780 A2383C0F00EA3E1E137C6C5AEA0FE013297AA818>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi6 6 15 /Fj 15 119 df11 D13 D<0007131F390F807F80381F81FF903883 CF00EB0F0F011EC7FCEA3F785B13FEEBFFC0387E0FF01303387C01F80100133000FC1470 0101136000F813F015E0903800F9C048EB7F80EC3F001C157C9424>20 D26 D<0003B512E0120F5A4814C0D8780EC7FC12E012C0C65AA3133CA2 137CA2137813F8A4485A12001B157D941D>28 D34 D<127C12FEA212FFA3127F1203A212071206120EA2 121C12381270126008117B8613>59 D<141CA2143CA21478A2147014F0A2EB01E0A2EB03 C0A214801307A2EB0F00A2130E131EA25BA25BA2137013F0A2485AA25B1203A2485AA290 C7FC5AA2121EA25AA212381278A25AA25AA216317CA420>61 D<1304A3130EA600F0EB01 E0397FDF7FC0001FB51200000713FC000113F038007FC06D5A497E137BEBF1E000017F13 E03803C078EB803848487E0006130C1B197E9820>63 D75 D107 D<3901F03F803903FCFFE039077FE1F0390E3F80F8000CEB0078013E 137CEA1C7E1200137CA201FC13FC15F85BA20001EB01F0EC03E0A29038FC07C03903FE1F 00EBFFFEEBE3F001E0C7FC1207A25BA2120FA2EA7FF8A212FF1E1F819420>112 D<137813F8A31201A25BA21203A2B51280A33807E0005BA2120FA25BA2121FA290C7FCEB 0180EA3F031400EA3E07130EEA3F3CEA1FF8EA07E0111F7D9E18>116 D<3907E001C0391FF003E0381CF807003814C01270126138E1F00F00011480120313E014 1F0007140013C0150CEC3F1CEC3E18147EECFE383903E1FF703901FF9FF039007F07C01E 157E9425>I<3807E007391FF00F80383CF81F12380070130F0061130738E1F003000114 00120313E05C0007130613C0140E140C141C5C6D5A6C6C5A6CB45A6C6C5A19157E941F> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy6 6 6 /Fk 6 113 df0 D48 D65 D<903807FFFE017FEBFFE048B612F83A07F1F03FFED80F01EB03FF001E02001380003CED 3FC0007C151F007849EB0FE0D8E003140712001603A35C130717C016075C010F1580EE0F 0091C75A49141E5E011E5C013E5CED03E0013CEB0780017C011FC7FC017813FC9038F80F F048B512C04849C8FC4813E02B227EA130>68 D83 D<171C173CA21778A217F0A2EE01E0160317C0EE0780A2EE0F00A2161E163E163C5EA25E A24B5A15035E6D495A1203D80FC049C7FC001F5CD87FE0131E00F35CEAE3F000015C7F00 005CEBFC01017C5B90387E03C0133E90383F0780131F028FC8FCEB0F9F14DEEB07FCA26D 5AA26D5AA26D5A2E327C8232>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmmi9 9 1 /Fl 1 29 df<90B612F85A12075A16F0261FC070C7FC383F00F0127C1278EAF80112F000 005BA21303A3495AA3130FA25C131FA449C8FCA5133E25207E9F22>28 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmsy9 9 1 /Fm 1 66 df<1707171F5F5FA25F5EA25EA25EA24C7EA2161EA2163CA2167804F87F16F0 150116E0ED03C0173FED0780150F16005D153E153C157C15785D14014A488092B6FC5C5C A25C023EC7121F5C8400105B383801F0EA7803267C07E081267F0FC0140FB5481660F0FB E091C813FF6C486F13C04917006C485E6C48ED03F0D8078092C8FC3B397EB53D>65 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn msam10 10.95 2 /Fn 2 63 df<180F183F18FF1703EF1FFCEF7FF0933801FFC004071300EE1FF8EE7FE092 3801FF80DB07FEC7FCED3FF8EDFFE002031380DA0FFEC8FCEC3FF0ECFFC0010390C9FCEB 0FFCEB7FF03801FFC0000790CAFCEA1FFCEA7FE0EAFF8048CBFC6C7EEA7FE0EA1FFCEA07 FF000113C038007FF0EB0FFCEB03FF010013C0EC3FF0EC0FFE913803FF80020013E0ED3F F8ED07FE923801FF8000F89138007FE000FEED1FF86C6CEC07FFD87FE0020113C0D81FFC 9138007FF0D807FFED1FFC000101C0EC03FF26007FF01400D90FFC153FD903FF150F0100 01C01400EC3FF0EC0FFE913803FF80020013E0ED3FF8ED07FE923801FF809238007FE0EE 1FF8EE07FF040113C09338007FF0EF1FFCEF03FEEF00FF183F180F384779B947>54 D<12F012FCB4FC13C0EA3FF8EA0FFE3803FF80C613E0EB1FF8EB07FE903801FF80903800 7FE0EC1FFCEC07FF020113C09138007FF0ED0FFCED03FF030013C0EE3FF0EE0FFE933803 FF80040013E0EF3FF8EF07FEEF01FFEF007FEF01FFEF07FEEF3FF8EFFFE0040313809338 0FFE00EE3FF0EEFFC0030390C7FCED0FFCED7FF0913801FFC0020790C8FCEC1FFCEC7FE0 903801FF80D907FEC8121FD91FF8157FD9FFE0EC01FF00030180EC07FED80FFEC8EA3FF8 D83FF8EDFFE0D87FC00203138048C8380FFE0000FCED3FF000F0EDFFC0C8000390C7FCED 0FFCED7FF0913801FFC0020790C8FCEC1FFCEC7FE0903801FF80D907FEC9FCEB1FF8EBFF E000031380D80FFECAFCEA3FF8EA7FC048CBFC12FC12F0384779B947>62 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo msbm10 10.95 4 /Fo 4 83 df69 D<267FFFFC91381FFFFC8080270FC00F800100 13C02703E007C0EC3F003801F0036C6C6C6C141E816D6C7E6D13786D7F153E9038F7801F 6E7E9039F3E00780D9F1F07F01F06D7EEC780191387C00F06E7F021E137C6E133C6F7E91 3807C01F0203EB0F80913801E0079238F003C0DA00F813E092387C01F0ED3C006F137803 1F137C92380F803E0307131E923803C00F04E0139E923901F007DE0300130393387801FE 167CEE3E0070137E70133EEE0780EFC01E933803E00E1601EE00F017F8177C173E171E17 0F188EEF07CE1703486CED01EE18FED807FE1500B500F0157E183EA2CB121E180E3E407E BD35>78 D80 D82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp ecbx1200 12 33 /Fp 33 119 df46 D49 D<903801FFF0011F13FF90B612C04815F0000715FC260FFC077F261FF0017F 273FC0007F1380496D13C0D87FF06D13E06D7F00FF6E13F07F8117F881A46C5A6C5A6C5A C8FC5D17F0A24B13E0A24B13C0A24B138017004B5A4B5A4B5A4A13E05E4A5B4A48C7FC4A 5A4A5AEC3FE04A5A4A4813F84990C7FCEB03FC49481301494814F0495AEB3F8049C7FC01 FE14034848140748B7FC4816E05A5A5A5AB8FCA217C0A42D417BC038>I<903801FFF001 0F13FF013F14E090B612F84801C17F4848C66C7ED807F86D7E496D1380D80FFC7F6D15C0 487EA31480A21400A36C4A13806C5AEA01F8C84813005E157F4B5A4A5B4A13E0021F5B01 07B5C7FC15FCEDFF8016E0D9000113F89138007FFE6F7E6F13806F13C06F13E0A217F081 A217F8A2EA1FF0487E487E487EA517F05DA26C4815E0495B494913C06C48491380D81FF8 4913006CB448485A6C90B55A6C15F0C65D011F91C7FC010313F02D427BC038>I<163F5E 5E5DA25D5D5D5DA25D5D92B5FC5CEC03F715E71407EC0FC7EC1F87EC3F07143E147C14FC EB01F8EB03F0EB07E014C0130FEB1F80EB3F00137E137C5B1201485A485A485A5B121F48 C7FC127E5AB91280A5C8000F90C7FCAC027FB61280A531407DBF38>I65 D68 DI78 D80 D<902601FFF0131C010F01 FF133C017FECC07C90B6EAF0FC000315F9489038C03FFF48EB0007D81FFC130149EB007F 4848143F49141F007F150F160749140312FF1601A26D1400A27F177C7F7F01FF92C7FC14 C06C13FCECFFE015FE6CECFFE016FC6C816C6F7E836C826C826C826C6C81131F010781EB 007F020780EC003F150303001480163F828200F881A282A26C81A37E18006C5DA26D5D6D 140701F05D6D140F01FE4A5AD9FFC0EB7FF09139FC03FFE0486CB65A486C5DD8F80F4AC7 FCD8F00114F826E0000F13C031467AC43E>83 D<003FBA12E0A59026FE000FEBC003D87F F09338007FF049173F0180170F190790C7FC007E1803A3007C1801A400FC19F8481800A5 C81700B3B3A20107B87EA545437CC24E>II<90380FFFF890B67E000315E04815F849C67F486CEB3FFEED 0FFF6F7FA281836C487FA2EA01F8C8FCA40207B5FC49B6FC131F90B512F90003EBFE0148 13F04813C0481300485A485AA2485AA25BA25D7F5D6C6C5B6D497F6C6C017F13F8DAC1FE EBFFC0000FEBFFFC6CECF07F0001ECC03F3A001FFE000F322C7DAB36>97 DI<91387FFF800107B512F8011F14FE01 7F809038FFF8030003D9E00713804813C04813801400485AA248486D1300A2007FEC00FC 4991C7FCA212FFAC6C7EA3123F6DEC07C0001F150F7F6C6DEB1F806E133F6C6DEB7F006C 01F813FEC69038FE07FC6DB55A011F5C010714809026007FFCC7FC2A2C7CAB32>I101 D<913803FFC0023F13F091B512F8010314FC4913CF011FEB1FFE14FCEB3FF8137F14F013 FF14E0ED0FFCED07F892C7FCABB612F8A5C601E0C7FCB3B0007FEBFFE0A527457DC422> I<17FE903B01FFF007FF80011FEBFF0F017F91B512C048B8FC4801E0EBFE7F489038803F FC489039001FFE3F49130F001F03FF1380496DEB0E00003F1680A9001F93C7FC6D5B000F 5D6D131F6C6D485A6C9038E0FFF891B55A4815C001DF91C8FC018113F0D80F80CAFCA37F A27F13F890B612E016FF6C16C017F06C82836C82120F4882D83FF8C77ED87FE002071380 49804848140090C9FC177FA36D15FFA26C6C4A13006D5CD83FF8EC0FFED81FFEEC3FFC3B 0FFFE003FFF8000390B612E0C61680011F02FCC7FC010114C032417DAC38>II<13FE487E4813804813 C04813E0A76C13C06C13806C13006C5A90C7FCABEB7FC0EA7FFFA512037EB3AEB6FCA518 467CC520>I107 DI<90287F800F FF80EB1FFFB5017F01E090B512C00281B5D8F80314F002836E4880913D87F83FFE0FF07F FC913D8FC01FFF1F803FFE000390269F000F90383E001F6C01BE6D49130F02FC4B804A6D 497FA24A5D4A5DA34A5DB3A6B60081B60003B512FEA5572C7CAB5E>I<90397F800FFFB5 017F13E00281B57E02838091398FF07FFC91399FC01FFE000314006C01BE130F02FC804A 7F5CA25CA35CB3A6B60083B512FEA5372C7CAB3E>II<90397FC03FFEB500C1B512C002CF14F091B612FC03E07F9238003FFF000301FC01 0F13806C496D13C04A15E04A7F4A6D13F0A218F882A318FC177FAA17FF18F8A34C13F0A2 5E6E15E06E4913C06E5B6E49138002FF4913009238E1FFFE02DFB512F802CF14E002C314 809126C03FF8C7FC92C9FCAEB67EA5363F7DAB3E>II<90397F80FF80B5008313E0028713F8028F13FCEC9FCF9138BF1FFE00 0313BE6C13FCA214F814F0ED0FFCA29138E003F092C7FCA35CB3A4B612E0A5272C7DAB2E >I<90393FFF078048B6FC12075A381FF807383FC0004848137F90C7123F00FE141FA215 0F7E7F6D90C7FC13F8EBFFC014FE6CEBFFC015F06C14FC6C806C806C158012016C6C14C0 1307D9001F13E0140100F87F153F6C141FA26C140FA27EED1FC07F6D133F01F0EBFF80D9 FE03130090B55A5DD8F87F13F0D8F00F90C7FC232C7CAB2C>IIII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq ecbx1095 10.95 49 /Fq 49 122 df28 D<141F143F147F14FEEB01FCEB03F8 EB07F0EB0FE0131FEB3FC0A2EB7F80EBFF00A2485AA212035B12075B120FA25B121FA348 5AA4127FA25BA412FFB2127FA47FA2123FA46C7EA3120F7FA212077F12037F1201A26C7E A2EB7F80EB3FC0A2EB1FE0130FEB07F0EB03F8EB01FCEB00FE147F143F141F185A77C328 >40 D<12F87E7E127F6C7E6C7E6C7E6C7E7F6C7EA26C7E6C7EA2EB7F80A214C0133F14E0 131F14F0A2130F14F8A3EB07FCA414FEA21303A414FFB214FEA41307A214FCA4EB0FF8A3 14F0131FA214E0133F14C0137F1480A2EBFF00A2485A485AA2485A5B485A485A485A48C7 FC12FE5A5A185A7AC328>I46 D48 DI<903807FFC0017F13 F848B6FC4815C0000F81261FF81F7F263FE0077FD980017F267FE0007F6D137F00FF6E7E 7F81178081A46C5A6C5AEA0F80C8FC4B1300A34B5AA24B5A4B5A5E4A5B4A5B4A5B4BC7FC 4A5A4A5AEC3FE04A5A4A5A903A01FE000F80495A495A4948EB1F00495AEB3F8049C7FC01 FE5C485A48B7FC485D5A5A5A5AB7FCA25EA4293C7BBB34>I<903801FFF0011F13FE017F EBFFC090B67E48018313F82603FC007F4848133F000F8101FE131F001F817FA76C48495A 6C5A6C5AC8485A5E15FF4A5B020713C0021F5B902607FFFEC7FC15F815FF16C0D9000313 F09138007FFC6F7E6F7E17808117C0A28117E0EA0FC0EA3FF0487EA2487EA417C05DA26C 484913805B6C484913006D495A3A1FFF03FFFC6C90B55A000315E06C5DD8003F49C7FC01 0313E02B3D7CBB34>II<000F15 3801E0EB01F801FE133F90B6FCA25E5E5E5E4BC7FC5D15F015C04AC8FC0180C9FCA99038 81FFE0018F13FC01BF13FF90B612C002817FD9FC007F01F0EB3FF84980496D7E5BC87F81 A21780A3120FEA3FE0487E12FF7FA41700495BA26C485C49133F495C6C6C495A391FF001 FFD9FE0713E06CB65A00035D6C4AC7FC6C6C13F8010F1380293D7BBB34>II<121F7F13F890B712F0A45A17E017 C01780A217005E485D007EC7EA03F85E007C4A5A150F4B5A4B5A484AC7FC15FE5DC7485A 14034A5AA24A5A141F5D143F5D147F14FFA24990C8FCA35BA25BA25CA2130FA5131FAA6D 5A6D5A6D5A2C3F7ABD34>I<903801FFF0010F13FF013F14C04914F02601FFC07F3A03FE 001FFC49130F48486D7E48481303496D7EA2001F80A27F7F7F6D5BEBFF8002E05BECF003 9138FC07FC6CEBFF0FED9FF86CECFFF05E6C15806C92C7FC6C15C06D14F06D80498048B6 7E488148486C1480380FFC1F261FF80714C048487E49C6FC4848013F13E0814848130715 0190C8FC167F163FA2161FA26D15C0163F6C7E6DEC7F807F6C6C903801FF00D81FFEEB07 FE3A0FFFC03FFC6C90B55A6C5DC615C0013F91C7FC010313F02B3D7CBB34>I<903801FF F0011F13FE017F6D7E90B67E4801E07F00079038803FF048496C7E496D7E484880003F14 0782485A6F1380A212FF17C0A517E0A55D127FA35D6C7E5D121F6C6C5B6C6C5B6CEBC1FB 6CEBFFF36C14E3013F01C313C00107130390C7FCA25DD803F01580EA0FFCA2486C15005D 5EA24B5A4B5A49137F4B5A260FF0035BD9FC0F5B6CB6C7FC6C5C6C14F86C6C13E0D90FFE C8FC2B3D7CBB34>II<16F84B7EA2 4B7EA34B7EA24B7FA24B7FA34B7FA24B7FA2157D03FD7F15F80201805D167F020380EDE0 3F0207804B7EA2020F814B7E021F814B7E4A81143E82027E81027C7F02FC815C82010182 91B7FC4982A3498202C0C7121F010F834A80011F8391C8FC834983013E81017E83137C83 B500FC49B612F8A5453E7CBD4E>65 DI<92260FFFC0131C4AB500F8137C020F02FF13FC027F15C149B7 12E301079138803FF7499039F80007FF4901E01301017F01807F4948C8127F4849153F4A 151F4849150F48491507485B18034849150191C9FC5A18005B127F197CA348481700AE6C 7E197CA3123F6D17FCA26C18F86E15016C7F18036C6D16F06C6D15076C6DED0FE06EED1F C06C6D153F6D6C6CEC7F80011F01E0903801FF006D01F8EB07FE6D9039FF803FFC010191 B512F06D6C5D020F1580020102FCC7FCDA000F13C03E407ABE4B>III76 D78 D80 D82 D<903A03FFC001C0013FEBFC0390B6120748158F000715FF481380391FFC001F01F01307 48487F15004848147FA249143F00FF151FA2160F7FA26D14077F7F01FE91C7FC6D7E6C13 FCECFFC015FE6CECFFC016F06C816C15FE6C816C16807E6C16C0013F15E0130F010015F0 1407EC003F030713F8150181167F00F8153F161FA2160F7EA36C16F0161F7E7F6DEC3FE0 01F0147F6DECFFC001FF491380DAF00F130091B55AD8FC7F5CD8F81F5CD8F00714C027E0 003FFEC7FC2D407ABE3A>I<003FB912F8A5903BFC007FFC007FD87FF0EE1FFC01C01607 49160390C71501007E1700A3007C187CA400FC187E48183EA5C81600B3AF011FB712F0A5 3F3D7CBC48>II<90383FFFE048 B512FC0007ECFF804881261FFC0313F06DC67F157F6F7E151F82150F6C5AA2EA03F0C8FC A2EC1FFF0107B5FC133F48B512EF0007EBF80F4813C0481300485AEA7FF8A2485AA25BA2 151F7F153F6C6C137F6D48B51280273FFF07F713FC6CEBFFE76C14C3000314013A003FF8 007F2E287DA732>97 DIII<903801FFF0010F13FE013FEBFF8090B612C0 4801E013E000079038803FF0489038001FF849EB0FFC48481307484814FEA2007F140316 FF5B12FF8190B7FCA401F0C8FCA5127F7FA2123F6D141F121F6D143F6C6C147F6C6D13FE 6C9038E003FC6C9038F80FF86C90B512F0013F14E0010F14800100EBF80028287DA72F> I IIII107 DI<01FFD93F FC903801FFE0B548B5D8800F13FC0207DAE03F13FF4A6E4880DA1FE1ECFF0F91293F007F F9F8037F0007017E90393FFBF0016C01F8DAFFC0804A6D497EA24A92C7FC4A5CA34A5CB3 A3B5D8FE07B5D8F03FEBFF80A551287CA758>I<01FFEB3FFCB548B5FC020714C04A80DA 1FC17F4AC6FC0007017C137F6C49804A133F5CA25CA35CB3A3B5D8FE0FB512E0A533287C A73A>II<9039FF80FFF8B5008F13FF02BF14C091B612F0 03837F9139FC007FFC000301F06D7E4A6D7E4A7F4A15808218C082A218E082A95EA218C0 A25E18805E6E4913006E5C6E137F02FC495ADAFF8313F092B55A02BF1480028F49C7FC02 8013E092C9FCADB512FEA5333A7DA73A>I<9038FF01FEB53807FFC0021F13E04A13F0ED BFF8EC7E3F000713FC6C13F814F014E0A2ED1FF09138C007C092C7FCA25CB3A2B67EA525 287DA72B>114 D<90383FFE1E0003B512FE120F5A383FF00F387FC00390C7FC157E12FE 153E7E7F01F090C7FC13FF14FCECFF806C14F0816C806C806C8000031580C6FC010F14C0 EB003F140300F81300157F6C143F151F7E7E6DEB3F806D137F6DEBFF00EBFC0790B55A15 F800F814E0D8F01F90C7FC22287DA729>IIIII121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmex10 10.95 34 /Fr 34 116 df<141E143E147E14FC14F8EB01F0EB03E01307EB0FC01480131FEB3F00A2 137EA25BA2485AA2485AA212075BA2120F5BA2121FA25BA2123FA448C7FCA812FEB3A412 7FA86C7EA4121FA27FA2120FA27F1207A27F1203A26C7EA26C7EA2137EA27FA2EB1F8013 0F14C0EB07E01303EB01F0EB00F814FC147E143E141E176C72832A>0 D<12F07E7E127E123E7E6C7E7F6C7E12037F6C7EA26C7EA2137EA27FA2EB1F80A214C013 0FA214E01307A214F0A21303A214F8A4EB01FCA8EB00FEB3A4EB01FCA8EB03F8A414F0A2 1307A214E0A2130F14C0A2131F1480A2EB3F00A2137EA25BA2485AA2485A5B1207485A5B 48C7FC123E127E5A5A5A176C7C832A>III<151F15FF1403EC0F FEEC3FF0EC7FC0ECFF80491300495A5C495AA25CB3AD130F5C131F133F495A495A000390 C7FCEA07FCEA1FF8EAFFC090C8FCA213C0EA1FF8EA07FC6CB4FCC67F6D7E6D7E131F130F 801307B3AD80A26D7E806D7E6D1380EC7FC0EC3FF0EC0FFEEC03FF1400151F206C768335 >8 D<12F8B4FC13E0EA3FF8EA0FFEEA03FFC67F6D7E6D7E131F6D7EA21307B3AD801303 80806D7E6D1380EC7FC0EC3FF0EC0FFCEC03FF1400A21403EC0FFCEC3FF0EC7FC0ECFF80 491300495A5C5C13075CB3AD130FA2495A133F495A495A000390C7FCEA0FFEEA3FF8EAFF E090C8FC12F8206C768335>I<12F0B3B3B3A5043B73811E>12 D16 D<12F87E7E127F6C7E6C7E120F6C7E6C7E7F6C7E1200137E137F6D7EA26D7E6D7EA26D7E A26D7EA26D7EA26D7EA2147FA281143F81141FA281140FA281A2140781A21403A281A314 0181A46E7EA6ED7F80AAED3FC0B3A9ED7F80AAEDFF00A64A5AA45D1403A35DA21407A25D 140FA25DA2141F5DA2143F5D147F92C7FCA214FEA2495AA2495AA2495AA2495AA2495A49 5AA249C8FC137E5B1201485A5B485A485A121F485A48C9FC12FE5A5A22A37D8336>II<12F87E7E12 7F6C7E6C7E6C7E6C7E12036C7E6C7E7F137F6D7E131F806D7E1307806D7E6D7EA26D7EA2 147F6E7EA26E7EA26E7EA26E7EA26E7EA26E7EA28114008181A282153F82A2151F82A215 0F82A2150782A21503A282A2150182A4150082A4EE7F80A6EE3FC0A7EE1FE0AB17F0160F B3AC161F17E0ABEE3FC0A7EE7F80A6EEFF00A45E1501A45E1503A25EA21507A25E150FA2 5E151FA25E153FA25E157F93C7FCA25D5D14015DA24A5AA24A5AA24A5AA24A5AA24A5AA2 4AC8FC14FEA2495AA2495A495A5C130F495A5C133F49C9FC13FE5B485A485A1207485A48 5A485A48CAFC12FE5A5A2CDA7D8343>II I24 DI<17F81603160F163FEE FFF04B13C0030713004B5AED1FF84B5A4B5A4B5A4A5B4A90C7FC5D4A5AA24A5AA24A5AA2 5D143FA25DB3B3B3A7147FA25DA214FFA292C8FC5B495AA2495A5C130F495A495A495A49 5A4890C9FC485AEA0FFCEA1FF0EA7FE0EAFF8048CAFCA26C7EEA7FE0EA1FF0EA0FFCEA03 FE6C7E6C7F6D7E6D7E6D7E6D7E1307806D7EA26D7E7F81A2147FA281A2143FB3B3B3A781 A2141F81A26E7EA26E7EA26E7E816E7F6E7F6F7E6F7E6F7EED0FFE6F7E030113C06F13F0 EE3FF8160F160316002DDA758344>I<12F812FE6C7E13E0EA7FF8EA1FFCEA07FE6C7E6C 13C06C6C7E133F6D7E6D7E8013076D7E6D7EA27F81A2147F81A2143FB3B3B3A781A2141F A281140F81A26E7EA26E7E6E7E82806F7E6F7E6F7EED0FFCED03FE6F7E6F13C0EE3FF0EE 0FF81603A2160FEE3FF0EEFFC04B13004B5AED0FFCED1FF04B5A4B5A4B5A5C93C7FC4A5A 4A5AA24A5AA25D141F5DA2143FA25DB3B3B3A7147FA25D14FFA292C8FC5BA2495A495A13 0F5C495A495A137F48485A4890C9FC485AEA1FFCEA7FF8EAFFE0138048CAFC12F82DDA75 8344>I[<17FC1603160F161F163FEEFFF84B13E04B13C04B138092380FFE004B5A4B5A4B 5A5E4B5A5C5E4A90C7FCA24A5AA24A5AA25D141FA25DA2143FA25DB3B3B3B3A5147F5DA3 14FFA25D5B92C8FCA25B5C13075C130F495A5C133F495A495A485B4890C9FC485A485A48 5A485AB45A138090CAFCA27F13E0EA3FF06C7E6C7E6C7E6C7E6C7F6C7F6D7E6D7E131F80 6D7E1307801303807FA2817F81A2147FA381143FB3B3B3B3A581A2141FA281A2140F81A2 6E7EA26E7EA26E7F82806F7E826F7E6F7E6F7E923807FF806F13C06F13E06F13F8EE3FFC 161F160F16031600>46 272 115 131 73 40 D80 D82 D88 DII<933807FFF8047FEBFF800303B612F0031F15FE037F6F7E4AB8 12E0020717F8021F17FE4A9026FE001F7F91B500E0010114C04991C8003F7F4901FC030F 7F4901F003037F4901C003007F4990CAEA3FFE4948717ED97FF805077F4A834948717F48 49717F4A187F4890CC6C7E4848737EA24848737E491907001F87491903003F874985A248 48731380A3491A7FA200FF1CC0A390CE123FB3B3B3B3AE007EF31F80A25A7F7B7F65>92 D<15C0EC03F0EC0FFC4A7E91387FFF8049B512E00107EB3FF890391FFC0FFE90393FE001 FF903AFF80007FC0D803FEC7EA1FF0D80FF8EC07FCD83FC0EC00FF48C9EA3F8000FCEE0F C000F01603321080C233>98 D104 DI108 DI<161F16FF1503150FED3FFEED7FF8913801FFE04A13804A1300EC0FFC 4A5A5D4A5A147F5D5D14FFA292C7FCB3B3AA5BA25C13035C1307495A131F495A495A495A 485B4890C8FCEA0FFCEA3FF8EAFFE0138048C9FC6C7E13E0EA3FF8EA0FFCEA03FF6C7F6C 7F6D7E6D7E6D7E130F6D7E130380130180A27FB3B3AA81A2147F8181143F6E7E816E7E6E B4FC6E13806E13E09138007FF8ED3FFEED0FFF15031500161F28A376833D>I<12F8B4FC 13C013F0EA7FFCEA1FFE3807FF8000017F6C7FEB3FF06D7E130F6D7E801303130180A27F B3B3AA81A2147F81143F816E7E816E7E6E7E6E7E6E13806E13C0ED3FF0ED1FFCED07FF15 01ED007FED01FF1507ED1FFCED3FF0EDFFC04A13804A13004A5A4A5A4A5A5D4A5A5D147F 5D14FFA292C7FCB3B3AA5BA25C130313075C495A131F495AEBFFE0485B00075BD81FFEC8 FCEA7FFCEAFFF013C090C9FC12F828A376833D>I<1CE01B011B03A2F307C0A2F30F80A2 F31F00A21B3EA263A263A2505AA2631A03A2505AA2505AA250C7FCA21A3EA262A262A24F 5AA24F5AA24F5AA24F5AA24FC8FCA2193EA261A261A24E5AA24E5AA2611807131801384C 5A137CD801FC4CC9FC487E0007173EEA1FFF003E5F5A26F07F805D1260C66C6C4A5AA26D 6C4A5AA24D5A6D7E4D5A6D7E4DCAFC6D7E173EA26D6C5CA26D6C5CA26E6C485AA24C5AEC 3FC04C5AEC1FE05F91380FF00FA24CCBFCEC07F8163EEC03FC5EEC01FE5EA26E6C5AA26F 5AA26F5AA25E151F93CCFC81536D76835B>I<1CE01B01A21B03A21CC01B07A21C801B0F A21C0063A21B1E1B3EA21B3C1B7CA21B781BF8A263A21A01A2631A03A2631A07A2631A0F A298C7FC62A21A1E1A3EA21A3C1A7CA21A781AF8A262A21901A2621903A2621907A26219 0FA297C8FC61A2191E193EA2193C197CA21978A219F8A2611801A2611803A26118071308 01185F0138160F133C017C94C9FC01FC5E1201486C161E0007173E120FD80EFF163C121E 48177C5A486C6C1578006017F812006D6C5D1701A2606D6C1403A2606D6C1407A260170F 6D7E95CAFC5F6D7E171EA2173E6D7E173C177C6D7E177817F8A26E6C5B1601A2DA3FC05B 1603A25F91381FE007A25F91380FF00FA294CBFC5EEC07F8161EA2913803FC3EA2163C16 7CEC01FE167816F8EC00FF5EA36F5AA36F5AA35E151FA293CCFC81A253A476835B>I[<1C E01B01A31B03A21CC0A31B07A21C80A31B0FA21C00A363A21B1EA41B3EA21B3CA31B7CA2 1B78A31BF8A263A41A01A263A31A03A263A31A07A263A31A0FA298C7FCA462A21A1EA31A 3EA21A3CA31A7CA21A78A31AF8A262A41901A262A31903A262A31907A262A4190FA297C8 FCA361A2191EA3193EA2193CA3197CA21978A419F8A261A31801A261A31803A261A31807 A201085F1318A21338180F137C96C9FC13FCA200015FA2486C161EA21207183E120F183C EA1EFF121C123C0038177C5A1878486C7EA2004017F8120060A26D7E1701A260A26D7E17 03A260A36D7E1707A260A36D6C140FA295CAFCA35F6D7E171EA4173E6D7E173CA3177C6D 7E1778A317F8A26E6C5BA31601A25FEC3FC0A31603A25FEC1FE0A21607A25FA2EC0FF016 0FA294CBFCA2EC07F85EA2161EA3EC03FC163EA2163CA3913801FE7CA21678A316F8EC00 FF5EA5157F5EA4153F5EA56F5AA593CCFC81A3>83 273 118 131 91 115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmsy7 7 2 /Fs 2 84 df<0103B512E0013F14FE90B71280000316E0270FF8FC0713F0D81FC0903800 7FF8D83F00EC1FFC003E150748ED03FE00FC1501EE00FF00F05BD80001157FA2173FA25C 1303173E177EA24A147C010715FC17F84A1301010F15F016034AEB07E0011FEC0FC0EE1F 8091C7EA7F004914FEED03F8013EEB0FF0017EEB3FC090267C07FFC7FC48B512FC4814F0 48148002F0C8FC30287EA734>68 D<913803FF80021F13F0027F13F849B5FC903903F81F FC903807E00390380FC001148049C712F849130116E06EC7FCA28080EB1FF814FE6D6C7E 6D13E0010113F89038007FFEEC1FFF0207138014019138007FC0D80780133F001F141F48 C7FC007E140F127C12FC1680151F6C15006C5C6D137E9038E001FC397FFC0FF86CB512E0 6C1480000749C7FCC613E0262A7DA829>83 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft msam10 10 1 /Ft 1 55 df54 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu ecrm0900 9 35 /Fu 35 122 df<390380038000071307390F000F00001E131E001C131C485BA2485BA200 60136000E013E0A2485BA300DF13DF39FF80FF806D13C0A3007F137FA2393F803F80391F 001F001A1881B31A>16 D<003E133E007F137F39FF80FF80A26D13C0A3007F137F003E13 3E00001300A300011301A20180138000031303A23907000700A2000E130E001E131E485B 485B007013701A1880B31A>I<91397FF01FF0903A07FFFC7FFC903A1FF07FFE7E903A3F C03FF8FF49C6485A01FE5C48485C485AA249EC807E033F1300151FA9B812C0A33B03F000 1F8000B3A7486C497EB50083B5FCA330347FB32D>27 DI<123E127FEAFF80A213 C0A3127F123E1200A31201A213801203A2EA0700A2120E121E5A5A12700A187A8815>44 DI<123E127FEAFF80A5EA7F00123E09097A8815>I<15E04A7EA3 4A7EA34A7EA24A7E140EA2EC1EFFEC1C7FA2023C7FEC383FA24A6C7EA34A6C7EA2010180 ECC007A2010380EC8003A249486C7EA3010E6D7E010FB5FC4980A2011CC77E013C810138 143FA2496E7EA3496E7EA200018200031507486C81D81FF8EC1FFCB549B512E0A333347D B33A>65 DI70 D85 D91 D93 D<3803FFE0001F13FC383F81FE9038C07F80EC1FC0A26E7EA2381F80 07C7FCA3EB03FF137F3801FFC73807FC07EA1FF0EA3FE013C0EA7F80EAFF00EDE1C05AA2 140FA26C131F387F803F91387BF380393FE1F1FF260FFFE013000001EB807C22207D9F26 >97 DII<153FEC0F FFA3EC007F81AFEB0FF8EB7FFF3901FE1FFF3803F803380FE001497E48487F003F8090C7 FC5AA212FEAA127FA36C6C5B001F5C7F380FE0036C6C4813803A01FC3FBFFC3900FFFE3F EB1FF826347DB32B>IIIIII108 D<3C03F03FF001FF8000FF9026FFFC0713E0903BF3 F0FE1F87F0903BF7C07F3E03F83B07FF003F78016C4802F07F4990381FE000A2495CA249 5CB2486C496C487EB53BC7FFFE3FFFF0A33C207E9F41>I<3903F03FE039FFF1FFF89038 F3F1FC9038F780FE3907FF007E6C48137F497FA25BA25BB2486CEB7F80B538C7FFFCA326 207E9F2B>II<3903F0FFC0D8FFF313F89038FFE1FC EC00FFD803FEEB3F804914C049131F49EB0FE016F01507A216F81503A8150716F0A2ED0F E0A26DEB1FC06DEB3F806D137F6DEBFE00ECC3FC9038F3FFF001F0138091C8FCAB487EB5 12C0A3252F7E9F2B>I<3803E0FE39FFE1FF809038E3DFC0EBE79F3807EF1FEA03EE13FC EC0F8049C7FCA35BB1487EB512E0A31A207F9F1E>114 D<3801FF8E000F13FEEA3F8138 3E007E0078133E00F8131E5A140E7E7EB490C7FC13F06CB47E14F06C7F6C7F00077FC67F 01071380EB007F00E0133F141F6C130F14077EA26CEB0F006C5B6C133EEBC0FC38F7FFF8 00E013C019207E9F1E>I<1370A513F0A41201A212031207120F381FFFFEB5FCA23803F0 00AF1407A7140F13F80001130EEBFC1E3800FE3CEB7FF8EB0FF0182E7FAD1E>IIIII<3A7FFF80FFF8A33A07FC003FE06C48EB1F 8000011500151E7F0000141CA2017E5BA2017F13786D1370A26D6C5AA214C1010F5B14E3 01075BA2D903F7C7FCA214FF6D5AA26D5AA31478A21470A214F05CA2495A127CEAFE035C 49C8FC5BEA7C1EEA783CEA3FF0EA0FC0252F7F9F29>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv ecrm0600 6 2 /Fv 2 51 df<137013F0120712FFA212F91201B3A6B512E0A313217AA01E>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr6 6 10 /Fw 10 118 df<131C1338137013E01201EA03C0EA0780A2EA0F00121EA35AA2127C1278 A312F8A25AA97EA21278A3127C123CA27EA37EEA0780A2EA03C0EA01E012001370133813 1C0E317AA418>40 D<12E012707E7E121E7EEA0780A2EA03C0EA01E0A3EA00F0A213F813 78A3137CA2133CA9137CA21378A313F813F0A2EA01E0A3EA03C0EA0780A2EA0F00121E12 1C5A5A5A0E317CA418>I<1438B2B712FEA3C70038C7FCB227277C9F2F>43 D<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F8131FAD0078131E A2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227DA01E>48 D<137013F0120712FFA212F91201B3A6B512E0A313217AA01E>II61 D<48B4FC000713C0381FC7E0383F01F0EA3E004813F814785AB512F8A200F8C7FCA37E12 7C14186C1338003F1378380FC1F03807FFE00001138015157D941B>101 D<380FF980EA3FFFEA780FEAF003EAE001A200F8C7FCB47E13FC6CB4FC6C1380120F3800 7FC0EAC0071303EAE00112F038F80380EAFC0FB51200EAC7F812157D9418>115 D<39078003C000FF137FA3001F130F000F1307AA140F141FEC3FE03907E07FFC3803FFE7 C613871E157E9422>117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx msam8 8 2 /Fx 2 63 df54 D<12E012F812FEEA7F80EA1FE0 EA07F8EA01FE38007F80EB1FE0EB07F8EB01FE9038007F80EC1FE0EC07F8EC01FE913800 7F80ED1FE0ED07F8ED01FE9238007F80EE1FC0160F163FEEFF00ED03FCED0FF0ED3FC003 FFC7FCEC03FCEC0FF0EC3FC002FFC8FCEB03FCEB0FF0EB3FC001FFC7EA03C0D803FC140F D80FF0143FD83FC0ECFF00B4C7EA03FC00FCEC0FF000F0EC3FC0C8B4C7FCEC03FCEC0FF0 EC3FC002FFC8FCEB03FCEB0FF0EB3FC001FFC9FCEA03FCEA0FF0EA3FC0B4CAFC12FC12F0 2A397AAB37>62 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmmi8 8 38 /Fy 38 122 df11 D13 D<903803FF804913F090380FBFF8EB0E0F90381C07F0140391C7FC131E 131FA26D7E80A2806D7E806D7EA280EB1FFF137F13FE3903F87F80EA07F0380FE03F13C0 EA1F80123FEB001F5AA2127E12FEA248EB3F00A4147E127C5C127E6C485A381F83F0380F C7E06CB45AC690C7FC1D2F7EAE1F>I<390FC01FF0393FF07FFC393DF9F8FE3978FBE07F 9038FFC03F0070130012F15BEAE1FC5B00E3147F120349137EA2000714FEA24913FCA200 0F1301A201C013F8A2001F1303A2018013F0A21407A2010013E0C7FC140FA215C0A2141F A21580A2143FA21500A2143E202B7E9C23>17 D20 D<13FF14C0EB3FE0131F130F80130780A21303801301A280130080A2808114 3FA281141F81A2143F4A7E14FF5B903803F7F8EB0FE390381FC3FCEB3F81EB7F0101FE7F 3801FC00000380485A48487F48481480D87FC0133F5B48C713C048141F4815E00078140F 232E7DAD29>I<013EEB07C0013F14E049130FA2017E14C0A201FE131FA2491480A20001 143FA2491400A200035CA249137EA20007ECFE38A24913FC1401000F1578EBF003913807 F8709039F81FFCF03A1FFC3EFDE09038FFFC7F90399FF01FC00180C8FC123FA290C9FCA2 5AA2127EA212FEA25AA25A252B7E9C2A>I<146014E0A5ECFFF0010313F890380FF07890 383FFFF890387FBFF001FEC7FC485AA2485A5B1207A25BA56C7E90B5FC7E3800FE0F48B5 FC3803EFFED807C0C7FC485A48C8FC123E123C127C127812F8A47EA2B4FC13C013F0EA7F FE383FFFC06C13F06C13FE00037FC67E131F1303EB007F141FA21378EB7E3EEB1FFCEB07 F81D3C7EAE1F>24 D<48B612F812075A5A5A90390781C000127C00F81303EAF00F12E012 00131E1407133EA2133C137CA213FC13F8000180A200038013F01207A2EBE003A2496C5A 251D7E9C29>II<0107B512F0133F5B48B6FC5A3A07FC3FC0 00380FF00F01E07F381FC007EA3F80A2EA7F00A2127E00FE130FA2485C141FA25D143F92 C7FC147E007C5BEA7E01383E03F0381F8FE0380FFF80D803FEC8FC241D7D9C28>I<48B6 1280000715C05A4815805AD901E0C7FC127C48485A12F012E0EA0007A25C130FA3131FA2 91C8FC5BA4137EA5137C221D7E9C1F>I<157015F0A25DA21401A25DA21403A25DA21407 A292C7FCA2ECFFF0010713FC90381FEE7F90397F0E1FC03A01FC1E0FE0D803F813073A07 F01C03F0D80FE01301261FC03C13F8EA3F801438EA7F001478127E00FEEB7003A24801F0 13F0150702E013E0150FD87C01EB1FC0007E15809138C03F00003F14FE391F83C1FC390F C3C7F83907FBBFE00001B5C7FC38003FF8EB078091C8FCA25BA2130EA2131EA2131CA213 3CA2253B7DAD2A>30 D33 DI<127E12FFA6127E08087A8714>58 D<127EB4FCA21380A4127F1201A21203A213005AA2120EA25A123C5A127009157A8714> III<14C0A4497EA800F8EC07C039FFC3F0FF6CB61280001FECFE00000314F0C6 14C0013F90C7FCEB0FFCA2497EA2497E143F90387E1F80496C7EEBF80748486C7EEBE001 48486C7E49137090C71230222180A023>63 D<167016F015018215031507A2150FA2151D 153DED39FC157115F015E0EC01C0A2EC038014074B7E140E021E137E141C5CA25C14F04A 137F495A49B6FCA25B91C7123F130E131E011C15805B0178141F5B5B12011207D81FF0EC 7FC0D8FFFE903807FFFEA32F2E7DAD35>65 D<013FB7FCA3903900FE00014AEB007F173F 0101151F171E4A140EA21303A25C16700107ECF01EA24AEBE0001501130F15039138C00F C091B5FC5BA29138801F801507133FA2020090C7FCA2495BA2017E90C8FCA213FEA25BA2 1201A25B487EB512F8A3302D7DAC2D>70 D<90263FFFFC90383FFF805F5D010090C7381F F8004A15E04AEC3F800101037EC7FC5F4AEB03F04C5A0103EC0F804CC8FC4A137E16F801 07495AED07E09138E00F804BC9FC010F137F5D02C17F14C7D91FCF7FECDF1F02FE7FECFC 0FD93FF07FECE00702C07FEC000349801501017E80150001FE80A24980830001153F8349 141F486CEC3FF0B539E001FFFEA3392D7CAC3C>75 D<0007B81280A25A903AF801FC007F 01C049131FD81F80ED0F00EB0003121E485CA200380107140E12785D127000F0010F141E A2C74990C7FCA2141FA25DA2143FA292C9FCA25CA2147EA214FEA25CA21301A25CA21303 A25CEB0FF8000FB512F8A3312D80AC29>84 D97 D99 D<137FEA07FF5AA2C67E137EA213FEA25BA21201A25BA21203A249B47E01F313E03907FF C7F0EC03F8EBFE0113F8120F13F013E013C0001F1303A201805B1407123F5DEB000FA248 ECC380141F127E158712FE1600485C151E15BEEC0FFC48EB07F0212E7CAD29>104 D106 D<137FEA07FF5AA2C67E137EA213FEA25BA21201A25BA21203A29038F003F0EC0FF80007 EB1F3CEC7CFCEBE0F9EBE1E1380FE3C1EBE7819038CF01F801DE13F0D81FFCC7FC5B7FEB FFC0487FEB8FF0EB07F81303D87F011338A2127EA200FE1478157000FC14F015E0EB00F9 ECFFC048EB3F801E2E7CAD25>I<390FC00FF8393FF07FFE393DF8FC7F3978F9E03F9039 FFC01F800070138000F113005BEAE1FCA2D8E3F8133F12034914005D1207157E4913FEA2 000FECFC38140113C0EDF878121F16700180EBF0F0EDF1E015FB913800FFC090C7EA7F00 251D7E9C2B>110 D<90387E01FC3901FF87FF9039EFDF8FC03903C7FE079138FC03E0EB 87F826078FF013F0ECE001EB0FC01503131F12001480A2013F1307A2020013E0150F5B16 C0017E131F168001FEEB3F006D137E5DEC81F848EBC7F09038FDFFC001F890C7FC91C8FC 1203A25BA21207A25BA2120FB5FCA25B242A839C24>112 D<90380FE07090383FF9F0EB FE7F3801F81FD803F013E03807E00FEA0FC0121F018013C0123FEB001F5A007E1480A200 FE133FA2481400A25CA2147E14FEA2EA7C01495AEA3E07EA1F1FEA0FFF3807F9F8EA0001 1303A25CA21307A25CA2130F48B5FCA25C1C2A7D9C20>I115 D<131F1480133FA21400A25BA2137EA213FEA2B512F8A33801FC005BA21203A25B A21207A25BA2120FA25BA2001F1370A2138014F0EB81E0A2EB83C0EB8780139F380FFE00 EA03F815297EA819>III<90387F81FE3901FFE3FF3A03E3F7CF803A0781FF0F C0390F01FE1F121E14FC123C02F8138000381500010390C7FC12005CA21307A25CA2010F 1307123E007F13C05DD8FF1F5B151ED8FE3F5B397C7FE07C9038FBF1F0393FF1FFE0391F C07F80221D7E9C28>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz ecrm0800 8 2 /Fz 2 51 df<130E131E137EEA01FE12FFA2EAFE7E1200B3AF13FF007F13FFA3182C7BAB 23>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA eccc1000 10 29 /FA 29 256 df<123E127FEAFF80A5EA7F00123E090977881B>46 D49 DII<15 0E151FA34B7EA24B7EA34B7EA3913801DFF015CF02037F1587A202077F1503A2020E7F15 01A24A6C7EA2023C800238137FA24A80163FA24A6D7EA20101814A130FA249488091B6FC A2498191C712034981010E1401A2498182A2496F7EA2017882173F13F8000183486C82D8 0FFE4B7EB500C0010FB512E0A33B3A7CB944>65 D70 D<912601FFE01370020F 13FC027F01FF13F0903A01FFE03FC1010790380007E3D90FFCEB01F3D93FF0EB00FFD97F C0147F4948143F4890C8121F4848150F49150700071603485A491501121F5B003F1600A2 485A1870A300FF94C7FC5BAA047FB512C07F127F9339003FF800EF0FF0A26C7EA26C7EA2 120F7F6C7E12037F6C7E6C6D141FD97FE0143F6D6C147FD90FFC14FF6DB4EB03FB010190 38E01FF16D6CB512C0020FEC0070020101F090C7FC3A3B7AB947>I73 D83 D<003FB812FCA3D9E001EB800790C790C7FC 007E177E007C173E0078171EA30070170EA400F0170F481707A4C81500B3B0020313C001 0FB612F0A338397CB841>I87 D<14074A7EA34A7EA24A7EA34A7E1477A2ECE3 F8A201017F14C1A290380380FEA201077F4A7EA24980010E133FA2496D7EA2013C80013F B5FCA249809038700007A2496D7EA200018115010003811207D81FF0497ED8FFFC011F13 F8A32D2B7DAA33>97 D<91387FE00E903903FFFC1E011F13FE90393FF03FBE9039FFC00F FE48EB0003D803FC130148481300485A167E4848143E485A161E5B127F160EA248C8FC16 00AA6C6C140EA3123F6D141EA26C6C143C6C7E16786C6C14F86C6CEB01F06CB4EB03E06C EBC00790393FF03F806DB51200010313FC9038007FE0272D7BAB31>99 DIII<91387FF007903903FFFC0F010F13FF90393FF83FDF9039 7FC007FF4848C67E48481300D807F880120F49804848805B003F81A2485A82A248C8FC93 C7FCA892387FFFF8A26C7E03001380EE7F006C7EA26C7EA26C7E7F6C7E6C6C5C6C6C5B39 007FC00390393FF81FDF010FB5128F010314079026007FF8C7FC2D2D7BAB35>III108 D110 DII114 D<9038FF80E0000313F1000F13FD381FC1FF383F003F003E130F48130714035A1401A214 007EA26C14007E13C0EA7FFCEBFFC06C13F814FE6C7F000714806C14C0C67E010713E0EB 007FEC1FF0140F1407140312E01401A27EA2EC03E07E6C13076CEB0FC039FF801F80EBF0 7F00F7B5120000F113FC38E03FF01C2D7BAB26>I<007FB712C0A39039003F801F007C15 070078150300701501A200F016E0A2481500A5C71500B3A64A7E013FB57EA32B2B7DAA31 >II121 D<9026FFC0E0903807FE07000301F191381FFF8F000F01FD027F13EF261FC0FF9138FE07 FF263F803F903801FC01263E000F9138F0007F48010749487F020382484B5A020182A202 00826C82A26C02006D90C7FC6C8201C015FED87FFC913803FFE0D9FFC015FE6C01FC6DEB FFE06C01FF6D14F86F8100076E013F7F6C6E6D7FC66C150301076DD9003F1380D9003F15 01DA0FF89138007FC00207163F0203161F0201160F00E0150702001607A26C82A2DA01F0 ED0F806C826C01036E131F6CD907E06DEB3F0027FF800FC001FC137E9026F03F809038FF 81FC00F7B500009038BFFFF800F149028F5B26E03FF002011380422D7BAB4B>255 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FB ecti1095 10.95 61 /FB 61 123 df27 D<93380FFFE0047F13FC4BB512FE923903FE007FDB07 F0133FDB0FC0137F4B5A153F93C7FC187E037E147C1800A215FE5DA31401A25DA31403A2 0103B712F0A218E0903A0007F00007170F5D18C0A2020F141FA24B1480A2173FA2021F15 00A24B5BA2177E143F17FE92C7FC5FA24A1301181C027E14F8A20403133C14FEEFF0384A 1578A218700101EDF8F0EE01F94AECFFE07013C0EF3F80010392C7FC5CA3495AA35CEA3E 0F127E00FE5B131F91CAFC133EEAFC7EEA7FFC5BEA1FE0385183BE33>I<93390FFF07F0 047F13E74BB5EAF7E0923807FC03DB0FE013FF4B485A048014C0ED3F00A24B7F037E9038 001F80A203FE143FA24B1500A2601401187E5D18FEA202035DA20103B7FCA260903A0007 F0000117035D60A2020F1407A24B5CA2170F141F605D171FA2023F5DA292C7123FA295C7 FC5C5F1807027E147EA2EFFE0F14FEEFFC0E4A151EA2181C0101EDFE3C4AEC7E7CEF7FF8 715AEF0FE0010392C8FC5CA3495AA35CEA3E0F127E00FE5B131F91CBFC133EEAFC7EEA7F FC5BEA1FE03C5183BE35>II39 DII44 D<387FFFFCA3B5FCA21605799520>II48 D<150E151EA2153C157C15FC1401EC03F81407141F147F903803FFF0EB1FFB14E7140701 0013E0A2140FA215C0A2141FA21580A2143FA21500A25CA2147EA214FEA25CA21301A25C A21303A25CA21307A25CA2130FA25CA2131FA2133FEBFFE0B612E0A215C01F3C76BB2E> I51 D<167C16FCA41501A216F8A2150316F0A2150716E0A2150F16C0A2151F1680 153F1600A2157EA215FE5DA24A5A5D14035D14074A5AA24A5A92C7FC5C91387E07809138 7C0FC014FC903801F81F14F0D903E01380130790380FC03F1480D91F001300133E495B5B 0001147E485A484813FE380FFFC0003FEBF8FC4813FFD8FE01141C3AF8001FFFFC007013 03C76C13F0913803F800A25DA21407A25DA2140FA25DA2141FA25DA292C7FC264E7DBB2E >I<4AB4FC020F13C0023F13F091387F87F8903801FE01903903FC00FC495A4948137E49 5A495AA2495A01FF147F91C7FC5A5B16FF1203A25BED01FE1207A25B1503A2150716FC49 130FA20003141F6DEB3FF8157F000114FF6D5A3A00FE07EFF090387FFFCF6DEB1FE0EB1F FEEB00C09138003FC0A2ED7F80A216005D5D4A5A1403007E5C00FE495A4A5A4A5A4A5A48 49C7FC00F013FE38F803FC38FE0FF8387FFFE06C1380D80FFCC8FC283D77BB2E>57 D<133FEB7F80EBFFC0A25AA21480A26C1300137E90C7FCB3EA1F80487E127F12FFA56C5A 007EC7FC122777A61C>II63 D<1738177817F8A21601A216031607A24C7EA2161DA216391679167116E1A2ED01C1A2ED 038115071601150E83ED1C00A25D157815705DA24A5AA24A5A140792C7FC140E834A8002 1FB6FC5C5C0270C77E5CA2495AA2495A130791C8FC130E188049153F133C137C5B120148 6C157FD80FFE913801FFE0B500C0013F13FFA3383F7BBE43>65 D<49B77E18F018FC903B 0007FE0001FE6E486D7E4BEC7F80F03FC01407A24B15E0181F140FA24B143FA2141F19C0 4B147FA2023F1680F0FF004B5B4D5A027F5D4D5A92C7EA1FF0EF3FC04AECFF8092B548C7 FC17F017FE4948C76C7EEF3FC04A6E7E717E01036F7EA24A6E7EA21307845CA2130F604A 1407A2011F5E170F5C4D5A013F5E173F4A4A5A4D5A017F4A90C7FC4C5A49C7EA0FF848ED 7FF0B812C094C8FC16F03B3E7BBD3F>I<923A01FFC001C0031FEBF00392B538FC078002 039038C0FE0F913A0FFE003F1FDA1FF8EB0FBFDA7FE0903807FF00DAFF807F4990C7FC49 488049485D49481400495A495A4948157C495AA24890C9FC48481678A2485A120F491670 121F5B95C7FC123F5B127FA25BA212FFA25BA590CCFC170FA2170E171EA2171C173C5FA2 6C6C5D16015F6C6C4A5A16076C6C4A5A6C6C4AC8FC163ED807FC14FC6C6C495A3A01FF80 0FE03A007FE07FC06DB5C9FC010F13FC010113E03A4072BE41>I<49B77E18F018FC903B 0007FE0003FE6E48903800FF804B143FF01FC00207ED0FE0F007F05DF003F8140FA24BEC 01FCA2141FA24B15FEA2143FA25DA2147F180392C8FCA24A16FCA24A1507A2130119F84A 150FA2010317F0A24A151F19E01307F03FC05CF07F80130FF0FF004A5D1701011F5E4D5A 4A4A5AA2013F4B5A4D5A4A4A5A05FFC7FC017FEC01FCEE07F849C7EA1FE0486DEBFFC0B8 C8FC16FC16C03F3E7BBD44>I<49B812F8A390260007FEC7127F6E48140F4B1403180102 0716F018005DA2140FA25D19E0141FA24B1338A2023FEC7801A24B0170C7FC17F0147F16 014B485A16074A131F92B5FC5FA2903901FE003F160F4A6D5AA21303A24A91C8FC180701 074A5BA24A010E131E93C7FC010F161C183C5C60131F605C1701013F4B5AA24A14074D5A 017F151FEF7F8049C7EA01FF48151FB9C7FCA25F3D3E7BBD3D>I<49B812F0A390260007 FEC7127F6E48141F4B14071803020716E018015DA2140FA25D19C0141FA25D1770023FEC F003A24B49C7FC1601147FA24B485A16074A130F163F92B55AA25B9138FE007F4A011FC8 FC821303A24A130EA20107141EA24A131CA2010F91C9FCA25CA2131FA25CA2133FA25CA2 137FA213FF487FB67EA33C3E7BBD3B>I<49B5D8FC01B512FCA3D9000790C700071300DA 03FCEC03FCA24B5D02071507A24B5DA2020F150FA24B5DA2021F151FA24B5DA2023F153F A24B5DA2027F157FA292C890C7FCA24A5D92B7FC60A24948C71201A24A5DA201031503A2 4A5DA201071507A24A5DA2010F150FA24A5DA2011F151FA24A5DA2013F153FA24A5DA201 7F157FA249C848C8FC486D497FB690B6FCA24A5D463E7BBD43>72 D<49B512FEA216FCD900071300EC03FCA25D1407A25DA2140FA25DA2141FA25DA2143FA2 5DA2147FA292C7FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA2 5CA2133FA25CA2137FA249C8FC487FB6FCA3273E7BBD23>I<49B6FCA3D9000790C8FCEC 03FCA25D1407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA213 01A25CA21303A25C18E001071501A24A15C01703130FEF07805CA2011FED0F00A24A5C17 3E133F177E4A5C1601017F4A5A160F49C7123F486D48B45AB8FCA25F333E7BBD39>76 D<902601FFFE93381FFFC062A2D9000794387FF0000203EFFFC0A203BF923801DF800207 17FFF103BF033FDB073FC7FCA2020FEE0E7FA2020EEE1C7E1938021E17FE197091261C1F 805D19E0023C16E1F001C1023892380381F8A20278ED0703A20270030E5B181C02F01607 6F6C133802E05F18700101EEE00FA202C0DA01C05BA20103923803801FA20280DA07005B 170E0107D907E0143F5F020094C8FC5F494B5BA2010E4B137EA2011EDAE1C013FEEEE380 011C5F04E7C7FC013CD903F7130116FE01385F01785C4C130313F8486C4A495AD80FFF4C 7EB500F0D9E007B512F016C0DAE0015E523E7ABD51>I<902601FFFC91381FFFFC81A290 260003FF020113C0721300197E70143C4A5E15BFDB3FC01470A2DA0F1F15F082020E5E15 0F021E6D1301A2DA1C075D82023C1503150302386D5CA291267801FE1307A2027093C7FC 6F7E02F05D824AEC800EA20101023F131E17C04A151C161F0103EDE03CA24A010F133817 F001071678160791C7EBF870A249913803FCF0A2010E5EEE01FE011E15FF82011C5EA201 3C157FA201385E0178153FA213F8486C6FC8FCEA0FFFB512F0834A140E463E7BBD43>I< 4BB47E030F13F0037F13FC913901FF01FF913A07F8007F80DA0FE0EB1FC0DA3F806D7E4A C76C7E02FC6E7EEB03F849486E7E495A011F6F7E495A4A8149C9FC5B4848821980485A12 075B120F4916FF121FA25B123F190048485DA448484B5AA34D5AA390C9485AA24D5AA260 173F604D5A17FF6D93C7FC007F4B5A4C5A003F4B5A6D5D4C5A6C6C4A5A000F4B5A6D02FF C8FC6C6C495A6C6CEB07F86C6CEB0FE03A007FC07FC06DB5C9FC010F13F8010013803940 73BE45>I<49B77E18F018FC903B0007FE0007FE6E48EB00FF4BEC7F80F03FC00207151F 19E05DA2020F16F0A25DA2141FF03FE05DA2023F16C0187F4B1580A2027FEDFF004D5A92 C75B17034AEC07F0EF1FE04A4A5ADC03FFC7FC49B612FC17F094C8FC02FCCAFC1303A25C A21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA249CBFC487FB512FEA33C3E7B BD3D>I<49B612FCEFFFC018F0903B0007FE000FF86E48EB01FE4B6D7EF07F800207153F 19C05D19E0140FA25DA2141FF07FC05DA2023FEDFF8019005D4D5A027F4A5A4D5A92C748 5A4D5A4AEC7F80DC03FEC7FC92B512F817C05B9139FE001FE04AEB07F0707E0103140183 4A1300A2010781A25CA2010F1401A25CA2011F1403A25CA2013F140719E05C1801137F04 03EB03C049C7FC486D9138FF0780B66D139F70EBFF004A6E5ACAEA0FF83B3F7BBD42>82 D<92391FF801C09238FFFE0302039038FF878091390FF81FCF91391FC007FF4A487E9126 7E000113004A7F494880495A4A147E0107153E5C130F4A143CA2131FA21738A28094C7FC 808080EB0FFEECFFE015FC6DEBFF806D14F06D806D14FE143F02077FDA007F7F150F1501 6F6C7E163FA2161FA2160F12075AA2000E5EA2001E151F94C7FCA25E003F157E167C16FC 6D495A486C495A6D495A6D495AD87DFCEB3F8027F8FF81FFC8FC90383FFFFC486C13F0D8 E00113C032407ABE33>I<48B9FCA25A9139801FF003D9FC009038E0007FD807F04A133F 49161F49013F141E485A90C75BA2001E147FA24892C7FC181C5D5A5D127000F00101153C A2C7491400A21403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA292C9 FCA25CA25CA21301A25CA21303A21307EB1FFE003FB6FCA3383E71BD41>I86 D<267FFFF890B500F890B512C0 B5FCA200070180010F90C7381FFC006C48C7D807FCEC07F0496E485D00016263A2040793 C7FC1A0E040F151E1A1C041F5DA2043B5D167B04735D16E34F5AED01C34F5AED03834FC8 FC92260703FC5B0401140E030E151E191C031C5D153C03385D157803705D6D13E000004D 5AEC01C04E5AEC03804EC9FCDA07005C180E020E5DA24A5D143C02385D5C605CEFFDC06D 5AEFFF805C7090CAFC91C8FC5F495DA2495DA2495D5B01705D523F6EBD5A>I<14FF0107 EBC7C04913EFEB3FC790387F81FFEBFE004848EB7F80485AA24848133F48481400A24848 5BA2003F147E5B007F14FEA290C75AA2481301A2485CA21403163815F05A02071378140F 007E90381FE07016F0143F6C01FF13E0018113F1391FC7F3F33A0FFFE3FFC06C01811380 3A01FE007F00252777A62E>97 DII II<16FE923803FF804B13C0ED0FC7ED1F8FED3F1FA2157EA2EE0F 8003FEC7FCA25DA31401A25DA31403A25DA20103B512F8A390260007E0C7FCA3140FA25D A3141FA25DA3143FA292C8FCA35CA2147EA314FEA25CA31301A25CA31303A25CA313075C A3495AA2123E007E5BEAFE1F91C9FC5B133EEAFCFCEA7FF85BEA1FE02A5183BE1C>III<14F8EB01FCA414 F814F090C7FCAE13FEEA03FF481380380FDFC0130F121E123CA21238EA781F1480EA703F 12F01400C65AA2137E13FEA25B1201A2485AA25B00071338A213E0000F1378A2EBC0F0A2 EBC1E01381EBC3C013CF3807FF806C1300EA01FC163C79BB1C>I107 DIIII<903907F003FC90390FF80FFF90261FFC3F1380903A3EFE7F1FC0 903A3C7EFC0FE09039783FF0079238E003F0017013C09038F07F80A201E0010013F8147E 000113FEA2C75AA20101140717F05CA20103140FA24A14E0161F130717C04A133F178001 0F147F170016FE6E485A131F9138F807F86E485A9138BE3FC090393FBFFF80DA9FFEC7FC EC07F891C9FC5BA2137EA213FEA25BA21201A25BA21203487E387FFFE0B5FCA22D3980A6 2E>I<9138FE01C0903807FF83011FEBCF80EB3FC790387F03FFEBFE0148486C1300485A 48487FA24848137EA2484813FEA2003F5C5B007F1301A201005BA2481303A2485CA21407 A25D5A140F141F5D007E133F147F007F13FFD83F035BEA1FCF90B5FC0007133FD801FC90 C7FCC7FC5CA2147EA214FEA25CA21301A25CA21303497E48B512E0A3223977A629>I<39 03F803FE3A07FC0FFF80486C4813C03A1F7F3F07E0381E3F7C393C1FF80F14F0003813E0 38783FC016C000700180138092C7FCD8F07FC8FCA2EA007EA213FEA25BA21201A25BA212 03A25BA21207A25BA2120FA25BA45B232779A626>III<01FE147CD803FF14FC487F380FDFC0010F 1301121E003C5DA200381403EA781F5E38703F8000F014071400C6485CA2017E130F13FE 5E5B0001141FA2495CA20003143FEE8380491403A2ED7F07A203FE13005EEBF801000101 03130E9039FC0FFF1E3A00FE3FBF3E90B5EA3FFC90393FFC1FF890390FF007F0292779A6 30>I<01FEEB07C0D803FFEB0FE048EB801F380FDFC0130F121E003C140F1507D8381F13 0312781480D8703F130112F01400C64814C0A2017E130313FE16805B00011407A2491400 5D1203151E5B151C153C5DA25D3801F8014A5A6D485A3900FF1F806DB4C7FC6D5AEB0FF8 232779A629>I<01FE02F813F8D803FF0101EB01FC4801801403380FDFC0D81F0F130312 1E003CEDF001170000380207147CEA781F5E26703F80153C00F0140F1400C6484A1338A2 017E011F147813FE4C13705B0001023F14F0A24991C712E017015D000317C049017E1303 A2EF0780A203FEEB0F006D7F000149141E6D48EB803E6C6C486D5A903AFF0F9FE1F8903A 7FFF8FFFF090261FFE075B902607FC001380362779A63C>I<90391FE00FE090397FF83F F89039FFFC7FFC3A01F8FEFC7E3903F07FF03A07C03FE0FE018013C0120F01001380001E 15FCED00F8001C1500003C5BA2C7127EA214FEA25CA21301A25CA2130316705C16F0EA7C 0700FEEC01E0A290390FE003C0ECF007D8FC1FEB0F803AF83FF81F0039FCFDFC7E39FFF8 FFFC6C486C5A391FC01FE027277BA629>I<01FE147CD803FF14FC487F380FDFC0D81F0F 1301121E003C15F8A200381403EA781F16F038703F8000F014071400C64814E0A2017E13 0F13FE16C05B0001141FA2491480A20003143FA2491400A25DA215FEA2EBF80100011303 9038FC0FFC3800FE3F90B5FCEB3FFD90380FF1F8EB00011403A25D140700065C381F800F 003F5C141F4A5A4AC7FCEB007E383C01FEEB03F8383F0FF06CB45A6C1380D803FCC8FC26 3979A62C>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: FC cmr8 8 28 /FC 28 127 df<15F0A24A7E4A7EA24A7EA24A7E140E91381E7F80141C91383C3FC01438 9138781FE014704A6C7E1507D901C07F01036D7E148001076D7E1400496D7E130E011E6E 7E131C013C6E7E133801786E7E137001F06E7E5B48486E7E160348488100076F7E90C8FC 486F7E120E001EEE7F80001FB8FC4817C0A24817E0A2B912F0342E7DAD3B>1 D<007FB612FEA600F0C8120FA2481507A4CAFCA4D801C0EB0380A390B6FCA69038C00003 A3CAFCA500E01507A46C150FA2007FB612FEA6282D7DAC2F>4 D<013FB57EA39026007F C0C7FC6E5AA6903801FFE0011F13FE9039FF3FBFC0D803F8EB87F0D807E0EB81F8D80FC0 EB80FCD81F80147E003F157F007F16800100143F4816C0A56C16800180147F003F160000 1F157ED80FC05CD807E0EB81F8D803F8EB87F0D800FFEBBFC090261FFFFEC7FC010113E0 9038003F80A64A7E013FB57EA32A2D7CAC33>8 D22 D<1307130F131F133E137C13F813F01201EA03E0A2EA07C0A2EA0F80A2EA1F00A3123EA3 127E127CA412FCA35AAB7EA3127CA4127E123EA37EA3EA0F80A2EA07C0A2EA03E0A2EA01 F0120013F8137C133E131F130F130710437AB11B>40 D<12E07E7E127C7E7E7E1380EA07 C0A2EA03E0A2EA01F0A2EA00F8A3137CA3137E133EA4133FA3131FAB133FA3133EA4137E 137CA313F8A3EA01F0A2EA03E0A2EA07C0A2EA0F8013005A123E5A5A5A5A10437CB11B> I43 D48 D<130E131E137EEA01FE12FFA2EAFE7E1200B3AF13FF007F13FFA3182C7BAB23>I< EBFFC0000313F8000F13FE381FC3FF48C61380007CEB3FC00078EB1FE012FC6CEB0FF07E 1407A4127EC7FC140F15E0A2141F15C0EC3F80EC7F005C5C495AEB03F0495A495A495A49 C7FC137E01F81370485A485A484813E0485A48C7FC48B5FC5AB6FC15C0A31C2C7DAB23> II<14075C5C5CA25C5CA25BEB03BFEB073FA2130E131C13181338 137013E0A2EA01C0EA0380EA0700A2120E5AA25A5A5AB612FCA3C7EA3F00A8EC7F809038 1FFFFCA31E2C7EAB23>I<001EEB0380381FC03F90B5FC15005C5C14F014C0D81C7CC7FC 90C8FCA7EB3FE0EBFFF8381FF0FEEBC03F9038001F80001E14C0001C130FC713E0A2EC07 F0A4123C12FEA5EC0FE05A0078EB1FC0127C6CEB3F80003FEB7F00381FC1FE6CB45A0003 13F0C613801C2D7DAB23>II56 D61 D91 D93 D<380FFF80003F13F0387F03FCEB00FE147FEC3F80A2003E131FC7FCA3EB0FFF90B5FC00 07131FEA0FF8EA3FC01380EA7F0012FE158E5AA2143F7E147F387F01FF393FC7E7FC391F FF87F83903FE01E01F1D7D9C23>97 D99 D<157CEC1FFCA314011400AC EB1FF0EBFFFE3803FC3F3807F00F380FC003381F800148C7FCA2127EA212FE5AA8127EA2 127F6C1301381F8003380FC0076C6C487E3A03F87FFFE03800FFFCEB3FF0232E7EAD27> II108 D<3803FF38001F13F8EA3F03EA7C00007813785A14387E7EB4130013F8EBFF806C13E06C 13F06C13F8000713FC38007FFE1303EAE000147E6C133E141E7E143E6C133C6C137C38FF 81F838F7FFE000E11380171D7E9C1C>115 D<13E0A51201A31203A21207120F381FFFF0 B5FCA23807E000AE1438A6147813F0000313F0EA01FD3800FFE0EB3FC015297FA81B>I< D803E0133E00FFEB0FFEA3000F13000007147EAF15FEA31401EBF0030003497E3A01FC1F 7FF03800FFFEEB1FF8241D7F9C27>I<3AFFFE03FFC0A33A0FF000FE00000714F812035D 6C6C5BA2EBFC0100005CEBFE03017E5BA26D48C7FCA2148FEB1F8EA2EB0FDCA214FC6D5A A26D5AA36D5AA25CA213035C1238D87C07C8FC12FE130E131E485AEA7878EA3FF0EA0FC0 222A7F9C25>121 D<3807C008380FE01C381FF878383FFFF038787FE038E01FC038400F 8016077AAC23>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FD cmsy8 8 16 /FD 16 113 df0 D6 D<170EA2170FA28384A2717E84170184717E187C183FF01F80F0 0FE0BA12FCA3CBEA0FE0F01F80F03F00187C604D5A601703604D5AA295C7FC5FA2170EA2 3E237CA147>33 D<13FCEA01FEA41203A213FCA2EA07F8A313F0120FA213E0A2EA1FC0A3 1380A2123F1300A2123E127EA2127CA25AA35A0F227EA413>48 DI< 91B512C01307131FD97FC0C7FC01FEC8FCEA01F8EA03E0485A485A90C9FC121E123E123C 5AA2127012F0A25AA2B712C0A300E0C9FCA27EA212701278A27E123E121E7E7F6C7E6C7E EA01F8EA00FEEB7FC0011FB512C013071300222B7AA52F>I<1770EE01F01603A2160FA2 161FA2163FA2167B16FB16F3150116E3ED03C3A2ED078383ED0F03151F151EED3E01153C 157815F815F01401EC03E015C01407DA0F807F15004AB5FC5CA25C02F8C7FC4948801210 383003E0387007C026780F80147FEAFE3FB5C8131C4916FC49153F6C4816F84916F06C48 ED1F80D80F8092C7FC36337EAF38>65 D<9039078003FE011F90381FFF80017F017F13C0 2601FF81B512E048903887F87F91380FE01F9039BF1F800F3A003F3F0007147E5C4A14C0 4AEB0F80161F4948EB3F004A13FE4B5A91388007F0ED1FE0913800FF80D9FF0790C7FCD9 FE1F13E016F849487F9138003FFE0001EC07FF4913016F1380167F4848143FA2161F5B12 07A3491500000F5D163E49147ED81F985C01FC495A48B4EB07F09138E07FE04890B51280 007C4AC7FCD8FC3F13F0D8F00790C8FC2B2F7EAD2E>I<91B512F0010F14FF017F15E048 B77E2707FE3F0313FC3B0FE03E003FFE261F807EEB07FFD83F006E1380007E1500007CEE 7FC000FC163F48EE1FE05AC7007C140F14FC1707A35C1301A218C04A140F13031880171F 4A150001075D173E4A5C010F15FC4C5A4A495A011F4A5A4C5A91C7EA3F80494AC7FCED01 FC013EEB07F8017EEB3FE090397C03FF8048B548C8FC4814F048148002F8C9FC332D7EAC 37>68 D<020EEBFF80DA7E0713F0902601FE1F7F902603F87F7F903A0FE0FE1FFE903A1F C1F803FF90393F03F00190277E07E0001380494848137FD801F849133F484848C713C049 48141F3807C03E000F137ED9807C140F001F13FC495A4813E0003E90C8FC127EA2188000 7C161F12FCA218005F173E177E177C6C16FC5F4C5A6C15036C4B5A6D4A5A4C5A6C6C4AC7 FC6D14FE6C6C495A6DEB07F0D80FFEEB1FE03A07FFC0FF806CD9FFFEC8FC6C14F86C6C13 E0D90FFEC9FC322F7CAD38>79 D83 D<1478A314FCA2497EA2EB03CFA29038078780A290380F03C0A2EB0E01011E7FA2496C7E A2491378A2497FA248487FA248487FA2497F00071580A248C7EA03C0A2001EEC01E0A248 EC00F0A2481578A248153CA248151CA226297CA72F>94 D<147FEB03FFEB0FF8EB1FE0EB 3F80A2EB7F00137EB3A213FEA2485A1203EA0FF0B45A138013E0EA0FF0EA03FC12016C7E A2137EB3A2137FEB3F80A2EB1FE0EB0FF8EB03FFEB007F18437BB123>102 D<12FEEAFFC0EA1FF0EA07F8EA01FCA26C7E137EB3A2137FA2EB3F8014C0EB0FF0EB07FF 13011307EB0FF0EB3FC01480EB7F00A2137EB3A213FE485AA2EA07F8EA1FF0EAFFC048C7 FC18437BB123>I<12E0B3B3B3AD034378B114>106 D<1807180FA2181EA2183CA21878A2 18F0A2EF01E0A2EF03C0A2EF0780A2EF0F00A2171EA25F177C17785FA24C5AA24C5AA24C 5AA2D801C04AC7FC487E000F151E487E003F5DEAFBF800F35DEAC1FC00015D6C7E4B5A01 7F13035E6D6C485AA26D6C48C8FCA290380FE01EA26D6C5AA26D6C5AA26D6C5AA26DB45A A26E5AA26E5AA26EC9FCA2140E38427C823B>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FE cmsy10 10.95 37 /FE 37 113 df0 D<123FEA7F80EAFFC0A6EA7F80EA3F000A0A79 9B19>I<0070167000F816F06C1501007E15036CED07E06C6CEC0FC06C6CEC1F806C6CEC 3F006C6C147E6C6C5C6C6C495A017E495A6D495A6D6C485A6D6C485A6D6C48C7FC903803 F07E6D6C5A903800FDF8EC7FF06E5A6E5AA24A7E4A7EECFDF8903801F8FC903803F07E49 487E49486C7E49486C7E49486C7E017E6D7E496D7E48486D7E4848147E4848804848EC1F 804848EC0FC048C8EA07E0007EED03F0481501481500007016702C2C73AC47>I<15075D B3A9BA12F0A4C8000FC9FCB3A6BA12F0A43C3C7BBB47>6 D15 D24 D<0207B7FC143F49B8FC13074948C9FC EB3FF0EB7F8001FECAFC485AEA03F0485A485A5B48CBFCA2123EA25AA21278A212F8A25A A87EA21278A2127CA27EA27EA26C7E7F6C7E6C7EEA01FC6C7EEB7F80EB3FF0EB0FFF6D90 B7FC1301EB003F1403383679B147>26 D<007FB612C0B712FCEEFF806C16E0C97FEE0FFC EE01FEEE007FEF3F80EF0FC0EF07E0EF03F01701EF00F8A2187CA2183EA2181EA2181FA2 180FA8181FA2181EA2183EA2187CA218F8A2EF01F01703EF07E0EF0FC0EF3F80EF7F00EE 01FEEE0FFCEE7FF0007FB75AB8128004FCC7FC6C15E0383679B147>I<06F0140F050315 3F0507157F051FEC01FFDD7FC0EB07FC4C48C7EA1FF0DC07FEEC7FE0DC0FF8ECFF80DC3F E0903803FE00DCFF80EB0FF8DB03FEC7EA3FE04B484A5ADB1FF04948C7FCDB7FC0EB07FC 4A48C7EA1FF0DA07FEEC7FE0DA0FF8ECFF80DA3FE0D903FEC8FCDAFF80EB0FF8D903FEC7 EA3FE049484A5AD91FF04948C9FCD97FC0EB07FC4848C7EA1FF0D807FEEC7FE0D80FF8EC FF80D83FE0D903FECAFCD8FF80EB0FF848C713E0A26C6C14F8D83FE0EB03FED80FF89038 00FF80D807FEEC7FE0D801FFEC1FF026007FC0EB07FCD91FF0EB01FFD907FC9038007FC0 6D6C6E7E902600FF80EB0FF8DA3FE0EB03FEDA0FF8903800FF80DA07FEEC7FE0DA01FFEC 1FF09126007FC0EB07FCDB1FF0EB01FFDB07FC9038007FC06F6C6E7E922600FF80EB0FF8 DC3FE0EB03FEDC0FF8903800FF80DC07FEEC7FE0DC01FFEC1FF09326007FC0EB07FCDD1F F0EB01FF0507EC007F0503153F0500150F503A7BB35B>I<00F0140F00FC15C06C816C6C 14F8D83FE0EB03FED80FF8903800FF80D807FEEC7FE0D801FFEC1FF026007FC0EB07FCD9 1FF0EB01FFD907FC9038007FC06D6C6E7E902600FF80EB0FF8DA3FE0EB03FEDA0FF89038 00FF80DA07FEEC7FE0DA01FFEC1FF09126007FC0EB07FCDB1FF0EB01FFDB07FC9038007F C06F6C6E7E922600FF80EB0FF8DC3FE0EB03FEDC0FF8903800FF80DC07FEEC7FE0DC01FF EC1FF09326007FC0EB07FCDD1FF0EB01FF0507EC007FA2051FEC01FFDD7FC0EB07FC4C48 C7EA1FF0DC07FEEC7FE0DC0FF8ECFF80DC3FE0903803FE00DCFF80EB0FF8DB03FEC7EA3F E04B484A5ADB1FF04948C7FCDB7FC0EB07FC4A48C7EA1FF0DA07FEEC7FE0DA0FF8ECFF80 DA3FE0D903FEC8FCDAFF80EB0FF8D903FEC7EA3FE049484A5AD91FF04948C9FCD97FC0EB 07FC4848C7EA1FF0D807FEEC7FE0D80FF8ECFF80D83FE0D903FECAFCD8FF80EB0FF848C7 13E0485D00F092CBFC503A7BB35B>I<1978A3197CA2193C193EA285A2737EA2737E8673 7E737E86737E1A7FF23F80F21FE0F20FF8BC12FE1BFFA21BFECDEA0FF8F21FE0F23F80F2 7F001AFE4F5A624F5A4F5A624F5AA24FC7FCA2193EA2193C197CA21978A350307BAE5B> 33 D39 D49 D<0207B512FC143F49B6FC 13074948C8FCEB3FF0EB7F8001FEC9FC485AEA03F0485A485A5B48CAFCA2123EA25AA212 78A212F8A25AA2B812FCA400F0CAFCA27EA21278A2127CA27EA27EA26C7E7F6C7E6C7EEA 01FC6C7EEB7F80EB3FF0EB0FFF6D90B512FC1301EB003F14032E3679B13D>I<173CA217 7CA217F8A2EE01F0160317E0EE07C0A2EE0F80A2EE1F00A2163EA25EA25E15015E4B5AA2 4B5AA24B5AA24BC7FCA2153EA25D15FC5D4A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA2 5C14FC5C495AA2495AA2495AA2495AA249C9FCA2133E137E137C5BA2485AA2485AA2485A A2485AA248CAFC5A123E5AA25AA25A12702E5274BF00>54 D<12F0AF12FCA412F0AF0622 7BA700>I<0070EE01C000F016036C1607A200781780007C160FA2003C1700003E5EA26C 163EA26C163C6D157CA26C6C5DA200035E6D1401A200015E6D140390B7FC6C5EA26D5D01 7CC7120FA26D4AC7FCA2011E141E011F143EA26D143C6E137CA26D6C5BA201035CECE001 A26D6C485AA201005CECF807A202785BEC7C0FA26E48C8FCA2EC1E1EEC1F3EA2EC0FFCA2 6E5AA36E5AA36E5AA2324080BE33>II65 D<021EEC7FC0027E903807FFF8D901FE011F7F0107027F7F 011F49B6FC013F903803F83F90287FFC0FE007138001F990381F8003010149487E010301 7E7F5DDAFDF8147F4A5A02FF16005D4949147E18FE4B5C92C7485A4D5A4A4A5A010FED1F C04A027FC7FCEE03FC4AEB1FF8EEFFE0DAF0075B011F011F13F84B13FE4A486D7E92B67E 013FD9003F7F4A01077F1601706C7E4A143F017F6F7E170F91C8FC17075B17035BA34848 5EA348484B5AA2604848150F604D5A260FE1C04AC7FCD9EFE0147E496C495A261FFFFCEB 07F0903ABFFF807FE0003F91B51280011F4AC8FCD87E0F14F8D87C0314C027F0007FF8C9 FC39407DBE3C>I<4AB512FE023FECFFE049B712FC010716FF011F17C090277FE1FC0114 F0D9FE01D9001F7FD803F803037F2607E00302007FD80FC0EE3FFFD81F80160FD83F0070 138048834B6E13C000FE835A480107EE7FE012C0C7173F5DA2191F140FA25DA21AC0141F A24B153F1A80A24A5AF17F00A2027F167E92C912FE61180102FE5E4E5A180701015F4A4B 5A4E5A4A4BC7FC0103167E6049484A5AEF07F0EF0FC04948EC3F8005FEC8FC4AEB03FC01 1FEC1FF0EEFFC090263F800F90C9FC49B512FC48B612E04892CAFC4814F892CBFC433E7E BD46>68 D<047FB612FE0307B81280151F157F4AB9FC913B03F80FF00001DA0FE0499038 007F00DA1F80167CDA3F001670027E011F92C7FC5C01015D5C1303163F49485C14C00106 C7FC90C8127F94CAFCA316FEA34B5AA34B5A93B612F0A25D616192270FE0000FC8FC95C9 FC5E151FA24B5AA293CBFC5D157E15FE5D14015D14035D14075D140F00075C381F801FD8 3FC05B387FE03FD8FFF090CCFCEBFE7E6C6C5A6C5B6C13E06C5BD801FECDFC49407FBD41 >70 D<0438F0018004F8180303031907A270F00F0065030761656570601C011C03030F18 071C0F704D5A150E1C3F031E187F7116FF047FEE01FB031C943803F3F81B07033C6D16E3 043FEE0FC70338EF1F87F33F0703786D037E5BDB701F16FC1A0103F0EFF80F03E06DEC03 F0040FED07E00201EF0FC003C0EE1F8071DA3F005B020301075D038016FE714948131F02 075FDB00034A5A4F5A4A6E495A020E01014A485C021E6E49C7FC61021C04FE143F023C6D EB81FC0238ED83F8027891387FC7F0F0CFE04AEDFFC0715B494893C8FC715A49485D0030 705A267C07805DD87F0F6F4882B5C849EEE1C0714816F74992CA13FF1E0049745A491BF8 6C48F20FE0D81FE097C8FCEA078062437DBE6D>77 DI<031EEB0FFC037E90B5FC912601FC0314C0912603F80F8091260FE01F8091271F80 7F037F91277F00FC007F9126FC01F8133F494848486D7E49484848130F494848486D7E90 260FC01F80D91F805B494848C77E49491680017E017E804913FE48485B0003130101F049 157F0007130301E05B000FEB07C001C090C9FC001F90CAFCA248481800A3007F60A290CB 5AA3484D5AA34E5AA2611807616D4C5A181F614EC7FC6D5E007F177E6D5E4D5A6D4B5A00 3F4C5A6D4B5A6C6C4B5A6D037FC8FC6C6C15FC02C0EB03F86C01F0EB1FE06C01FEEBFF80 6C90B6C9FC6C15FC013F14E0010F91CAFC010113F041407BBE48>I83 D<0070EE01C000F01603B3B317077E1880007C160FA2 6CEE1F00003F5E6C6C157ED80FE04A5A6C6C4A5AD803FCEC0FF06CB46CEB7FE03B007FF8 07FF806DB6C7FC010F14FC010114E09026001FFEC8FC32377BB53D>91 DI94 D<00F0EE01C017036C1607A2007CEE0F80A2003C1700003E5EA26C163EA26C6C5DA26C6C 5DA26C6C4A5AA200015E6D1403A26C6C4A5AA2017C4A5AA26D4AC7FCA2011E141E011F14 3EA26D6C5BA26D6C5BA26D6C485AA26D6C485AA201005CECF807A291387C0F80A26E48C8 FCA2EC1F3EA2EC0FFCA26E5AA36E5AA26E5AA232377BB53D>I<387FFFFCB5FCA300F0C7 FCB3B3B3B3AD1270165A71C328>100 D I<15FF1407143FEC7FF0ECFFC0491300495A5C495AA25CB3A8130FA2495AA2495AEBFF80 4890C7FCEA0FFEEAFFF813C013F8EA0FFEEA01FF6C7FEB3FC06D7EA26D7EA21307B3A880 A26D7E806D7E6D13C0EC7FF0EC3FFF14071400205B7AC32D>II<12F0B3B3B3B3B3045B76C319>106 D<12F0A27EA21278127CA2123C123EA2121E121FA27E7FA212077FA212037FA212017FA2 12007FA21378137CA27FA2131E131FA27F80A2130780A2130380A2130180A2130080A214 78147CA2143C143EA2141E141FA26E7EA2140781A2140381A2140181A2140081A2157815 7CA2153C153EA2151E151FA2811680A2150716C0A215031501225B7BC32D>110 D<1A071A0F1A1FA21A3EA21A7CA21AF8A2F101F0A2F103E0A2F107C0A2F10F80A2F11F00 A2193EA261A261A24E5AA24E5AA24E5AA24E5AA24EC7FCA2183EA260A260A24D5AA20118 4B5A137801F85ED803FC15071207486C4B5A123F486C4BC8FC12FD00F86D143E12E0C66C 6C5CA2013F5D804C5A6D7E4C5A6D7E4C5A6D7E4C5A6D7E4CC9FC6D7E163E7F6F5AA26E6C 5AA291383FE1F0A291381FF3E0A26EB45AA26E5BA26E90CAFCA25D14015D14005D157848 5B7A834C>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FF cmmi10 10.95 59 /FF 59 123 df11 D13 DI<4A7E 4A7EA25DA5EDFFF016F85C5C143F91387E7FE04AC7FC495A495A495A495A495A49C8FC13 3E137E5B5B485A12035B12075B120F5B121F90C9FC5AA2123E127EA2127CA212FCA35AA5 7EA47E7EA27FEA7FE013F8EA3FFEEBFFC06C13F86C13FF6C14C00001806C6C7F011F7F13 039038007FFC140F14031401A21400A35DEB1C0190383E03F090383FC7E06DB45A01075B D901FEC7FC25537CBF28>16 DI I<133E017E147F01FE903803FF80150F5D00015C15FC4948481300EC03E00003903807C0 FCDA1F80C7FC4948C8FC147C00075BEBFBF0EBFFE014804813FEECFFC015F09038E07FFC 001FEB0FFE1403496C7E80003F8017E05BA2007FECFF01A290C700FE13C016035A92387E 07805A92387F0F00ED3F9F16FE486E5A48EC07F02B297BA734>20 DII<4A7E4A7EA25DA5EDFF F0020F13F8143F91B5FC1303903907FE7FE0D90FF8C7FC495A495A495AA2495AA24890C8 FCA25BA76C7EA290387FBFFCECFFFE7F7F5B90387F7FF8D801FCC8FC485A5B485A485A48 5A90C9FC5A123E127E127C12FCA25AA27EA37E7E7F13E0EA7FF813FF6C13C06C13F86C13 FE6CEBFFC0000114F06C6C13FC130F01037FEB007F141F14031400157EA3131E90381F80 FCECE1F86DB45A01035B9038007F8025537EBF28>24 D<013FB7FC90B8128012035A4817 00A2281FE03803C0C7FC383F0078127E007C5D48EB70074813F05AEA000114E0150F0103 91C8FCA214C01307A2010F5B02807FA2131FA2EB3F00A24980137E13FEA20001815B8200 03140F5BA26F5A6C485C31287DA634>II<021FB6FC 91B712801303130F491600137F9027FFF07FE0C7FC489038801FF03903FE000F496D7E48 481303485A5B121F4848130115035B127F90C7FCA2481407A2485D150FA25E151F5E153F 5E4BC8FC007E5C007F495A6C495A6D485A391FC01FE0390FF07FC06CB5C9FC000113FC38 007FE031287DA634>I<013FB612C090B712E012035A4816C05AD9C00FC8FC383F001E12 7E127C48133E5A48133CC7127CA45CA31301A25C1303A31307A25C130FA4495AA3133FA2 5C6DC9FC2B287DA628>I<1678A21670A216F0A25EA21501A25EA21503A25EA21507A293 C7FCA25DA2150E913807FFF8023F13FF91B61280010390381C3FE0D90FF8EB0FF0903A1F E03C07F8D93FC0EB01FC9038FF00384848EC00FE48480178137E49157F48481370120F48 4801F0133FA248485B177F387F8001A201005BA248010314FFA2484A13FE1601020714FC 1603030013F8007E15074AEB0FF0007FED1FE0020E14C0D83F80EC3F80021EEBFF00D81F C0495A3A0FE01C07FCD807F8495A3A03FE3C7FE06CB612806C6C49C7FC010F13F0D90078 C8FCA21470A214F0A25CA21301A25CA21303A25CA21307A230527CBE36>30 D32 DII39 D<123FEA7F80EAFFC0A6EA7F80EA3F000A0A798919>58 D<123FEA7F80EAFFC0A313E0A3127F123F1200A41201A213C01203A2EA0780A2120F1300 121E123E5A127812700B1C798919>I<1806181F187FEF01FFEF07FEEF3FF8EFFFE00403 138093380FFE00EE3FF0EEFFC0030390C7FCED0FFCED7FF0913801FFC0020790C8FCEC1F FCEC7FE0903801FF80D907FEC9FCEB1FF8EBFFE000031380D80FFECAFCEA3FF8EAFFC090 CBFCA213C0EA3FF8EA0FFE3803FF80C613E0EB1FF8EB07FE903801FF809038007FE0EC1F FCEC07FF020113C09138007FF0ED0FFCED03FF030013C0EE3FF0EE0FFE933803FF800400 13E0EF3FF8EF07FEEF01FFEF007F181F1806383679B147>II<126012F812FE6C7EEA7F E0EA1FFCEA07FF000113C038007FF0EB0FFCEB03FF010013C0EC3FF0EC0FFE913803FF80 020013E0ED3FF8ED07FE923801FF809238007FE0EE1FF8EE07FF040113C09338007FF0EF 1FFCEF03FF1700A21703EF1FFCEF7FF0933801FFC004071300EE1FF8EE7FE0923801FF80 DB07FEC7FCED3FF8EDFFE002031380DA0FFEC8FCEC3FF0ECFFC0010390C9FCEB0FFCEB7F F03801FFC0000790CAFCEA1FFCEA7FE0EAFF8048CBFC12F81260383679B147>I<4AB47E 020F13F091383F83FC9138FC007ED901F0EB3F804948EB1FC04A130F4948EB07E049C7EA 03F080D91FE0EB01F88017FC013F1400A25C17FE6D5A90C9FCA515E091383FFF0191B512 81010314C190390FF803E190391FE000F14948137B495AD801FEC7EA3BFC161F485A485A 4848EC0FF8A2485A17F0003F151F5B007F16E0A249143F17C012FFEE7F8090C8FC17005E 5E15015E6C4A5A15075E6C6C495A4B5A6C6CEB7F806C6C49C7FC9038F003FE3907FC0FF8 0001B512E06C1480D91FFCC8FC2F417CBF30>64 D<17075F5FA24D7EA2177F17FFA25EA2 5E5E177F160E84161C173F16381678167016E0A2ED01C015031680DB07007FA2030E131F 151E151C5DA25D15F05D4A4880A24A48130F140792C7FC4AB6FCA25C023CC7120F14385C 845C1707495A495AA249C8FC5B131EA2013E8213FE0001160F260FFF80EC3FFFB500F001 07B512FCA219F83E407DBF44>I<49B712F8F0FF8019E0D9000390C7EA1FF06E48EC07F8 4B6E7E727E020382844B1680A20207167FA24B15FFA2140F1A004B5C61021F1503614B4A 5A4E5A023F4B5A4E5A4BECFF80DD03FEC7FC027FEC0FFC92B612E095C8FC18E04AC7EA0F F8EF03FE4AEC00FF727E010183183F4A82181F1303A24A82A21307614A153FA2010F5F18 7F4A5E18FF011F4B90C7FC604A14034D5A013FED1FF04D5A4948ECFFC0496C010790C8FC B812FC17F094C9FC413E7DBD45>II<49B9FCA3D90003903880000F6E90C712014BEC007F193F0203163E191E5DA21407 A25D191C140FA25D1707021F4A133CA24B010E1300171E143FA24B5B177C027F14FC1607 92B55AA291B6FCED800792380001F016005BA24A5CA201031401A24A5CA2010791C9FCA2 5CA2130FA25CA2131FA25CA2133FA2137F497EB612F8A3403E7DBD3A>70 DI<0103B612 C0A3D90003EBC0006E90C7FCA25D1403A25DA21407A25DA2140FA25DA2141FA25DA2143F A25DA2147FA25DA214FFA292C8FCA25BA25CA21303A25CA21307A25CA2130FA25CA2131F A25CA2133FA2137F48487EB612E0A32A3E7DBD28>73 D<49B600C090387FFFF896B5FC5F D9000301C0C7001F13806E90C8380FFC001AF04BED1FC002034CC7FC197E4B5DF001F002 07ED07E04E5A4B4AC8FC183E020F15FC4D5A4BEB03E04D5A021FEC1F80053EC9FC4B5B4C 5A023F495A4C5A4B487E163F027F497E16FF1583923887EFFC9138FF8F8792389F07FE15 7E4B6C7E4913F04B7E03C0804B7E4948C77FA24A6E7EA20107153F844A141F84130F717E 5C1707011F8217034A81A2013F6F7FA249484A7FD9FFF04A13F0B600E0017F13FFA34D3E 7DBD4D>75 D<49B56C93383FFFF06398B512E0D90003F1F0006E6D4B13C003DFEE03BF64 0203EF077FA2039F040E90C7FC1A1C9126078FE05E62030F4C5AA2020F17E1F101C1020E 606F6CEC0381021E1783F10703021C040E5BA2023CEE1C0719380238606F6C14700278EE E00FA20270DB01C05BA202F0923803801FF007004A6C6C5E180E01014C133FA202C04B5C 600103187F6F6C5B4A95C8FC4D5A01074B485BA291C749C75A170E49027F14015F010E4B 5CA2011E4B1303A2011C4B5C013C5D043F1407017E5DD801FE92C7485A2607FF80EE1FFC B500FC013E011FB512F8163C161C5C3E7DBD58>77 D<49B56C49B512F8A27016F0D90001 DB001F1300F107F8706E5A62912603DFF05DA2DB9FF85D158F02071607ED87FC030793C7 FC82DA0F035DA291260E01FF140EA2DA1E006D131EA2021C6E131C167F023C163C707E02 381638830278011F1478A202706D6C1370A202F06D6C13F0A24A6E5B160301011601EE01 FF4A5E188101036E1383A24A91387FC380A20107ED3FE7A291C801F7C8FC171F4916FF83 010E5EA2011E1507A2011C6F5A133C1701137ED801FE5E2607FF801400B512FC18781870 4D3E7DBD49>I<49B712F018FF19C0D90003903980007FE06E90C7EA0FF84BEC03FC1801 020382727E5DA214071A805DA2140F4E13005DA2021F5E18034B5D1807023F5E4E5A4B4A 5A4E5A027F4B5A4EC7FC4BEB03FCEF3FF891B712E095C8FC17F092CBFC5BA25CA21303A2 5CA21307A25CA2130FA25CA2131FA25CA2133FA2495AEBFFF0B612E0A3413E7DBD3A>80 DI<48B912FC5AA2DA 8003EB001F2607FC0049130301F04AEB01F8490107140048481778495C90C7FC48140F12 1E4C14705A151F5A5E19F048143F19E0485DC81600157FA25EA215FFA293C9FCA25CA25D A21403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA214FF010313F000 1FB612FCA25E3E3D7FBC35>84 D86 DI<153FEDFFC05CEC03F3 913807E1E0EC0FC11580141FEC3F01A2027E13C0A214FC0101130314F801031480150714 F0010714005DEB0FE0151EA249485AA25D133F4A5A1481017F5BEC83C014075D4948C7FC 5CEBFE3E5C1478000113F8EBFDF0EBFFE05C5C91C8FC5B5B5BA212031207120F121F123F 127F12FD00F9EC078000F0140F0000141F6DEB3F00017C13FC4A5A90383E07F090383FFF E06D13806D48C7FCEB00C023427FBF26>96 DIIII<167F923801FFC04B13E0923807F3F0ED0FC7ED1FCF169F153FA2ED7F 1FA217E0EE0FC04BC7FC5DA41401A25DA31403A20107B512FEA390260007F8C7FCA25DA3 140FA25DA4141FA25DA3143FA25DA3147FA292C8FCA35CA25CA31301A25CA313035CA449 5AEA1F87007F5B13C712FFEBCFC0138FEB9F80131F4848C9FCEA7C7EEA7FFC6C5AEA0FE0 2C537CBF2D>102 DII107 D 110 D112 D<91387FC01C903901FFF07C0107EBF8FC90391FF1F9F890383FC07F90387F803F9038FF 001F484814F04848130F5B1207484814E0A24848131FA2003F15C05B007F143FA2491480 A200FF147FA290C71300A25DA25D5A1401A2007E495A007F1307140F6C131F6D485A381F C07FEBE3FF380FFFF70003EBE7F0C61307EB000FA25DA2141FA25DA2143FA25DA2147F4A 7E013F13FE5BA2263A7DA729>III<14F8130113 03A41307A25CA2130FA25CA2131FA25CA2B612F0A215E039003F8000137FA291C7FCA25B A25BA21201A25BA21203A25BA21207A25BA2120FEC01C013E01403121FEC078013C0EC0F 005C143E5C000F5BEBE3F06CB45A6C5BC690C7FC1C3A7EB821>I<137F2601FFC0EB0F80 486DEB1FC0EA07E7380F87F00103143FEA1E07003C1680A20038157FEA780F170000705B D8F01F5CA2C648485BA2EC8001137F5E1400491303A2495CA200011407171C4914F0A203 0F133CA2EEE038031F1378153F6C6C017F137003FF13F09039FF01F7E1D97F87EBF3E090 3A3FFFC3FFC06D1381903A03FE007F002E297EA734>I<017F14F83A01FFC001FC489038 E003FEEA07E7380F87F01303EA1E07003E1401003C14000038157ED8780F143EA200705B D8F01F141EA2C64848131CA24A133C137F163891C7FC49147816704914F0A2000115E015 015BED03C0A2ED0780150F1600151E6C6C133E5D017F5BECC3F06DB45A010F5BD903FEC7 FC27297EA72C>I120 D<137F2601FFC0EB0F80486DEB1FC0EA07E7 380F87F00103143FEA1E07003C1680A20038157FEA780F170000705BD8F01F5CA2C64848 5BA2EC8001137F5E1400491303A2495CA200011407A2495CA2150FA25E151F153F6C6C13 7F4B5A6D5AEB7F876DB5FC6DEBBF80903803FE3F90C7127FA293C7FC00035CD81FC05B38 3FE0015D14034A5A5D49485A49485A393C003F80003E01FFC8FC381F83FEEBFFF8000713 E0000190C9FC2A3B7EA72D>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: FG cmr10 10.95 37 /FG 37 127 df<16E04B7EA24B7EA24B7EA24B7EA24B7EA203397FA203707FA24B6C7EA2 4A486C7EA24A486C7EA24A486C7EA2020E6D7EA24A6D7EA24A6D7FA24A6D7FA24A6E7EA2 49486E7EA249486E7EA249C86C7EA2010E6F7EA2496F7EA2496F7FA2496F7FA249707EA2 4848707EA24848707EA248CA6C7EA2000E717E000FB9FC4884A2481980A24819C0A2BB12 E043407CBF4C>1 D<007FB812E0A70070CAFC00F017F0A2481770A5CCFCA601E01570A5 90B712F0A701E0C81270A590CBFCA700E01770A66C17F0A20078EE01E0007FB8FCA7343D 7CBC3D>4 DI<011FB612E0A3D900070180C7FCDA01FEC8FCA8913807FF C0027F13FC0103B67E903A0FFDFE7FE0D93FE1EB0FF8D9FF01EB01FED801FE6E7ED803FC ED7F80D807F8ED3FC0D80FF0ED1FE0001F17F0D83FE0ED0FF8A2007F17FC01C0150700FF 17FEA8007F17FC01E0150F003F17F8A2D81FF0ED1FF0000F17E0D807F8ED3FC0D803FCED 7F80D801FEEDFF00D800FF4A5AD93FE1EB0FF8D90FFDEB7FE00103B61280D9007F01FCC7 FC020713C0DA01FEC8FCA8913807FF80011FB612E0A3373E7BBD42>8 D<913803FFE0023F13FE49B612C00107018013F0903A1FFC001FFCD97FF0EB07FFD9FFC0 01017F48496D7F4890C86C7E48486F7E48486F7EA248486F7EA248486F7EA3007F834981 A96C6C4B5AA46C6C4B5AA2000F5FA26C6C4B5AA200035F6D153F00015F00005F6D157F6D 93C7FCA26D6C14FE011F5DA226E00FC0903901F80380A2010715F0267003E0903903E007 00A2010115C0A200785F6C6C6C903807801ED83FFFEDFFFEA46C5FA3393F7CBE42>10 D22 D<147814F8EB01F01303EB07E0EB0FC01480EB1F005B137E13 7C13FC5B1201485AA2485AA25B120FA25B121FA348C7FCA4127EA612FE5AB27E127EA67E A46C7EA3120F7FA212077FA26C7EA26C7E12007F137C137E7F7FEB0F8014C0EB07E0EB03 F01301EB00F81478155A78C323>40 D<12F07E127C127E7E6C7E120F6C7E7F6C7E12017F 12007F137EA27FA27F1480A2130F14C0A3EB07E0A4EB03F0A614F81301B2130314F0A6EB 07E0A4EB0FC0A31480131FA214005BA2137EA25B5B12015B1203485A5B485A121F48C7FC 127E127C5A5A155A7BC323>I<150FB3AABA12F0A4C8000FC9FCB3AA3C3C7BB447>43 D48 D<14E013011303130F137FEA07FFB5FC13 9FEAF81F1200B3B3ABEB7FF8B612FCA31E3C78BB2D>II I<151E153EA2157E15FEA21401A214031407A2140E141E141C143C1478147014F0EB01E0 14C01303EB078014005B131E131C133C5B137013F05B485A12035B48C7FC5A120E121E5A 123812785AB8FCA3C73801FE00AC4A7E0103B6FCA3283D7EBC2D>I54 D56 D<123FEA7F80EAFFC0A6EA7F80EA3F00C7FCB3123FEA7F80EA FFC0A6EA7F80EA3F000A2779A619>58 D<123FEA7F80EAFFC0A6EA7F80EA3F00C7FCB312 3FEA7F8012FF13C0A5127F123F1201A41203A21380A21207A2EA0F00A2121EA25A127C12 7812700A3979A619>I61 D91 D93 D<1318133C137E13FF4813803803E7C03807C3E0380F81F0 381F00F8003C133C48131E48130F00601306180D76BD2D>I97 D100 DIII105 D108 D<2701F803FF49B47E00FF010FD9E00713F0023FD9F01F7F913B7E0FF83F07FC0007903B F807FC7C03FE2703F9E003EBF0013C01FBC001FDE0000280D9FFC07F01FF6D8191C75B49 92C7FCA3495CB3A5486C496CECFF80B5D8F87FD9FC3F13FEA347287DA74C>I<3901F803 FF00FF010F13E0023F7F91387E0FF800079038F807FC3903F9E0033901FBC00102807F01 FF130091C7FC5BA35BB3A5486C497EB5D8F87F13FCA32E287DA733>I<14FF010713E001 1F13F890387F81FE9038FC003F4848EB1F804848EB0FC04848EB07E04848EB03F0001F15 F8491301003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA36C6CEB03F8000F15 F06D13076C6CEB0FE06C6CEB1FC06C6CEB3F803A007F81FE006DB45A010F13F0010090C7 FC282A7EA82D>I<3901FC07FE00FF90383FFFC04A13F09039FDFE1FFC3A03FFF007FE6C 9038C001FF4A6C138091C7127F4915C0EE3FE049141FA217F0160FA217F81607A9160F17 F0A2161F17E0163F6D15C0167F6DECFF806E4813006E485A9138F007FC9039FDFC1FF801 FCB512E0023F5BDA07FEC7FC91C9FCAD487EB512F8A32D3A7EA733>I<90387FE0E03803 FFFD000F13FF381FE07F383F001F007E1307007C130312FC481301A214007E7E7E01C013 0013F8387FFFC014FC6C7F6CEBFF806C14C0000314E0C6FC010713F0EB007FEC0FF800E0 130714036C1301A214007EA26C130115F06C13036C14E0EBC00F9038F03FC000FBB51280 00F1EBFE0038E03FF01D2A7DA824>115 D117 D120 D126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FH ecrm1095 10.95 85 /FH 85 256 df<00E01470A36C14F0007014E0007813016CEB03C0003E13076CEB0F8039 0FE07F003807FFFE6C5BC613F0EB3FC01C0E78B92D>8 D<39038001C001C013E0000713 03390F8007C0391F000F80001E1400003E5B003C131EA2485BA20070133800F01378A248 1370A439FF807FC001C013E001E013F0A3007F133FA3393FC01FE0391F800FC01C1C80BE 1E>16 D<003FEB1F80397F803FC039FFC07FE0A301E013F0A3007F133F003F131F0000EB 0070A4000114F0A24913E000031301A239078003C0A2000F13070100138048130F003EEB 1F0048133E0078133C0038131C1C1C7FBE1E>I21 D25 D<913A03FFC01FF0021F9038F87FFC91B538FDFFFE0103903980FFFC7F903B07FC007FF0 FFD91FF0495A49485C5C49485C91C7147E49027F13004992C7FCADB91280A3C648C76CC7 FCB3AE486C4A7E007FD9FC3FEBFF80A3383F7FBE35>27 D<913803FF80023F13E091B512 F801031303903907FC007CD91FF013FE90383FE00114C0EB7F8014005B5BED007893C7FC ABB712FEA33900FE000315011500B3AC486C497E007FD9F83F13FCA32E3F7FBE32>I<91 3803FFDE023F13FE91B5FC01031301EB0FFCEB1FF0EB3FE014C0EB7F8091C7FC5B5BADB7 FCA3C648C7FCB3AE486C497E007FD9FC7F13FCA32E3F7FBE32>I<913B03FFC001FFC002 3FD9F01F13F091B5D8F87F13FC0103903981FDFF81903C07FC007FFE003ED91FF002F813 7F494849485B4A5C49485C91C75B5B4992C7FC70143C96C7FCABBBFCA3C648C7387F0001 8485B3AC486C4A6CEBFF80007FD9FC3FD9FE3F13FEA3473F7FBE4B>I<123FEA7F80EAFF C0AAEA7F80ABEA3F00AB121EACC7FCA8123FEA7F80EAFFC0A6EA7F80EA3F000A4079BF19 >33 D<123FEA7F80EAFFC0A313E0A3127F123F1200A41201A213C01203A2EA0780A2120F 1300121E123E5A127812700B1C79BE19>39 D<147814F8EB01F01303EB07E0EB0FC01480 EB1F005B137E137C13FC5B1201485AA2485AA25B120FA25B121FA348C7FCA4127EA612FE 5AB27E127EA67EA46C7EA3120F7FA212077FA26C7EA26C7E12007F137C137E7F7FEB0F80 14C0EB07E0EB03F01301EB00F81478155A78C323>I<12F07E127C127E7E6C7E120F6C7E 7F6C7E12017F12007F137EA27FA27F1480A2130F14C0A3EB07E0A4EB03F0A614F81301B2 130314F0A6EB07E0A4EB0FC0A31480131FA214005BA2137EA25B5B12015B1203485A5B48 5A121F48C7FC127E127C5A5A155A7BC323>I<123FEA7F80EAFFC0A313E0A3127F123F12 00A41201A213C01203A2EA0780A2120F1300121E123E5A127812700B1C798919>44 DI<123FEA7F80EAFFC0A6EA7F80EA3F000A0A798919>I48 DIII<151E153EA2157E15FEA214011403A21407140F140E141E 143C1438147814F014E01301EB03C014801307EB0F00130E131E5B133813785B5B120148 5A5B120748C7FC120E121E5A123812785AB8FCA3C8EAFE00AC4A7E49B6FCA3283C7EBB2D >I<0007140701C0133F9038FC01FF90B6FC5D5D15F05D158002FCC7FC90C9FCADEB03FF 011F13C0017F13F09038FF07FC9038F801FE9038E000FF497F49EB3F8090C713C0C8121F 16E0A3ED0FF0A5123C127F5AA4ED1FE0A212FC007015C00078143F007CEC7F807E6CECFF 006D485A390FE007FC3907FC1FF00001B55A6C6C1380D91FFCC7FC243D7CBB2D>II<1238123C123F90B612FCA44815F8 16F0A20078C7EA03E0007015C0ED0780150F160048141E153E153C5DC812F84A5A5D4A5A 14075D4AC7FCA2141E143EA25CA214FC5CA21301A2495AA41307A3495AA6131FAC6D5A26 3F7BBD2D>III<123FEA7F80EAFFC0A6EA7F80EA3F00C7 FCB3123FEA7F80EAFFC0A6EA7F80EA3F000A2779A619>I<123FEA7F80EAFFC0A6EA7F80 EA3F00C7FCB3123FEA7F80EAFFC0A313E0A3127F123F1200A41201A213C01203A2EA0780 A2120F1300121E123E5A127812700B3979A619>I63 D<15074B7EA34B7EA34B7EA34B7EA24B7E15E7A20201 7F15C3A202037F1581A291380701FF81A2020E6D7EA2021E80021C133FA2023C80023813 1FA20278800270130FA24A801607A249486D7EA291B6FC4981A2903A07800001FF91C8FC A24982010E157FA24982173FA2496F7EA21378717E13F8000183487ED80FFFED3FFEB500 E00107B512F8A33D3F7DBE44>65 DIIIIIIII<01 1FB512F0A39039000FFE00EC03FCB3B3A7123E127FEAFF80A314075DA2EB000F0078495A 007E495A6C495A391FE1FF806CB5C7FC000313FC38007FE0243F7CBD2E>IIIIIII82 DI<003FB9FCA3D9F000EB C00301806D48C67E48C7ED3F80007C170FA200781707A300701703A548EF01C0A5C892C7 FCB3B24B7E4A7F0107B612F8A33A3E7DBD41>IIII<007FB5D88003B512E0A3C649C7387FFE00D97FF8EC 3FF06D48EC1FC0011F5E6D6C92C7FC6D6C143E173C6D6C14386D6C14785F6D6D5B91387F C0014C5ADA3FE05B91381FF0074CC8FC91380FF80E6E6C5A163C913803FE386E6C5A16F0 6E5B6F5AA26F7E6F7EA28282153FED7BFEED71FF15F1DA01E07F4B6C7E14034A486C7E4B 6C7E5C021E6D7E021C6D7E143C4A6D7E02706D7E14F049486D7F4A6E7E130349486E7E01 0F6F7E49C8FC496F7E496C8101FF4B7E000701E091383FFF80B500F849B512FEA33F3E7E BD44>I I91 D93 D<3801FFFC0007EBFF804814E0391FE01FF0EC07F86E7E6E7E14 006C487F6C487FC8FCA5EC7FFF010FB5FC90387FFE7F3801FFC000071300EA0FFCEA1FF0 485A485AA2485AEE038090C7FCA215FFA26D5A5C397FC007BF263FE00FEB87003A1FF83F 1FFF260FFFFE5B00039038F80FF83A007FE003E029277DA62D>97 DI<903803FFF0011F13FE017F7F3A01FF807F803803FE00 EA07F8485A5B001FEC3F004848131E92C7FC485AA348C9FCAA7F127FA27F003FEC01C06D 1303121F6C6CEB07806D130F6C6CEB1F006CB4133E6CEBC1FC6C6CB45A011F13E0010390 C7FC22277DA628>IIII<16FE903907FC03FF90263FFF8F138090B612CF4890381FFE1F3903FC 07F83A07F001FC0F93C7FC48486C7EA24848137FA86C6C13FEA26C6C485AA23903FC07F8 9038FF1FF0ECFFE0D8073F1380D907FCC8FC48CAFCA47F7F7F6CB512F815FF6C15E08282 000F813A1FE0003FFED83F80130748C71201007E6E7E00FE815A82A36C5D6C5D6C5D6C6C 495A01E01307D80FF8EB1FF06CB4EBFFE0000190B512806C6C49C7FC010713E0293B7EA7 2D>I IIII< EA01FC12FFA3120712031201B3B3B1487EB512F8A3153F7EBE1A>I<2701FC03FF49B47E 00FF011FD9E00F13F04A6D487F913B7E0FF83F07FC0007903BF807FC7C03FE2703FDE003 EBF0016CB4486CB4486C7E02805C91C7497FA24992C7FCA2495CB3A5486C496CECFF80B5 D8F87FD9FC3F13FEA347277EA64C>I<3901FC03FF00FF011F13C04A7F9138FE1FF03A07 FDF007F8000313E06CB4486C7E1480EC0001A25BA25BB3A5486C497EB500F8B512F8A32D 277EA632>I<14FF010F13F0013F13FC49C67ED801FCEB3F80D803F0EB0FC04848EB07E0 4848EB03F0A24848EB01F8003F15FCA248C812FEA34815FFA96C15FEA26D1301003F15FC A26C6CEB03F8A26C6CEB07F06C6CEB0FE06C6CEB1FC06C6CEB3F80D8007FEBFE006DB45A 010F13F00101138028277EA62D>I<3901FC07FE00FF90383FFFC091B512F09039FDFC1F F83A03FFF007FC6C9038C003FE4A6C7E91C7138049147F4915C0163F17E0161FA217F016 0FA8161FA217E0A2163F17C0167F6DECFF806D15006E5A6E485A9138E00FFC9039FDFC3F F001FCB55A023F1380DA0FFCC7FC91C9FCAE487EB512F8A32C397EA632>II<3901F81FE000FFEB7FF0ECFFF89038F9FBFC000713E3 3803FBC30001138313FFEC01F8EC00F0491300A35BB3A4487EB512FCA31E277EA623>I< 9038FFF0E0000713FD4813FF383FE07FEB000F007E1307007C1303481301A214007E7E6C 140013C013FE387FFFF014FC6C13FF6C1480000714C0000114E0D8003F13F01301903800 1FF800E0130714036C1301A214007EA26CEB01F07E6C13039038800FE09038F03FC000FB B5128000F1EBFE0038E03FF01D277DA624>I<131CA5133CA4137CA213FCA21201120312 07121FB612C0A3D801FCC7FCB3A215E0A8EBFE01A20000EB03C013FF90387FCF8090383F FF006D5AEB07FC1B387EB723>IIIIII<001FB61280A3EBE000D98001130049485A001E495A121C 003C495A4A5A143F00385C4A5A4A5AA24990C7FCC6485A13075C495A495A013FEB038014 E0EB7FC0EBFF80A24813004848130712075B4848140048485BA248485B4848137F484848 5A90B6FCA321277EA628>I<00401420006014600078EB01E06CEB03C0001FEB0F80390F 801F003807E07E6C6C5A6CB45A6C5B6D5A6D5A6D5A6DC7FC130690C8FCA59038FFF0E000 0713FD001F13FF383FC07F387F000F007E1307007C1303481301A214007E7E6C140013C0 13FE387FFFF06C13FC14FF6C1480000714C0000114E0D8003F13F013019038001FF800E0 130714036C1301A214007EA26C13016C14F06C130390388007E0EBF03F00FBB5128000F1 140038E03FF81D3B7DBA24>178 D<1318133C137E13FF4813803803E7C03807C3E01381 380F00F0001E137848133C48131E48130F0060130600201304C8FCA313FE123FA3120312 011200B3AB487E007F13F8A318397FB81A>238 D<1418143C147E14FF497F903803E7C0 903807C3E0148149C67E011E1378497F497F497F0160130690C9FCA414FF010F13F0013F 13FC49C67ED801FCEB3F80D803F0EB0FC04848EB07E04848EB03F0A24848EB01F8003F15 FCA248C812FEA34815FFA96C15FEA26D1301003F15FCA26C6CEB03F8A26C6CEB07F06C6C EB0FE06C6CEB1FC06C6CEB3F80D8007FEBFE006DB45A010F13F00101138028397EB82D> 244 D255 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FI ecbx1440 14.4 31 /FI 31 121 df<151E153E15FE1403140F147FEB07FF0003B5FCB6FCA3EBF87FEAFC00C7 FCB3B3B3A6007FB712FCA52E4E76CD42>49 DI<91 380FFFE091B512FE0107ECFFC0011F15F090263FFC077F90267FE00013FE49C76C7E4848 80D803F86E138048B416C06E7F6E15E0487FA86C5B4C13C0000190C7FCEA007C90C84813 80A218004C5A4C5A4B5B4B5B4B5B031F5B92B5C7FC027F13FC16E016FEEEFFC0DA000713 F0030013FC70B4FC041F138018C07013E07013F0A27013F8A218FCA28218FEA3EA07E0EA 1FF8487E487EB5FCA418FC5EA218F8495C007F17F0495CD83FF016E0494A13C0D81FF802 7F13806CB491B512006CD9F0075B000190B612F86C5E011F158001034AC7FCD9003F13C0 374F7BCD42>I<17FC1601A216031607160FA2161F163F167F16FFA25D5D5DA25D5DED3F 7F153E157C15FCEC01F815F0EC03E01407EC0FC0EC1F801500143E147E5C5C495A130349 5A5C495A131F49C7FC133E5B13FC485A485A5B485A120F485A90C8FC123E127E5ABA1280 A5C901FCC7FCAF021FB71280A5394E7CCD42>I<173FA24D7EA34D7EA24C7FA34C7FA24C 7FA34C7FA24C7FA34C7FEE3E7F047E80167C8304FC804C7E03018116F08303038116E003 076D7F16C083030F81168083031F8193C7FC4B81153E84037E82037C8003FC825D840201 834B800203835D8402078392B8FC4A83A34A8392C9FC4A83143E85027E84027C8202FC84 5C850101855C85494884850107855C85010F85D93FF082B600F0020FB712C0A55A537CD2 63>65 D73 D80 D82 DI<003FBB12FCA59126C0007FEB000301FCC7ED 003FD87FF0F00FFE49180749180349180190C81600A2007E1A7EA3007C1A3EA500FC1A3F 481A1FA6C91700B3B3AC49B912C0A550517BD05B>I<0103B57E013F14F848B7FC4816C0 48D9800F7F03037F486D6C7F6F6C7E83163F707EA2846C497FA2D801FEC7FCC9FCA6030F B5FC020FB6FC91B7FC1307013FEBF80F90B512800003EBFC004813F0485B4813804890C7 FCA2485A5B00FFEF81F0A25BA25EA26D5CA26C6C5C6D91B5FC6C90268003F7EBC3E06C90 26C007E313FF6C9026F83FC314C06C90B5008114800001030014006C6C01F8EB3FFC0103 01C0EB0FE03C357CB442>97 DI<913807FFFC027FEBFFC00103B612F8010F81013F9038 801FFE90387FFE00D9FFF8497E485B485B485B485BA24890C7FC705A485AEE07F8007F92 C8FCA25BA212FFAD6C7EA46C7EA26CEE0F806E141F7E6EEC3F006C7F6C6D147E6C6D14FE 6C01FE495A6D6CEB07F86D9038F03FF0010F90B512C001035DD9007F01FCC7FC020713E0 31357CB43A>I<943803FF80040FB5FCA5EE003F170FB3A5913807FFC0027F13FC0103B6 FC010F15CF013FD9C07FB5FC90397FFE000FD9FFF8130348491300484980484980484980 834890C8FCA2485AA2127FA25BA212FFAD127FA27FA2123FA36C7E5F6C6D5C6C6D5C6C93 B5FC6E5B6C01F8130726007FFE011F14E090283FFF80FFCFEBFF806D90B5128F0107ECFE 0F010014F0020F138041537CD249>I<91380FFFC0027F13FC0103B6FC010F15C0013F01 8313F090267FFC007FD9FFF0EB3FFC4849131F48496D7E48496D7E5A91C76C13805A4916 C0003F81A2127F4916E0A28212FFA290B8FCA401FCCAFCA7127F7FA2123FA26DED03E07E 17076C7F6C6DEC0FC0171F6C6DEC3F806C6DEC7F006C01FCEB01FED93FFFEB07FC6D9038 E03FF8010790B55A010115C0D9003F49C7FC020313F033357CB43C>II<18FFDA3FFF010713800103B5D8F01F13C0011FDAFE7F13E0017F91B512BF90 26FFFC0FEBFC3F48D9F003EBF07F48EBC000484990387FF83F4890C7EA3FFC19C0489338 FE0F8049021F90C7FC003F82AA001F5E6D143F6C5EA26C6D495A6C6D495A6CD9F0035BDA FC0F5B91B65A484BC8FCD807C314F0D9C03F90C9FC91CBFC485A7FA37F7F7F13FE90B7FC 17F86C16FF18C0846C836C836C83841203000F834848C71207D83FF8EC007F4848031F13 8049150700FF825B83A46D5DA26C6C4B13006D5D6C6C4B5A6C6C4B5A6C6C6CECFFF86C01 E001035B6C01FE013F5BC690B71280011F03FCC7FC010315E0D9001F01FCC8FC3B4E7CB5 42>II<137F3801FFC0487F487FA2487FA76C5BA26C5B6C5B6C6CC7FC90C8 FCAEEB1FF8B5FCA512017EB3B3A5B612F0A51C547CD324>I107 DIII<913803FFE0023F13FE49B612C0010715F0011F9038007FFCD93FF8EB 0FFE49486D7ED9FFC001017F48496D7F48834890C86C7E488349153F001F83A2003F8349 151F007F83A400FF1880AB007F1800A46C6C4B5AA3001F5F6D157F6C5F6C5F6C6D4A5A6E 5B6C6D495B6C6D495BD93FFCD91FFEC7FC6DB4EB7FFC010790B512F0010115C0D9003F49 C8FC020313E039357CB442>II<912607FFC0EB0F80027F01F8131F0103B500FE133F010FEC FF80499039E07FC07F017F9039001FE0FF4948EB07F04801F8EB03F94849EB01FD48496D B5FC48824A80485B834890C8FC835A5BA312FF5BAB7F127FA46C7EA2806C5E6C6D5C5F6C 6D91B5FC6C6D5B6E5B6C6D5B6C6CB4EB1FEF6D9038C0FFCF010F90B5120F010314FC0100 14F0020F138091C8FCB2040FB61280A5414C7CB445>I<90393FF007FCB590381FFF8003 7F13E092B512F09139F1FE7FF8ECF3F800019138E0FFFC6CEBF7C0A2ECFF805DA25CEE7F F8A24AEB1FE093C7FCA45CB3AAB612FEA52E357DB435>I<903907FFF01E017FEBFE3E48 B612FE120748EB007FD81FF8130FD83FE0130749130148481300A290C8127E5A163E7F7F 7F01F891C7FC13FF14F8ECFFC06C14FCEDFF806C15E06C8116FC6C810003816C16806C7E 010F15C013001407DA003F13E0150700F81401816C157F163FA26C151FA26C16C06D143F 7FEE7F8013F06D903801FF0001FE495A9039FFC01FFC91B55AD8FC3F5CD8F80F1480D8F0 0101F8C7FC2B357CB434>I<147CA614FCA41301A31303A21307A2130F131F133F137F13 FF1203000F90B512FEB7FCA426007FFCC8FCB3A9EE0F80AA161F80013F15006E5B6D147E ED80FE6DEBE1FC6DEBFFF801015C6D6C13C0020F90C7FC294C7ECB33>II120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FJ ecti1000 10 29 /FJ 29 122 df<93B5FC030714C092391FF007E092393F8003F092387F000715FE5D4A5A 18E0020315C04B90C7FCA21407A25DA3140FA25DA3010FB7FCA3903A001F80007EA2023F 14FEA292C75AA21601A24A5CA2027E1303A25F14FE16075C5FA20101140FA24AECC1C0A2 EE1FC31303EF83805C1787A20107ED8F00160F4A14DEEE07FEEE01F8010F91C8FC5CA313 1F5CA2003E90CAFC127EEAFE3F133E5BA2485AEA7FF0EA1FC0344A82B92F>28 D<387FFFF8A2B5FCA214F0150579941E>45 D<121FEA7F80A212FFA41300127E09097788 19>I<137E13FF5AA413FEA2EA00F81300B2121FEA7F80A212FFA41300127E102477A319> 58 D67 D<0103B612FEEFFFC018F0903B000FF8000FF80207EC03FC4BEB00FE020F157FF03F 804B141F19C0021F150F19E05D1807143F19F05DA2147FA292C8FCA25C180F5CA2130119 E04A151FA2130319C04A153FA201071780187F4A1600A2010F16FEA24A4A5A60011F1503 4D5A4A5D4D5A013F4B5A173F4A4AC7FC17FC017FEC03F84C5A49C7EA1FC0484AB45A007F 90B548C8FCB712F016803C397CB83F>I<0107B8FCA3903A001FF00007020F14004B147F 021F153E181E5DA2143FA25D181C147FA29238000380A24A130718004A91C7FC5E13015E 4A133E16FE49B5FCA25EECF8010107EB007CA24A1338A2010F147818E04A13701701131F 93380003C05C1880013F1507A24AEC0F005F017F151E173E91C8127E5F4914014C5A4848 140F000315FFB85AA25F38397BB838>I<0103B500F890387FFFE0A39026000FFCC7381F FE006E48EC0FF04B5D020FED1F804EC7FC4B147C60021FEC03F0EF07C04B495A4DC8FC02 3F143E17FC92388001F04C5A027F495A4C5ADB003FC9FC167E4A5B1501ECFE034B7E0101 131F153F9138FC7DFF15F8903803FDF09139FFE07F80158003007F4948133F5C4A804A13 1F130F707E5C83011F1407A24A801603133F834A1301A2017F6E7EA249C7487F486D497F 007F01FE011F13FCB55CA243397CB840>75 D<902607FFF8923807FFF06161D9001FEFF8 00020F4C5A62021F167F19EF021D5FDA1CFCEC01CF023C16DFF0039F02389238071F80A2 0278ED0E3FA20270031C90C7FCA202F04B5A187002E0167E037E14E0010117FEEF01C002 C04A485AA20103ED0701A20280020E5BA20107ED1C03173802005E6F137049160717E001 0EDA01C05BA2011E913803800FA2011CDA07005BA2013C020E131F5E01385FED1FB80178 163F16F001704A91C8FC13F04C5B1201486C4A13FED80FFC4B7EB500C0D9007F13FC151E 150E4C397AB84A>77 D<9238FFC00C0203EBF81C020FEBFC3C91393FE0FE7C91397F003F FC02FCEB0FF849481307495A494813034A14F0010F14015C131F91C713E0A25BA217C0A2 6E90C7FC80A214F0EB1FFEECFFC015F86DEBFF806D806D806D806D6C7F140F02007F151F 1507150315011500A2167C120EA3001E15FCA25E1501003E5D1503003F5D1507486C495A 6D495A6D49C7FC01F813FE39F9FF03FC00F0EBFFF0013F13C0D8E00790C8FC2E3B7AB92F >83 D97 D<137FEA1FFFA3C6FC137EA213FEA25BA21201A25BA21203A2 5BA21207A25BEBE3FC380FEFFF9038FF9F809038FE0FC09038F807E0EA1FF0EBE00301C0 13F0A2EA3F80A21300A2481307A2127EA200FE130FA24814E0A2141F15C0A248EB3F80A2 EC7F00A214FE6C485A007C5B495A6C485A381F1FC06CB4C7FCEA03FC1C3A77B926>III< 903801FF80010713E090381FC3F090387F01F8EBFE0048481378485A485A485A001F14F8 EBC001003FEB03F090388007E0007FEB1FC0903883FF809038FFFE0014C090C8FC5A5AA6 5A007E1438157815F8EC01F06CEB03E06CEB0FC09038803F80390FC1FE003803FFF8C613 C01D2477A326>I104 DI<151F153F157FA4 153E1500ACEC3F80ECFFE0903801F3F0EB03E1903807C1F81481EB0F01131EA2EB1C0313 3C15F01338EB0007A215E0A2140FA215C0A2141FA21580A2143FA21500A25CA2147EA214 FEA25CA21301A25CA21303A2383E07F0127E38FE0FE05C495A49C7FCEAFCFEEA7FF8EA1F E0204783B619>I108 D<2707F001FFEB0FF8260FFC0F9038C07FFE3C1F7E1F8FE0FC7F3D1E3E7E03F3F01F8026 3C3FF8EBF7C0003801F014800078903AE001FF000F02C05B0070494848131FEAF07F0200 5B12E0017E5CD800FE0107143FA2494A1400600001140F187E495C18FE0003141F4D5A49 0280141C18F80007023F0103133CA2490200EBF0381978000F5CF0E0F049137EF0E1E094 3801F3C0F0FF8049017C9038007F003E2479A344>I<3907F001FF260FFC0F13C03A1F7E 1F8FE03A1E3E7E03F0383C3FF8003813F00078EBE00114C00070EB8003EAF07F140012E0 137ED800FE1307A2495C150F12015E5B151F12034B5A49EC838016030007EC7F07A24901 7E13005E120FED7C1E5B5E6F5AED3FF049EB0FE0292479A32F>II<90390FC01FE090391F F07FF890393DF8FCFC903978F9F07E9138FFC03F4913804B7E49481480A23801E1FCA2EB C1F8A2D80003143FA25CA20107147FA24A1400A2010F5C5E5C4B5A131F4B5A14806E485A 013F495A5E6E485A6E48C7FC90387F78FEEC7FF890387E1FE091C9FC13FEA25BA21201A2 5BA21203A25B487EB512C0A3293480A32A>I<3907E00FF8391FF83FFE391EFC7E3F393C 7CF81F90397FE03F800078EBC07F1480007015001400D8F0FE133E92C7FC485AA21201A2 5BA21203A25BA21207A25BA2120FA25BA2121FA25BA490C9FC212479A323>114 DIII119 D121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FK cmex10 10 1 /FK 1 113 df<1B381B781BF8A2F201F0A2F203E0A2F207C0A2F20F80A2F21F00A21A3E A262A262A24F5AA24F5AA24F5AA262190FA24FC7FCA2193EA261A261A24E5AA24E5AA24E 5AA24E5AA24EC8FCA2183EA260130801185E137C01FC4B5A1201D807FE4B5A120FD81CFF 4B5A127826E07F804A5A124000004CC9FC6D7E173E6D7E5F6D7E5FA26D6C495AA26D6C49 5AA26D6C5C1607A26D6C495AA26E6C48CAFCA291383FC03EA25EEC1FE05EEC0FF0EDF1F0 EC07F9EDFBE0A26EB45AA26E5BA26E90CBFCA25D157E157C153C4D64788353>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FL cmr10 10 10 /FL 10 112 df<14E01301EB03C01307EB0F80EB1F00133E133C137C5B5B1201485AA248 5AA3485AA248C7FCA35A123EA3127E127CA512FCA35AAC7EA3127CA5127E123EA3123F7E A36C7EA26C7EA36C7EA26C7E12007F137C133C133E7FEB0F80EB07C01303EB01E0130013 5278BD20>40 D<12E07E1278127C7E7E6C7E12077F6C7E12017F6C7EA2137CA37FA27FA3 1480130FA314C01307A514E0A31303AC1307A314C0A5130F1480A3131F1400A3133EA25B A35BA2485A5B1203485A5B120F48C7FC123E5A12785A5A13527CBD20>I 48 DI61 D91 D93 D103 D108 D111 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FM cmmi7 7 6 /FM 6 117 df<39038003E03907C00FF0000F133F1477903881E7E090388383C0D81F87 C7FC139E133813F0EA3FFEEBFFE0383E0FF0EB01F8387E00FC147C007C141CA200FC143C 1538481378EC7C7015F048EB3FE0EC0F801E197B9827>20 D<48B512FC000714FE5A4814 FC393F038000123CEA780712F0120049C7FCA35BA2131EA2133EA35BA41378A21F197E98 1F>28 D<127C12FEA212FFA3127F1203A21207A21206120E120C121C1238127012600812 7A8614>59 D61 D<1303A4497EA600E0141C39FF8FC7FC90B5FC003F14F0000F14C00001EBFE0038007FF8 6D5AA2497EA2EBFCFCEBF87C48487EEBE01E0003131F3907C00F80EB8007EB00031E1D7E 9C22>63 D<133CA2137CA413FC5BA31201B512E0A33803F0005BA312075BA3120F5BA312 1FEB00E0130114C0EA3F03383E0780381F0F00131EEA0FFCEA07F013247EA319>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FN cmr7 7 6 /FN 6 117 df48 D<13381378EA01F8121F12FF12FE12E01200B3AAB512F8 A315267BA521>I<1438147814F81301A213031307130EA2131C13381370A213E0EA01C0 EA0380A2EA0700120E5AA25A5A5AB612E0A3C7EAF800A890383FFFE0A31B267EA521>52 D100 DI116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FO cmsy10 10 8 /FO 8 113 df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ndDVIPSBitmapFont %DVIPSBitmapFont: FP cmmi10 10 10 /FP 10 121 df<91B612E01307131F5B5B9026FFC3FEC7FC4848C67ED803F87F48487F48 48804848131FA2485AA248C7FCA2127E00FE143FA24892C7FCA25D157E15FE485C14015D 6C13034A5A007C495A007E495A6C495A261F807EC8FC380FC3FC3807FFF0C613802B247D A32F>27 D<017FB6FC48B7FC5A5A4815FE261FC01CC7FCEB003C123E5A0078137C481378 A2C7FC14F8A25C1301A313035CA31307A25C130FA3131F5CA591C8FC28247EA324>I34 D<123E127FEAFF80A213C0A3127F123E1200A31201A213801203A213005A5A120E5A123C 5A12700A19798817>59 DI<150E151E153EA215 3C157CA2157815F8A215F01401A215E01403A215C01407A21580140FA215005CA2141E14 3EA2143C147CA2147814F8A25C1301A25C1303A2495AA25C130FA291C7FC5BA2131E133E A2133C137CA2137813F8A25B1201A25B1203A25B1207A25B120FA290C8FC5AA2121E123E A2123C127CA2127812F8A25AA21F537BBD2A>I99 D104 D116 D<90391FE00FF090397FF83FFC3A01F8FC7C7E3903E07E F83A07C03FF0FE9038801FE1D80F0013C1121E158148133FED80F8003891C7FCA2C75AA2 147EA214FEA25CA21301A24A1338A2D83E03147812FF4A13F01307ED01E0010FEB03C039 FE1FF8073AFC3EFC0F8090397C7E3F00397FF83FFC391FE00FF027247DA32F>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FQ ecrm1000 10 59 /FQ 59 256 df<91391FFC01FE49B5388FFF80903B07FC0FFFC7C090260FE003EB1FE090 393FC007FE90397F000FFC5B4914F8485A9338F00FC049010790C7FC1503ABB812F8A328 01F80003F0C7FCB3AB486C497E267FFFE0B512F0A3333A7FB930>27 DI<14E01301EB03C01307EB0F80EB1F00133E133C137C5B5B1201485AA2485AA3485AA2 48C7FCA35A123EA3127E127CA512FCA35AAC7EA3127CA5127E123EA3123F7EA36C7EA26C 7EA36C7EA26C7E12007F137C133C133E7FEB0F80EB07C01303EB01E01300135278BD20> 40 D<12E07E1278127C7E7E6C7E12077F6C7E12017F6C7EA2137CA37FA27FA31480130F A314C01307A514E0A31303AC1307A314C0A5130F1480A3131F1400A3133EA25BA35BA248 5A5B1203485A5B120F48C7FC123E5A12785A5A13527CBD20>I<123E127FEAFF80A213C0 A3127F123E1200A31201A213801203A213005A5A120E5A123C5A12700A19798817>44 DI<123E127FEAFF80A5EA7F00123E0909798817>I48 DIII<1538157815F8A214011403 1407A2140F141FA2143B1473146314E3EB01C3EB0383A2EB0703130EA2131C1338133013 7013E0EA01C0A2EA0380EA0700A2120E5A5AA25A5AB712F8A3C73803F800AA4A7E0103B5 12F8A325367EB52A>I<000E141CD80F80137C9038F807FC90B55A5D5D5D5D4AC7FC14F0 000EC9FCAAEB0FFCEB7FFF9038FC1FC0390FE00FE09038C003F001807F496C7E000E6D7E C8FC81A2811680A4123E127F487EA4160090C75A5A00785C1401007C495A6C5C6C130739 0FC01FF03907F07FC06CB55A6C49C7FC38003FF021377CB52A>II<1238123C123F90B612E0A44815C0168016000078C7121F0070141E5D157C00F0 5C485C4A5A1403C7485A5D4AC7FC5C141E5C147C147814F85C1301A2495AA213075C130F A3131FA3495AA5137FAB6DC8FC23397BB72A>I57 D<123E127FEAFF80A5EA7F00123EC7FCB2123E127FEAFF80A5EA7F00123E092479A317> I<1538157CA315FEA34A7EA34A7FA39138073FC0A391380E1FE0A34A6C7EA34A6C7EA202 787FEC7003A202F07FECE001A20101804A7EA20103814A137FA201078191B6FCA2498101 0EC7121FA24981160FA2496E7EA3496E7EA213F0707E1201486C4A7ED80FFE4A1380B500 C090B512FEA3373A7DB93E>65 DI68 DII<913A01FFC001C0021F13F891 B5EAFE0301039038C07F07903A07FE001FCFD91FF8EB07EFD93FE0EB01FF49487F494814 7F4890C8123F485A4848151F49150F120F491507121F491503123FA2485A1701A3484892 C7FCAB93B6FCA26C7E9338007FE0EF3FC0A26C7EA2121F7F120F7F12077F6C7E6C7E6C6D 147F6D6C14FF6D7ED91FF81303D907FEEB0FEF903A03FFE03FC7010090B51203021FEBFC 01020101E0C7FC383B7CB941>III76 DII80 D<90391FFC01C090387FFF8148B512C33907FC0FF7390FF003FF381FC00049137F48C712 3F48141F007E140F15075AA21503A27E15017EA26D90C7FC7FEA7FF013FC383FFFC014FC 6CEBFFC06C14F06C14FC6C806C806C6C1480010F14C0010014E0140F020013F0153F151F ED0FF81507A200E01403A21501A27EA36CEC03F0A27E6CEC07E06C140F6D14C001E0EB1F 8001F8EB7F0039F9FF01FE00F0EBFFFCD8E01F13F0010313C0253B7CB92E>83 D<003FB812E0A3D9E003EB003F90260001FE1307007EEE03F0007C160100781600A30070 1770A400F01778481738A4C71600B3B0913807FF80011FB612E0A335397DB83C>II 87 D97 DI<90380FFF80013F13F09038FF03F8D803FC13FCEA07F0120FEA1FE013 C0123F90388001F8007F90C7FCA290C8FC5AAA6C7EA3003F140E6D131E121F6D133C6C6C 137C6C6C13F83903FC01F03900FF0FE090383FFF8090380FFE001F247DA325>III<14FF 010713C090381FE3E090383F8FF0EB7F0F137E13FE13FC12019038F807E091C7FCACB512 FCA3D801F8C7FCB3AB487E387FFFF8A31C3A7FB919>IIII107 DI<2703F01FF8 EB3FF000FFD97FFEEBFFFC903BF1F87F83F0FF903BF3E01F87C03F3D0FF7800FCF001F80 2603FF0013DE4902FC14C0496D48130FA2495CA2495CB3A3486C496CEB1FE0B500C1B500 83B5FCA340247EA345>I<3903F01FF800FFEB7FFE9039F1F87F809038F3E01F3A0FF780 0FC03803FF004980491307A25BA25BB3A3486C497EB500C1B51280A329247EA32E>II< 3903F03FF039FFF1FFFC9038F7F0FF9039FFC03FC000079038001FE06C48EB0FF049EB07 F85B49EB03FCA2ED01FEA316FF81A85D16FEA3ED03FC7FED07F86DEB0FF06D14E06DEB1F C09138807F809039F7F0FF009038F1FFFC9038F07FE091C8FCAC487EB512C0A328347EA3 2E>I<903907FC01C090383FFF839038FF0FC33901FE03E33903F800F74848137F000F14 3F485A4848131FA24848130FA312FF90C7FCA87F127FA36C6C131FA26C6C133F157F6C6C 13FF6C6C5A3903FC03EF3900FF0FCF90383FFF0FEB0FFC90C7FCACED1FE00203B5FCA328 347DA32C>I<3907E07F8039FFE1FFC09038E3E7E09038E78FF0000F130FEA03EEA213FC EC07E09038F803C091C7FCA25BB3A2487EB512F0A31C247EA321>I<3801FFC7000F13FF EA1F81383E007F48131F1278487FA2807E7E6C90C7FC6C7E13F8387FFFC06C13F06C13FC 6C7F00037FC66C138013039038007FC000E0131F140F6C1307A214037EA26CEB0780A26C EB0F006C6C5AEBE0FE38F3FFF800E013E01A247DA321>I<1338A51378A413F8A21201A2 12031207001FB5FCB6FCA2D801F8C7FCB2EC01C0A8EBFC03000014801407137E90387F9F 00EB1FFEEB07F81A337FB220>IIIIII<003FB512FCA2EBC00390380007F8003C14F0140F0038EB1FE0007814C0143F0070EB 7F80ECFF00A2495A495A00005B1307495A495AA2495A495AEC000E5B485A485AA2484813 1E485A49131C121F4848133C49137C007F14FC38FF000790B5FCA21F247EA325>I255 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FR ecbx1000 10 7 /FR 7 117 df65 D97 D<13FFB5FCA512077EAE913801FFC0021F13FC027F7F91B61280030713C0DAF80113E091 39E0007FF04A133F91C713F8161F17FCA2160F17FEAB17FC161FA217F86E133F6EEB7FF0 6EEBFFE0ECF801DAFE0F13C001F9B61200D9F87F5BD9F01F13F8D9E00313802F3A7EB935 >I<903803FFE0011F13FE90B6FC4815804801C113C0000F130113FEEA1FFCEA3FF813F0 007F6D1380A249EB3E0000FF91C7FCAA7F127FA27F003FEC03E06D13076C7E6C6CEB0FC0 EC801F00039038F07F806C90B512006C5C011F13F8010313C023257DA42A>I<9038FE0F F800FFEB3FFE4A7E91B5128002FD13C0EBFFF1000713E17E14C1A202801380A2ED3E0092 C7FC91C8FCB2B512FEA522257EA427>114 D<90387FF8780007B512F85A123FEBE01F38 7F8007EB000300FE130114007E7F6D130013FCEBFFF014FE6CEBFF8015E06C14F06C14F8 000714FC1201D8003F13FE1300140F00F8130314017E14007E6C1301018013FCEBC00390 38F81FF890B512F015E000F9148039F03FFC001F257DA426>I<131FA55BA45BA25BA25A 5A5A001FEBFFE0B6FCA4000390C7FCB115F8A7140114816CEB83F014E76CEBFFE06D13C0 6D1380903807FE001D357EB425>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FS ecrm1200 12 16 /FS 16 123 df66 D71 D78 D97 DI<167FED3FFFA315018182B3ECFFC0010713F8011F13FC90387FE0FF9039 FF801FFF4848487E4848130348487F485A48487F82485AA2485AA448C8FCAB7E7FA3123F 7F001F5D6D5B6C7E00075C6D5BD803FE4913806C6C90383F7FC027007FC1FE13FE90383F FFF8010F13F00101018013002F467DC436>100 DI103 D105 D 108 D<3901FC03FF00FF010F13E04A13F8EC7F0F00079038F803FC0003496C7E3901FDE0 00EBFFC04A7F91C77EA25BA35BB3A8486CECFF80B5D8F83F13FEA32F2C7DAB36>110 D<3903F807F800FFEB1FFCEC7FFEECFEFF000713F83803F9E000015B13FB5C9038FF007E 1500A25BA45BB3A648B4FCB512FEA3202C7DAB26>114 D<90387FF0383903FFFE7848EB FFF8381FF03F383F8007EB0003007E1301007C1300481478A315387E7E7E6D130013E0EA 7FFEEBFFF06C13FE6CEBFF806C14C06C14E0000114F0D8003F13F8010113FCEB001F1407 00E0EB01FE14007E157EA2153E7EA27E157C7E6C14FC90388001F89038C007F039FBF81F E000F1B512C000F0140038E01FF81F2E7DAC26>I<130EA5131EA4133EA2137EA213FE12 0112031207121FB612F0A3C648C7FCB3A4151CA9153C7F6D13381578148090383FC0F0EB 1FF390380FFFE06D13C0010013001E3E7EBC26>II<003FB612E0A39038E0003F0180EB7F C0003EC7EAFF80123C4A13000038495A0078495AA24A5A0070495AA24A5A4A5A14FFC75B 4990C7FC495AA2495A495AA2494813E0495A137F5C9038FF800148010013C0A2485A485A 1503485A48481307485A150F4848EB3F8039FF8001FF90B6FCA3232B7DAA2B>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FT ecrm1728 17.28 21 /FT 21 122 df80 D97 DI<9138 03FFC0021F13FC027F13FF902601FFC013C0903A07FE000FF0D90FF8EB03F84948EB00FC 4948147E4948143E4948804890C85A4848913801FF80A248485C120F5B121F5BA2003F6F 1300496E5A007F93C7FCA45B12FFAE6C7EA46C7EA3001FEE01C07F000F16037F0007EE07 807F6C6CED0F007E6C6D141E6E143E6D6C5CD91FF05CD90FFCEB03F06DB4EB0FE0010190 38E07F806D6CB5C7FC021F13FC020313C032417BBF3C>I<181EEF3FFEEE07FFA4EE000F 1703A21701B3AA913801FFC0021F13FC027F13FF903A01FFE07F8101079038000FE1D90F FCEB03F1D91FF0EB00F94948147D4948143F4948141F4890C8120F48481507A248481503 485A1701121F5B123FA25B127FA4485AAE127F7FA3123FA27F121FA26C6C1503A26C6C15 07170F6C7E0001161F6D153F6C6D147D6D6CECF9FF6D6CEB01F1D91FF8D907E113C0D907 FED91FC1EBFF80903A03FF80FF010100EBFFFE023F13F00203018091C7FC41657BE34B> I<913803FF80023F13F891B512FE0103903883FF80902607FC007FD91FF0EB3FE049486D 7E49486D7E49486D7E4890C76C7E5B00036F7E485A701380485A177F484816C0A2123F49 153FA2007F17E0A35B12FFA290B8FCA301C0CAFCA9127F7FA4123F7FA2001F17E07F000F 16017F0007EE03C06C7E6DED07806C160F6C6D15006D6C141E6D6C147ED91FF85C6D6C49 5AD903FFEB07E06D9038E03FC06D6CB5C7FC020F13FC020113E033417CBF3C>II104 D<13FCEA03FFA2487FA66C90C7FCA2EA00FC90C8FCB3A2EB0780 EA0FFFB5FCA41203C6FCA2137FB3B3AC497E487FB61280A4195F7BDE25>I107 D109 DIII<010FEB0FFCD80FFFEB3FFFB590 B51280913901FC1FC0913903F03FE0913807C07F0003EB0F80C6EB1F00141E6D5AA24AEB 3FC0EE0F004A90C7FCA25CA35CA55CB3B0497E4813F0B612F8A42B3F7BBE34>114 D<903901FFE007011FEBFC0F90B6FC489038807FDF3A07F8000FFFD80FE0130348487F48 487F90C87E007E81A2825A82A37E827EA27F7FD87FF091C7FC13FC6CB4FC14F06CEBFF80 6C14F86C14FF6C15C0C615F0013F80010F80010180D9000F7FDA007F1380030F13C01503 030013E000E0157F163FEE1FF06C150FA21607A26C1503A37EA26C16E016077E6D140F17 C06D141F6DEC3F806DEC7F00D8FCFC14FED8F87EEB03FC90393FC03FF8486CB512E00107 148048C601F8C7FC2C417CBF35>I<1470A714F0A51301A31303A21307A2130FA2131F13 3F137F13FF1203000F90B6FCB8FCA326000FF0C8FCB3AEEE01C0AD16038001071580A26E 1307010315006E5B01015C6E133E6D6D5A91387FF1F891381FFFF06E5B020113802A597E D734>IIII121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 278 691 a FT(P)l(ath)l(wise)47 b(description)f(of)g(dynamic)f (pitc)l(hfork)h(bifurcations)1332 874 y(with)g(additiv)l(e)g(noise)1170 1126 y FS(Nils)31 b(Berglund)h(and)h(Barbara)f(Gen)m(tz)1707 1446 y FR(Abstract)345 1584 y FQ(The)h(slo)n(w)f(drift)h(\(with)h(sp)r (eed)f FP(")p FQ(\))g(of)g(a)f(parameter)g(through)g(a)g(pitc)n(hfork)h (bifurcation)f(p)r(oin)n(t,)345 1684 y(kno)n(wn)27 b(as)g(the)g (dynamic)g(pitc)n(hfork)g(bifurcation,)g(is)g(c)n(haracterized)e(b)n(y) i(a)g(signi\034can)n(t)g(dela)n(y)f(of)345 1783 y(the)f(transition)f (from)g(the)h(unstable)f(to)g(the)h(stable)f(state.)36 b(W)-7 b(e)25 b(describ)r(e)f(the)g(e\033ect)h(of)g(an)f(addi-)345 1883 y(tiv)n(e)e(noise,)g(of)g(in)n(tensit)n(y)f FP(\033)s FQ(,)i(b)n(y)f(giving)e(precise)h(estimates)g(on)g(the)h(b)r(eha)n (viour)f(of)g(the)h(individual)345 1983 y(paths.)35 b(W)-7 b(e)23 b(sho)n(w)e(that)h(un)n(til)h(time)1488 1923 y FO(p)p 1558 1923 39 4 v 1558 1983 a FP(")f FQ(after)f(the)i (bifurcation,)g(the)f(paths)g(are)f(concen)n(trated)g(in)345 2082 y(a)h(region)f(of)h(size)f FP(\033)s(=")1024 2052 y FN(1)p FM(=)p FN(4)1150 2082 y FQ(around)g(the)i(bifurcating)e (equilibrium.)35 b(With)23 b(high)f(probabilit)n(y)-7 b(,)22 b(they)345 2182 y(lea)n(v)n(e)e(a)h(neigh)n(b)r(ourho)r(o)r(d)f (of)h(this)g(equilibrium)g(during)f(a)h(time)h(in)n(terv)-5 b(al)20 b FL([)2663 2122 y FO(p)p 2732 2122 V 60 x FP(";)14 b(c)2844 2111 y FK(p)p 2927 2111 257 4 v 71 x FP(")p FO(j)p FL(log)g FP(\033)s FO(j)g FL(])p FQ(,)23 b(after)345 2281 y(whic)n(h)34 b(they)h(are)e(lik)n(ely)h(to)g(sta)n(y)f(close)h (to)g(the)g(corresp)r(onding)e(deterministic)j(solution.)56 b(W)-7 b(e)345 2381 y(deriv)n(e)26 b(exp)r(onen)n(tially)g(small)h(upp) r(er)g(b)r(ounds)g(for)f(the)i(probabilit)n(y)d(of)i(the)h(sets)e(of)h (exceptional)345 2481 y(paths,)h(with)g(explicit)g(v)-5 b(alues)27 b(for)g(the)h(exp)r(onen)n(ts.)118 2661 y FJ(Date.)37 b FQ(August)28 b(4,)f(2000.)118 2774 y(2000)f FJ(Mathematics)31 b(Subje)l(ct)e(Classi\034c)l(ation.)39 b FQ(37H20,)26 b(60H10)g(\(primary\),)h(34E15,)f(93E03)g (\(secondary\).)118 2887 y FJ(Keywor)l(ds)31 b(and)f(phr)l(ases.)39 b FQ(Dynamic)28 b(bifurcation,)f(pitc)n(hfork)g(bifurcation,)h(additiv) n(e)f(noise,)g(bifurcation)h(de-)118 3000 y(la)n(y)-7 b(,)26 b(singular)f(p)r(erturbations,)h(sto)r(c)n(hastic)g(di\033eren)n (tial)g(equations,)g(random)g(dynamical)g(systems,)g(path)n(wise)118 3113 y(description,)h(concen)n(tration)f(of)i(measure.)118 3397 y FI(1)131 b(In)l(tro)t(duction)118 3600 y FH(Ph)m(ysical)25 b(systems)f(are)i(often)g(describ)s(ed)f(b)m(y)h(ordinary)g(di\033eren) m(tial)e(equations)h(\(ODEs\))i(of)e(the)h(form)1630 3772 y FG(d)p FF(x)p 1630 3812 103 4 v 1635 3895 a FG(d)o FF(s)1768 3833 y FG(=)f FF(f)10 b FG(\()p FF(x;)15 b(\025)p FG(\))p FF(;)1317 b FH(\(1.1\))118 4049 y(where)35 b FF(x)f FH(is)f(the)i(state)f(of)g(the)g(system,)g FF(\025)g FH(a)h(parameter,)g(and)g FF(s)f FH(denotes)h(time.)50 b(The)35 b(mo)s(del)e(\(1.1\))118 4162 y(ma)m(y)24 b(ho)m(w)m(ev)m(er)j (b)s(e)d(to)s(o)h(crude,)h(since)e(it)f(neglects)i(all)d(kinds)i(of)g (p)s(erturbations)h(acting)f(on)h(the)g(system.)118 4275 y(W)-8 b(e)40 b(are)f(in)m(terested)h(here)g(in)e(the)h(com)m(bined)h (e\033ect)f(of)g(t)m(w)m(o)h(p)s(erturbations:)59 b(a)39 b(slo)m(w)g(drift)f(of)h(the)118 4388 y(parameter,)31 b(and)g(an)f(additiv)m(e)g(noise.)259 4501 y(A)d(slo)m(wly)f(drifting)g (parameter)i FF(\025)e FG(=)e FF("s)p FH(,)k(\(with)g FF(")d FE(\034)g FG(1)p FH(\),)k(ma)m(y)e(mo)s(del)f(the)i (deterministic)d(c)m(hange)118 4614 y(in)e(time)f(of)h(some)f(exterior) h(in\035uence,)i(suc)m(h)f(as)f(the)h(climate)d(acting)i(on)h(an)g (ecosystem)e(or)i(a)f(magnetic)118 4727 y(\034eld)31 b(acting)g(on)g(a)g(ferromagnet.)44 b(Ob)m(viously)-8 b(,)30 b(non)m(trivial)g(dynamics)f(can)j(only)e(b)s(e)h(exp)s(ected)h (when)118 4840 y FF(\025)j FH(is)e(allo)m(w)m(ed)i(to)g(v)-5 b(ary)34 b(b)m(y)h(an)g(amoun)m(t)g(of)f(order)i FG(1)p FH(,)g(and)f(th)m(us)g(the)g(system)e(has)i(to)g(b)s(e)f(considered)118 4953 y(on)j(the)f(time)f(scale)g FF(")887 4920 y FD(\000)p FC(1)982 4953 y FH(.)58 b(This)35 b(is)g(usually)f(done)j(b)m(y)g(in)m (tro)s(ducing)f(the)g FB(slow)i(time)43 b FF(t)34 b FG(=)h FF("s)p FH(,)j(whic)m(h)118 5066 y(transforms)30 b(\(1.1\))h(in)m(to)f (the)h(singularly)c(p)s(erturb)s(ed)k(equation)1609 5299 y FF(")1661 5237 y FG(d)p FF(x)p 1661 5278 V 1671 5361 a FG(d)o FF(t)1799 5299 y FG(=)25 b FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))p FF(:)1306 b FH(\(1.2\))118 5509 y(It)30 b(is)e(kno)m(wn)j(that)g(solutions)d(of)i(this)f(system)f(tend) j(to)f(sta)m(y)g(close)f(to)h(stable)g(equilibrium)c(branc)m(hes)118 5622 y(of)k FF(f)40 b FH([Gr,)30 b(Ti],)g(see)g(Fig.)25 b(1a.)41 b(New,)30 b(and)h(sometimes)d(surprising)h(phenomena)i(o)s (ccur)g(when)g(suc)m(h)g(an)1867 5871 y(1)p eop %%Page: 2 2 2 1 bop 354 236 a 24240837 7999473 6578176 27036303 32496189 35719495 startTexFig doclip 354 236 a %%BeginDocument: fig1.ps %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: determ.dvi %%Pages: 1 %%PageOrder: Ascend %%BoundingBox: 100 411 494 543 %%EndComments %DVIPSCommandLine: dvips determ -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.04:1528 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup 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 /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore 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 /IE 256 array N 2 string 0 1 255{IE S dup 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (determ.dvi) @start %DVIPSBitmapFont: Fa ecrm1095 10.95 1 /Fa 1 50 df49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb eccc1000 10 8 /Fb 8 118 df<123E127FEAFF80A5EA7F00123E090977881B>46 D49 D70 D101 D<91387FF007903903FFFC0F010F13FF9039 3FF83FDF90397FC007FF4848C67E48481300D807F880120F49804848805B003F81A2485A 82A248C8FC93C7FCA892387FFFF8A26C7E03001380EE7F006C7EA26C7EA26C7E7F6C7E6C 6C5C6C6C5B39007FC00390393FF81FDF010FB5128F010314079026007FF8C7FC2D2D7BAB 35>103 D105 D114 D117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy10 10 1 /Fc 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmr7 7 3 /Fd 3 117 df100 DI116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr10 10 2 /Fe 2 42 df<14E01301EB03C01307EB0F80EB1F00133E133C137C5B5B1201485AA2485A A3485AA248C7FCA35A123EA3127E127CA512FCA35AAC7EA3127CA5127E123EA3123F7EA3 6C7EA26C7EA36C7EA26C7E12007F137C133C133E7FEB0F80EB07C01303EB01E013001352 78BD20>40 D<12E07E1278127C7E7E6C7E12077F6C7E12017F6C7EA2137CA37FA27FA314 80130FA314C01307A514E0A31303AC1307A314C0A5130F1480A3131F1400A3133EA25BA3 5BA2485A5B1203485A5B120F48C7FC123E5A12785A5A13527CBD20>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi7 7 2 /Ff 2 117 df<1303A4497EA600E0141C39FF8FC7FC90B5FC003F14F0000F14C00001EB FE0038007FF86D5AA2497EA2EBFCFCEBF87C48487EEBE01E0003131F3907C00F80EB8007 EB00031E1D7E9C22>63 D<133CA2137CA413FC5BA31201B512E0A33803F0005BA312075B A3120F5BA3121FEB00E0130114C0EA3F03383E0780381F0F00131EEA0FFCEA07F013247E A319>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi10 10 2 /Fg 2 121 df116 D<90391FE00FF090397FF83FFC3A01F8FC7C7E3903E07EF83A07C03FF0FE9038801FE1D8 0F0013C1121E158148133FED80F8003891C7FCA2C75AA2147EA214FEA25CA21301A24A13 38A2D83E03147812FF4A13F01307ED01E0010FEB03C039FE1FF8073AFC3EFC0F8090397C 7E3F00397FF83FFC391FE00FF027247DA32F>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh ecrm1000 10 29 /Fh 29 123 df<14E01301EB03C01307EB0F80EB1F00133E133C137C5B5B1201485AA248 5AA3485AA248C7FCA35A123EA3127E127CA512FCA35AAC7EA3127CA5127E123EA3123F7E A36C7EA26C7EA36C7EA26C7E12007F137C133C133E7FEB0F80EB07C01303EB01E0130013 5278BD20>40 D<12E07E1278127C7E7E6C7E12077F6C7E12017F6C7EA2137CA37FA27FA3 1480130FA314C01307A514E0A31303AC1307A314C0A5130F1480A3131F1400A3133EA25B A35BA2485A5B1203485A5B120F48C7FC123E5A12785A5A13527CBD20>I66 D68 D<913A01FFC001C0021F13F891B5EAFE0301039038C07F07903A07FE001FCFD91F F8EB07EFD93FE0EB01FF49487F4948147F4890C8123F485A4848151F49150F120F491507 121F491503123FA2485A1701A3484892C7FCAB93B6FCA26C7E9338007FE0EF3FC0A26C7E A2121F7F120F7F12077F6C7E6C7E6C6D147F6D6C14FF6D7ED91FF81303D907FEEB0FEF90 3A03FFE03FC7010090B51203021FEBFC01020101E0C7FC383B7CB941>71 D78 D97 DI<90380FFF80013F13F09038FF03F8D803FC13FCEA07F0120FEA1FE013C012 3F90388001F8007F90C7FCA290C8FC5AAA6C7EA3003F140E6D131E121F6D133C6C6C137C 6C6C13F83903FC01F03900FF0FE090383FFF8090380FFE001F247DA325>III<14FF0107 13C090381FE3E090383F8FF0EB7F0F137E13FE13FC12019038F807E091C7FCACB512FCA3 D801F8C7FCB3AB487E387FFFF8A31C3A7FB919>IIII107 DI<2703F01FF8EB3F F000FFD97FFEEBFFFC903BF1F87F83F0FF903BF3E01F87C03F3D0FF7800FCF001F802603 FF0013DE4902FC14C0496D48130FA2495CA2495CB3A3486C496CEB1FE0B500C1B50083B5 FCA340247EA345>I<3903F01FF800FFEB7FFE9039F1F87F809038F3E01F3A0FF7800FC0 3803FF004980491307A25BA25BB3A3486C497EB500C1B51280A329247EA32E>II<3903 F03FF039FFF1FFFC9038F7F0FF9039FFC03FC000079038001FE06C48EB0FF049EB07F85B 49EB03FCA2ED01FEA316FF81A85D16FEA3ED03FC7FED07F86DEB0FF06D14E06DEB1FC091 38807F809039F7F0FF009038F1FFFC9038F07FE091C8FCAC487EB512C0A328347EA32E> I<3907E07F8039FFE1FFC09038E3E7E09038E78FF0000F130FEA03EEA213FCEC07E09038 F803C091C7FCA25BB3A2487EB512F0A31C247EA321>114 D<3801FFC7000F13FFEA1F81 383E007F48131F1278487FA2807E7E6C90C7FC6C7E13F8387FFFC06C13F06C13FC6C7F00 037FC66C138013039038007FC000E0131F140F6C1307A214037EA26CEB0780A26CEB0F00 6C6C5AEBE0FE38F3FFF800E013E01A247DA321>I<1338A51378A413F8A21201A2120312 07001FB5FCB6FCA2D801F8C7FCB2EC01C0A8EBFC03000014801407137E90387F9F00EB1F FEEB07F81A337FB220>IIII121 D<003FB512FCA2EBC003 90380007F8003C14F0140F0038EB1FE0007814C0143F0070EB7F80ECFF00A2495A495A00 005B1307495A495AA2495A495AEC000E5B485A485AA24848131E485A49131C121F484813 3C49137C007F14FC38FF000790B5FCA21F247EA325>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 264 1914 a 11663564 8391059 4604723 21707980 36837785 44731596 startTexFig doclip 264 1914 a %%BeginDocument: fig_stab.ps %! %%BoundingBox: 70 330 560 680 /l {lineto} def /m {moveto} def /n {newpath} def /p {pnt} def /s {stroke} def /t {translate} def /h {scale} def newpath 2 setlinewidth s n 1.5 setlinewidth 121.5 636.2 m 119 636.2 t 1 0.8929 h 0 0 2.45 360 719.8 arc 1 1.12 h -119 -636.2 t 119 636.2 m 119.5 631.4 l 120 626.7 l 120.5 622.1 l 121 617.7 l 121.5 613.3 l 122.2 606.8 l 122.9 600.6 l 123.7 594.6 l 124.4 588.8 l 125.1 583.2 l 125.9 577.8 l 126.6 572.6 l 127.3 567.5 l 128.1 562.7 l 128.8 557.9 l 129.5 553.4 l 130.3 549 l 131.2 543.3 l 132.2 538 l 133.2 532.8 l 134.2 527.9 l 134.2 527.9 m 132.9 534.2 l s n 135.2 523.3 m 126.5 543.7 l 130.6 545.1 t 1 0.8929 h 0 0 4.32 200.6 360.6 arc 1 1.12 h -130.6 -545.1 t 134.9 545.1 l 135.2 523.3 l closepath fill s n 135.2 523.3 m 136.2 518.8 l 137.4 513.6 l 138.6 508.6 l 139.8 504 l 141.1 499.6 l 142.5 494.7 l 144 490.2 l 145.5 486 l 147.2 481.5 l 149.1 476.9 l 151.1 472.8 l 153.3 468.7 l 155.8 464.7 l 158.4 461 l 161.6 457.5 l 165.3 454.3 l 169.5 451.7 l 174.1 450 l 179 449.1 l 183.9 449.2 l 188.8 449.9 l 193.7 451.1 l 198.4 452.7 l 203 454.7 l 207.4 456.7 l 211.9 459 l 216.3 461.4 l 220.4 463.8 l 224.6 466.2 l 228.8 468.8 l 232.9 471.3 l 237.1 474 l 241.3 476.6 l 245.4 479.3 l 249.6 481.9 l 253.8 484.6 l 257.9 487.3 l 262.1 490 l 266.2 492.7 l 270.4 495.4 l 274.6 498 l 278.7 500.7 l 282.9 503.3 l 287.1 505.9 l 291.2 508.5 l 295.4 511.1 l 299.6 513.7 l 303.7 516.2 l 307.9 518.8 l 312.1 521.3 l s n 312.1 521.3 m 316.2 523.8 l 320.4 526.3 l 324.6 528.8 l 328.7 531.2 l 332.9 533.7 l 337 536.1 l 341.2 538.5 l 345.4 540.9 l 349.5 543.2 l 353.7 545.6 l 357.9 547.9 l 362 550.2 l 366.2 552.6 l 370.6 555 l 375 557.4 l 379.4 559.8 l 383.8 562.2 l 388.3 564.5 l 392.7 566.9 l 397.1 569.2 l 401.5 571.5 l 405.9 573.9 l 410.3 576.1 l 414.7 578.4 l 419.1 580.7 l 423.5 582.9 l 427.9 585.2 l 432.4 587.4 l 436.8 589.6 l 441.2 591.8 l 445.6 594 l 450 596.2 l 454.4 598.3 l 458.8 600.5 l 463.2 602.6 l 467.6 604.8 l 472 606.9 l 476.5 609 l 480.9 611.1 l 485.3 613.2 l 489.7 615.2 l 494.1 617.3 l 498.5 619.4 l 502.9 621.4 l 507.3 623.4 l 511.7 625.5 l 516.1 627.5 l 520.6 629.5 l 525 631.5 l 529.4 633.5 l s n 529.4 633.5 m 533.8 635.4 l 538.2 637.4 l 542.6 639.4 l 547 641.3 l 551.4 643.3 l 555.8 645.2 l 560.2 647.1 l 170.4 636.2 m 168 636.2 t 1 0.8929 h 0 0 2.45 360 719.8 arc 1 1.12 h -168 -636.2 t 168 636.2 m 168.7 630.4 l 169.5 624.7 l 170.2 619.2 l 170.9 613.9 l 171.7 608.8 l 172.4 603.8 l 173.1 599 l 173.9 594.4 l 174.6 589.9 l 175.3 585.6 l 176.3 580 l 177.3 574.7 l 178.3 569.7 l 179.3 564.8 l 180.2 560.2 l 181.2 555.8 l 182.5 550.6 l 183.7 545.7 l 183.7 545.7 m 182 551.9 l s n 184.9 541.1 m 175.2 561.2 l 179.2 562.7 t 1 0.8929 h 0 0 4.32 203.3 363.3 arc 1 1.12 h -179.2 -562.7 t 183.5 562.9 l 184.9 541.1 l closepath fill s n 184.9 541.1 m 186.1 536.7 l 187.6 531.9 l 189.1 527.3 l 190.5 523.1 l 192.3 518.6 l 194 514.5 l 195.9 510.3 l 198.1 506 l 200.6 501.9 l 203.3 498 l 206.2 494.5 l 209.7 491.1 l 213.6 488.2 l 218 486 l 222.6 484.6 l 227.5 484 l 232.4 484.1 l 237.3 484.8 l 242.2 486 l 246.9 487.4 l 251.5 489.2 l 256.2 491.1 l 260.6 493.2 l 265 495.3 l 269.4 497.6 l 273.8 499.9 l 278.2 502.3 l 282.4 504.7 l 286.6 507 l 290.7 509.4 l 294.9 511.8 l 299.1 514.2 l 303.2 516.7 l 307.4 519.1 l 311.6 521.5 l 315.7 523.9 l 319.9 526.4 l 324.1 528.8 l 328.2 531.2 l 332.4 533.6 l 336.6 536 l 340.7 538.3 l 344.9 540.7 l 349.1 543.1 l 353.2 545.4 l 357.4 547.7 l 361.5 550 l 366 552.5 l 370.4 554.9 l 374.8 557.3 l s n 374.8 557.3 m 379.2 559.7 l 383.6 562.1 l 388 564.4 l 392.4 566.8 l 396.8 569.1 l 401.2 571.4 l 405.6 573.7 l 410.1 576 l 414.5 578.3 l 418.9 580.6 l 423.3 582.8 l 427.7 585.1 l 432.1 587.3 l 436.5 589.5 l 440.9 591.7 l 445.3 593.9 l 449.7 596.1 l 454.2 598.2 l 458.6 600.4 l 463 602.5 l 467.4 604.6 l 471.8 606.8 l 476.2 608.9 l 480.6 611 l 485 613.1 l 489.4 615.1 l 493.8 617.2 l 498.3 619.2 l 502.7 621.3 l 507.1 623.3 l 511.5 625.4 l 515.9 627.4 l 520.3 629.4 l 524.7 631.4 l 529.1 633.4 l 533.5 635.3 l 537.9 637.3 l 542.4 639.3 l 546.8 641.2 l 551.2 643.2 l 555.6 645.1 l 560 647 l 564.4 648.9 l s n 2.5 setlinewidth 70 337.1 m 70 337.1 l 74.9 343.9 l 79.8 350.3 l 84.7 356.5 l 89.6 362.4 l 94.5 368.2 l 99.4 373.7 l 104.3 379.1 l 109.2 384.4 l 114.1 389.5 l 119 394.4 l 123.9 399.2 l 128.8 404 l 133.7 408.6 l 138.6 413.1 l 143.5 417.5 l 148.4 421.8 l 153.3 426.1 l 158.2 430.3 l 163.1 434.4 l 168 438.4 l 172.9 442.3 l 177.8 446.2 l 182.7 450.1 l 187.6 453.9 l 192.5 457.6 l 197.4 461.3 l 202.3 464.9 l 207.2 468.4 l 212.1 472 l 217 475.4 l 221.9 478.9 l 226.8 482.3 l 231.7 485.6 l 236.6 489 l 241.5 492.2 l 246.4 495.5 l 251.3 498.7 l 256.2 501.9 l 261.1 505 l 266 508.1 l 270.9 511.2 l 275.8 514.2 l 280.7 517.3 l 285.6 520.2 l 290.5 523.2 l 295.4 526.1 l 300.3 529.1 l 305.2 531.9 l 310.1 534.8 l s n 310.1 534.8 m 315 537.6 l 319.9 540.4 l 324.8 543.2 l 329.7 546 l 334.6 548.8 l 339.5 551.5 l 344.4 554.2 l 349.3 556.9 l 354.2 559.5 l 359.1 562.2 l 364 564.8 l 368.9 567.4 l 373.8 570 l 378.7 572.6 l 383.6 575.1 l 388.5 577.6 l 393.4 580.2 l 398.3 582.7 l 403.2 585.1 l 408.1 587.6 l 413 590.1 l 417.9 592.5 l 422.8 594.9 l 427.7 597.3 l 432.6 599.7 l 437.5 602.1 l 442.4 604.5 l 447.3 606.8 l 452.2 609.2 l 457.1 611.5 l 462 613.8 l 466.9 616.1 l 471.8 618.4 l 476.7 620.7 l 481.6 622.9 l 486.5 625.2 l 491.4 627.4 l 496.3 629.6 l 501.2 631.9 l 506.1 634.1 l 511 636.2 l 515.9 638.4 l 520.8 640.6 l 525.7 642.8 l 530.6 644.9 l 535.5 647.1 l 540.4 649.2 l 545.3 651.3 l 550.2 653.4 l 555.1 655.5 l s n 555.1 655.5 m 560 657.6 l s showpage %%EndDocument endTexFig 2038 1914 a 11663564 8391059 4604723 21707980 36837785 44731596 startTexFig doclip 2038 1914 a %%BeginDocument: fig_pitch.ps %! %%BoundingBox: 70 330 560 680 /l {lineto} def /m {moveto} def /n {newpath} def /p {pnt} def /s {stroke} def /t {translate} def /h {scale} def newpath 2 setlinewidth s n 1.5 setlinewidth 123.9 643.6 m 121.3 643.6 t 1 0.6754 h 0 0 2.565 360 719.8 arc 1 1.481 h -121.3 -643.6 t 121.3 643.6 m 121.5 638.2 l 121.8 631.6 l 122.2 625.5 l 122.5 620 l 122.8 613.8 l 123.2 608.1 l 123.6 602.8 l 124.1 597.2 l 124.6 591.2 l 125.2 585.8 l 125.8 580.2 l 125.8 580.2 m 125.1 585.9 l s n 126.5 574.6 m 119.1 596.6 l 123.7 597.3 t 1 0.6754 h 0 0 4.687 192.8 356.8 arc 1 1.481 h -123.7 -597.3 t 128.3 597.1 l 126.5 574.6 l closepath fill s n 126.5 574.6 m 123.9 643.6 m 121.3 643.6 t 1 0.6754 h 0 0 2.565 360 719.8 arc 1 1.481 h -121.3 -643.6 t 121.3 643.6 m 121.5 638.2 l 121.8 631.6 l 122.2 625.5 l 122.5 620 l 122.8 613.8 l 123.2 608.1 l 123.6 602.8 l 124.1 597.2 l 124.6 591.2 l 125.2 585.8 l 125.8 580.2 l 126.5 574.6 l 127.2 569.1 l 128.1 563.6 l 129 558.4 l 130.1 553 l 131.3 547.7 l 132.7 542.5 l 134.3 537.4 l 136.2 532.4 l 138.6 527.3 l 141.6 522.5 l 145.3 517.9 l 150.2 513.8 l 156.3 510.6 l 163.3 508.4 l 170.7 507 l 178.3 506.2 l 186 505.8 l 193.7 505.5 l 201.4 505.3 l 209.1 505.2 l 216.8 505.2 l 224.5 505.1 l 232.2 505.1 l 239.9 505.1 l 247.6 505.1 l 255.3 505 l 263 505 l 270.7 505 l s n 270.7 505 m 278.4 505 l 286.1 505 l 293.8 505 l 301.5 505 l 309.2 505 l 316.9 505 l 324.6 505 l 332.3 505 l 340 505 l 347.7 505 l 355.4 505.1 l 363.1 505.1 l 370.7 505.1 l 378.4 505.1 l 386.1 505.2 l 393.8 505.2 l 401.5 505.4 l 409.2 505.5 l 416.9 505.8 l 424.6 506.3 l 432.2 507.2 l 439.7 508.6 l 446.6 511 l 452.6 514.3 l 457.3 518.4 l 461.1 523.1 l 464.1 528 l 466.6 533 l 468.6 538 l 470.5 543.3 l 472.1 548.5 l 473.6 553.7 l 475 559.2 l 476.2 564.4 l 477.3 569.5 l 478.5 575 l 479.6 580.2 l 480.7 585.6 l 480.7 585.6 m 479.5 579.9 l s n 481.7 591.1 m 482 568.6 l 477.3 568.6 t 1 0.6754 h 0 0 4.687 0.4622 164.5 arc 1 1.481 h -477.3 -568.6 t 472.8 569.4 l 481.7 591.1 l closepath fill s n 481.7 591.1 m 482.8 596.7 l 483.9 602.2 l 485 607.5 l 486 612.7 l 487.2 618 l 488.4 623.2 l 489.8 628.6 l 491.3 633.9 l 493.1 639 l 495.3 644.1 l 498 648.9 l 501.9 653.5 l 507 657.4 l 513.2 660.5 l 519.8 663.3 l 526.5 665.9 l 533.3 668.4 l 540.1 670.8 l 547 673.3 l 553.8 675.7 l 560.7 678.1 l s n 2.5 setlinewidth 562.6 330 m 562.6 330 l 547.3 335.2 l 532.4 340.4 l 518 345.6 l 504.1 350.8 l 490.6 356 l 477.6 361.2 l 465.1 366.4 l 453 371.6 l 441.4 376.8 l 430.2 382 l 419.5 387.2 l 409.3 392.4 l 399.5 397.6 l 390.2 402.8 l 381.3 408 l 373 413.2 l 365 418.4 l 357.6 423.6 l 350.6 428.8 l 344 434 l 337.9 439.2 l 332.3 444.4 l 327.2 449.6 l 322.5 454.8 l 318.2 460 l 314.5 465.1 l 311.2 470.3 l 308.3 475.5 l 305.9 480.7 l 304 485.9 l 302.5 491.1 l 301.5 496.3 l 301 501.5 l 300.9 506.7 l 301.3 511.9 l 302.1 517.1 l 303.5 522.3 l 305.2 527.5 l 307.5 532.7 l 310.2 537.9 l 313.3 543.1 l 316.9 548.3 l 321 553.5 l 325.5 558.7 l 330.5 563.9 l 336 569.1 l 341.9 574.3 l 348.3 579.5 l 355.2 584.7 l s n 355.2 584.7 m 362.5 589.9 l 370.3 595.1 l 378.5 600.3 l 387.2 605.5 l 396.4 610.7 l 406 615.9 l 416.1 621.1 l 426.6 626.3 l 437.6 631.5 l 449.1 636.7 l 461 641.9 l 473.4 647.1 l 486.2 652.3 l 499.6 657.5 l 513.3 662.7 l 527.6 667.9 l 542.3 673.1 l 557.4 678.3 l 562.6 680 l s n 2.5 setlinewidth 300.9 505 m 70 505 l s n [10.26] 0 setdash 300.9 505 m 560 505 l s n [] 0 setdash s showpage %%EndDocument endTexFig 243 1956 a Fh(\(a\))1665 b(\(b\))408 2854 y Fg(x)455 2823 y Ff(?)494 2854 y Fe(\()p Fg(t)p Fe(\))609 2169 y Fg(x)656 2138 y Fd(det)656 2189 y Ff(t)2912 2039 y Fg(x)2959 2008 y Ff(?)2998 2039 y Fe(\()p Fg(t)p Fe(\))2262 2287 y Fg(x)2309 2256 y Fd(det)2309 2307 y Ff(t)2841 2877 y Fc(\000)p Fg(x)2953 2847 y Ff(?)2991 2877 y Fe(\()p Fg(t)p Fe(\))268 3829 y Fb(Figure)32 b(1.)40 b Fh(Nils)28 b(Berglund)f(and)g(Barbara)e(Gen)n(tz)268 3928 y(Dynamic)i(pitc)n (hfork)g(bifurcations)g(with)h(additiv)n(e)f(noise)1867 5871 y Fa(1)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF %%EndDocument endTexFig 268 1403 a FA(Figure)27 b(1.)41 b FQ(Solutions)23 b(of)g(the)h(slo)n (wly)e(time-dep)r(enden)n(t)i(equation)e(\(1.2\))h(represen)n(ted)f(in) i(the)f FL(\()p FP(t;)14 b(x)p FL(\))p FQ(-)268 1503 y(plane.)46 b(\(a\))30 b(Stable)h(case:)42 b(A)31 b(stable)g (equilibrium)f(branc)n(h)g FP(x)2243 1473 y FM(?)2282 1503 y FL(\()p FP(t)p FL(\))h FQ(attracts)f(nearb)n(y)f(solutions)h FP(x)3392 1473 y FN(det)3392 1523 y FM(t)3489 1503 y FQ(.)268 1603 y(T)-7 b(w)n(o)21 b(solutions)g(with)i(di\033eren)n(t)f (initial)h(conditions)e(are)h(sho)n(wn.)34 b(They)22 b(con)n(v)n(erge)e(exp)r(onen)n(tially)h(fast)268 1702 y(to)k(eac)n(h)g(other,)h(as)f(w)n(ell)h(as)f(to)h(a)f(neigh)n(b)r (ourho)r(o)r(d)g(of)h(order)e FP(")i FQ(of)g FP(x)2404 1672 y FM(?)2442 1702 y FL(\()p FP(t)p FL(\))p FQ(.)37 b(\(b\))27 b(Pitc)n(hfork)e(bifurcation:)268 1802 y(The)36 b(stable)f(equilibrium)h FP(x)i FL(=)f(0)e FQ(b)r(ecomes)h(unstable)g (at)g FP(t)h FL(=)g(0)e FQ(\(brok)n(en)h(line\))g(and)g(exp)r(els)g(t)n (w)n(o)268 1901 y(stable)24 b(equilibrium)g(branc)n(hes)f FO(\006)p FP(x)1395 1871 y FM(?)1433 1901 y FL(\()p FP(t)p FL(\))p FQ(.)37 b(A)25 b(solution)f FP(x)2032 1871 y FN(det)2032 1922 y FM(t)2153 1901 y FQ(is)g(sho)n(wn,)h(whic)n(h)f(is)g (attracted)g(b)n(y)g FP(x)g FL(=)e(0)p FQ(,)268 2001 y(and)31 b(sta)n(ys)g(close)g(to)h(the)h(origin)e(for)g(a)h(\034nite)g (time)h(after)e(the)i(bifurcation.)50 b(This)32 b(phenomenon)f(is)268 2101 y(kno)n(wn)26 b(as)h(bifurcation)g(dela)n(y)-7 b(.)118 2374 y FH(equilibrium)25 b(branc)m(h)30 b(undergo)s(es)g(a)f (bifurcation.)39 b(These)29 b(phenomena)g(are)g(usually)e(called)h FB(dynamic)118 2487 y(bifur)-5 b(c)g(ations)26 b FH([Ben)q(])801 2454 y Fz(1)840 2487 y FH(.)39 b(In)24 b(the)i(case)f(of)g(the)g(Hopf)g (bifurcation,)h(when)f(the)h(equilibrium)21 b(gets)k(unstable)118 2600 y(while)33 b(exp)s(elling)f(a)i(stable)g(p)s(erio)s(dic)f(orbit,)i (the)g(bifurcation)e(is)g(substan)m(tially)g(dela)m(y)m(ed:)49 b(solutions)118 2713 y(of)i(\(1.2\))45 b(trac)m(k)g(the)g(unstable)f (equilibrium)c(\(for)45 b(a)f FB(non-vanishing)g FH(time)f(in)m(terv)-5 b(al)43 b(in)h(the)h(limit)118 2826 y FF(")35 b FE(!)f FG(0)p FH(\))i(b)s(efore)f(jumping)g(to)h(the)g(limit)c(cycle)j([Sh,)h (Ne].)56 b(A)36 b(similar)c(phenomenon)k(exists)e(for)i(the)118 2939 y(dynamic)28 b(pitc)m(hfork)h(bifurcation)f(of)h(an)g(equilibrium) d(without)j(drift,)f(the)i(simplest)c(example)i(b)s(eing)118 3052 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))31 b(=)f FF(tx)22 b FE(\000)f FF(x)751 3019 y FC(3)824 3052 y FH(\(Fig.)28 b(1b\).)49 b(The)34 b(dela)m(y)f(has)g(b)s(een)h(observ)m (ed)g(exp)s(erimen)m(tally)-8 b(,)32 b(for)h(instance,)h(in)118 3165 y(lasers)29 b([ME)q(])h(and)h(in)e(a)i(damp)s(ed)e(rotating)i(p)s (endulum)e([BK)q(].)259 3277 y(These)j(phenomena)f(ha)m(v)m(e)i(the)e (adv)-5 b(an)m(tage)33 b(of)e(pro)m(viding)f(a)h(gen)m(uinely)g (dynamic)f(p)s(oin)m(t)g(of)h(view)118 3390 y(for)44 b(the)h(concept)g(of)f(a)g(bifurcation.)81 b(Although)44 b(one)h(often)f(sa)m(ys)g(that)g(a)h(bifurcation)e(diagram)118 3503 y(\(represen)m(ting)22 b(the)f(asymptotic)e(states)h(of)g(the)h (system)e(as)i(a)f(function)g(of)h(the)g(parameter\))g(is)e(obtained) 118 3616 y(b)m(y)26 b(v)-5 b(arying)24 b(the)i(con)m(trol)g(parameter)g FF(\025)p FH(,)g(the)g(impatien)m(t)e(exp)s(erimen)m(talist)f(taking)h (this)h(literally)d(ma)m(y)118 3729 y(ha)m(v)m(e)38 b(the)e(surprise)g (to)g(disco)m(v)m(er)h(unstable)f(stationary)g(states)g(of)g(the)g (system)f(\(s\)he)i(in)m(v)m(estigates.)118 3842 y(The)30 b(asymptotic)d(state)i(of)g(the)g(system)f(\(1.1\))h(with)g(slo)m(wly)e (v)-5 b(arying)28 b(parameter)h FF(\025)p FG(\()p FF("s)p FG(\))d(=)f FF(\025)p FG(\()p FF(t)p FG(\))30 b FH(ma)m(y)118 3955 y(dep)s(end)e(not)g(only)f(on)h(the)g(initial)c(condition)k FG(\()p FF(x)1799 3969 y FC(0)1838 3955 y FF(;)15 b(t)1911 3969 y FC(0)1951 3955 y FG(\))p FH(,)28 b(but)h(also)d(on)i(the)g (history)f(of)g(v)-5 b(ariation)27 b(of)g(the)118 4068 y(parameter)k FE(f)p FF(\025)p FG(\()p FF(t)p FG(\))p FE(g)796 4082 y Fy(t)p Fx(>)p Fy(t)901 4091 y Fw(0)942 4068 y FH(.)259 4181 y(The)j(p)s(erturbation)f(of)39 b(\(1.1\))34 b(b)m(y)f(an)g(additiv)m(e)f(noise)h(can)g(b)s(e)g(mo)s (deled)e(b)m(y)i(a)g(sto)s(c)m(hastic)g(di\033er-)118 4294 y(en)m(tial)d(equation)g(\(SDE\))h(of)f(the)h(form)1362 4498 y FG(d)o FF(x)1464 4512 y Fy(s)1526 4498 y FG(=)25 b FF(f)10 b FG(\()p FF(x)1764 4512 y Fy(s)1801 4498 y FF(;)15 b(\025)p FG(\))g(d)p FF(s)20 b FG(+)g FF(\033)e FG(d)p FF(W)2356 4512 y Fy(s)2392 4498 y FF(;)1059 b FH(\(1.3\))118 4702 y(where)39 b FF(W)473 4716 y Fy(s)548 4702 y FH(denotes)g(the)f(standard)h(Wiener)f(pro)s(cess,)i(and)e FF(\033)j FH(measures)d(the)g(noise)g(in)m(tensit)m(y)-8 b(.)64 b(A)118 4815 y(widespread)25 b(approac)m(h)i(is)d(to)h(analyse)f (the)h(probabilit)m(y)f(densit)m(y)h(of)f FF(x)2532 4829 y Fy(s)2569 4815 y FH(,)i(whic)m(h)f(satis\034es)f(the)h(F)-8 b(okk)m(er\025)118 4928 y(Planc)m(k)27 b(equation.)39 b(In)27 b(particular,)g(if)f FE(\000)p FF(f)36 b FH(can)27 b(b)s(e)g(written)g(as)f(the)i(gradien)m(t)f(of)g(a)g FB(p)-5 b(otential)30 b(function)118 5041 y FF(F)13 b FH(,)45 b(then)e(there)g(is)e(a)h(unique)g(stationary)g(densit)m(y)f FF(p)p FG(\()p FF(x;)15 b(\025)p FG(\))47 b(=)d(e)2445 5008 y FD(\000)p Fy(F)10 b FC(\()p Fy(x;\025)p FC(\))p Fy(=\033)2787 4984 y Fw(2)2841 5041 y FF(=)-5 b(N)10 b FH(,)46 b(where)d FF(N)52 b FH(is)41 b(the)118 5154 y(normalization.)e(This)29 b(form)m(ula)g(sho)m(ws)h(that)g(for)g (small)d(noise)i(in)m(tensit)m(y)-8 b(,)30 b(the)h(stationary)e(densit) m(y)h(is)118 5267 y(sharply)g(p)s(eak)m(ed)g(around)i(stable)d (equilibria)f(of)i FF(f)10 b FH(.)p 118 5346 1418 4 v 222 5400 a Fv(1)256 5431 y Fu(Unfortunately)-6 b(,)27 b(the)f(term)g(\020dynamical)f(bifurcation\021)34 b(is)27 b(used)f(in)h(a)g(di\033eren)n(t)f(sense)h(in)g(the)f(con)n(text)g(of)h (random)118 5523 y(dynamical)g(systems,)i(namely)e(to)h(describ)r(e)h (a)f(bifurcation)h(of)g(the)e(family)h(of)h(in)n(v)l(arian)n(t)f (measures)g(as)g(opp)r(osed)h(to)f(a)118 5614 y(\020phenomenological)e (bifurcation\021,)h(see)f(for)g(instance)g([Ar].)1867 5871 y FH(2)p eop %%Page: 3 3 3 2 bop 259 328 a FH(That)37 b(metho)s(d)f(has,)i(ho)m(w)m(ev)m(er,)i (t)m(w)m(o)d(ma)5 b(jor)36 b(limitations.)54 b(The)37 b(\034rst)f(one)h(is)e(that)i(the)f(F)-8 b(okk)m(er-)118 441 y(Planc)m(k)38 b(equation)g(is)e(di\036cult)h(to)h(solv)m(e,)h (except)f(in)f(the)h(linear)e(and)j(in)d(the)j(gradien)m(t)f(case.)63 b(The)118 553 y(second)46 b(limitation)41 b(is)j(more)h(serious:)69 b(the)45 b(densit)m(y)g(giv)m(es)f(no)i(information)d(on)j (correlations)e(in)118 666 y(time,)31 b(and)i(ev)m(en)f(when)h(the)g (densit)m(y)e(is)g(strongly)g(lo)s(calized,)f(individual)g(paths)i(can) g(p)s(erform)g(large)118 779 y(excursions.)51 b(This)33 b(is)f(wh)m(y)j(other)g(approac)m(hes)g(are)g(imp)s(ortan)m(t.)51 b(A)33 b(classical)e(one)k(is)d(based)j(on)f(the)118 892 y(computation)c(of)g(\034rst)h(exit)e(times)g(from)g(the)i(neigh)m (b)s(ourho)s(o)s(d)f(of)g(stable)g(equilibria)d([FW,)k(FJ].)259 1005 y(The)41 b(e\033ect)f(of)g(bifurcations)f(has)h(b)s(een)g(studied) g(more)g(recen)m(tly)g(b)m(y)h(metho)s(ds)e(based)h(on)h(the)118 1118 y(concept)27 b(of)d(random)i(attractors)g([CF94)q(,)f(Sc)m(hm)q(,) g(Ar].)39 b(In)24 b(particular,)i(Crauel)g(and)f(Flandoli)f(sho)m(w)m (ed)118 1231 y(that)k(according)f(to)g(their)g(de\034nition,)g(\020A)m (dditiv)m(e)g(noise)f(destro)m(ys)h(a)h(pitc)m(hfork)e(bifurcation\021) 34 b([CF98)q(].)118 1344 y(The)43 b(ph)m(ysical)e(in)m(terpretation)i (of)f(random)g(attractors)i(is,)g(ho)m(w)m(ev)m(er,)j(not)c(straigh)m (tforw)m(ard,)k(and)118 1457 y(alternativ)m(e)37 b(c)m (haracterizations)h(of)g(sto)s(c)m(hastic)f(bifurcations)f(are)i (desirable.)60 b(In)37 b(the)h(same)f(w)m(a)m(y)h(a)118 1570 y(slo)m(wly)24 b(v)-5 b(arying)24 b(parameter)i(helps)e(our)i (understanding)g(of)f(bifurcations)f(in)h(the)g(deterministic)e(case,) 118 1683 y(it)30 b(can)g(pro)m(vide)h(a)f(new)h(p)s(oin)m(t)f(of)g (view)f(in)h(the)g(case)h(of)f(random)g(dynamical)f(systems.)259 1795 y(Let)d(us)e(consider)h(the)g(com)m(bined)g(e\033ect)g(of)g(a)f (slo)m(wly)g(drifting)f(parameter)j(and)f(an)g(additiv)m(e)f(noise)118 1908 y(on)h(the)h(ODE)f(\(1.1\))q(.)39 b(W)-8 b(e)25 b(will)e(fo)s(cus)h(on)h(the)h(case)f(of)g(a)g(pitc)m(hfork)g (bifurcation,)g(where)h(the)f(questions)118 2021 y FB(How)31 b(do)-5 b(es)32 b(the)g(additive)f(noise)g(a\033e)-5 b(ct)33 b(the)e(bifur)-5 b(c)g(ation)32 b(delay?)41 b FH(and)29 b FB(Wher)-5 b(e)33 b(do)-5 b(es)31 b(the)h(p)-5 b(ath)32 b(go)f(after)118 2134 y(cr)-5 b(ossing)38 b(the)h(bifur)-5 b(c)g(ation)38 b(p)-5 b(oint?)58 b FH(are)36 b(of)g(ma)5 b(jor)35 b(ph)m(ysical)g(in)m(terest.)58 b(The)37 b(situation)e(of)h (the)g(drift)118 2247 y(term)g FF(f)45 b FH(in)36 b(\(1.3\))g(dep)s (ending)h(explicitly)32 b(on)37 b(time)e(is)f(considerably)i(more)f (di\036cult)h(to)g(solv)m(e)g(than)118 2360 y(the)h(autonomous)f(case,) i(and)f(th)m(us)g(m)m(uc)m(h)f(less)f(understo)s(o)s(d.)58 b(One)37 b(can)g(exp)s(ect,)g(ho)m(w)m(ev)m(er,)j(that)d(a)118 2473 y(slo)m(w)31 b(time)g(dep)s(endence)i(mak)m(es)e(the)h(problem)f (accessible)g(to)h(p)s(erturbation)g(theory)-8 b(,)33 b(and)f(that)h(one)118 2586 y(ma)m(y)c(tak)m(e)g(adv)-5 b(an)m(tage)31 b(of)d(tec)m(hniques)h(dev)m(elop)s(ed)g(to)g(study)g (singularly)e(p)s(erturb)s(ed)i(equations)g(suc)m(h)118 2699 y(as)h(\(1.2\))q(.)40 b(With)30 b FF(\025)25 b FG(=)g FF("s)p FH(,)30 b(Equation)h(\(1.3\))g(b)s(ecomes)1346 2903 y FG(d)o FF(x)1448 2917 y Fy(s)1510 2903 y FG(=)25 b FF(f)10 b FG(\()p FF(x)1748 2917 y Fy(s)1785 2903 y FF(;)15 b("s)p FG(\))g(d)p FF(s)20 b FG(+)g FF(\033)e FG(d)p FF(W)2372 2917 y Fy(s)2408 2903 y FF(:)1043 b FH(\(1.4\))118 3107 y(If)32 b(w)m(e)i(in)m(tro)s(duce)f(again)g(the)g (slo)m(w)g(time)e FF(t)f FG(=)f FF("s)p FH(,)k(the)h(Bro)m(wnian)f (motion)f(is)g(rescaled,)h(resulting)f(in)118 3220 y(the)f(SDE)1313 3349 y FG(d)p FF(x)1416 3363 y Fy(t)1471 3349 y FG(=)1577 3287 y(1)p 1577 3328 46 4 v 1579 3411 a FF(")1632 3349 y(f)10 b FG(\()p FF(x)1774 3363 y Fy(t)1803 3349 y FF(;)15 b(t)p FG(\))g(d)q FF(t)20 b FG(+)2163 3287 y FF(\033)p 2132 3328 119 4 v 2132 3346 a FE(p)p 2207 3346 43 4 v 2207 3411 a FF(")2275 3349 y FG(d)o FF(W)2411 3363 y Fy(t)2441 3349 y FF(:)1010 b FH(\(1.5\))118 3563 y(Our)35 b(analysis)d(of)40 b(\(1.5\))35 b(is)d(restricted)i(to)h (one-dimensional)d FF(x)p FH(.)51 b(The)35 b(noise)e(in)m(tensit)m(y)h FF(\033)j FH(should)d(b)s(e)118 3676 y(considered)26 b(as)g(a)h(function)e(of)h FF(")p FH(.)40 b(Indeed,)27 b(since)f(w)m(e)h(no)m(w)g(consider)f(the)g(equation)g(on)h(the)f(time) f(scale)118 3789 y FF(")160 3756 y FD(\000)p FC(1)255 3789 y FH(,)j(a)g(constan)m(t)h(noise)f(in)m(tensit)m(y)f(w)m(ould)h (lead)g(to)g(an)g(in\034nite)f(spreading)h(of)g(tra)5 b(jectories)27 b(as)h FF(")e FE(!)f FG(0)p FH(.)118 3902 y(In)30 b(the)h(case)f(of)g(the)h(pitc)m(hfork)f(bifurcation,)g(w)m(e)h (will)d(need)i(to)h(assume)e(that)i FF(\033)e FE(\034)3036 3837 y(p)p 3112 3837 V 65 x FF(")p FH(.)259 4015 y(V)-8 b(arious)24 b(particular)g(cases)f(of)h(equation)g(\(1.5\))h(ha)m(v)m (e)g(b)s(een)f(studied)g(b)s(efore,)h(from)f(a)g(non-rigorous)118 4128 y(p)s(oin)m(t)40 b(of)g(view.)70 b(In)40 b(the)h(linear)e(case)h FF(f)10 b FG(\()p FF(x;)15 b(\025)p FG(\))43 b(=)e FF(\025x)p FH(,)i(the)e(distribution)d(of)i(\034rst)h(exit)e(times)g(w)m(as)118 4241 y(in)m(v)m(estigated)g(and)h(compared)g(with)f(exp)s(erimen)m(ts)f (in)g([TM)q(,)h(SMC,)g(SHA],)j(while)c([JL)q(])h(deriv)m(ed)g(a)118 4354 y(form)m(ula)i(for)g(the)h(last)e(crossing)h(of)g(zero.)74 b(In)41 b(the)h(case)g FF(f)10 b FG(\()p FF(x;)15 b(\025)p FG(\))44 b(=)g FF(\025x)27 b FE(\000)h FF(x)2892 4321 y FC(3)2931 4354 y FH(,)44 b([Ga])e(studied)f(the)118 4467 y(dep)s(endence)j(of)f(the)g(dela)m(y)f(on)h FF(")g FH(and)g FF(\033)j FH(n)m(umerically)-8 b(,)44 b(while)e([Ku])h (considered)g(the)g(asso)s(ciated)118 4580 y(F)-8 b(okk)m(er-Planc)m(k) 32 b(equation,)e(the)h(solution)e(of)h(whic)m(h)g(she)g(appro)m (ximated)h(b)m(y)g(a)f(Gaussian)f(Ansatz.)259 4693 y(In)d(the)h(presen) m(t)g(w)m(ork,)h(w)m(e)f(analyse)e(\(1.5\))i(for)f(a)h(general)f(class) f(of)h(o)s(dd)g(functions)g FF(f)10 b FG(\()p FF(x;)15 b(\025)p FG(\))26 b FH(under-)118 4806 y(going)34 b(a)g(pitc)m(hfork)h (bifurcation.)51 b(W)-8 b(e)35 b(use)f(a)g(di\033eren)m(t)h(approac)m (h,)i(based)d(on)h(a)f(precise)g(con)m(trol)h(of)118 4918 y(the)26 b FB(whole)i(p)-5 b(aths)26 b FE(f)p FF(x)836 4932 y Fy(s)873 4918 y FE(g)918 4932 y Fy(t)943 4941 y Fw(0)979 4932 y Fx(6)p Fy(s)p Fx(6)p Fy(t)1176 4918 y FH(of)f(the)h(pro)s(cess.)38 b(The)26 b(results)f(th)m(us)h(con)m (tain)g(m)m(uc)m(h)g(more)f(information)118 5031 y(than)38 b(the)f(probabilit)m(y)e(densit)m(y)-8 b(.)60 b(It)37 b(also)f(turns)h(out)g(that)g(the)g(tec)m(hnique)h(w)m(e)f(use)g(allo)m (ws)f(to)h(deal)118 5144 y(with)27 b(nonlinearities)f(in)h(quite)g(a)g (natural)h(w)m(a)m(y)-8 b(.)41 b(Our)28 b(results)f(can)h(b)s(e)g (summarized)e(in)h(the)h(follo)m(wing)118 5257 y(w)m(a)m(y)j(\(see)g (Fig.)24 b(2\):)181 5394 y FE(\017)47 b FH(Solutions)28 b(of)h(the)g(deterministic)e(equation)i(\(1.2\))h(starting)e(near)i(a)f (stable)f(equilibrium)e(branc)m(h)273 5507 y(of)31 b FF(f)41 b FH(are)31 b(kno)m(wn)h(to)g(reac)m(h)g(a)g(neigh)m(b)s(ourho) s(o)s(d)f(of)g(order)h FF(")g FH(of)e(that)i(branc)m(h)h(in)d(a)i(time) e(of)h(order)273 5619 y FF(")p FE(j)p FG(log)18 b FF(")p FE(j)p FH(.)73 b(W)-8 b(e)42 b(sho)m(w)g(that)g(the)g(paths)f(of)g(the) h(SDE)g(\(1.5\))g(with)e(the)i(same)f(initial)d(condition)1867 5871 y(3)p eop %%Page: 4 4 4 3 bop 435 236 a 22972849 14917438 7499120 23484088 31838371 39271710 startTexFig doclip 435 236 a %%BeginDocument: fig2.ps %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: paths.dvi %%Pages: 1 %%PageOrder: Ascend %%BoundingBox: 114 357 484 597 %%EndComments %DVIPSCommandLine: dvips paths -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.04:1533 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup 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 /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore 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 /IE 256 array N 2 string 0 1 255{IE S dup 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (paths.dvi) @start %DVIPSBitmapFont: Fa ecrm1095 10.95 1 /Fa 1 50 df49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb ecrm1000 10 27 /Fb 27 123 df66 D68 D<913A01FFC001C0021F13F891B5EAFE03010390 38C07F07903A07FE001FCFD91FF8EB07EFD93FE0EB01FF49487F4948147F4890C8123F48 5A4848151F49150F120F491507121F491503123FA2485A1701A3484892C7FCAB93B6FCA2 6C7E9338007FE0EF3FC0A26C7EA2121F7F120F7F12077F6C7E6C7E6C6D147F6D6C14FF6D 7ED91FF81303D907FEEB0FEF903A03FFE03FC7010090B51203021FEBFC01020101E0C7FC 383B7CB941>71 D78 D97 DI<90380FFF80013F13F09038FF03F8D803FC13FCEA07F0120FEA1FE013C012 3F90388001F8007F90C7FCA290C8FC5AAA6C7EA3003F140E6D131E121F6D133C6C6C137C 6C6C13F83903FC01F03900FF0FE090383FFF8090380FFE001F247DA325>III<14FF0107 13C090381FE3E090383F8FF0EB7F0F137E13FE13FC12019038F807E091C7FCACB512FCA3 D801F8C7FCB3AB487E387FFFF8A31C3A7FB919>IIII107 DI<2703F01FF8EB3F F000FFD97FFEEBFFFC903BF1F87F83F0FF903BF3E01F87C03F3D0FF7800FCF001F802603 FF0013DE4902FC14C0496D48130FA2495CA2495CB3A3486C496CEB1FE0B500C1B50083B5 FCA340247EA345>I<3903F01FF800FFEB7FFE9039F1F87F809038F3E01F3A0FF7800FC0 3803FF004980491307A25BA25BB3A3486C497EB500C1B51280A329247EA32E>II<3903 F03FF039FFF1FFFC9038F7F0FF9039FFC03FC000079038001FE06C48EB0FF049EB07F85B 49EB03FCA2ED01FEA316FF81A85D16FEA3ED03FC7FED07F86DEB0FF06D14E06DEB1FC091 38807F809039F7F0FF009038F1FFFC9038F07FE091C8FCAC487EB512C0A328347EA32E> I<3907E07F8039FFE1FFC09038E3E7E09038E78FF0000F130FEA03EEA213FCEC07E09038 F803C091C7FCA25BB3A2487EB512F0A31C247EA321>114 D<3801FFC7000F13FFEA1F81 383E007F48131F1278487FA2807E7E6C90C7FC6C7E13F8387FFFC06C13F06C13FC6C7F00 037FC66C138013039038007FC000E0131F140F6C1307A214037EA26CEB0780A26CEB0F00 6C6C5AEBE0FE38F3FFF800E013E01A247DA321>I<1338A51378A413F8A21201A2120312 07001FB5FCB6FCA2D801F8C7FCB2EC01C0A8EBFC03000014801407137E90387F9F00EB1F FEEB07F81A337FB220>IIII121 D<003FB512FCA2EBC003 90380007F8003C14F0140F0038EB1FE0007814C0143F0070EB7F80ECFF00A2495A495A00 005B1307495A495AA2495A495AEC000E5B485A485AA24848131E485A49131C121F484813 3C49137C007F14FC38FF000790B5FCA21F247EA325>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc eccc1000 10 8 /Fc 8 118 df<123E127FEAFF80A5EA7F00123E090977881B>46 D50 D70 D101 D<91387FF007903903FFFC0F010F13FF90393FF83FDF90397FC007 FF4848C67E48481300D807F880120F49804848805B003F81A2485A82A248C8FC93C7FCA8 92387FFFF8A26C7E03001380EE7F006C7EA26C7EA26C7E7F6C7E6C6C5C6C6C5B39007FC0 0390393FF81FDF010FB5128F010314079026007FF8C7FC2D2D7BAB35>103 D105 D114 D117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmmi7 7 4 /Fd 4 117 df<48B512FC000714FE5A4814FC393F038000123CEA780712F0120049C7FC A35BA2131EA2133EA35BA41378A21F197E981F>28 D<127C12FEA212FFA3127F1203A212 07A21206120E120C121C12381270126008127A8614>59 D<1303A4497EA600E0141C39FF 8FC7FC90B5FC003F14F0000F14C00001EBFE0038007FF86D5AA2497EA2EBFCFCEBF87C48 487EEBE01E0003131F3907C00F80EB8007EB00031E1D7E9C22>63 D<133CA2137CA413FC5BA31201B512E0A33803F0005BA312075BA3120F5BA3121FEB00E0 130114C0EA3F03383E0780381F0F00131EEA0FFCEA07F013247EA319>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmsy7 7 1 /Fe 1 69 df<0103B512E0013F14FE90B71280000316E0270FF8FC0713F0D81FC0903800 7FF8D83F00EC1FFC003E150748ED03FE00FC1501EE00FF00F05BD80001157FA2173FA25C 1303173E177EA24A147C010715FC17F84A1301010F15F016034AEB07E0011FEC0FC0EE1F 8091C7EA7F004914FEED03F8013EEB0FF0017EEB3FC090267C07FFC7FC48B512FC4814F0 48148002F0C8FC30287EA734>68 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmr10 10 4 /Ff 4 127 df<14E01301EB03C01307EB0F80EB1F00133E133C137C5B5B1201485AA248 5AA3485AA248C7FCA35A123EA3127E127CA512FCA35AAC7EA3127CA5127E123EA3123F7E A36C7EA26C7EA36C7EA26C7E12007F137C133C133E7FEB0F80EB07C01303EB01E0130013 5278BD20>40 D<12E07E1278127C7E7E6C7E12077F6C7E12017F6C7EA2137CA37FA27FA3 1480130FA314C01307A514E0A31303AC1307A314C0A5130F1480A3131F1400A3133EA25B A35BA2485A5B1203485A5B120F48C7FC123E5A12785A5A13527CBD20>I61 D126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmsy10 10 4 /Fg 4 113 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr7 7 4 /Fh 4 117 df48 D100 DI116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi10 10 5 /Fi 5 121 df<017FB6FC48B7FC5A5A4815FE261FC01CC7FCEB003C123E5A0078137C48 1378A2C7FC14F8A25C1301A313035CA31307A25C130FA3131F5CA591C8FC28247EA324> 28 D34 D104 D116 D<90391FE00FF090397FF83FFC3A01F8FC7C7E3903E07EF83A07C03FF0FE9038801FE1D8 0F0013C1121E158148133FED80F8003891C7FCA2C75AA2147EA214FEA25CA21301A24A13 38A2D83E03147812FF4A13F01307ED01E0010FEB03C039FE1FF8073AFC3EFC0F8090397C 7E3F00397FF83FFC391FE00FF027247DA32F>120 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 354 1447 a 24240837 15756538 2302361 3486433 43810652 30785863 startTexFig doclip 354 1447 a %%BeginDocument: paths.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: paths.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta2 %%CreationDate: Thu Aug 3 19:30:48 2000 %%For: berglund@hilbert.wias-berlin.de (Nils Berglund,0518,,) %%Orientation: Portrait %%BoundingBox: 35 53 666 468 %%Pages: 0 %%BeginSetup %%IncludeFeature: *PageSize Letter %%EndSetup %%Magnification: 1.00 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -35.0 577.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n 0 9641 m 0 0 l 16841 0 l 16841 9641 l cp clip 0.06000 0.06000 sc % Polyline n 600 9600 m 16800 9600 l 16800 1200 l 600 1200 l cp gs col7 1.00 shd ef gr % Polyline n 1800 4199 m 2085 4439 l 2333 4634 l 2603 4837 l 2805 4942 l 3120 5024 l 3547 5039 l 3990 5032 l 4455 4979 l 5018 4889 l 5498 4807 l 6045 4709 l 6637 4597 l 7207 4484 l 7192 6292 l 6360 6149 l 5730 6044 l 5010 5924 l 4193 5789 l 3705 5714 l 3352 5677 l 2948 5639 l 2700 5579 l 2460 5459 l 2265 5287 l 2055 5077 l 1800 4799 l cp gs col12 0.40 tnt ef gr % Polyline n 7200 6593 m 7200 4208 l 7673 3968 l 8318 3728 l 9083 3533 l 9705 3413 l 10298 3353 l 10800 3315 l 10800 7485 l 10073 7433 l 9428 7335 l 8723 7193 l 8100 7005 l 7545 6773 l cp gs col4 0.60 tnt ef gr % Polyline 15.000 slw n 8100 4425 m 8100 3150 l gs col0 s gr % Polyline 0.000 slw n 8100 3150 m 8370 2920 l 8708 2685 l 9023 2490 l 9398 2325 l 9765 2213 l 10170 2108 l 10500 2063 l 10793 2025 l 10793 2378 l 10523 2415 l 10208 2483 l 9945 2550 l 9548 2693 l 9233 2843 l 8948 3023 l 8715 3240 l 8565 3420 l 8423 3675 l 8288 3990 l 8100 4380 l cp gs col1 0.40 tnt ef gr % Polyline 30.000 slw n 8100 3795 m 8101 3793 l 8103 3789 l 8106 3782 l 8111 3770 l 8119 3754 l 8128 3734 l 8140 3709 l 8153 3680 l 8168 3649 l 8185 3614 l 8202 3579 l 8220 3542 l 8238 3505 l 8256 3470 l 8275 3435 l 8293 3402 l 8310 3370 l 8327 3341 l 8345 3313 l 8362 3288 l 8379 3263 l 8397 3240 l 8415 3218 l 8433 3198 l 8451 3178 l 8470 3158 l 8490 3139 l 8510 3119 l 8530 3100 l 8551 3081 l 8572 3062 l 8593 3043 l 8615 3024 l 8637 3005 l 8659 2986 l 8681 2968 l 8704 2949 l 8726 2931 l 8749 2912 l 8772 2894 l 8796 2876 l 8820 2858 l 8844 2840 l 8868 2823 l 8893 2805 l 8919 2788 l 8945 2771 l 8972 2754 l 9000 2738 l 9025 2724 l 9050 2710 l 9076 2697 l 9102 2684 l 9129 2671 l 9157 2658 l 9184 2645 l 9213 2632 l 9241 2619 l 9270 2606 l 9299 2593 l 9329 2581 l 9358 2568 l 9388 2556 l 9418 2543 l 9448 2531 l 9478 2519 l 9508 2507 l 9538 2495 l 9569 2483 l 9599 2471 l 9630 2460 l 9660 2448 l 9691 2437 l 9722 2427 l 9752 2416 l 9783 2406 l 9815 2396 l 9846 2387 l 9878 2378 l 9908 2370 l 9939 2362 l 9971 2355 l 10004 2347 l 10039 2340 l 10075 2333 l 10113 2326 l 10153 2319 l 10195 2312 l 10239 2305 l 10285 2298 l 10332 2291 l 10380 2284 l 10429 2276 l 10479 2269 l 10528 2263 l 10576 2256 l 10624 2250 l 10669 2243 l 10711 2238 l 10750 2233 l 10786 2228 l 10817 2224 l 10843 2220 l 10866 2218 l 10883 2215 l 10897 2214 l 10906 2212 l 10912 2212 l 10916 2211 l 10917 2211 l gs col0 s gr % Polyline 15.000 slw n 1800 4799 m 1801 4800 l 1803 4803 l 1807 4807 l 1814 4815 l 1823 4825 l 1836 4839 l 1851 4856 l 1870 4877 l 1892 4901 l 1916 4928 l 1944 4958 l 1974 4991 l 2006 5025 l 2040 5061 l 2075 5098 l 2111 5135 l 2147 5172 l 2184 5209 l 2220 5244 l 2256 5279 l 2292 5312 l 2327 5343 l 2361 5373 l 2395 5400 l 2428 5426 l 2461 5450 l 2493 5471 l 2525 5491 l 2558 5510 l 2591 5526 l 2625 5542 l 2656 5555 l 2688 5567 l 2720 5578 l 2753 5589 l 2787 5598 l 2821 5607 l 2855 5616 l 2890 5623 l 2925 5630 l 2961 5636 l 2997 5642 l 3033 5647 l 3070 5651 l 3107 5656 l 3144 5659 l 3181 5663 l 3218 5667 l 3256 5670 l 3294 5673 l 3332 5676 l 3370 5680 l 3409 5683 l 3448 5686 l 3487 5690 l 3526 5694 l 3566 5698 l 3607 5703 l 3648 5708 l 3689 5713 l 3731 5718 l 3773 5724 l 3817 5731 l 3860 5737 l 3905 5744 l 3951 5752 l 3997 5759 l 4036 5765 l 4075 5772 l 4115 5778 l 4156 5785 l 4198 5792 l 4240 5799 l 4284 5806 l 4328 5813 l 4373 5820 l 4418 5828 l 4464 5835 l 4511 5843 l 4559 5851 l 4607 5859 l 4656 5867 l 4705 5875 l 4755 5883 l 4805 5891 l 4855 5900 l 4906 5908 l 4956 5916 l 5007 5925 l 5058 5933 l 5108 5941 l 5158 5950 l 5209 5958 l 5258 5966 l 5308 5974 l 5357 5982 l 5405 5990 l 5453 5998 l 5500 6006 l 5546 6014 l 5592 6021 l 5637 6029 l 5681 6036 l 5724 6043 l 5766 6050 l 5807 6057 l 5848 6064 l 5887 6070 l 5926 6077 l 5963 6083 l 6000 6089 l 6046 6097 l 6092 6104 l 6136 6112 l 6181 6119 l 6225 6127 l 6269 6134 l 6314 6142 l 6360 6150 l 6406 6157 l 6453 6165 l 6501 6174 l 6549 6182 l 6598 6190 l 6648 6199 l 6697 6207 l 6747 6216 l 6795 6224 l 6843 6232 l 6889 6240 l 6933 6248 l 6974 6255 l 7013 6261 l 7048 6267 l 7079 6273 l 7106 6277 l 7130 6281 l 7149 6285 l 7164 6287 l 7175 6289 l 7183 6290 l 7188 6291 l 7191 6292 l 7192 6292 l gs col0 s gr % Polyline n 7200 6600 m 7200 4200 l gs col0 s gr % Polyline n 1800 4800 m 1800 4200 l gs col0 s gr % Polyline n 8093 3150 m 8095 3149 l 8098 3146 l 8104 3141 l 8113 3133 l 8126 3123 l 8143 3109 l 8163 3093 l 8187 3074 l 8213 3053 l 8241 3030 l 8271 3006 l 8301 2981 l 8332 2957 l 8362 2933 l 8392 2909 l 8421 2887 l 8448 2866 l 8475 2846 l 8500 2827 l 8524 2809 l 8548 2792 l 8572 2776 l 8595 2760 l 8617 2745 l 8639 2731 l 8662 2716 l 8685 2702 l 8708 2687 l 8731 2673 l 8754 2658 l 8777 2644 l 8801 2629 l 8824 2614 l 8848 2599 l 8872 2585 l 8896 2570 l 8920 2555 l 8944 2541 l 8968 2526 l 8993 2512 l 9017 2498 l 9042 2484 l 9067 2470 l 9093 2456 l 9118 2443 l 9144 2430 l 9171 2417 l 9198 2405 l 9225 2393 l 9251 2382 l 9277 2372 l 9304 2361 l 9331 2351 l 9358 2341 l 9386 2331 l 9415 2322 l 9443 2312 l 9472 2303 l 9501 2293 l 9531 2284 l 9560 2275 l 9590 2266 l 9620 2257 l 9650 2248 l 9680 2239 l 9709 2231 l 9739 2222 l 9769 2213 l 9798 2205 l 9828 2197 l 9857 2189 l 9886 2181 l 9914 2173 l 9943 2166 l 9972 2159 l 10000 2152 l 10028 2145 l 10058 2138 l 10089 2132 l 10121 2126 l 10153 2120 l 10188 2114 l 10224 2108 l 10262 2102 l 10302 2096 l 10343 2090 l 10386 2085 l 10430 2079 l 10474 2073 l 10518 2067 l 10562 2062 l 10603 2057 l 10642 2052 l 10678 2048 l 10709 2044 l 10736 2041 l 10757 2038 l 10774 2036 l 10786 2035 l 10794 2034 l 10798 2033 l 10800 2033 l gs col0 s gr % Polyline n 8100 4388 m 8101 4387 l 8102 4383 l 8105 4377 l 8109 4367 l 8115 4354 l 8123 4336 l 8133 4313 l 8145 4286 l 8160 4254 l 8176 4218 l 8194 4179 l 8213 4136 l 8233 4091 l 8255 4043 l 8277 3994 l 8300 3945 l 8323 3896 l 8346 3847 l 8368 3799 l 8390 3753 l 8412 3709 l 8433 3666 l 8453 3626 l 8473 3588 l 8492 3553 l 8510 3519 l 8527 3489 l 8545 3460 l 8562 3433 l 8578 3407 l 8595 3383 l 8616 3355 l 8637 3328 l 8658 3302 l 8679 3278 l 8699 3255 l 8720 3233 l 8741 3212 l 8762 3192 l 8782 3173 l 8803 3155 l 8823 3136 l 8844 3119 l 8864 3102 l 8885 3085 l 8906 3068 l 8927 3051 l 8949 3034 l 8971 3017 l 8994 3000 l 9017 2983 l 9041 2966 l 9066 2949 l 9091 2931 l 9118 2914 l 9145 2897 l 9173 2880 l 9198 2866 l 9224 2852 l 9251 2838 l 9279 2824 l 9307 2810 l 9336 2797 l 9366 2784 l 9397 2770 l 9428 2757 l 9460 2744 l 9492 2731 l 9525 2718 l 9557 2705 l 9591 2692 l 9624 2679 l 9657 2667 l 9690 2654 l 9723 2642 l 9756 2629 l 9789 2617 l 9821 2605 l 9853 2594 l 9884 2582 l 9915 2571 l 9945 2561 l 9975 2550 l 10004 2540 l 10033 2531 l 10061 2522 l 10088 2513 l 10119 2504 l 10150 2495 l 10182 2487 l 10214 2479 l 10247 2471 l 10282 2464 l 10317 2457 l 10355 2450 l 10393 2443 l 10433 2436 l 10473 2429 l 10513 2422 l 10554 2416 l 10593 2410 l 10631 2404 l 10666 2399 l 10698 2394 l 10726 2390 l 10750 2386 l 10770 2383 l 10785 2381 l 10796 2380 l 10802 2379 l 10806 2378 l 10808 2378 l gs col0 s gr % Polyline n 1808 4192 m 1809 4193 l 1812 4195 l 1816 4199 l 1824 4206 l 1834 4215 l 1848 4227 l 1866 4243 l 1887 4261 l 1913 4283 l 1941 4308 l 1974 4336 l 2009 4366 l 2047 4398 l 2087 4432 l 2129 4468 l 2172 4504 l 2216 4540 l 2260 4576 l 2304 4612 l 2348 4647 l 2391 4681 l 2433 4713 l 2474 4744 l 2514 4773 l 2553 4800 l 2590 4825 l 2626 4848 l 2662 4869 l 2696 4889 l 2729 4906 l 2762 4922 l 2795 4936 l 2828 4949 l 2861 4961 l 2894 4971 l 2928 4981 l 2962 4989 l 2995 4997 l 3029 5003 l 3062 5009 l 3096 5014 l 3129 5019 l 3163 5023 l 3196 5026 l 3229 5029 l 3262 5031 l 3295 5033 l 3328 5034 l 3361 5035 l 3395 5036 l 3428 5037 l 3462 5038 l 3496 5038 l 3531 5038 l 3566 5038 l 3602 5038 l 3638 5038 l 3675 5038 l 3712 5037 l 3751 5037 l 3790 5036 l 3831 5035 l 3872 5033 l 3915 5032 l 3959 5030 l 4004 5027 l 4050 5024 l 4087 5021 l 4124 5018 l 4162 5014 l 4201 5010 l 4242 5006 l 4282 5001 l 4324 4996 l 4367 4990 l 4411 4984 l 4455 4978 l 4500 4972 l 4546 4965 l 4593 4958 l 4641 4950 l 4688 4943 l 4737 4935 l 4786 4927 l 4835 4919 l 4885 4910 l 4935 4902 l 4985 4894 l 5035 4885 l 5085 4876 l 5135 4868 l 5185 4859 l 5235 4850 l 5284 4842 l 5333 4833 l 5381 4825 l 5429 4817 l 5476 4809 l 5523 4801 l 5569 4793 l 5614 4785 l 5658 4778 l 5701 4770 l 5744 4763 l 5785 4756 l 5826 4749 l 5865 4742 l 5904 4736 l 5942 4729 l 5979 4723 l 6015 4717 l 6060 4709 l 6105 4701 l 6149 4693 l 6192 4686 l 6236 4678 l 6279 4669 l 6323 4661 l 6368 4652 l 6413 4643 l 6459 4634 l 6506 4625 l 6554 4615 l 6602 4606 l 6651 4596 l 6700 4586 l 6748 4576 l 6796 4566 l 6842 4556 l 6888 4547 l 6931 4538 l 6971 4529 l 7009 4521 l 7043 4514 l 7074 4507 l 7101 4502 l 7124 4497 l 7142 4493 l 7157 4490 l 7168 4488 l 7176 4486 l 7181 4485 l 7184 4484 l 7185 4484 l gs col0 s gr % Polyline 30.000 slw n 1800 4500 m 1801 4501 l 1804 4504 l 1808 4508 l 1815 4516 l 1825 4526 l 1839 4540 l 1856 4557 l 1876 4577 l 1900 4601 l 1927 4628 l 1957 4658 l 1990 4690 l 2025 4724 l 2062 4760 l 2100 4796 l 2138 4833 l 2177 4870 l 2217 4906 l 2256 4941 l 2294 4975 l 2332 5007 l 2368 5038 l 2404 5067 l 2439 5093 l 2473 5118 l 2507 5141 l 2540 5161 l 2572 5180 l 2605 5197 l 2637 5213 l 2670 5227 l 2702 5239 l 2734 5250 l 2767 5261 l 2800 5270 l 2834 5278 l 2868 5286 l 2903 5292 l 2938 5298 l 2974 5304 l 3010 5308 l 3046 5312 l 3082 5316 l 3119 5319 l 3155 5322 l 3192 5324 l 3230 5326 l 3267 5328 l 3304 5330 l 3342 5331 l 3380 5333 l 3418 5334 l 3456 5336 l 3494 5338 l 3532 5339 l 3571 5341 l 3610 5343 l 3649 5345 l 3688 5348 l 3727 5350 l 3767 5352 l 3807 5355 l 3848 5357 l 3889 5360 l 3930 5362 l 3968 5364 l 4005 5366 l 4044 5367 l 4083 5369 l 4122 5370 l 4161 5371 l 4201 5372 l 4241 5373 l 4282 5373 l 4322 5374 l 4363 5375 l 4404 5375 l 4446 5375 l 4487 5376 l 4529 5376 l 4571 5376 l 4613 5376 l 4655 5376 l 4697 5376 l 4740 5376 l 4782 5376 l 4825 5376 l 4867 5376 l 4909 5375 l 4952 5375 l 4994 5375 l 5037 5375 l 5079 5375 l 5122 5375 l 5164 5376 l 5207 5376 l 5249 5376 l 5292 5376 l 5334 5376 l 5377 5376 l 5419 5377 l 5462 5377 l 5505 5377 l 5546 5377 l 5587 5377 l 5629 5378 l 5672 5378 l 5717 5378 l 5762 5378 l 5810 5378 l 5859 5378 l 5910 5378 l 5962 5378 l 6017 5378 l 6073 5378 l 6131 5378 l 6191 5378 l 6252 5378 l 6314 5378 l 6378 5378 l 6441 5378 l 6505 5378 l 6568 5378 l 6631 5378 l 6692 5378 l 6752 5378 l 6809 5378 l 6863 5378 l 6913 5378 l 6960 5378 l 7003 5378 l 7041 5378 l 7075 5378 l 7104 5378 l 7129 5378 l 7149 5378 l 7164 5378 l 7176 5378 l 7184 5378 l 7189 5378 l 7192 5378 l 7193 5378 l gs col0 s gr 15.000 slw % Ellipse n 11400 5400 5100 2100 0 360 DrawEllipse gs col0 s gr 30.000 slw % Ellipse n 11400 5400 5100 3300 0 360 DrawEllipse gs col0 s gr % Polyline 0.000 slw n 10875 9000 m 16800 9000 l 16800 1800 l 10875 1800 l cp gs col7 1.00 shd ef gr % Polyline 1 slj 7.500 slw n 1800 4500 m 1845 4445 l 1875 4350 l 1905 4750 l 1945 4520 l 1950 4800 l 2025 4500 l 2060 4910 l 2100 4650 l 2135 4940 l 2175 4575 l 2185 4805 l 2220 4705 l 2250 5175 l 2325 4950 l 2400 5250 l 2440 4885 l 2475 4990 l 2510 4845 l 2530 5275 l 2585 4920 l 2625 5135 l 2670 4970 l 2700 5325 l 2775 5025 l 2800 5175 l 2820 5080 l 2850 5475 l 2925 5100 l 2950 5195 l 2985 5115 l 3000 5550 l 3035 5465 l 3055 5605 l 3075 5475 l 3115 5355 l 3150 5625 l 3165 5160 l 3200 5260 l 3225 5175 l 3300 5475 l 3375 5100 l 3415 5255 l 3450 5175 l 3490 5240 l 3525 5100 l 3590 5610 l 3620 5250 l 3675 5550 l 3750 5175 l 3810 5525 l 3855 5260 l 3900 5700 l 3950 5480 l 3975 5550 l 4005 5270 l 4050 5550 l 4095 5150 l 4125 5250 l 4175 5155 l 4200 5475 l 4255 5240 l 4275 5625 l 4310 5470 l 4350 5550 l 4375 5120 l 4425 5175 l 4450 5070 l 4455 5470 l 4495 5180 l 4500 5775 l 4545 5445 l 4575 5550 l 4605 5440 l 4650 5775 l 4675 5520 l 4725 5700 l 4750 5615 l 4770 5765 l 4800 5250 l 4850 5490 l 4875 5325 l 4915 5470 l 4950 5025 l 4980 5200 l 5005 5050 l 5025 5850 l 5060 5290 l 5100 5475 l 5155 5735 l 5175 5625 l 5200 5755 l 5230 5090 l 5265 5330 l 5285 5045 l 5320 5210 l 5345 5005 l 5360 5555 l 5400 5475 l 5435 5620 l 5475 4875 l 5515 5160 l 5550 5100 l 5585 5290 l 5625 5100 l 5670 5450 l 5710 5125 l 5775 5625 l 5805 5545 l 5835 5675 l 5850 5175 l 5890 5330 l 5925 5175 l 6000 5700 l 6035 5040 l 6075 5175 l 6105 5025 l 6150 5325 l gs col0 s gr % Polyline n 8250 3225 m 8325 3600 l 8360 3370 l 8400 3450 l 8435 3340 l 8475 3525 l 8505 2910 l 8550 3000 l 8575 2870 l 8625 3150 l 8650 3065 l 8675 3165 l 8700 2700 l 8775 3075 l 8805 2925 l 8835 2965 l 8850 2625 l 8890 2765 l 8925 2700 l 8980 2800 l 9000 2550 l 9075 2850 l 9100 2720 l 9135 2795 l 9150 2550 l 9200 2600 l 9240 2475 l 9300 2775 l 9330 2650 l 9345 2730 l 9375 2400 l 9425 2605 l 9450 2550 l 9495 2610 l 9525 2400 l 9600 2625 l 9630 2405 l 9675 2550 l 9710 2485 l 9750 2550 l 9765 2340 l 9810 2450 l 9845 2230 l 9900 2325 l 9975 2175 l 10050 2475 l 10080 2265 l 10125 2400 l 10175 2350 l 10200 2475 l 10285 2130 l 10350 2325 l 10410 2220 l 10425 2400 l 10500 2175 l 10575 2400 l 10605 2190 l 10645 2300 l 10650 2100 l 10725 2325 l 10800 2100 l gs col0 s gr % Polyline n 6150 5325 m 6180 5055 l 6215 5485 l 6225 4875 l 6260 5230 l 6300 5100 l 6375 4875 l 6380 5220 l 6410 5075 l 6450 5625 l 6480 5210 l 6525 5475 l 6560 5250 l 6600 5925 l 6645 5590 l 6675 5700 l 6700 5560 l 6750 6075 l 6765 5605 l 6810 5880 l 6825 5025 l 6870 5325 l 6900 5025 l 6935 5195 l 6955 4670 l 6990 4855 l 7010 4590 l 7075 5230 l 7125 5010 l 7160 5480 l 7200 4950 l 7235 4735 l 7290 5575 l 7320 5275 l 7350 5850 l 7385 5510 l 7425 5625 l 7450 5510 l 7500 6525 l 7525 6215 l 7550 6535 l 7575 5475 l 7620 5880 l 7650 5775 l 7670 5925 l 7687 4878 l 7715 5220 l 7742 4533 l 7780 5310 l 7790 5010 l 7875 6225 l 7890 5685 l 7930 5960 l 7940 4880 l 7975 5340 l 7995 4190 l 8022 4573 l 8032 3913 l 8060 4180 l 8095 3815 l 8122 3533 l 8160 3865 l 8192 3793 l 8215 4035 l 8250 3225 l gs col0 s gr % Polyline 0 slj 15.000 slw gs clippath 6240 2085 m 6300 1844 l 6360 2085 l 6360 1770 l 6240 1770 l cp clip n 6300 8700 m 6300 1800 l gs col0 s gr gr % arrowhead n 6240 2085 m 6300 1844 l 6360 2085 l 6300 2045 l 6240 2085 l cp gs col7 1.00 shd ef gr col0 s % Polyline gs clippath 11406 5350 m 11646 5400 l 11406 5450 l 11730 5450 l 11730 5350 l cp clip n 1200 5400 m 11700 5400 l gs col0 s gr gr % arrowhead n 11406 5350 m 11646 5400 l 11406 5450 l 11446 5400 l 11406 5350 l cp gs col7 1.00 shd ef gr col0 s % Ellipse n 8100 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 1800 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 7200 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 6300 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr $F2psEnd rs %%EndDocument endTexFig 503 2564 a Fi(t)533 2576 y Fh(0)1991 2504 y Fg(p)p 2060 2504 39 4 v 60 x Fi(")258 b(\034)33 b Ff(=)22 b Fi(\034)2549 2576 y Fe(D)3290 2564 y Fi(t)1329 2293 y Fg(B)s Ff(\()p Fi(h)p Ff(\))2983 2021 y Fg(D)2652 1525 y(A)2718 1495 y Fd(\034)2760 1525 y Ff(\()p Fi(h)p Ff(\))1991 1856 y Fi(x)2038 1825 y Fd(?)2076 1856 y Ff(\()p Fi(t)p Ff(\))1778 1596 y Fi(x)1370 b(x)3242 1556 y Fh(det)p Fd(;\034)3242 1616 y(t)3213 1903 y Ff(~)-48 b Fi(x)q Ff(\()p Fi(t)p Ff(\))2369 2375 y Fi(x)2416 2387 y Fd(t)268 4295 y Fc(Figure)32 b(2.)40 b Fb(Nils)28 b(Berglund)f(and)g(Barbara)e(Gen)n(tz)268 4395 y(Dynamic)i(pitc)n(hfork)g(bifurcations)g(with)h(additiv)n(e)f (noise)1867 5871 y Fa(1)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF %%EndDocument endTexFig 268 2280 a FA(Figure)32 b(2.)41 b FQ(A)28 b(t)n(ypical)f(path)h FP(x)1301 2292 y FM(t)1358 2280 y FQ(of)g(the)g(sto)r(c)n(hastic)f (di\033eren)n(tial)g(equation)g(\(1.5\))h(near)e(a)i(pitc)n(hfork)268 2379 y(bifurcation.)61 b(W)-7 b(e)36 b(pro)n(v)n(e)e(that)i(with)g (probabilit)n(y)f(exp)r(onen)n(tially)g(close)g(to)h FL(1)p FQ(,)h(the)g(path)e(has)h(the)268 2479 y(follo)n(wing)d(b)r(eha) n(viour.)58 b(F)-7 b(or)35 b FP(t)1252 2491 y FN(0)1325 2479 y Ft(6)g FP(t)g Ft(6)1590 2419 y FO(p)p 1660 2419 39 4 v 1660 2479 a FP(")o FQ(,)i(it)f(sta)n(ys)e(in)h(a)g(strip)g FO(B)s FL(\()p FP(h)p FL(\))f FQ(constructed)h(around)f(the)268 2579 y(deterministic)28 b(solution)f(with)i(the)f(same)f(initial)i (condition.)38 b(After)28 b FP(t)c FL(=)2620 2519 y FO(p)p 2689 2519 V 60 x FP(")p FQ(,)k(it)h(lea)n(v)n(es)d(the)j(domain)268 2678 y FO(D)39 b FQ(at)e(a)f(random)g(time)h FP(\034)48 b FL(=)38 b FP(\034)1295 2690 y Fs(D)1353 2678 y FQ(,)i(whic)n(h)c(is)h (t)n(ypically)f(of)h(the)g(order)2586 2607 y FK(p)p 2669 2607 257 4 v 71 x FP(")p FO(j)p FL(log)14 b FP(\033)s FO(j)p FQ(.)65 b(Then)37 b(it)g(sta)n(ys)268 2778 y(\(up)e(to)g(times)g (of)f(order)g FL(1)g FQ(at)h(least\))g(in)g(a)f(strip)h FO(A)1958 2748 y FM(\034)1999 2778 y FL(\()p FP(h)p FL(\))h FQ(constructed)e(around)g(the)h(deterministic)268 2877 y(solution)30 b FP(x)632 2838 y FN(det)p FM(;\034)632 2898 y(t)817 2877 y FQ(starting)g(at)h(time)h FP(\034)41 b FQ(on)31 b(the)g(b)r(oundary)f(of)i FO(D)r FQ(.)47 b(The)32 b(widths)f(of)g FO(B)s FL(\()p FP(h)p FL(\))g FQ(and)g FO(A)3358 2847 y FM(\034)3400 2877 y FL(\()p FP(h)p FL(\))268 2977 y FQ(are)26 b(prop)r(ortional)g(to)h(a)g (parameter)f FP(h)i FQ(satisfying)f FP(\033)f FO(\034)d FP(h)g FO(\034)2253 2917 y(p)p 2323 2917 39 4 v 2323 2977 a FP(")o FQ(.)273 3254 y FH(are)31 b(t)m(ypically)d(concen)m (trated)k(in)d(a)i(neigh)m(b)s(ourho)s(o)s(d)f(of)g(order)h FF(\033)i FH(of)c(the)i(deterministic)d(solution)273 3367 y(\(Theorem)j(2.3\).)181 3480 y FE(\017)47 b FH(A)32 b(particular)f(solution)g(of)g(the)h(deterministic)e(equation)h (\(1.2\))i(is)d(kno)m(wn)j(to)f(exist)f(in)g(a)g(neigh-)273 3593 y(b)s(ourho)s(o)s(d)k(of)g(order)h FF(")f FH(of)f(eac)m(h)j (unstable)d(equilibrium)e(branc)m(h)k(of)f FF(f)10 b FH(.)53 b(P)m(aths)36 b(that)g(start)f(in)f(a)273 3706 y(neigh)m(b)s(ourho)s(o)s(d)g(of)e(order)i FF(\033)i FH(of)d(this)f(solution)f(are)j(lik)m(ely)c(to)j(lea)m(v)m(e)g(that)h (neigh)m(b)s(ourho)s(o)s(d)f(in)f(a)273 3819 y(time)d(of)h(order)h FF(")p FE(j)p FG(log)17 b FF(")p FE(j)31 b FH(\(Theorem)g(2.5\).)181 3932 y FE(\017)47 b FH(When)25 b(a)g(pitc)m(hfork)f(bifurcation)g(o)s (ccurs)g(at)h FF(x)g FG(=)g(0)p FH(,)h FF(t)f FG(=)g(0)p FH(,)h(the)f(t)m(ypical)f(paths)h(are)g(concen)m(trated)273 4045 y(in)30 b(a)h(neigh)m(b)s(ourho)s(o)s(d)g(of)f(order)i FF(\033)s(=")1555 4012 y FC(1)p Fy(=)p FC(4)1696 4045 y FH(of)e(the)i(deterministic)c(solution)h(with)h(the)i(same)d(initial) 273 4158 y(condition)h(up)h(to)f(time)1117 4092 y FE(p)p 1193 4092 43 4 v 66 x FF(")g FH(\(Theorem)h(2.8\).)181 4270 y FE(\017)47 b FH(After)31 b(the)g(bifurcation)g(p)s(oin)m(t,)g (the)g(paths)g(are)h(lik)m(ely)c(to)j(lea)m(v)m(e)g(a)g(neigh)m(b)s (ourho)s(o)s(d)g(of)g(order)3553 4197 y FE(p)p 3629 4197 33 4 v 73 x FF(t)273 4383 y FH(of)f(the)h(unstable)f(equilibrium)d(b)s (efore)j(a)g(time)f FF(c)1965 4306 y Fr(p)p 2056 4306 281 4 v 77 x FF(")p FE(j)p FG(log)18 b FF(\033)s FE(j)30 b FH(\(Theorem)h(2.9\).)181 4496 y FE(\017)47 b FH(Once)35 b(they)e(ha)m(v)m(e)i(left)e(this)g(neigh)m(b)s(ourho)s(o)s(d,)i(the)f (paths)g(remain)e(with)i(high)f(probabilit)m(y)f(in)h(a)273 4609 y(region)26 b(of)g(size)f FF(\033)s(=)906 4535 y FE(p)p 982 4535 33 4 v 74 x FF(t)h FH(around)h(the)f(corresp)s(onding)g (deterministic)e(solution,)h(whic)m(h)i(approac)m(hes)273 4722 y(a)k(stable)e(equilibrium)e(branc)m(h)32 b(of)e FF(f)39 b FH(lik)m(e)29 b FF("=t)1862 4689 y FC(3)p Fy(=)p FC(2)2003 4722 y FH(\(Theorem)i(2.10\).)118 4859 y(These)41 b(results)f(sho)m(w)i(that)f(the)g(bifurcation)f(dela)m(y)-8 b(,)44 b(whic)m(h)d(is)f(observ)m(ed)h(in)f(the)h(dynamical)e(sys-)118 4972 y(tem)c(\(1.2\))q(,)h(is)e(destro)m(y)m(ed)i(b)m(y)g(additiv)m(e)e (noise)g(as)h(so)s(on)g(as)g(the)g(noise)f(is)g(not)i(exp)s(onen)m (tially)d(small.)118 5084 y(Do)g(they)h(mean)f(that)g(the)h(dynamic)e (bifurcation)g(itself)f(is)h(destro)m(y)m(ed)j(b)m(y)e(additiv)m(e)f (noise?)48 b(This)33 b(is)118 5197 y(mainly)e(a)h(matter)h(of)g (de\034nition.)48 b(On)33 b(one)g(hand,)i(w)m(e)e(will)e(see)h(that)i (indep)s(enden)m(tly)f(of)f(the)h(initial)118 5310 y(condition,)k(the)g (probabilit)m(y)e(of)h(reac)m(hing)g(the)h(upp)s(er,)h(rather)f(than)g (the)f(lo)m(w)m(er)h(branc)m(h)h(emerging)118 5423 y(from)24 b(the)h(bifurcation)f(p)s(oin)m(t,)h(is)e(close)h(to)1597 5387 y FC(1)p 1597 5402 36 4 v 1597 5455 a(2)1642 5423 y FH(.)38 b(The)25 b(asymptotic)f(state)h(is)e(th)m(us)i(selected)f(b)m (y)h(the)g(noise,)118 5536 y(and)i(not)f(b)m(y)h(the)f(initial)d (condition.)39 b(Hence,)27 b(the)g(bifurcation)e(is)g(destro)m(y)m(ed)j (in)d(the)i(sense)f(of)f([CF98)q(].)1867 5871 y(4)p eop %%Page: 5 5 5 4 bop 118 328 a FH(On)27 b(the)f(other)h(hand,)h(individual)23 b(paths)j(are)h(concen)m(trated)i(near)d(the)h(stable)e(equilibrium)e (branc)m(hes)118 441 y(of)35 b FF(f)10 b FH(,)36 b(whic)m(h)f(means)f (that)i(the)g(bifurcation)e(diagram)g(will)f(b)s(e)i(made)g(visible)d (b)m(y)j(the)h(noise,)g(m)m(uc)m(h)118 553 y(more)26 b(so)g(than)h(in)e(the)h(deterministic)e(case.)39 b(So)27 b(w)m(e)g(do)f(observ)m(e)h(a)f(qualitativ)m(e)e(c)m(hange)k(in)d(b)s (eha)m(viour)118 666 y(when)31 b FF(\025)f FH(c)m(hanges)i(its)d(sign,) g(whic)m(h)i(can)g(b)s(e)f(considered)g(as)g(a)h(bifurcation.)259 779 y(The)k(precise)e(statemen)m(ts)h(and)g(a)g(discussion)e(of)h (their)h(consequences)g(are)h(giv)m(en)e(in)g(Section)h(2.)118 892 y(In)39 b(Section)g(2.2,)i(w)m(e)f(analyse)e(the)h(motion)f(near)h (equilibrium)c(branc)m(hes)41 b(a)m(w)m(a)m(y)f(from)e(bifurcation)118 1005 y(p)s(oin)m(ts.)53 b(The)36 b(actual)e(pitc)m(hfork)h(bifurcation) f(is)g(discussed)f(in)h(Section)h(2.3.)55 b(A)34 b(few)h(consequences) 118 1118 y(are)e(deriv)m(ed)g(in)f(Section)h(2.4.)49 b(Section)33 b(3)f(con)m(tains)i(the)f(pro)s(ofs)f(of)h(the)g(\034rst)g (t)m(w)m(o)h(theorems)f(on)g(the)118 1231 y(motion)g(near)h(non)m (bifurcating)g(equilibria,)e(while)h(the)h(pro)s(ofs)f(of)h(the)g(last) f(three)h(theorems)g(on)g(the)118 1344 y(pitc)m(hfork)c(bifurcation)g (are)h(giv)m(en)f(in)f(Section)h(4.)118 1582 y Fq(A)m(c)m(kno)m (wledgemen)m(ts:)118 1754 y FH(It's)45 b(a)g(great)h(pleasure)f(to)h (thank)f(An)m(ton)i(Bo)m(vier)e(for)g(sharing)g(our)g(en)m(th)m (usiasm.)85 b(W)-8 b(e)46 b(enjo)m(y)m(ed)118 1867 y(liv)m(ely)29 b(discussions)h(and)i(his)f(constan)m(t)i(in)m(terest)f(in)f(the)h (progress)h(of)e(our)h(w)m(ork.)46 b(The)33 b(cen)m(tral)f(ideas)118 1980 y(w)m(ere)46 b(dev)m(elop)s(ed)g(during)f(m)m(utual)g(visits)d(in) j(Berlin)f(resp.)h(A)m(tlan)m(ta.)86 b(N.)15 b(B.)45 b(thanks)g(the)h(WIAS)118 2092 y(and)37 b(B.)15 b(G.)36 b(thanks)g(T)-8 b(urga)m(y)37 b(Uzer)f(and)h(the)f(Sc)m(ho)s(ol)g(of)g (Ph)m(ysics)f(at)h(Georgia)g(T)-8 b(ec)m(h)38 b(for)d(their)h(kind)118 2205 y(hospitalit)m(y)-8 b(.)56 b(N.)15 b(B.)36 b(w)m(as)g(partially)e (supp)s(orted)j(b)m(y)f(the)g(F)-8 b(onds)37 b(National)d(Suisse)h(de)h (la)f(Rec)m(herc)m(he)118 2318 y(Scien)m(ti\034que,)i(and)f(b)m(y)f (the)h(Nonlinear)f(Con)m(trol)h(Net)m(w)m(ork)g(of)f(the)h(Europ)s(ean) h(Comm)m(unit)m(y)-8 b(,)36 b(Gran)m(t)118 2431 y(ERB)30 b(FMRX)m(CT\025970137.)118 2716 y FI(2)131 b(Statemen)l(t)44 b(of)g(results)118 2922 y Fp(2.1)112 b(Preliminaries)118 3094 y FH(W)-8 b(e)31 b(consider)f(nonlinear)g(It\364)g(SDEs)g(of)g (the)h(form)1038 3302 y FG(d)p FF(x)1141 3316 y Fy(t)1196 3302 y FG(=)1302 3240 y(1)p 1302 3281 46 4 v 1304 3364 a FF(")1357 3302 y(f)10 b FG(\()p FF(x)1499 3316 y Fy(t)1528 3302 y FF(;)15 b(t)p FG(\))g(d)q FF(t)20 b FG(+)1888 3240 y FF(\033)p 1857 3281 119 4 v 1857 3299 a FE(p)p 1932 3299 43 4 v 1932 3365 a FF(")2000 3302 y FG(d)o FF(W)2136 3316 y Fy(t)2166 3302 y FF(;)196 b(x)2439 3316 y Fy(t)2464 3325 y Fw(0)2528 3302 y FG(=)25 b FF(x)2676 3316 y FC(0)2716 3302 y FF(;)735 b FH(\(2.1\))118 3543 y(where)34 b FE(f)p FF(W)513 3557 y Fy(t)543 3543 y FE(g)588 3557 y Fy(t)p Fx(>)p Fy(t)693 3566 y Fw(0)766 3543 y FH(is)e(the)i(standard)g(Wiener)f(pro)s(cess)g(on)h(some)e(probabilit)m (y)g(space)i FG(\(\012)p FF(;)15 b FE(F)9 b FF(;)15 b Fo(P)p FG(\))p FH(.)51 b(Ini-)118 3656 y(tial)36 b(conditions)g FF(x)777 3670 y FC(0)853 3656 y FH(are)i(alw)m(a)m(ys)f(assumed)g(to)g (b)s(e)g(square-in)m(tegrable)g(with)g(resp)s(ect)g(to)h Fo(P)f FH(and)h(in-)118 3769 y(dep)s(enden)m(t)g(of)e FE(f)p FF(W)801 3783 y Fy(t)831 3769 y FE(g)876 3783 y Fy(t)p Fx(>)p Fy(t)981 3792 y Fw(0)1021 3769 y FH(.)57 b(All)34 b(sto)s(c)m(hastic)i(in)m(tegrals)g(are)g(considered)g(as)g (It\364)g(in)m(tegrals,)h(but)g(note)118 3882 y(that)h(It\364)g(and)g (Stratono)m(vic)m(h)h(in)m(tegrals)e(agree)h(for)g(in)m(tegrands)g(dep) s(ending)g(only)e(on)i(time)e(and)i FF(!)s FH(.)118 3995 y(Without)32 b(further)h(men)m(tioning)f(w)m(e)h(alw)m(a)m(ys)g(assume) e(that)i FF(f)42 b FH(satis\034es)31 b(the)i(usual)f(\(lo)s(cal\))f (Lipsc)m(hitz)118 4107 y(and)e(b)s(ounded-gro)m(wth)i(conditions)c (whic)m(h)i(guaran)m(tee)h(existence)e(and)h(\(path)m(wise\))g (uniqueness)e(of)h(a)118 4220 y(\(strong\))k(solution)e FE(f)p FF(x)906 4234 y Fy(t)936 4220 y FE(g)981 4234 y Fy(t)1042 4220 y FH(of)38 b(\(2.1\))q(.)43 b(Under)31 b(these)h(conditions,)e(there)i(exists)e(a)h(con)m(tin)m(uous)i(v)m (ersion)118 4333 y(of)c FE(f)p FF(x)317 4347 y Fy(t)347 4333 y FE(g)392 4347 y Fy(t)423 4333 y FH(.)40 b(Therefore)30 b(w)m(e)h(ma)m(y)e(assume)g(that)h(the)g(paths)g FF(!)e FE(7!)d FF(x)2385 4347 y Fy(t)2415 4333 y FG(\()p FF(!)s FG(\))30 b FH(are)g(con)m(tin)m(uous)h(for)e Fo(P)p FH(-almost)118 4446 y(all)g FF(!)f FE(2)d FG(\012)p FH(.)259 4559 y(W)-8 b(e)41 b(in)m(tro)s(duce)f(the)h(notation)g Fo(P)1435 4526 y Fy(t)1460 4535 y Fw(0)1495 4526 y Fy(;x)1555 4535 y Fw(0)1633 4559 y FH(for)f(the)h(la)m(w)f(of)g(the)g(pro)s(cess)g FE(f)p FF(x)2820 4573 y Fy(t)2850 4559 y FE(g)2895 4573 y Fy(t)p Fx(>)p Fy(t)3000 4582 y Fw(0)3040 4559 y FH(,)j(starting)d(in) f FF(x)3622 4573 y FC(0)118 4672 y FH(at)g(time)d FF(t)483 4686 y FC(0)523 4672 y FH(,)k(and)f(use)f Fo(E)996 4639 y Fy(t)1022 4648 y Fw(0)1062 4639 y Fy(;x)1122 4648 y Fw(0)1199 4672 y FH(to)g(denote)h(exp)s(ectations)f(with)g(resp)s(ect)g (to)h Fo(P)2858 4639 y Fy(t)2883 4648 y Fw(0)2918 4639 y Fy(;x)2978 4648 y Fw(0)3016 4672 y FH(.)65 b(Note)38 b(that)h(the)118 4785 y(sto)s(c)m(hastic)d(pro)s(cess)f FE(f)p FF(x)959 4799 y Fy(t)989 4785 y FE(g)1034 4799 y Fy(t)p Fx(>)p Fy(t)1139 4808 y Fw(0)1215 4785 y FH(is)f(an)j (\(inhomogeneous\))f(Mark)m(o)m(v)g(pro)s(cess.)57 b(W)-8 b(e)36 b(are)h(in)m(terested)f(in)118 4898 y(\034rst)41 b(exit)f(times)g(of)g FF(x)922 4912 y Fy(t)993 4898 y FH(from)g(space\025time)g(sets.)73 b(Let)42 b FE(A)g(\032)h Fo(R)52 b FE(\002)27 b FG([)p FF(t)2602 4912 y FC(0)2641 4898 y FF(;)15 b(t)2714 4912 y FC(1)2754 4898 y FG(])41 b FH(b)s(e)g(Borel-measurable.)118 5011 y(Assuming)28 b(that)j FE(A)f FH(con)m(tains)h FG(\()p FF(x)1274 5025 y FC(0)1313 5011 y FF(;)15 b(t)1386 5025 y FC(0)1426 5011 y FG(\))p FH(,)31 b(w)m(e)g(de\034ne)g(the)g(\034rst)f(exit)f (time)g(of)h FG(\()p FF(x)2827 5025 y Fy(t)2857 5011 y FF(;)15 b(t)p FG(\))31 b FH(from)e FE(A)h FH(b)m(y)1231 5204 y FF(\034)1271 5218 y FD(A)1356 5204 y FG(=)25 b(inf)1563 5130 y Fr(\010)1616 5204 y FF(t)g FE(2)g FG([)p FF(t)1818 5218 y FC(0)1858 5204 y FF(;)15 b(t)1931 5218 y FC(1)1970 5204 y FG(])10 b(:)31 b(\()p FF(x)2148 5218 y Fy(t)2178 5204 y FF(;)15 b(t)p FG(\))26 b FE(62)f(A)2471 5130 y Fr(\011)2523 5204 y FF(;)928 b FH(\(2.2\))118 5396 y(and)30 b(agree)g(to)f(set)g FF(\034)817 5410 y FD(A)878 5396 y FG(\()p FF(!)s FG(\))c(=)g FE(1)k FH(for)g(those)h FF(!)e FE(2)d FG(\012)k FH(whic)m(h)g(satisfy)e FG(\()p FF(x)2513 5410 y Fy(t)2543 5396 y FG(\()p FF(!)s FG(\))p FF(;)15 b(t)p FG(\))27 b FE(2)e(A)j FH(for)h(all)f FF(t)d FE(2)g FG([)p FF(t)3459 5410 y FC(0)3498 5396 y FF(;)15 b(t)3571 5410 y FC(1)3611 5396 y FG(])p FH(.)118 5509 y(F)-8 b(or)24 b(con)m(v)m(enience,)i(w)m(e)d(shall)e(call)h FF(\034)1315 5523 y FD(A)1398 5509 y FH(the)h FB(\034rst)i(exit)h(time) f(of)h FF(x)2243 5523 y Fy(t)2298 5509 y FB(fr)-5 b(om)25 b FE(A)p FH(.)38 b(T)m(ypically)-8 b(,)22 b(w)m(e)i(will)c(consider)118 5622 y(sets)34 b(of)g(the)h(form)f FE(A)e FG(=)g FE(f)p FG(\()p FF(x;)15 b(t)p FG(\))34 b FE(2)e Fo(R)47 b FE(\002)23 b FG([)p FF(t)1615 5636 y FC(0)1654 5622 y FF(;)15 b(t)1727 5636 y FC(1)1767 5622 y FG(])10 b(:)32 b FF(g)1902 5636 y FC(1)1942 5622 y FG(\()p FF(t)p FG(\))h FF(<)f(x)h(<)f(g)2412 5636 y FC(2)2451 5622 y FG(\()p FF(t)p FG(\))p FE(g)k FH(with)e(con)m(tin)m(uous)i(functions)1867 5871 y(5)p eop %%Page: 6 6 6 5 bop 118 328 a FF(g)161 342 y FC(1)244 328 y FF(<)43 b(g)401 342 y FC(2)441 328 y FH(.)72 b(Note)42 b(that)f(in)f(this)h (case,)i FF(\034)1547 342 y FD(A)1649 328 y FH(is)c(a)i(stopping)g (time)2391 295 y Fz(2)2470 328 y FH(with)g(resp)s(ect)g(to)g(the)h (canonical)118 441 y(\034ltration)30 b(of)g FG(\(\012)p FF(;)15 b FE(F)9 b FF(;)15 b Fo(P)p FG(\))32 b FH(generated)g(b)m(y)e FE(f)p FF(x)1609 455 y Fy(t)1639 441 y FE(g)1684 455 y Fy(t)p Fx(>)p Fy(t)1789 464 y Fw(0)1830 441 y FH(.)259 553 y(Before)c(turning)f(to)h(the)g(precise)e(statemen)m(ts)i(of)f(our) h(results,)f(let)g(us)g(in)m(tro)s(duce)h(some)e(notations.)118 666 y(W)-8 b(e)31 b(shall)d(use)181 803 y FE(\017)47 b(d)p FF(y)s FE(e)31 b FH(for)f FF(y)e Fn(>)d FG(0)31 b FH(to)f(denote)i(the)e(smallest)e(in)m(teger)j(whic)m(h)f(is)f (greater)j(than)f(or)f(equal)g(to)g FF(y)s FH(,)g(and)181 916 y FE(\017)47 b FF(y)18 b FE(_)d FF(z)32 b FH(and)c FF(y)18 b FE(^)d FF(z)32 b FH(to)c(denote)h(the)f(maxim)m(um)d(or)j (minim)m(um,)e(resp)s(ectiv)m(ely)-8 b(,)27 b(of)g(t)m(w)m(o)j(real)d (n)m(um)m(b)s(ers)273 1029 y FF(y)33 b FH(and)e FF(z)t FH(.)181 1142 y FE(\017)47 b FH(By)26 b FF(g)s FG(\()p FF(u)p FG(\))h(=)e FE(O)s FG(\()p FF(u)p FG(\))i FH(w)m(e)g(indicate)f (that)h(there)h(exist)d FF(\016)k(>)c FG(0)i FH(and)g FF(K)32 b(>)25 b FG(0)h FH(suc)m(h)i(that)f FF(g)s FG(\()p FF(u)p FG(\))f Fn(6)f FF(K)7 b(u)26 b FH(for)273 1255 y(all)g FF(u)f FE(2)g FG([0)p FF(;)15 b(\016)s FG(])p FH(,)31 b(where)d FF(\016)j FH(and)d FF(K)35 b FH(of)27 b(course)h(do)g(not)g(dep)s(end)g(on)g FF(")g FH(or)f FF(\033)s FH(.)40 b(Similarly)-8 b(,)25 b FF(g)s FG(\()p FF(u)p FG(\))h(=)f FD(O)6 b FG(\(1\))273 1367 y FH(is)30 b(to)g(b)s(e)h(understo)s(o)s(d)g(as)f FG(lim)1309 1381 y Fy(u)p FD(!)p FC(0)1475 1367 y FF(g)s FG(\()p FF(u)p FG(\))e(=)d(0)p FH(.)42 b(F)-8 b(rom)30 b(time)g(to)h(time,)e(w)m(e)i (write)g FF(g)s FG(\()p FF(u)p FG(\))c(=)f FD(O)3379 1381 y Fy(T)3434 1367 y FG(\(1\))32 b FH(to)273 1480 y(indicate)27 b(that)i(c)m(ho)s(osing)f FB(a)i(priori)d FH(a)h(su\036cien)m(tly)f(small)f FF(T)40 b FH(allo)m(ws)27 b(to)h(mak)m(e)g(the)g(corresp)s(onding)273 1593 y(term)i(arbitrarily)f (small)e(for)j(all)f FF(u)h FH(from)g(some)f FF(T)13 b FH(-dep)s(enden)m(t)32 b(in)m(terv)-5 b(al.)118 1730 y(Finally)d(,)29 b(let)i(us)f(p)s(oin)m(t)h(out)g(that)h(most)e (estimates)g(hold)g(for)h(small)d(enough)k FF(")g FH(only)-8 b(,)30 b(and)i(often)f(only)118 1843 y(for)f Fo(P)p FH(-almost)g(all)e FF(!)g FE(2)d FG(\012)p FH(.)40 b(W)-8 b(e)31 b(will)d(stress)h(these)i (facts)f(only)f(when)i(confusion)f(migh)m(t)g(arise.)118 2083 y Fp(2.2)112 b(Non)m(bifurcating)36 b(equilibria)118 2255 y FH(W)-8 b(e)23 b(start)g(b)m(y)h(considering)e(the)h(nonlinear)f (SDE)i(\(2.1\))g(in)e(the)h(case)g(of)f FF(f)32 b FH(admitting)22 b(a)h(non)m(bifurcating)118 2368 y(equilibrium)k(branc)m(h.)44 b(W)-8 b(e)31 b(will)d(assume)i(that)i(there)f(exists)e(an)j(in)m(terv) -5 b(al)30 b FF(I)j FG(=)26 b([0)p FF(;)15 b(T)e FG(])32 b FH(or)f FG([0)p FF(;)15 b FE(1)p FG(\))32 b FH(suc)m(h)118 2481 y(that)f(the)g(follo)m(wing)d(prop)s(erties)i(hold:)181 2617 y FE(\017)47 b FH(there)31 b(exists)e(a)h(function)g FF(x)1235 2584 y Fy(?)1300 2617 y FG(:)c FF(I)32 b FE(!)25 b Fo(R)18 b FH(,)36 b(called)29 b FB(e)-5 b(quilibrium)34 b(curve)p FH(,)e(suc)m(h)f(that)1464 2803 y FF(f)10 b FG(\()p FF(x)1606 2765 y Fy(?)1645 2803 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))26 b(=)f(0)181 b FE(8)p FF(t)25 b FE(2)g FF(I)7 b FG(;)1005 b FH(\(2.3\))181 2988 y FE(\017)47 b FF(f)38 b FH(is)28 b(t)m(wice)h(con)m(tin)m(uously)g(di\033eren)m (tiable)f(with)h(resp)s(ect)g(to)g FF(x)g FH(and)g FF(t)p FH(,)g(with)g(uniformly)d(b)s(ounded)273 3101 y(deriv)-5 b(ativ)m(es,)29 b(for)i(all)d FF(t)d FE(2)g FF(I)37 b FH(and)31 b(all)e FF(x)h FH(in)f(a)i(neigh)m(b)s(ourho)s(o)s(d)f(of)g FF(x)2570 3068 y Fy(?)2610 3101 y FG(\()p FF(t)p FG(\))p FH(;)181 3214 y FE(\017)47 b FH(the)31 b(linearization)d(of)i FF(f)40 b FH(at)30 b FF(x)1296 3181 y Fy(?)1336 3214 y FG(\()p FF(t)p FG(\))p FH(,)g(de\034ned)i(as)1576 3399 y FF(a)p FG(\()p FF(t)p FG(\))26 b(=)e FF(@)1896 3413 y Fy(x)1941 3399 y FF(f)10 b FG(\()p FF(x)2083 3362 y Fy(?)2122 3399 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))p FF(;)1118 b FH(\(2.4\))273 3585 y(is)29 b(b)s(ounded)i(a)m(w)m (a)m(y)h(from)e(zero,)h(that)g(is,)e(there)i(exists)d(a)j(constan)m(t)g FF(a)2669 3599 y FC(0)2734 3585 y FF(>)25 b FG(0)31 b FH(suc)m(h)f(that)1583 3770 y FE(j)p FF(a)p FG(\()p FF(t)p FG(\))p FE(j)d Fn(>)e FF(a)1955 3784 y FC(0)2085 3770 y FE(8)p FF(t)f FE(2)h FF(I)7 b(:)1125 b FH(\(2.5\))118 3955 y(In)30 b(the)h(deterministic)d(case)i FF(\033)f FG(=)24 b(0)p FH(,)31 b(the)g(follo)m(wing)e(result)g(is)g(kno)m(wn)i (\(see)g(Fig.)24 b(1a\):)118 4127 y Fq(Theorem)34 b(2.1)g (\(Deterministic)j(case)e FH([Ti,)30 b(Gr])p Fq(\).)42 b FB(Consider)32 b(the)h(e)-5 b(quation)1579 4355 y FF(")1631 4294 y FG(d)p FF(x)1734 4308 y Fy(t)p 1631 4335 133 4 v 1656 4418 a FG(d)p FF(t)1799 4355 y FG(=)25 b FF(f)10 b FG(\()p FF(x)2037 4369 y Fy(t)2066 4355 y FF(;)15 b(t)p FG(\))p FF(:)1277 b FH(\(2.6\))118 4562 y FB(Ther)-5 b(e)33 b(ar)-5 b(e)34 b(c)-5 b(onstants)32 b FF(")969 4576 y FC(0)1009 4562 y FF(;)15 b(c)1088 4576 y FC(0)1128 4562 y FF(;)g(c)1207 4576 y FC(1)1273 4562 y FF(>)25 b FG(0)p FB(,)32 b(dep)-5 b(ending)34 b(only)d(on)i FF(f)10 b FB(,)31 b(such)i(that)f(for)g FG(0)26 b FF(<)f(")h Fn(6)f FF(")3239 4576 y FC(0)3279 4562 y FB(,)181 4698 y FE(\017)48 b FH(\(2.6\))33 b FB(admits)f(a)g(p)-5 b(articular)34 b(solution)h Fr(b)-54 b FF(x)1677 4665 y FC(det)1669 4721 y Fy(t)1811 4698 y FB(such)33 b(that)1408 4884 y FE(j)s Fr(b)-54 b FF(x)1493 4846 y FC(det)1485 4906 y Fy(t)1615 4884 y FE(\000)20 b FF(x)1758 4846 y Fy(?)1797 4884 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FF(c)2086 4898 y FC(1)2126 4884 y FF(")93 b FE(8)p FF(t)24 b FE(2)h FF(I)7 b FG(;)949 b FH(\(2.7\))181 5069 y FE(\017)48 b FB(if)23 b FE(j)p FF(x)430 5083 y FC(0)471 5069 y FE(\000)r FF(x)596 5036 y Fy(?)635 5069 y FG(\(0\))p FE(j)k Fn(6)e FF(c)937 5083 y FC(0)1000 5069 y FB(and)f FF(a)p FG(\()p FF(t)p FG(\))i Fn(6)f FE(\000)p FF(a)1559 5083 y FC(0)1622 5069 y FB(for)f(al)5 b(l)23 b FF(t)i FE(2)g FF(I)31 b FB(\(that)24 b(is,)h(when)f FF(x)2698 5036 y Fy(?)2761 5069 y FB(is)g(a)g(stable)g(e)-5 b(quilibrium\),)274 5182 y(then)32 b(the)h(solution)f FF(x)1017 5149 y FC(det)1017 5204 y Fy(t)1152 5182 y FB(of)51 b FH(\(2.6\))33 b FB(with)f(initial)f (c)-5 b(ondition)32 b FF(x)2402 5149 y FC(det)2402 5206 y(0)2530 5182 y FG(=)25 b FF(x)2678 5196 y FC(0)2749 5182 y FB(satis\034es)1116 5367 y FE(j)p FF(x)1193 5330 y FC(det)1193 5390 y Fy(t)1316 5367 y FE(\000)e Fr(b)-55 b FF(x)1466 5330 y FC(det)1458 5390 y Fy(t)1568 5367 y FE(j)26 b Fn(6)f FE(j)p FF(x)1792 5381 y FC(0)1852 5367 y FE(\000)e Fr(b)-54 b FF(x)2003 5330 y FC(det)1995 5390 y(0)2105 5367 y FE(j)15 b FG(e)2185 5330 y FD(\000)p Fy(a)2277 5339 y Fw(0)2312 5330 y Fy(t=)p FC(2)p Fy(")2553 5367 y FE(8)p FF(t)24 b FE(2)h FF(I)7 b(:)657 b FH(\(2.8\))p 118 5445 1418 4 v 222 5499 a Fv(2)256 5531 y Fu(F)-6 b(or)24 b(a)g(general)g(Borel-measurable)g(set)g Fm(A)p Fu(,)h(the)e(\034rst)g(exit)g(time)g Fl(\034)2186 5539 y Fk(A)2263 5531 y Fu(is)h(still)g(a)g(stopping)g(time)f(with)g(resp)r (ect)h(to)g(the)118 5622 y(canonical)j(\034ltration,)f(completed)f(b)n (y)g(the)g(n)n(ull)h(sets.)1867 5871 y FH(6)p eop %%Page: 7 7 7 6 bop 118 328 a Fq(Remark)48 b(2.2.)g FH(The)c(particular)f(solution) i Fr(b)-54 b FF(x)1781 295 y FC(det)1925 328 y FH(is)42 b(often)i(called)e(a)h FB(slow)h(solution)f FH(or)g FB(adiab)-5 b(atic)118 441 y(solution)30 b FH(of)g(equation)g(\(2.6\))q(.)40 b(It)30 b(is)f(not)i(unique)f(in)f(general,)i(as)f(suggested)h(b)m(y)g (\(2.8\))q(.)259 614 y(W)-8 b(e)36 b(return)h(no)m(w)f(to)g(the)g(SDE)h (\(2.1\))f(with)f FF(\033)j(>)33 b FG(0)p FH(.)57 b(W)-8 b(e)36 b(need)g(no)g(additional)f(assumption)f(on)118 727 y FF(\033)k FH(in)33 b(this)h(section.)52 b(Ho)m(w)m(ev)m(er,)38 b(the)d(results)e(are)i(only)e(in)m(teresting)i(when)g FF(\033)g FG(=)d FD(O)2988 741 y Fy(")3025 727 y FG(\(1\))p FH(.)54 b(Let)35 b(us)f(\034rst)118 840 y(consider)d(the)g(stable)g (case,)g(that)h(is,)d(w)m(e)j(assume)e(that)i FF(a)p FG(\()p FF(t)p FG(\))27 b Fn(6)f FE(\000)p FF(a)2458 854 y FC(0)2524 840 y FF(<)g FG(0)31 b FH(for)g(all)e FF(t)d FE(2)g FF(I)7 b FH(.)43 b(W)-8 b(e)31 b(assume)118 953 y(that)39 b(at)f FF(t)g FG(=)g(0)p FH(,)i FF(x)782 967 y Fy(t)849 953 y FH(starts)e(at)g(some)f(\(deterministic\))f FF(x)2134 967 y FC(0)2212 953 y FH(su\036cien)m(tly)h(close)g(to)h FF(x)3066 920 y Fy(?)3105 953 y FG(\(0\))p FH(.)65 b(Theorem)118 1066 y(2.1)34 b(tells)e(us)i(that)g(the)g(deterministic)d(solution)i FF(x)1884 1033 y FC(det)1884 1088 y Fy(t)2020 1066 y FH(with)g(the)h(same)f(initial)e(condition)i FF(x)3335 1033 y FC(det)3335 1090 y(0)3468 1066 y FG(=)e FF(x)3622 1080 y FC(0)118 1179 y FH(reac)m(hes)h(a)e(neigh)m(b)s(ourho)s(o)s(d)h (of)f(order)h FF(")f FH(of)g FF(x)1692 1146 y Fy(?)1731 1179 y FG(\()p FF(t)p FG(\))h FH(exp)s(onen)m(tially)e(fast.)259 1292 y(W)-8 b(e)26 b(are)g(in)m(terested)f(in)g(the)h(sto)s(c)m(hastic) e(pro)s(cess)h FF(y)1984 1306 y Fy(t)2039 1292 y FG(=)g FF(x)2187 1306 y Fy(t)2227 1292 y FE(\000)10 b FF(x)2360 1259 y FC(det)2360 1314 y Fy(t)2461 1292 y FH(,)27 b(whic)m(h)e (describ)s(es)g(the)g(deviation)118 1405 y(due)31 b(to)f(noise)g(from)f (the)i(deterministic)d(solution)h FF(x)1932 1372 y FC(det)2034 1405 y FH(.)40 b(It)30 b(ob)s(eys)g(the)h(SDE)683 1625 y FG(d)o FF(y)778 1639 y Fy(t)833 1625 y FG(=)939 1564 y(1)p 939 1604 46 4 v 941 1688 a FF(")994 1552 y Fr(\002)1032 1625 y FF(f)10 b FG(\()p FF(x)1174 1588 y FC(det)1174 1648 y Fy(t)1296 1625 y FG(+)20 b FF(y)1432 1639 y Fy(t)1461 1625 y FF(;)15 b(t)p FG(\))21 b FE(\000)f FF(f)10 b FG(\()p FF(x)1823 1588 y FC(det)1823 1648 y Fy(t)1924 1625 y FF(;)15 b(t)p FG(\))2032 1552 y Fr(\003)2086 1625 y FG(d)p FF(t)20 b FG(+)2322 1564 y FF(\033)p 2290 1604 119 4 v 2290 1622 a FE(p)p 2366 1622 43 4 v 66 x FF(")2434 1625 y FG(d)o FF(W)2570 1639 y Fy(t)2600 1625 y FF(;)196 b(y)2866 1639 y FC(0)2930 1625 y FG(=)25 b(0)p FF(:)380 b FH(\(2.9\))118 1855 y(W)-8 b(e)21 b(will)e(pro)m(v)m(e)j(that)f FF(y)891 1869 y Fy(t)941 1855 y FH(remains)f(in)g(a)h(neigh)m(b)s (ourho)s(o)s(d)g(of)g FG(0)g FH(with)g(high)f(probabilit)m(y)-8 b(.)37 b(It)20 b(is)g(instructiv)m(e)118 1968 y(to)31 b(consider)f(\034rst)g(the)h(linearization)d(of)37 b(\(2.9\))31 b(around)g FF(y)d FG(=)d(0)p FH(,)31 b(whic)m(h)f(has)h(the)f(form)1331 2194 y FG(d)p FF(y)1430 2156 y FC(0)1427 2216 y Fy(t)1494 2194 y FG(=)1600 2132 y(1)p 1600 2173 46 4 v 1602 2256 a FF(")1657 2194 y FG(\026)-47 b FF(a)q FG(\()p FF(t)p FG(\))p FF(y)1855 2156 y FC(0)1852 2216 y Fy(t)1909 2194 y FG(d)p FF(t)20 b FG(+)2145 2132 y FF(\033)p 2114 2173 119 4 v 2114 2191 a FE(p)p 2190 2191 43 4 v 65 x FF(")2257 2194 y FG(d)p FF(W)2394 2208 y Fy(t)2423 2194 y FF(;)982 b FH(\(2.10\))118 2424 y(where)711 2537 y FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1031 2551 y Fy(x)1075 2537 y FF(f)10 b FG(\()p FF(x)1217 2499 y FC(det)1217 2559 y Fy(t)1319 2537 y FF(;)15 b(t)p FG(\))25 b(=)g FF(a)p FG(\()p FF(t)p FG(\))c(+)f FE(O)s FG(\()p FF(")p FG(\))h(+)f FE(O)2185 2463 y Fr(\000)2227 2537 y FE(j)p FF(x)2304 2551 y FC(0)2364 2537 y FE(\000)g FF(x)2507 2499 y Fy(?)2546 2537 y FG(\(0\))p FE(j)15 b FG(e)2743 2499 y FD(\000)p Fy(a)2835 2508 y Fw(0)2870 2499 y Fy(t=)p FC(2)p Fy(")3003 2463 y Fr(\001)3044 2537 y FF(:)361 b FH(\(2.11\))118 2695 y(T)-8 b(aking)40 b FF(")h FH(and)f FE(j)p FF(x)774 2709 y FC(0)841 2695 y FE(\000)26 b FF(x)990 2662 y Fy(?)1030 2695 y FG(\(0\))p FE(j)41 b FH(su\036cien)m(tly)e(small,)h(w)m(e)h(ma)m(y)f(assume)f(the) i(existence)e(of)h(constan)m(ts)119 2808 y FG(\026)-46 b FF(a)166 2822 y FC(+)264 2808 y Fn(>)39 b FG(\026)-46 b FF(a)421 2822 y FD(\000)519 2808 y FF(>)38 b FG(0)h FH(suc)m(h)g(that)f FE(\000)q FG(\026)-46 b FF(a)1246 2822 y FC(+)1344 2808 y Fn(6)39 b FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))39 b Fn(6)g FE(\000)q FG(\026)-46 b FF(a)1872 2822 y FD(\000)1969 2808 y FH(for)38 b(all)e FF(t)j FE(2)f FF(I)7 b FH(.)64 b(The)39 b(solution)e(of)45 b(\(2.10\))39 b(with)118 2921 y(arbitrary)30 b(initial)e(condition)h FF(y)1208 2888 y FC(0)1205 2945 y(0)1278 2921 y FH(is)g(giv)m(en)h(b)m (y)670 3166 y FF(y)718 3129 y FC(0)715 3189 y Fy(t)783 3166 y FG(=)25 b FF(y)927 3129 y FC(0)924 3189 y(0)981 3166 y FG(e)p 1028 3094 35 3 v -37 x Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1234 3166 y FG(+)1346 3105 y FF(\033)p 1315 3146 119 4 v 1315 3164 a FE(p)p 1391 3164 43 4 v 65 x FF(")1458 3043 y Fr(Z)1549 3069 y Fy(t)1509 3249 y FC(0)1594 3166 y FG(e)p 1641 3094 35 3 v -37 x Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1899 3166 y FG(d)p FF(W)2036 3180 y Fy(s)2073 3166 y FF(;)p 2304 3116 44 4 v 196 w(\013)p FG(\()p FF(t;)15 b(s)p FG(\))26 b(=)2660 3043 y Fr(Z)2751 3069 y Fy(t)2710 3249 y(s)2797 3166 y FG(\026)-46 b FF(a)p FG(\()p FF(u)p FG(\))15 b(d)p FF(u;)321 b FH(\(2.12\))118 3402 y(where)32 b(w)m(e)g(write)p 756 3352 V 31 w FF(\013)p FG(\()p FF(t;)15 b FG(0\))28 b(=)p 1127 3352 V 26 w FF(\013)p FG(\()p FF(t)p FG(\))k FH(for)f(brevit)m(y)-8 b(.)43 b(Note)31 b(that)p 2214 3352 V 32 w FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))27 b Fn(6)g FE(\000)q FG(\026)-46 b FF(a)2692 3416 y FD(\000)2750 3402 y FG(\()p FF(t)21 b FE(\000)g FF(s)p FG(\))31 b FH(whenev)m(er)h FF(t)26 b Fn(>)h FF(s)p FH(.)118 3515 y(If)j FF(y)257 3482 y FC(0)254 3539 y(0)326 3515 y FH(has)g(v)-5 b(ariance)30 b FF(v)885 3529 y FC(0)950 3515 y Fn(>)25 b FG(0)p FH(,)31 b(then)g FF(y)1401 3482 y FC(0)1398 3537 y Fy(t)1470 3515 y FH(has)f(v)-5 b(ariance)1162 3760 y FF(v)s FG(\()p FF(t)p FG(\))27 b(=)e FF(v)1479 3774 y FC(0)1533 3760 y FG(e)1574 3723 y FC(2)p 1616 3688 35 3 v Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1822 3760 y FG(+)1903 3699 y FF(\033)1958 3666 y FC(2)p 1903 3739 95 4 v 1929 3823 a FF(")2022 3637 y Fr(Z)2113 3663 y Fy(t)2073 3843 y FC(0)2158 3760 y FG(e)2198 3723 y FC(2)p 2240 3688 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2499 3760 y FG(d)o FF(s:)813 b FH(\(2.13\))118 3988 y(Since)37 b(the)g(\034rst)g(term)f(decreases)i(exp)s(onen)m(tially)d(fast,)j(the) f(initial)d(v)-5 b(ariance)36 b FF(v)2943 4002 y FC(0)3020 3988 y FH(is)f(\020forgotten\021)46 b(as)118 4101 y(so)s(on)35 b(as)h FG(e)489 4068 y FC(2)p 531 4033 V Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")758 4101 y FH(is)e(small)f(enough,)38 b(whic)m(h)e(happ)s(ens)g(already)f(for)g FF(t)f(>)g FE(O)s FG(\()p FF(")p FE(j)p FG(log)18 b FF(")p FE(j)p FG(\))p FH(.)57 b(F)-8 b(or)36 b FF(y)3412 4068 y FC(0)3409 4126 y(0)3486 4101 y FG(=)e(0)p FH(,)118 4214 y(\(2.12\))d(implies)c (in)j(particular)g(that)g(for)h(an)m(y)f FF(\016)f(>)c FG(0)p FH(,)1356 4413 y Fo(P)1417 4375 y FC(0)p Fy(;)p FC(0)1512 4339 y Fr(\010)1565 4413 y FE(j)p FF(y)1638 4375 y FC(0)1635 4435 y Fy(t)1678 4413 y FE(j)g Fn(>)g FF(\016)1867 4339 y Fr(\011)1946 4413 y Fn(6)g FG(e)2083 4375 y FD(\000)p Fy(\016)2171 4352 y Fw(2)2206 4375 y Fy(=)p FC(2)p Fy(v)r FC(\()p Fy(t)p FC(\))2398 4413 y FF(;)1007 b FH(\(2.14\))118 4600 y(and)34 b(th)m(us)h(the)f(probabilit) m(y)f(of)h(\034nding)g FF(y)1585 4567 y FC(0)1582 4623 y Fy(t)1624 4600 y FH(,)h(at)f(an)m(y)g(giv)m(en)g FF(t)d FE(2)g FF(I)7 b FH(,)35 b(outside)e(a)h(strip)f(of)h(width)g(m)m(uc)m (h)118 4713 y(larger)c(than)585 4635 y Fr(p)p 676 4635 197 4 v 78 x FG(2)p FF(v)s FG(\()p FF(t)p FG(\))i FH(is)c(v)m(ery)j (small.)259 4826 y(Our)21 b(\034rst)g(main)e(result)h(states)g(that)i (the)e FB(whole)k(p)-5 b(ath)22 b FE(f)p FF(x)2171 4840 y Fy(s)2208 4826 y FE(g)2253 4840 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)2481 4826 y FH(of)e(the)h(solution)e(of)i(the)f FB(nonline)-5 b(ar)118 4939 y FH(equation)40 b(\(2.1\))h(lies)d(in)i(a) g(similar)d(strip)i(with)h(high)g(probabilit)m(y)-8 b(.)70 b(W)-8 b(e)40 b(only)g(need)g(to)h(mak)m(e)f(one)118 5052 y(concession:)f(the)28 b(width)f(of)g(the)h(strip)e(has)i(to)f(b)s (e)g(b)s(ounded)h(a)m(w)m(a)m(y)i(from)c(zero.)40 b(Therefore,)29 b(w)m(e)f(de\034ne)118 5165 y(the)j(strip)e(as)959 5278 y FE(B)1019 5292 y FC(s)1051 5278 y FG(\()p FF(h)p FG(\))d(=)1295 5204 y Fr(\010)1348 5278 y FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f Fo(R)44 b FE(\002)20 b FF(I)d FG(:)31 b FE(j)p FF(x)20 b FE(\000)g FF(x)2200 5240 y FC(det)2200 5300 y Fy(t)2302 5278 y FE(j)26 b FF(<)f(h)2501 5195 y Fr(p)p 2592 5195 151 4 v 83 x FF(\020)7 b FG(\()p FF(t)p FG(\))2742 5204 y Fr(\011)2795 5278 y FF(;)610 b FH(\(2.15\))118 5436 y(where)1089 5583 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b(=)1478 5521 y(1)p 1370 5562 261 4 v 1370 5645 a(2)p FE(j)q FG(\026)-46 b FF(a)q FG(\(0\))p FE(j)1656 5583 y FG(e)1696 5545 y FC(2)p 1738 5510 35 3 v Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1944 5583 y FG(+)2025 5521 y(1)p 2025 5562 46 4 v 2027 5645 a FF(")2095 5459 y Fr(Z)2186 5485 y Fy(t)2146 5665 y FC(0)2231 5583 y FG(e)2272 5545 y FC(2)p 2314 5510 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2572 5583 y FG(d)o FF(s:)740 b FH(\(2.16\))1867 5871 y(7)p eop %%Page: 8 8 8 7 bop 118 328 a FF(\033)173 295 y FC(2)213 328 y FF(\020)46 b FH(can)39 b(b)s(e)h(in)m(terpreted)g(as)f(the)h(v)-5 b(ariance)40 b(\(2.13\))g(of)f(the)h(pro)s(cess)f(\(2.12\))h(starting)f (with)g(initial)118 441 y(v)-5 b(ariance)30 b FF(v)516 455 y FC(0)581 441 y FG(=)25 b FF(\033)732 408 y FC(2)772 441 y FF(=)p FG(\(2)p FE(j)q FG(\026)-46 b FF(a)q FG(\(0\))p FE(j)p FG(\))p FH(.)42 b(W)-8 b(e)31 b(shall)d(sho)m(w)j(in)f(Lemma)f (3.1)i(that)936 680 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b(=)1318 618 y(1)p 1217 659 248 4 v 1217 742 a(2)p FE(j)p FF(a)p FG(\()p FF(t)p FG(\))p FE(j)1495 680 y FG(+)19 b FE(O)s FG(\()p FF(")p FG(\))j(+)e FE(O)1960 606 y Fr(\000)2001 680 y FE(j)p FF(x)2078 694 y FC(0)2138 680 y FE(\000)g FF(x)2281 642 y Fy(?)2320 680 y FG(\(0\))p FE(j)15 b FG(e)2517 642 y FD(\000)p Fy(a)2609 651 y Fw(0)2644 642 y Fy(t=)p FC(2)p Fy(")2777 606 y Fr(\001)2819 680 y FF(:)586 b FH(\(2.17\))118 929 y(Let)31 b FF(\034)320 948 y FD(B)366 956 y Fw(s)394 948 y FC(\()p Fy(h)p FC(\))524 929 y FH(denote)h(the)e (\034rst)h(exit)e(time)g(of)h FF(x)1697 943 y Fy(t)1757 929 y FH(from)f FE(B)2030 943 y FC(s)2062 929 y FG(\()p FF(h)p FG(\))p FH(.)118 1114 y Fq(Theorem)34 b(2.3)h(\(Stable)j (case\).)43 b FB(Ther)-5 b(e)34 b(exist)f FF(")1894 1128 y FC(0)1934 1114 y FB(,)g FF(d)2042 1128 y FC(0)2114 1114 y FB(and)g FF(h)2342 1128 y FC(0)2382 1114 y FB(,)f(dep)-5 b(ending)34 b(only)f(on)g FF(f)10 b FB(,)32 b(such)h(that)118 1227 y(for)f FG(0)26 b FF(<)f(")g Fn(6)g FF(")634 1241 y FC(0)674 1227 y FB(,)32 b FF(h)26 b Fn(6)f FF(h)960 1241 y FC(0)1032 1227 y FB(and)32 b FE(j)p FF(x)1284 1241 y FC(0)1344 1227 y FE(\000)20 b FF(x)1487 1194 y Fy(?)1526 1227 y FG(\(0\))p FE(j)27 b Fn(6)e FF(d)1836 1241 y FC(0)1876 1227 y FB(,)725 1484 y Fo(P)786 1446 y FC(0)p Fy(;x)881 1455 y Fw(0)920 1410 y Fr(\010)973 1484 y FF(\034)1013 1502 y FD(B)1059 1510 y Fw(s)1088 1502 y FC(\()p Fy(h)p FC(\))1213 1484 y FF(<)g(t)1342 1410 y Fr(\011)1420 1484 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)1927 1383 y Fr(n)1988 1484 y FE(\000)2069 1422 y FG(1)p 2069 1463 46 4 v 2069 1546 a(2)2136 1422 y FF(h)2188 1389 y FC(2)p 2134 1463 95 4 v 2134 1546 a FF(\033)2189 1520 y FC(2)2239 1410 y Fr(\002)2277 1484 y FG(1)20 b FE(\000)g(O)s FG(\()p FF(")p FG(\))h FE(\000)f(O)s FG(\()p FF(h)p FG(\))2929 1410 y Fr(\003)2968 1383 y(o)3029 1484 y FF(;)376 b FH(\(2.18\))118 1705 y FB(wher)-5 b(e)1493 1843 y FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))26 b(=)1882 1781 y FE(j)p 1917 1731 44 4 v FF(\013)p FG(\()p FF(t)p FG(\))p FE(j)p 1882 1822 213 4 v 1947 1905 a FF(")1989 1879 y FC(2)2125 1843 y FG(+)20 b(2)p FF(:)1144 b FH(\(2.19\))259 2048 y(The)30 b(pro)s(of,)f(giv)m(en)g(in)f(Section)h(3.1,)h(is)d (divided)h(in)m(to)h(t)m(w)m(o)h(main)e(steps.)40 b(First,)28 b(w)m(e)i(sho)m(w)f(that)h(an)118 2161 y(estimate)c(of)g(the)h(form)f (\(2.18\))q(,)h(but)g(without)g(the)g(term)g FE(O)s FG(\()p FF(h)p FG(\))p FH(,)h(holds)e(for)g(the)i(solution)d(of)h(the)h(linear) 118 2274 y(equation)35 b(\(2.10\))q(.)55 b(Then)36 b(w)m(e)g(sho)m(w)g (that)g(whenev)m(er)h FE(j)p FF(y)2087 2241 y FC(0)2084 2296 y Fy(s)2127 2274 y FE(j)c FF(<)h(h)2342 2196 y Fr(p)p 2433 2196 160 4 v 78 x FF(\020)7 b FG(\()p FF(s)p FG(\))35 b FH(for)g FG(0)f Fn(6)f FF(s)g Fn(6)h FF(t)p FH(,)i(one)g(almost)118 2387 y(surely)29 b(also)h(has)g FE(j)p FF(y)795 2401 y Fy(s)831 2387 y FE(j)c FF(<)f(h)p FG(\(1)c(+)f FE(O)s FG(\()p FF(h)p FG(\)\))1454 2309 y Fr(p)p 1546 2309 V 1546 2387 a FF(\020)7 b FG(\()p FF(s)p FG(\))30 b FH(for)h FG(0)25 b Fn(6)g FF(s)g Fn(6)g FF(t)p FH(.)118 2572 y Fq(Remark)44 b(2.4.)i FH(The)40 b(result)e(of)h(the)h(preceding)g (theorem)g(remains)e(true)i(when)g FG(1)p FF(=)p FG(2)p FE(j)q FG(\026)-46 b FF(a)r FG(\(0\))p FE(j)40 b FH(in)f(the)118 2685 y(de\034nition)c(\(2.16\))i(of)f FF(\020)7 b FG(\()p FF(t)p FG(\))35 b FH(is)g(replaced)h(b)s(e)g(an)g(arbitrary)g FF(\020)2235 2699 y FC(0)2274 2685 y FH(,)h FB(pr)-5 b(ovide)g(d)38 b FF(\020)2736 2699 y FC(0)2809 2685 y FF(>)d FG(0)p FH(.)57 b(The)36 b(terms)g FE(O)s FG(\()p FE(\001)p FG(\))118 2798 y FH(ma)m(y)21 b(then)h(dep)s(end)g(on)g FF(\020)964 2812 y FC(0)1003 2798 y FH(.)37 b(Note)22 b(that)f FF(\020)7 b FG(\()p FF(t)p FG(\))22 b FH(and)f FF(\033)1854 2765 y FC(2)1894 2798 y FF(v)s FG(\()p FF(t)p FG(\))h FH(are)g(b)s(oth)f(solutions)f(of)h(the)h(same)e(di\033eren)m (tial)118 2911 y(equation)31 b FF("z)576 2878 y FD(0)626 2911 y FG(=)26 b(2)q(\026)-46 b FF(a)q FG(\()p FF(t)p FG(\))p FF(z)25 b FG(+)20 b(1)p FH(,)32 b(with)e(p)s(ossibly)f (di\033eren)m(t)i(initial)d(conditions.)41 b(If)30 b FF(x)2966 2925 y FC(0)3026 2911 y FE(\000)20 b FF(x)3169 2878 y Fy(?)3209 2911 y FG(\(0\))27 b(=)f FE(O)s FG(\()p FF(")p FG(\))p FH(,)118 3023 y FF(\020)7 b FG(\()p FF(t)p FG(\))41 b FH(is)e(an)i(adiabatic)f(solution)f(\(in)h(the)h(sense)f(of) h(Theorem)g(2.1\))g(of)f(the)h(di\033eren)m(tial)e(equation,)118 3136 y(sta)m(ying)30 b(close)f(to)i(the)g(equilibrium)26 b(branc)m(h)32 b FF(z)1737 3103 y Fy(?)1802 3136 y FG(=)25 b(1)p FF(=)p FE(j)p FG(2)q(\026)-46 b FF(a)r FG(\()p FF(t)p FG(\))p FE(j)p FH(.)259 3321 y(The)38 b(estimate)d(\(2.18\))j (has)f(b)s(een)g(designed)f(for)h(situations)e(where)j FF(\033)h FE(\034)d FG(1)p FH(,)j(and)e(is)f(useful)f(for)118 3434 y FF(\033)f FE(\034)d FF(h)h FE(\034)f FG(1)p FH(.)51 b(W)-8 b(e)34 b(exp)s(ect)g(the)g(exp)s(onen)m(t)h(to)f(b)s(e)f (optimal)f(in)h(this)g(case,)i(but)f(did)f(not)h(attempt)g(to)118 3547 y(optimize)h(the)i(prefactor)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))p FH(,)39 b(whic)m(h)e(leads)e(to)i(sub)s(exp)s (onen)m(tial)e(corrections.)59 b(If)36 b(w)m(e)h(assume,)118 3660 y(for)30 b(instance,)g(that)h FF(\033)e FG(=)c FF(")1048 3627 y Fy(q)1086 3660 y FH(,)30 b FF(q)f(>)24 b FG(0)p FH(,)31 b(and)g(tak)m(e)g FF(h)25 b FG(=)g FF(")1994 3627 y Fy(p)2065 3660 y FH(with)30 b FG(0)25 b FF(<)g(p)g(<)g(q)s FH(,)30 b(\(2.18\))h(can)g(b)s(e)f(written)h(as)164 3899 y Fo(P)225 3862 y FC(0)p Fy(;x)320 3871 y Fw(0)358 3826 y Fr(\010)411 3899 y FF(\034)451 3918 y FD(B)497 3926 y Fw(s)526 3918 y FC(\()p Fy(h)p FC(\))651 3899 y FF(<)25 b(t)780 3826 y Fr(\011)858 3899 y Fn(6)g FG(\()p FF(t)12 b FG(+)g FF(")1159 3862 y FC(2)1199 3899 y FG(\))j(exp)1388 3798 y Fr(n)1449 3899 y FE(\000)1660 3838 y FG(1)p 1530 3878 307 4 v 1530 3965 a(2)p FF(")1617 3939 y FC(2\()p Fy(q)r FD(\000)p Fy(p)p FC(\))1846 3826 y Fr(\002)1884 3899 y FG(1)d FE(\000)g(O)s FG(\()p FF(")p FG(\))g FE(\000)g(O)s FG(\()p FF(")2458 3862 y Fy(p)2499 3899 y FG(\))g FE(\000)g(O)s FG(\()p FF(")2781 3862 y FC(2\()p Fy(q)r FD(\000)p Fy(p)p FC(\))3000 3899 y FE(j)p FG(log)17 b FF(")p FE(j)p FG(\))3260 3826 y Fr(\003)3299 3798 y(o)3360 3899 y FF(:)45 b FH(\(2.20\))118 4124 y(The)38 b FF(t)p FH(-dep)s(endence)i(of)d(the)h(prefactor)h(is)e (to)h(b)s(e)f(exp)s(ected.)64 b(It)38 b(is)e(due)i(to)g(the)h(fact)f (that)g(as)g(time)118 4237 y(increases,)33 b(the)g(probabilit)m(y)f(of) g FF(x)1304 4251 y Fy(t)1366 4237 y FH(escaping)h(from)e(a)i(neigh)m(b) s(ourho)s(o)s(d)g(of)g FF(x)2802 4204 y FC(det)2802 4260 y Fy(t)2936 4237 y FH(also)f(increases,)h(but)118 4350 y(v)m(ery)d(slo)m(wly)e(if)g FF(\033)33 b FH(is)28 b(small.)38 b(The)30 b(estimate)e(\(2.18\))j(sho)m(ws)f(that)g(for)g(a)f(fraction)h FF(\015)k FH(of)c(tra)5 b(jectories)29 b(to)118 4463 y(lea)m(v)m(e)i(the)f(strip)g FE(B)769 4477 y FC(s)800 4463 y FG(\()p FF(h)p FG(\))p FH(,)i(w)m(e)f(ha)m(v)m(e)g(to)g(w)m(ait) f FB(at)j(le)-5 b(ast)31 b FH(for)f(a)g(time)f FF(t)2393 4477 y Fy(\015)2467 4463 y FH(giv)m(en)h(b)m(y)910 4723 y FE(j)p 945 4673 44 4 v FF(\013)p FG(\()p FF(t)1061 4737 y Fy(\015)1106 4723 y FG(\))p FE(j)c FG(=)f FF(\015)5 b(")1382 4686 y FC(2)1437 4723 y FG(exp)1576 4622 y Fr(n)1646 4662 y FG(1)p 1646 4702 46 4 v 1646 4786 a(2)1713 4662 y FF(h)1765 4629 y FC(2)p 1712 4702 95 4 v 1712 4786 a FF(\033)1767 4759 y FC(2)1816 4650 y Fr(\002)1854 4723 y FG(1)21 b FE(\000)f(O)s FG(\()p FF(")p FG(\))h FE(\000)f(O)s FG(\()p FF(h)p FG(\))2507 4650 y Fr(\003)2546 4622 y(o)2627 4723 y FE(\000)f FG(2)p FF(")2804 4686 y FC(2)2845 4723 y FF(;)560 b FH(\(2.21\))118 4945 y(whic)m(h)31 b(is)e(compatible)g (with)h(results)f(on)h(the)h(autonomous)g(case.)259 5058 y(Let)36 b(us)e(no)m(w)i(consider)f(the)g(unstable)f(case,)i(that)g (is,)e(w)m(e)i(no)m(w)g(assume)e(that)h(the)g(linearization)118 5171 y FF(a)p FG(\()p FF(t)p FG(\))26 b(=)f FF(@)439 5185 y Fy(x)483 5171 y FF(f)10 b FG(\()p FF(x)625 5138 y Fy(?)664 5171 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))26 b FH(satis\034es)e FF(a)p FG(\()p FF(t)p FG(\))i Fn(>)e FF(a)1549 5185 y FC(0)1614 5171 y FF(>)h FG(0)g FH(for)g(all)e FF(t)j FE(2)e FF(I)7 b FH(.)39 b(Theorem)25 b(2.1)h(sho)m(ws)f(the)g(existence)g(of)118 5283 y(a)j(particular)g (solution)j Fr(b)-55 b FF(x)1006 5250 y FC(det)998 5306 y Fy(t)1136 5283 y FH(of)28 b(the)h(deterministic)d(equation)i(\(2.6\)) h(suc)m(h)g(that)g FE(j)s Fr(b)-54 b FF(x)2994 5250 y FC(det)2986 5306 y Fy(t)3112 5283 y FE(\000)16 b FF(x)3251 5250 y Fy(?)3290 5283 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FF(c)3579 5297 y FC(1)3619 5283 y FF(")118 5396 y FH(for)30 b(all)f FF(t)c FE(2)g FF(I)7 b FH(.)40 b(W)-8 b(e)31 b(de\034ne)h FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1376 5410 y Fy(x)1420 5396 y FF(f)10 b FG(\()s Fr(b)-54 b FF(x)1570 5363 y FC(det)1562 5419 y Fy(t)1672 5396 y FF(;)15 b(t)p FG(\))25 b(=)g FF(a)p FG(\()p FF(t)p FG(\))c(+)f FE(O)s FG(\()p FF(")p FG(\))26 b FF(>)f FG(0)31 b FH(and)p 2735 5346 44 4 v 31 w FF(\013)p FG(\()p FF(t)p FG(\))26 b(=)3008 5323 y Fr(R)3068 5349 y Fy(t)3051 5428 y FC(0)3115 5396 y FG(\026)-47 b FF(a)p FG(\()p FF(s)p FG(\))15 b(d)p FF(s)p FH(.)259 5509 y(The)35 b(linearization)e(of)41 b(\(2.1\))35 b(around)k Fr(b)-54 b FF(x)1676 5476 y FC(det)1668 5532 y Fy(t)1812 5509 y FH(again)34 b(admits)f(a)i(solution)e(of)i(the)g(form)e(\(2.12\))q(.) 54 b(In)118 5622 y(this)30 b(case,)h(ho)m(w)m(ev)m(er,)j(the)d(v)-5 b(ariance)31 b(\(2.13\))h(gro)m(ws)g(exp)s(onen)m(tially)d(fast,)i(and) g(th)m(us)h(one)f(exp)s(ects)g(the)1867 5871 y(8)p eop %%Page: 9 9 9 8 bop 118 328 a FH(probabilit)m(y)34 b(of)h FF(x)745 342 y Fy(t)810 328 y FH(remaining)f(close)g(to)39 b Fr(b)-54 b FF(x)1634 295 y FC(det)1626 350 y Fy(t)1771 328 y FH(to)35 b(b)s(e)g(small.)53 b(This)34 b(is)g(the)i(con)m(ten)m(ts)h(of)e(the)h (second)118 441 y(main)29 b(result)h(of)g(this)f(section.)40 b(W)-8 b(e)31 b(in)m(tro)s(duce)f(the)h(set)940 645 y FE(B)1000 659 y FC(u)1043 645 y FG(\()p FF(h)p FG(\))26 b(=)1287 517 y Fr(\032)1355 645 y FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f Fo(R)44 b FE(\002)20 b FF(I)d FG(:)31 b FE(j)p FF(x)20 b FE(\000)k Fr(b)-55 b FF(x)2215 608 y FC(det)2207 668 y Fy(t)2317 645 y FE(j)26 b FF(<)2591 584 y(h)p 2474 624 288 4 v 2474 642 a Fr(p)p 2565 642 198 4 v 78 x FG(2)q(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))2772 517 y Fr(\033)3430 645 y FH(\(2.22\))118 889 y(and)31 b(the)g(\034rst)f(exit)f(time)g FF(\034)1060 907 y FD(B)1106 915 y Fw(u)1144 907 y FC(\()p Fy(h)p FC(\))1274 889 y FH(of)h FF(x)1429 903 y Fy(t)1488 889 y FH(from)g FE(B)1762 903 y FC(u)1805 889 y FG(\()p FF(h)p FG(\))p FH(.)118 1054 y Fq(Theorem)37 b(2.5)g(\(Unstable)i(case\).)44 b FB(Ther)-5 b(e)36 b(exist)f FF(")2026 1068 y FC(0)2101 1054 y FB(and)f FF(h)2330 1068 y FC(0)2370 1054 y FB(,)h(dep)-5 b(ending)36 b(only)e(on)h FF(f)10 b FB(,)34 b(such)h(that)118 1167 y(for)d(al)5 b(l)32 b FF(h)26 b Fn(6)f FF(\033)e FE(^)d FF(h)773 1181 y FC(0)813 1167 y FB(,)32 b(al)5 b(l)32 b FF(")26 b Fn(6)f FF(")1208 1181 y FC(0)1280 1167 y FB(and)32 b(al)5 b(l)32 b FF(x)1636 1181 y FC(0)1708 1167 y FB(satisfying)f FG(\()p FF(x)2196 1181 y FC(0)2235 1167 y FF(;)15 b FG(0\))27 b FE(2)e(B)2528 1181 y FC(u)2571 1167 y FG(\()p FF(h)p FG(\))p FB(,)33 b(we)g(have)1072 1399 y Fo(P)1133 1362 y FC(0)p Fy(;x)1228 1371 y Fw(0)1267 1326 y Fr(\010)1320 1399 y FF(\034)1360 1418 y FD(B)1406 1426 y Fw(u)1443 1418 y FC(\()p Fy(h)p FC(\))1568 1399 y Fn(>)25 b FF(t)1697 1326 y Fr(\011)1775 1399 y Fn(6)1871 1329 y FE(p)p 1947 1329 41 4 v 70 x FG(e)16 b(exp)2142 1298 y Fr(n)2202 1399 y FE(\000)p FF(\024)2335 1338 y(\033)2390 1305 y FC(2)p 2335 1378 95 4 v 2336 1462 a FF(h)2388 1435 y FC(2)p 2460 1288 44 4 v 2450 1338 a FF(\013)p FG(\()p FF(t)p FG(\))p 2450 1378 163 4 v 2510 1462 a FF(")2622 1298 y Fr(o)2682 1399 y FF(;)723 b FH(\(2.23\))118 1612 y FB(wher)-5 b(e)34 b FF(\024)25 b FG(=)568 1576 y Fy(\031)p 556 1591 67 4 v 556 1643 a FC(2e)633 1538 y Fr(\000)675 1612 y FG(1)20 b FE(\000)g(O)s FG(\()p FF(h)p FG(\))h FE(\000)f(O)s FG(\()p FF(")p FG(\))1327 1538 y Fr(\001)1370 1612 y FB(.)259 1777 y FH(The)33 b(pro)s(of,)g(giv)m(en)f(in)f(Section)h(3.2,)i(is)c(based)j(on)g(a)f (partition)f(of)h(the)h(in)m(terv)-5 b(al)31 b FG([0)p FF(;)15 b(t)p FG(])34 b FH(in)m(to)e(small)118 1890 y(in)m(terv)-5 b(als,)23 b(and)h(a)e(comparison)g(of)h(the)g(nonlinear)f(equation)h (with)f(its)f(linearization)g(on)i(eac)m(h)h(in)m(terv)-5 b(al.)259 2003 y(This)38 b(result)f(sho)m(ws)i(that)f FF(x)1256 2017 y Fy(t)1324 2003 y FH(is)f(unlik)m(ely)f(to)i(remain)f (in)h FE(B)2371 2017 y FC(u)2414 2003 y FG(\()p FF(h)p FG(\))h FH(as)f(so)s(on)g(as)g FF(t)g FE(\035)g FF("\033)3327 1970 y FC(2)3367 2003 y FF(=h)3464 1970 y FC(2)3505 2003 y FH(.)64 b(A)118 2116 y(ma)5 b(jor)36 b(limitation)d(of)43 b(\(2.23\))37 b(is)e(that)i(it)f(requires)g FF(h)g Fn(6)f FF(\033)s FH(.)59 b(Obtaining)36 b(an)h(estimate)e(for)i(larger)f FF(h)118 2228 y FH(is)30 b(p)s(ossible,)e(but)k(requires)e (considerably)g(more)g(w)m(ork.)43 b(W)-8 b(e)31 b(will)e(pro)m(vide)i (suc)m(h)g(an)g(estimate)f(in)g(the)118 2341 y(more)f(di\036cult,)f (but)h(also)f(more)g(in)m(teresting)h(case)g(of)f(the)h(pitc)m(hfork)g (bifurcation,)f(see)h(Theorem)g(2.9)118 2454 y(b)s(elo)m(w.)118 2693 y Fp(2.3)112 b(Pitc)m(hfork)36 b(bifurcation)118 2865 y FH(W)-8 b(e)35 b(no)m(w)h(consider)f(the)g(SDE)h(\(2.1\))f(in)f (the)i(case)e(of)h FF(f)44 b FH(undergoing)36 b(a)f(pitc)m(hfork)f (bifurcation.)54 b(W)-8 b(e)118 2978 y(will)28 b(assume)h(that)181 3115 y FE(\017)47 b FF(f)37 b FH(is)26 b(three)i(times)d(con)m(tin)m (uously)i(di\033eren)m(tiable)g(with)g(resp)s(ect)g(to)g FF(x)g FH(and)h FF(t)f FH(in)g(a)g(neigh)m(b)s(ourho)s(o)s(d)273 3227 y FE(N)348 3241 y FC(0)417 3227 y FH(of)j FG(\(0)p FF(;)15 b FG(0\))p FH(;)181 3340 y FE(\017)47 b FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)f FE(\000)p FF(f)10 b FG(\()p FE(\000)p FF(x;)15 b(t)p FG(\))29 b FH(for)i(all)d FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f(N)1712 3354 y FC(0)1751 3340 y FH(;)181 3453 y FE(\017)47 b FF(f)40 b FH(exhibits)28 b(a)j(sup)s(ercritical)d(pitc)m(hfork)i(bifurcation)g (at)g(the)h(origin,)e(i.e.)765 3631 y FF(@)813 3645 y Fy(x)858 3631 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))26 b(=)f(0)p FF(;)197 b(@)1550 3645 y Fy(tx)1619 3631 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))27 b FF(>)e FG(0)181 b FH(and)g FF(@)2597 3645 y Fy(xxx)2721 3631 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))27 b FF(<)e FG(0)p FF(:)261 b FH(\(2.24\))259 3808 y(The)34 b(assumption)e(that)i FF(f)42 b FH(b)s(e)34 b(o)s(dd)f(is)f(not)i(necessary)f(for)g(the)h (existence)e(of)h(a)h(pitc)m(hfork)f(bifur-)118 3921 y(cation.)48 b(Ho)m(w)m(ev)m(er,)35 b(the)e(deterministic)d(system)h(b) s(eha)m(v)m(es)j(v)m(ery)f(di\033eren)m(tly)e(if)h FF(x)d FG(=)g(0)k FH(is)e(not)i(alw)m(a)m(ys)118 4034 y(an)f(equilibrium.)41 b(The)32 b(most)f(natural)h(situation)e(in)h(whic)m(h)h FF(f)10 b FG(\(0)p FF(;)15 b(t)p FG(\))28 b(=)f(0)32 b FH(for)g(all)e FF(t)h FH(is)f(the)i(one)g(where)118 4147 y FF(f)40 b FH(is)28 b(o)s(dd.)259 4259 y(By)33 b(rescaling)g FF(x)g FH(and)h FF(t)p FH(,)g(w)m(e)g(ma)m(y)f(arrange)i (that)f FF(@)2048 4273 y Fy(tx)2117 4259 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))32 b(=)e(1)k FH(and)g FF(@)2811 4273 y Fy(xxx)2934 4259 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))32 b(=)e FE(\000)p FG(6)k FH(as)f(in)118 4372 y(the)g(standard)g(case)f FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))28 b(=)g FF(tx)21 b FE(\000)g FF(x)1475 4339 y FC(3)1515 4372 y FH(.)45 b(This)32 b(implies)c(in)j(particular)h (that)h(the)f(linearization)e(of)i FF(f)118 4485 y FH(at)f FF(x)25 b FG(=)g(0)30 b FH(satis\034es)1331 4598 y FF(a)p FG(\()p FF(t)p FG(\))c(=)f FF(@)1652 4612 y Fy(x)1696 4598 y FF(f)10 b FG(\(0)p FF(;)15 b(t)p FG(\))26 b(=)f FF(t)20 b FG(+)g FE(O)s FG(\()p FF(t)2348 4561 y FC(2)2388 4598 y FG(\))p FF(:)982 b FH(\(2.25\))118 4752 y(A)25 b(standard)h(result)f(of)g(bifurcation)g(theory)g([GH,)g(IJ])h(states)f (that)h(under)g(these)g(assumptions,)f(there)118 4864 y(is)j(a)g(neigh)m(b)s(ourho)s(o)s(d)i FE(N)38 b(\032)25 b(N)1182 4878 y FC(0)1250 4864 y FH(of)j FG(\(0)p FF(;)15 b FG(0\))31 b FH(in)d(whic)m(h)h(the)g(only)f(solutions)f(of)i FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))25 b(=)g(0)k FH(are)h(the)f (line)118 4977 y FF(x)c FG(=)g(0)31 b FH(and)g(the)f(curv)m(es)906 5155 y FF(x)c FG(=)f FE(\006)p FF(x)1203 5117 y Fy(?)1242 5155 y FG(\()p FF(t)p FG(\))p FF(;)196 b(x)1618 5117 y Fy(?)1658 5155 y FG(\()p FF(t)p FG(\))26 b(=)1883 5076 y FE(p)p 1958 5076 33 4 v 1958 5155 a FF(t)1991 5081 y Fr(\002)2029 5155 y FG(1)21 b(+)f FD(O)2243 5169 y Fy(t)2273 5155 y FG(\(1\))2388 5081 y Fr(\003)2427 5155 y FF(;)196 b(t)25 b Fn(>)g FG(0)p FF(:)558 b FH(\(2.26\))118 5332 y(If)36 b FE(N)50 b FH(is)35 b(small)f(enough,)40 b(the)d(equilibrium)c FF(x)k FG(=)f(0)h FH(is)e(stable)h(for)h FF(t)f(<)g FG(0)h FH(and)h(unstable)e(for)h FF(t)f(>)g FG(0)p FH(,)118 5445 y(while)29 b FF(x)c FG(=)g FE(\006)p FF(x)650 5412 y Fy(?)689 5445 y FG(\()p FF(t)p FG(\))31 b FH(are)g(stable)e(equilibria)f(with)i(linearization)1125 5622 y FF(a)1173 5585 y Fy(?)1213 5622 y FG(\()p FF(t)p FG(\))c(=)f FF(@)1486 5636 y Fy(x)1530 5622 y FF(f)10 b FG(\()p FF(x)1672 5585 y Fy(?)1711 5622 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))26 b(=)f FE(\000)p FG(2)p FF(t)2193 5549 y Fr(\002)2231 5622 y FG(1)c(+)e FD(O)2445 5636 y Fy(t)2475 5622 y FG(\(1\))2590 5549 y Fr(\003)2629 5622 y FF(:)776 b FH(\(2.27\))1867 5871 y(9)p eop %%Page: 10 10 10 9 bop 118 328 a FH(The)31 b(only)e(solutions)g(of)h FF(@)1031 342 y Fy(x)1075 328 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)f(0)30 b FH(in)g FE(N)43 b FH(are)30 b(the)h(curv)m(es)893 512 y FF(x)25 b FG(=)g FE(\006)6 b FG(\026)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FF(;)202 b FG(\026)-51 b FF(x)p FG(\()p FF(t)p FG(\))26 b(=)1790 430 y Fr(p)p 1881 430 124 4 v 82 x FF(t=)p FG(3)2005 439 y Fr(\002)2043 512 y FG(1)20 b(+)g FD(O)2257 526 y Fy(t)2287 512 y FG(\(1\))2402 439 y Fr(\003)2441 512 y FF(;)196 b(t)25 b Fn(>)g FG(0)p FF(:)544 b FH(\(2.28\))118 697 y(If)35 b FF(f)45 b FH(is)34 b(four)h(times)f(con)m(tin)m(uously)i (di\033eren)m(tiable,)g(the)g(terms)f FD(O)2421 711 y Fy(t)2450 697 y FG(\(1\))i FH(in)e(the)h(last)e(three)j(equations)118 810 y(can)31 b(b)s(e)f(replaced)g(b)m(y)h FE(O)s FG(\()p FF(t)p FG(\))p FH(.)259 923 y(W)-8 b(e)31 b(brie\035y)f(state)g(what)h (is)e(kno)m(wn)i(for)f(the)h(deterministic)d(equation)1579 1145 y FF(")1631 1084 y FG(d)p FF(x)1734 1098 y Fy(t)p 1631 1125 133 4 v 1656 1208 a FG(d)p FF(t)1799 1145 y FG(=)d FF(f)10 b FG(\()p FF(x)2037 1159 y Fy(t)2066 1145 y FF(;)15 b(t)p FG(\))p FF(;)1231 b FH(\(2.29\))118 1356 y(where)42 b(w)m(e)g(tak)m(e)g(an)g(initial)c(condition)i FG(\()p FF(x)1644 1370 y FC(0)1684 1356 y FF(;)15 b(t)1757 1370 y FC(0)1797 1356 y FG(\))44 b FE(2)f(N)54 b FH(with)41 b FF(x)2378 1370 y FC(0)2461 1356 y FF(>)i FG(0)f FH(and)g FF(t)2882 1370 y FC(0)2965 1356 y FF(<)h FG(0)p FH(,)h(see)d(Fig.)36 b(1b.)118 1469 y(Observ)m(e)31 b(that)g FF(\013)p FG(\()p FF(t;)15 b(t)860 1483 y FC(0)900 1469 y FG(\))26 b(=)1057 1396 y Fr(R)1118 1422 y Fy(t)1100 1501 y(t)1125 1510 y Fw(0)1179 1469 y FF(a)p FG(\()p FF(s)p FG(\))15 b(d)p FF(s)30 b FH(is)f(decreasing)h(for)h FF(t)2179 1483 y FC(0)2243 1469 y FF(<)25 b(t)g(<)g FG(0)31 b FH(and)f(increasing)g(for) g FF(t)25 b(>)g FG(0)p FH(.)118 1640 y Fq(De\034nition)36 b(2.6.)41 b FB(The)32 b FH(bifurcation)e(dela)m(y)i FB(is)g(de\034ne)-5 b(d)33 b(as)1249 1825 y FG(\005\()p FF(t)1385 1839 y FC(0)1425 1825 y FG(\))26 b(=)f(inf)1692 1752 y Fr(\010)1745 1825 y FF(t)g(>)g FG(0)10 b(:)32 b FF(\013)p FG(\()p FF(t;)15 b(t)2210 1839 y FC(0)2250 1825 y FG(\))25 b FF(>)g FG(0)2451 1752 y Fr(\011)2505 1825 y FF(;)900 b FH(\(2.30\))118 2010 y FB(with)32 b(the)h(c)-5 b(onvention)33 b FG(\005\()p FF(t)1054 2024 y FC(0)1094 2010 y FG(\))25 b(=)g FE(1)33 b FB(if)e FF(\013)p FG(\()p FF(t;)15 b(t)1660 2024 y FC(0)1700 2010 y FG(\))26 b FF(<)f FG(0)33 b FB(for)f(al)5 b(l)32 b FF(t)25 b(>)g FG(0)p FB(,)32 b(for)g(which)h FF(\013)p FG(\()p FF(t;)15 b(t)3064 2024 y FC(0)3104 2010 y FG(\))33 b FB(is)f(de\034ne)-5 b(d.)259 2181 y FH(One)33 b(easily)e(sho)m(ws)i(that)g FG(\005\()p FF(t)1300 2195 y FC(0)1340 2181 y FG(\))f FH(is)g(di\033eren)m(tiable)f(for)i FF(t)2219 2195 y FC(0)2290 2181 y FH(su\036cien)m(tly)f(close)g(to)h FG(0)p FH(,)g(and)h(satis\034es)118 2294 y FG(lim)244 2308 y Fy(t)269 2317 y Fw(0)304 2308 y FD(!)p FC(0)p FD(\000)484 2294 y FG(\005\()p FF(t)620 2308 y FC(0)660 2294 y FG(\))26 b(=)f(0)30 b FH(and)h FG(lim)1194 2308 y Fy(t)1219 2317 y Fw(0)1255 2308 y FD(!)p FC(0)p FD(\000)1435 2294 y FG(\005)1503 2261 y FD(0)1526 2294 y FG(\()p FF(t)1594 2308 y FC(0)1634 2294 y FG(\))25 b(=)g FE(\000)p FG(1)p FH(.)118 2466 y Fq(Theorem)33 b(2.7)g(\(Deterministic)j(case\).)42 b FB(L)-5 b(et)32 b FF(x)1916 2433 y FC(det)1916 2488 y Fy(t)2050 2466 y FB(b)-5 b(e)32 b(the)h(solution)e(of)50 b FH(\(2.29\))33 b FB(with)e(initial)f(c)-5 b(on-)118 2579 y(dition)32 b FF(x)431 2546 y FC(det)431 2601 y Fy(t)456 2610 y Fw(0)558 2579 y FG(=)25 b FF(x)706 2593 y FC(0)745 2579 y FB(.)41 b(Then)33 b(ther)-5 b(e)33 b(exist)g(c)-5 b(onstants)33 b FF(")1923 2593 y FC(0)1962 2579 y FB(,)f FF(c)2061 2593 y FC(0)2101 2579 y FB(,)g FF(c)2200 2593 y FC(1)2272 2579 y FB(dep)-5 b(ending)34 b(only)e(on)g FF(f)10 b FB(,)31 b(and)h(times)1259 2769 y FF(t)1292 2783 y FC(1)1356 2769 y FG(=)25 b FF(t)1485 2783 y FC(0)1545 2769 y FG(+)20 b FE(O)s FG(\()p FF(")p FE(j)p FG(log)d FF(")p FE(j)p FG(\))1259 2907 y FF(t)1292 2921 y FC(2)1356 2907 y FG(=)25 b(\005\()p FF(t)1588 2921 y FC(1)1628 2907 y FG(\))h(=)f(\005\()p FF(t)1921 2921 y FC(0)1961 2907 y FG(\))20 b FE(\000)g(O)s FG(\()p FF(")p FE(j)p FG(log)e FF(")p FE(j)p FG(\))1259 3045 y FF(t)1292 3059 y FC(3)1356 3045 y FG(=)25 b(\005\()p FF(t)1588 3059 y FC(0)1628 3045 y FG(\))c(+)f FE(O)s FG(\()p FF(")p FE(j)p FG(log)d FF(")p FE(j)p FG(\))3430 2908 y FH(\(2.31\))118 3238 y FB(such)33 b(that,)f(if)g FG(0)26 b FF(<)f(x)842 3252 y FC(0)906 3238 y Fn(6)g FF(c)1041 3252 y FC(0)1081 3238 y FB(,)32 b FG(0)26 b FF(<)f(")g Fn(6)g FF(")1513 3252 y FC(0)1585 3238 y FB(and)33 b FG(\()p FF(x)1848 3205 y FC(det)1848 3261 y Fy(t)1950 3238 y FF(;)15 b(t)p FG(\))26 b FE(2)f(N)13 b FB(,)1072 3345 y Fr(\()1145 3437 y FG(0)26 b FF(<)f(x)1364 3404 y FC(det)1364 3459 y Fy(t)1491 3437 y Fn(6)g FF(c)1626 3451 y FC(1)1666 3437 y FF(")15 b FG(e)1764 3404 y Fy(\013)p FC(\()p Fy(t;t)1906 3413 y Fw(1)1942 3404 y FC(\))p Fy(=")2134 3437 y FB(for)32 b FF(t)2311 3451 y FC(1)2375 3437 y Fn(6)25 b FF(t)g Fn(6)g FF(t)2658 3451 y FC(2)1145 3572 y FE(j)p FF(x)1222 3539 y FC(det)1222 3595 y Fy(t)1345 3572 y FE(\000)20 b FF(x)1488 3539 y Fy(?)1527 3572 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FF(c)1816 3586 y FC(1)1856 3572 y FF(")236 b FB(for)32 b FF(t)25 b Fn(>)g FF(t)2465 3586 y FC(3)2504 3572 y FB(.)3430 3500 y FH(\(2.32\))259 3757 y(The)32 b(pro)s(of)e(is)g(a)h(straigh)m(tforw)m(ard)h (consequence)g(of)e(di\033eren)m(tial)g(inequalities,)e(see)j(for)f (instance)118 3869 y([Ber)q(,)g(Prop)s(ositions)f(4.6)i(and)f(4.8].)259 3982 y(W)-8 b(e)35 b(no)m(w)g(consider)f(the)g(SDE)h(\(2.1\))g(for)f FF(\033)h(>)d FG(0)p FH(.)52 b(The)35 b(results)e(in)g(this)g(section)h (are)h(only)e(in)m(ter-)118 4095 y(esting)i(for)g FF(\033)i FG(=)c FD(O)r FG(\()814 4030 y FE(p)p 890 4030 43 4 v 65 x FF(")p FG(\))p FH(,)k(while)d(one)i(of)f(them)g(\(Theorem)h(2.9\)) g FB(r)-5 b(e)g(quir)g(es)37 b FH(a)f(condition)e(of)h(the)h(form)118 4208 y FF(\033)s FE(j)p FG(log)17 b FF(\033)s FE(j)411 4175 y FC(3)p Fy(=)p FC(2)547 4208 y FG(=)25 b FE(O)s FG(\()753 4143 y FE(p)p 829 4143 V 65 x FF(")p FG(\))31 b FH(\(where)g(w)m(e)g(ha)m(v)m(e)h(not)e(tried)g(to)h(optimize)d(the)j (exp)s(onen)m(t)g FG(3)p FF(=)p FG(2)p FH(\).)259 4321 y(Let)41 b(us)e(\034x)g(an)h(initial)d(condition)i FG(\()p FF(x)1592 4335 y Fy(t)1617 4344 y Fw(0)1656 4321 y FF(;)15 b(t)1729 4335 y FC(0)1769 4321 y FG(\))41 b FE(2)g(N)52 b FH(with)39 b FF(t)2322 4335 y FC(0)2402 4321 y FF(<)i FG(0)p FH(.)69 b(F)-8 b(or)40 b(an)m(y)g FF(T)54 b FE(2)40 b FG(\(0)p FF(;)15 b FE(j)p FF(t)3390 4335 y FC(0)3431 4321 y FE(j)p FG(\))p FH(,)42 b(w)m(e)118 4434 y(can)32 b(apply)f(Theorem)h(2.3)f(on)h(the)g(in)m(terv)-5 b(al)30 b FG([)p FF(t)1733 4448 y FC(0)1773 4434 y FF(;)15 b FE(\000)p FF(T)e FG(])31 b FH(to)h(sho)m(w)g(that)g FE(j)p FF(x)2617 4448 y FD(\000)p Fy(T)2727 4434 y FE(j)g FH(is)e(lik)m(ely)f (to)i(b)s(e)g(of)h(order)118 4547 y FF(\033)173 4514 y FC(1)p FD(\000)p Fy(\016)317 4547 y FG(+)14 b FF(c)441 4561 y FC(1)481 4547 y FF(")h FG(e)579 4514 y Fy(\013)p FC(\()p FD(\000)p Fy(T)6 b(;t)798 4523 y Fw(1)833 4514 y FC(\))p Fy(=")961 4547 y FH(for)27 b(an)m(y)i FF(\016)g(>)c FG(0)p FH(.)39 b(W)-8 b(e)29 b(can)f(also)f(apply)g(the)h(theorem)g (for)g FF(t)d(>)g(T)40 b FH(to)28 b(sho)m(w)h(that)118 4660 y(the)k(curv)m(es)f FE(\006)p FF(x)677 4627 y Fy(?)716 4660 y FG(\()p FF(t)p FG(\))h FH(attract)g(nearb)m(y)g(tra)5 b(jectories.)46 b(Hence)33 b(there)g(is)d(no)j(limitation)c(in)i (considering)118 4773 y(the)41 b(SDE)g(\(2.1\))g(in)f(a)h(domain)e(of)h (the)h(form)f FE(j)p FF(x)p FE(j)j Fn(6)f FF(d)p FH(,)h FE(j)p FF(t)p FE(j)g Fn(6)f FF(T)53 b FH(where)41 b FF(d)g FH(and)g FF(T)53 b FH(can)41 b(b)s(e)f(tak)m(en)118 4886 y(small)29 b(\(indep)s(enden)m(tly)i(of)g FF(")h FH(and)g FF(\033)i FH(of)d(course!\),)i(with)e(an)g(initial)e(condition)h FF(x)2931 4900 y FD(\000)p Fy(T)3068 4886 y FG(=)d FF(x)3218 4900 y FC(0)3289 4886 y FH(satisfying)118 4999 y FE(j)p FF(x)195 5013 y FC(0)235 4999 y FE(j)e Fn(6)g FF(d)p FH(.)259 5111 y(W)-8 b(e)37 b(\034rst)f(sho)m(w)h(that)g FF(x)1097 5125 y Fy(t)1163 5111 y FH(is)e(lik)m(ely)f(to)i(remain)g (small)d(for)j FE(\000)p FF(T)48 b Fn(6)35 b FF(t)h Fn(6)2768 5046 y FE(p)p 2843 5046 V 2843 5111 a FF(")q FH(.)58 b(A)m(ctually)-8 b(,)38 b(it)d(turns)118 5224 y(out)42 b(to)g(b)s(e)f(con)m(v)m(enien)m(t)j(to)d(sho)m(w)i(that)f FF(x)1621 5238 y Fy(t)1692 5224 y FH(remains)e(close)h(to)h(the)g (solution)e FF(x)2962 5238 y FC(0)3017 5224 y FG(e)3057 5191 y Fy(\013)p FC(\()p Fy(t;)p FD(\000)p Fy(T)10 b FC(\))p Fy(=")3422 5224 y FH(of)41 b(the)118 5337 y(linearization)28 b(of)37 b(\(2.29\))q(.)j(W)-8 b(e)31 b(de\034ne)g(the)g(\020v)-5 b(ariance-lik)m(e\021)36 b(function)961 5578 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b(=)1395 5517 y(1)p 1242 5558 352 4 v 1242 5641 a(2)p FE(j)p FF(a)p FG(\()p FE(\000)p FF(T)13 b FG(\))p FE(j)1619 5578 y FG(e)1659 5541 y FC(2)p Fy(\013)p FC(\()p Fy(t;)p FD(\000)p Fy(T)d FC(\))p Fy(=")2032 5578 y FG(+)2113 5517 y(1)p 2113 5558 46 4 v 2115 5641 a FF(")2184 5455 y Fr(Z)2275 5481 y Fy(t)2234 5661 y FD(\000)p Fy(T)2359 5578 y FG(e)2400 5541 y FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2700 5578 y FG(d)p FF(s:)611 b FH(\(2.33\))1845 5871 y(10)p eop %%Page: 11 11 11 10 bop 118 328 a FH(W)-8 b(e)38 b(shall)e(sho)m(w)i(in)f(Lemma)f (4.2)i(that)g(for)g(su\036cien)m(tly)f(small)d FF(")p FH(,)40 b(there)e(exist)e(constan)m(ts)j FF(c)3391 342 y FD(\006)3488 328 y FH(suc)m(h)118 441 y(that)851 568 y FF(c)890 582 y FD(\000)p 851 608 99 4 v 859 692 a FE(j)p FF(t)p FE(j)985 629 y Fn(6)25 b FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1362 568 y FF(c)1401 582 y FC(+)p 1362 608 V 1370 692 a FE(j)p FF(t)p FE(j)2194 629 y FH(for)30 b FE(\000)p FF(T)38 b Fn(6)25 b FF(t)g Fn(6)g FE(\000)2815 559 y(p)p 2890 559 43 4 v 2890 629 a FF(")q FH(,)472 b(\(2.34\))842 787 y FF(c)881 801 y FD(\000)p 832 827 119 4 v 832 846 a FE(p)p 907 846 43 4 v 907 911 a FF(")985 848 y Fn(6)25 b FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1372 787 y FF(c)1411 801 y FC(+)p 1362 827 119 4 v 1362 846 a FE(p)p 1438 846 43 4 v 65 x FF(")2194 848 y FH(for)30 b FE(\000)2403 778 y(p)p 2479 778 V 70 x FF(")25 b Fn(6)g FF(t)g Fn(6)2796 778 y FE(p)p 2872 778 V 70 x FF(")p FH(.)491 b(\(2.35\))118 1085 y(The)31 b(function)f FF(\020)7 b FG(\()p FF(t)p FG(\))30 b FH(is)f(used)h(to)h(de\034ne)g (the)g(strip)453 1274 y FE(B)s FG(\()p FF(h)p FG(\))26 b(=)760 1201 y Fr(\010)813 1274 y FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f FG([)p FE(\000)p FF(d;)15 b(d)10 b FG(])22 b FE(\002)e FG([)p FE(\000)p FF(T)8 b(;)1695 1204 y FE(p)p 1770 1204 V 1770 1274 a FF(")16 b FG(])10 b(:)31 b FE(j)p FF(x)20 b FE(\000)g FF(x)2159 1288 y FC(0)2214 1274 y FG(e)2254 1237 y Fy(\013)p FC(\()p Fy(t;)p FD(\000)p Fy(T)10 b FC(\))p Fy(=")2577 1274 y FE(j)26 b FF(<)f(h)2776 1192 y Fr(p)p 2867 1192 151 4 v 82 x FF(\020)7 b FG(\()p FF(t)p FG(\))3017 1201 y Fr(\011)3070 1274 y FF(:)335 b FH(\(2.36\))118 1463 y(Let)31 b FF(\034)320 1482 y FD(B)r FC(\()p Fy(h)p FC(\))498 1463 y FH(denote)g(the)g (\034rst)f(exit)f(time)g(of)h FF(x)1670 1477 y Fy(t)1730 1463 y FH(from)f FE(B)s FG(\()p FF(h)p FG(\))p FH(.)118 1638 y Fq(Theorem)g(2.8)g(\(Beha)m(viour)h(for)f FF(t)c Fn(6)1558 1572 y FE(p)p 1634 1572 43 4 v 66 x FF(")16 b Fq(\).)38 b FB(Ther)-5 b(e)29 b(exist)g(c)-5 b(onstants)28 b FF(")2694 1652 y FC(0)2762 1638 y FB(and)h FF(h)2986 1652 y FC(0)3025 1638 y FB(,)g(dep)-5 b(ending)30 b(only)118 1751 y(on)i FF(f)10 b FB(,)31 b FF(T)45 b FB(and)33 b FF(d)p FB(,)f(such)h(that)f(for)h FG(0)25 b FF(<)g(")h Fn(6)f FF(")1648 1765 y FC(0)1688 1751 y FB(,)32 b FF(h)25 b Fn(6)g FF(h)1973 1765 y FC(0)2013 1685 y FE(p)p 2089 1685 V 66 x FF(")p FB(,)32 b FE(j)p FF(x)2268 1765 y FC(0)2308 1751 y FE(j)25 b Fn(6)g FF(h=")2593 1718 y FC(1)p Fy(=)p FC(4)2737 1751 y FB(and)32 b FE(\000)p FF(T)38 b Fn(6)25 b FF(t)g Fn(6)3324 1685 y FE(p)p 3399 1685 V 3399 1751 a FF(")q FB(,)655 1995 y Fo(P)716 1958 y FD(\000)p Fy(T)6 b(;x)878 1967 y Fw(0)916 1921 y Fr(\010)969 1995 y FF(\034)1009 2014 y FD(B)r FC(\()p Fy(h)p FC(\))1182 1995 y FF(<)25 b(t)1311 1921 y Fr(\011)1389 1995 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)1897 1867 y Fr(\032)1965 1995 y FE(\000)2046 1934 y FG(1)p 2046 1974 46 4 v 2046 2057 a(2)2112 1934 y FF(h)2164 1901 y FC(2)p 2111 1974 95 4 v 2111 2057 a FF(\033)2166 2031 y FC(2)2216 1867 y Fr(\024)2264 1995 y FG(1)20 b FE(\000)g FF(r)s FG(\()p FF(")p FG(\))h FE(\000)f(O)2763 1867 y Fr(\022)2840 1934 y FF(h)2892 1901 y FC(2)p 2840 1974 92 4 v 2864 2057 a FF(")2942 1867 y Fr(\023)o(\025\033)3430 1995 y FH(\(2.37\))118 2228 y FB(wher)-5 b(e)1333 2366 y FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))26 b(=)1722 2305 y FE(j)p FF(\013)p FG(\()p FF(t;)15 b FE(\000)p FF(T)e FG(\))p FE(j)22 b FG(+)e FE(O)s FG(\()p FF(")p FG(\))p 1722 2345 689 4 v 2026 2429 a FF(")2068 2402 y FC(2)2421 2366 y FF(;)984 b FH(\(2.38\))118 2556 y FB(and)32 b(with)g FF(r)s FG(\()p FF(")p FG(\))26 b(=)f FE(O)s FG(\()p FF(")p FG(\))33 b FB(for)f FE(\000)p FF(T)38 b Fn(6)25 b FF(t)g Fn(6)g FE(\000)1614 2490 y(p)p 1690 2490 43 4 v 66 x FF(")p FB(,)32 b(and)g FF(r)s FG(\()p FF(")p FG(\))26 b(=)f FE(O)s FG(\()2355 2490 y FE(p)p 2431 2490 V 66 x FF(")p FG(\))33 b FB(for)f FE(\000)2756 2490 y(p)p 2832 2490 V 66 x FF(")25 b Fn(6)g FF(t)g Fn(6)3149 2490 y FE(p)p 3225 2490 V 66 x FF(")p FB(.)259 2731 y FH(The)33 b(pro)s(of)f(\(giv)m(en)g(in)f(Section)i(4.2\))f(and)h(the)f(in)m (terpretation)h(of)f(this)f(result)h(are)g(v)m(ery)h(close)e(in)118 2844 y(spirit)d(to)j(those)f(of)g(Theorem)g(2.3.)41 b(The)30 b(only)g(di\033erence)g(lies)e(in)h(the)h(kind)g(of)f FF(")p FH(-dep)s(endence)j(of)e(the)118 2957 y(error)i(terms.)43 b(The)32 b(estimate)e(\(2.37\))i(is)e(useful)h(when)h FF(\033)e FE(\034)d FF(h)g FE(\034)2429 2891 y(p)p 2505 2891 V 66 x FF(")p FH(,)32 b(and)g(sho)m(ws)g(that)f(the)h(t)m(ypical) 118 3069 y(spreading)f(of)f(paths)h(around)h(the)f(deterministic)d (solution)i(will)e(slo)m(wly)h(gro)m(w)j(un)m(til)e FF(t)c FG(=)3256 3004 y FE(p)p 3332 3004 V 65 x FF(")q FH(,)k(where)118 3182 y(it)g(is)e(of)i(order)h FF(\033)s(=")780 3149 y FC(1)p Fy(=)p FC(4)892 3182 y FH(,)f(see)g(Fig.)24 b(2.)259 3295 y(Let)32 b(us)f(no)m(w)h(examine)e(what)i(happ)s(ens)f(for)g FF(t)c Fn(>)1961 3230 y FE(p)p 2037 3230 V 65 x FF(")q FH(.)42 b(W)-8 b(e)32 b(\034rst)f(sho)m(w)h(that)g FF(x)2965 3309 y Fy(t)3025 3295 y FH(is)e(lik)m(ely)f(to)i(lea)m(v)m(e)118 3408 y(quite)j(so)s(on)h(a)f(suitably)f(de\034ned)j(region)f FE(D)i FH(con)m(taining)e(the)g(line)f FF(x)f FG(=)f(0)p FH(.)54 b(The)36 b(b)s(oundary)f(of)f FE(D)k FH(is)118 3521 y(de\034ned)31 b(through)g(a)f(function)36 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FH(,)31 b(whic)m(h)f(can)h(b)s(e)e(c)m (hosen)i(somewhat)f(arbitrarily)-8 b(,)29 b(but)i(should)e(lie)118 3634 y(b)s(et)m(w)m(een)39 b FG(\026)-51 b FF(x)p FG(\()p FF(t)p FG(\))32 b FH(and)g FF(x)884 3601 y Fy(?)924 3634 y FG(\()p FF(t)p FG(\))p FH(,)g(in)f(order)h(to)g(simplify)27 b(the)32 b(analysis)d(of)i(the)h(dynamics)e(after)i FF(x)3310 3648 y Fy(t)3371 3634 y FH(has)f(left)118 3747 y FE(D)s FH(.)40 b(A)30 b(con)m(v)m(enien)m(t)i(de\034nition)e(is)1575 3860 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))26 b(=)1846 3778 y FE(p)p 1922 3778 54 4 v 82 x FF(\025)15 b(x)2042 3822 y Fy(?)2081 3860 y FG(\()p FF(t)p FG(\))p FF(;)1221 b FH(\(2.39\))118 4019 y(where)30 b FF(\025)f FH(is)f(a)h(free)g (parameter.)41 b(W)-8 b(e)29 b(need)h(to)f(assume,)g(ho)m(w)m(ev)m(er,) j(that)d FF(\025)c FE(2)g FG(\()2847 3983 y FC(1)p 2847 3998 36 4 v 2847 4050 a(3)2893 4019 y FF(;)2943 3983 y FC(1)p 2943 3998 V 2943 4050 a(2)2989 4019 y FG(\))p FH(.)40 b(W)-8 b(e)29 b(no)m(w)h(de\034ne)1025 4208 y FE(D)e FG(=)1219 4134 y Fr(\010)1272 4208 y FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f FG([)p FE(\000)p FF(d;)15 b(d)10 b FG(])22 b FE(\002)d FG([)1981 4138 y FE(p)p 2057 4138 43 4 v 70 x FF(")q(;)c(T)e FG(])d(:)31 b FE(j)p FF(x)p FE(j)26 b FF(<)k FG(~)-50 b FF(x)p FG(\()p FF(t)p FG(\))2676 4134 y Fr(\011)2729 4208 y FF(:)676 b FH(\(2.40\))118 4397 y(Note)31 b(that)f FE(D)j FH(has)d(the)h(prop)s(ert)m(y)g(that)g (for)f(all)f FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f(D)33 b FH(with)d FF(x)25 b FE(6)p FG(=)g(0)p FH(,)1012 4562 y FG(1)p 1009 4603 52 4 v 1009 4686 a FF(x)1070 4624 y(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))26 b Fn(>)f FF(\024a)p FG(\()p FF(t)p FG(\))182 b FH(with)29 b FF(\024)d FG(=)f(1)c FE(\000)f FF(\025)g FE(\000)g FD(O)2585 4638 y Fy(T)2640 4624 y FG(\(1\))p FH(.)650 b(\(2.41\))118 4833 y(Let)31 b FF(\034)320 4847 y FD(D)411 4833 y FH(denote)g(the)g(\034rst)f(exit)f (time)g(of)h FF(x)1583 4847 y Fy(t)1643 4833 y FH(from)f FE(D)s FH(.)118 5008 y Fq(Theorem)i(2.9)h(\(Escap)s(e)h(from)d FE(D)s Fq(\).)40 b FB(L)-5 b(et)30 b FG(\()p FF(x)1774 5022 y FC(0)1814 5008 y FF(;)15 b(t)1887 5022 y FC(0)1927 5008 y FG(\))25 b FE(2)g(D)33 b FB(and)d(assume)h(that)f FF(\033)s FE(j)p FG(log)17 b FF(\033)s FE(j)3138 4975 y FC(3)p Fy(=)p FC(2)3274 5008 y FG(=)25 b FE(O)s FG(\()3480 4942 y FE(p)p 3556 4942 43 4 v 66 x FF(")p FG(\))p FB(.)118 5121 y(Then)32 b(for)g FF(t)530 5135 y FC(0)595 5121 y Fn(6)25 b FF(t)g Fn(6)g FF(T)13 b FB(,)367 5367 y Fo(P)428 5329 y Fy(t)453 5338 y Fw(0)489 5329 y Fy(;x)549 5338 y Fw(0)587 5293 y Fr(\010)640 5367 y FF(\034)680 5381 y FD(D)766 5367 y Fn(>)25 b FF(t)895 5293 y Fr(\011)973 5367 y Fn(6)g FF(C)1134 5381 y FC(0)1194 5367 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))1343 5285 y Fr(p)p 1435 5285 152 4 v 1435 5367 a FF(a)p FG(\()p FF(t)p FG(\))1606 5305 y FE(j)p FG(log)17 b FF(\033)s FE(j)p 1606 5346 239 4 v 1698 5429 a FF(\033)1855 5239 y Fr(\022)1922 5367 y FG(1)j(+)2088 5305 y FF(\013)p FG(\()p FF(t;)15 b(t)2287 5319 y FC(0)2327 5305 y FG(\))p 2088 5346 275 4 v 2204 5429 a FF(")2373 5239 y Fr(\023)2591 5305 y FG(e)2631 5273 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;t)2869 5282 y Fw(0)2905 5273 y FC(\))p Fy(=")p 2450 5346 697 4 v 2450 5364 a Fr(p)p 2541 5364 606 4 v 90 x FG(1)20 b FE(\000)g FG(e)2738 5427 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3011 5436 y Fw(0)3046 5427 y FC(\))p Fy(=")3156 5367 y FF(;)249 b FH(\(2.42\))118 5622 y FB(wher)-5 b(e)34 b FF(C)438 5636 y FC(0)502 5622 y FF(>)25 b FG(0)33 b FB(is)f(a)g(\(numeric)-5 b(al\))34 b(c)-5 b(onstant.)1845 5871 y FH(11)p eop %%Page: 12 12 12 11 bop 259 328 a FH(The)28 b(pro)s(of)f(of)g(this)f(result)g(\(giv)m (en)h(in)g(Section)g(4.3\))h(is)e(b)m(y)h(far)g(the)g(most)g(in)m(v)m (olv)m(ed)g(of)g(the)g(presen)m(t)118 441 y(w)m(ork.)73 b(W)-8 b(e)41 b(start)g(b)m(y)g(estimating,)h(in)e(a)h(similar)d(w)m(a) m(y)k(as)e(in)g(Theorem)h(2.5,)k(the)c(\034rst)g(exit)f(time)118 553 y(from)28 b(a)g(strip)f FE(S)35 b FH(of)28 b(width)g(sligh)m(tly)e (larger)i(than)h FF(\033)s(=)1935 476 y Fr(p)p 2027 476 162 4 v 2027 553 a FF(a)p FG(\()p FF(s)p FG(\))p FH(.)40 b(The)29 b(probabilit)m(y)e(of)h(returning)g(to)h(zero)118 666 y(after)38 b(lea)m(ving)e FE(S)44 b FH(can)38 b(b)s(e)f(estimated;) j(it)c(is)g(small)e(but)k(not)g(exp)s(onen)m(tially)d(small.)59 b(Ho)m(w)m(ev)m(er,)41 b(the)118 779 y(probabilit)m(y)26 b(of)i(neither)f(lea)m(ving)g FE(D)j FH(nor)e(returning)g(to)g(zero)g (is)e(exp)s(onen)m(tially)g(small.)37 b(This)26 b(fact)i(can)118 892 y(b)s(e)i(used)h(to)f(devise)f(an)i(iterativ)m(e)e(sc)m(heme)i (that)g(leads)e(to)i(the)f(exp)s(onen)m(tial)g(estimate)f(\(2.42\))q(.) 259 1005 y(W)-8 b(e)32 b(p)s(oin)m(t)e(out)i(that)f(for)g(ev)m(ery)g (subset)g FE(D)1739 972 y FD(0)1788 1005 y FE(\032)26 b(D)s FH(,)31 b(w)m(e)g(ha)m(v)m(e)i Fo(P)2418 972 y Fy(t)2443 981 y Fw(0)2478 972 y Fy(;x)2538 981 y Fw(0)2576 1005 y FE(f)p FF(\034)2661 1020 y FD(D)2718 1001 y Fk(0)2771 1005 y Fn(>)26 b FF(t)p FE(g)g Fn(6)g Fo(P)3130 972 y Fy(t)3155 981 y Fw(0)3191 972 y Fy(;x)3251 981 y Fw(0)3289 1005 y FE(f)p FF(\034)3374 1019 y FD(D)3461 1005 y Fn(>)g FF(t)p FE(g)p FH(,)118 1118 y(and)31 b(th)m(us)g(\(2.42\))g(still)c (pro)m(vides)j(an)h(upp)s(er)g(b)s(ound)f(for)g(the)h(\034rst)f(exit)g (time)f(from)g(smaller)f(sets.)259 1231 y(Let)36 b(us)f(\034nally)f (consider)h(what)h(happ)s(ens)f(after)h(the)g(path)g(has)f(left)f FE(D)k FH(at)d(time)f FF(\034)44 b FG(=)33 b FF(\034)3360 1245 y FD(D)3420 1231 y FH(.)56 b(One)118 1344 y(can)31 b(deduce)g(from)f(the)g(de\034nition)g(\(2.39\))h(of)36 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))31 b FH(that)g(for)2240 1278 y FE(p)p 2316 1278 43 4 v 66 x FF(")25 b Fn(6)g FF(t)g Fn(6)g FF(T)43 b FH(and)31 b FE(j)p FF(x)p FE(j)26 b Fn(>)k FG(~)-50 b FF(x)p FG(\()p FF(t)p FG(\))p FH(,)399 1531 y FF(@)447 1545 y Fy(x)491 1531 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))25 b Fn(6)i FG(~)-47 b FF(a)p FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1183 1545 y Fy(x)1227 1531 y FF(f)10 b FG(\()c(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))26 b Fn(6)f FE(\000)p FF(\021)s(a)p FG(\()p FF(t)p FG(\))182 b FH(with)30 b FF(\021)e FG(=)d(3)p FF(\025)c FE(\000)f FG(1)h FE(\000)e FD(O)2953 1545 y Fy(T)3008 1531 y FG(\(1\))p FH(.)282 b(\(2.43\))118 1727 y(Let)31 b FF(x)332 1682 y FC(det)q Fy(;\034)332 1751 y(t)523 1727 y FH(denote)h(the)f(solution)e(of)h(the)h(deterministic)d (equation)j(\(2.29\))g(starting)f(in)36 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))31 b FH(at)g(time)118 1839 y FF(\034)36 b FH(\(the)26 b(case)f(where)i(one)f(starts)f(at)h FE(\000)6 b FG(~)-51 b FF(x)o FG(\()p FF(t)p FG(\))26 b FH(is)e(obtained)i(b)m(y)g(symmetry\).)37 b(W)-8 b(e)26 b(shall)e(sho)m(w)i(in)f(Prop)s(osi-)118 1961 y(tion)d(4.11)h(that)g FF(x)719 1917 y FC(det)q Fy(;\034)719 1985 y(t)902 1961 y FH(alw)m(a)m(ys)f(remains)f(b)s(et)m(w)m(een)30 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))23 b FH(and)g FF(x)2251 1928 y Fy(?)2290 1961 y FG(\()p FF(t)p FG(\))p FH(,)i(and)e(approac)m (hes)h FF(x)3120 1928 y Fy(?)3159 1961 y FG(\()p FF(t)p FG(\))f FH(according)118 2074 y(to)984 2187 y FF(x)1036 2142 y FC(det)q Fy(;\034)1036 2211 y(t)1222 2187 y FG(=)i FF(x)1370 2149 y Fy(?)1410 2187 y FG(\()p FF(t)p FG(\))20 b FE(\000)g(O)1699 2086 y Fr(\020)1814 2125 y FF(")p 1763 2166 143 4 v 1763 2253 a(t)1796 2226 y FC(3)p Fy(=)p FC(2)1916 2086 y Fr(\021)1991 2187 y FE(\000)g(O)2157 2113 y Fr(\000)2198 2117 y FE(p)p 2274 2117 51 4 v 70 x FF(\034)25 b FG(e)2379 2149 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")2728 2113 y Fr(\001)2770 2187 y FF(:)635 b FH(\(2.44\))118 2369 y(Moreo)m(v)m(er,)32 b(deterministic)c(solutions)h(starting)h(at)g(di\033eren)m(t)h(times)d (approac)m(h)k(eac)m(h)g(other)f(lik)m(e)624 2570 y FG(0)26 b Fn(6)f FF(x)843 2523 y FC(det)q Fy(;)961 2475 y FD(p)p 1019 2475 33 3 v 1019 2523 a Fy(")843 2594 y(t)1076 2570 y FE(\000)20 b FF(x)1219 2526 y FC(det)q Fy(;\034)1219 2594 y(t)1405 2570 y Fn(6)1501 2496 y Fr(\000)1543 2570 y FF(x)1595 2532 y FC(det)q Fy(;)1713 2485 y FD(p)p 1771 2485 V 1771 2532 a Fy(")1595 2593 y(\034)1828 2570 y FE(\000)26 b FG(~)-51 b FF(x)p FG(\()p FF(\034)10 b FG(\))2091 2496 y Fr(\001)2149 2570 y FG(e)2189 2532 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)e FC(\))p Fy(=")2734 2570 y FE(8)p FF(t)24 b FE(2)h FG([)p FF(\034)5 b(;)15 b(T)e FG(])p FF(:)276 b FH(\(2.45\))118 2775 y(The)31 b(linearization)d(of)i FF(f)40 b FH(at)30 b FF(x)1171 2730 y FC(det)q Fy(;\034)1171 2799 y(t)1362 2775 y FH(satis\034es)780 2986 y FF(a)828 2948 y Fy(\034)871 2986 y FG(\()p FF(t)p FG(\))c(=)f FF(@)1144 3000 y Fy(x)1188 2986 y FF(f)10 b FG(\()p FF(x)1330 2942 y FC(det)p Fy(;\034)1330 3010 y(t)1490 2986 y FF(;)15 b(t)p FG(\))26 b(=)f FF(a)1768 2948 y Fy(?)1807 2986 y FG(\()p FF(t)p FG(\))c(+)f FE(O)2097 2885 y Fr(\020)2161 2924 y FF(")p 2161 2965 43 4 v 2166 3048 a(t)2214 2885 y Fr(\021)2288 2986 y FG(+)g FE(O)2454 2912 y Fr(\000)2496 2986 y FF(t)15 b FG(e)2584 2948 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")2933 2912 y Fr(\001)2975 2986 y FF(:)430 b FH(\(2.46\))118 3210 y(F)-8 b(or)31 b(giv)m(en)f FF(\034)10 b FH(,)30 b(w)m(e)h(construct)h(a)e(strip)g FE(A)1512 3177 y Fy(\034)1554 3210 y FG(\()p FF(h)p FG(\))i FH(around)f FF(x)2066 3177 y FC(det)q Fy(;\034)2257 3210 y FH(of)f(the)g(form)858 3405 y FE(A)931 3368 y Fy(\034)974 3405 y FG(\()p FF(h)p FG(\))c(=)1218 3332 y Fr(\010)1271 3405 y FG(\()p FF(x;)15 b(t)p FG(\))10 b(:)32 b FF(\034)j Fn(6)25 b FF(t)g Fn(6)g FF(T)8 b(;)15 b FE(j)p FF(x)21 b FE(\000)e FF(x)2199 3361 y FC(det)q Fy(;\034)2199 3429 y(t)2360 3405 y FE(j)26 b FF(<)f(h)2559 3323 y Fr(p)p 2650 3323 194 4 v 82 x FF(\020)2697 3379 y Fy(\034)2740 3405 y FG(\()p FF(t)p FG(\))2843 3332 y Fr(\011)2896 3405 y FF(;)509 b FH(\(2.47\))118 3592 y(where)31 b(the)g(function)f FF(\020)936 3559 y Fy(\034)979 3592 y FG(\()p FF(t)p FG(\))g FH(is)f(de\034ned)j(b)m(y)333 3836 y FF(\020)380 3798 y Fy(\034)422 3836 y FG(\()p FF(t)p FG(\))26 b(=)767 3775 y(1)p 657 3815 265 4 v 657 3898 a(2)p FE(j)q FG(~)-46 b FF(a)q FG(\()p FF(\034)10 b FG(\))p FE(j)947 3836 y FG(e)987 3798 y FC(2)p Fy(\013)1067 3775 y Fj(\034)1107 3798 y FC(\()p Fy(t;\034)e FC(\))p Fy(=")1333 3836 y FG(+)1414 3775 y(1)p 1414 3815 46 4 v 1416 3898 a FF(")1484 3712 y Fr(Z)1575 3739 y Fy(t)1535 3918 y(\034)1620 3836 y FG(e)1660 3798 y FC(2)p Fy(\013)1740 3775 y Fj(\034)1780 3798 y FC(\()p Fy(t;s)p FC(\))p Fy(=")2000 3836 y FG(d)o FF(s;)196 b(\013)2372 3798 y Fy(\034)2415 3836 y FG(\()p FF(t;)15 b(s)p FG(\))26 b(=)2723 3712 y Fr(Z)2814 3739 y Fy(t)2774 3918 y(s)2859 3836 y FF(a)2907 3798 y Fy(\034)2950 3836 y FG(\()p FF(u)p FG(\))15 b(d)q FF(u;)214 b FH(\(2.48\))118 4066 y(and)31 b(satis\034es)991 4195 y FF(\020)1038 4157 y Fy(\034)1080 4195 y FG(\()p FF(t)p FG(\))26 b(=)1436 4133 y(1)p 1315 4174 287 4 v 1315 4257 a(2)p FE(j)p FF(a)1433 4231 y Fy(?)1473 4257 y FG(\()p FF(t)p FG(\))p FE(j)1632 4195 y FG(+)20 b FE(O)1798 4094 y Fr(\020)1877 4133 y FF(")p 1862 4174 73 4 v 1862 4257 a(t)1895 4231 y FC(3)1945 4094 y Fr(\021)2019 4195 y FG(+)g FE(O)2185 4094 y Fr(\020)2249 4133 y FG(1)p 2249 4174 46 4 v 2255 4257 a FF(t)2320 4195 y FG(e)2360 4157 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")2709 4094 y Fr(\021)2763 4195 y FF(;)642 b FH(\(2.49\))118 4402 y(cf.)30 b(Lemma)g(4.12.)41 b(Let)31 b FF(\034)992 4420 y FD(A)1048 4401 y Fj(\034)1087 4420 y FC(\()p Fy(h)p FC(\))1217 4402 y FH(denote)g(the)g(\034rst)f(exit)f (time)g(of)h FF(x)2389 4416 y Fy(t)2449 4402 y FH(from)f FE(A)2735 4369 y Fy(\034)2778 4402 y FG(\()p FF(h)p FG(\))p FH(.)118 4575 y Fq(Theorem)36 b(2.10)g(\(Approac)m(h)g(to)h FF(x)1479 4542 y Fy(?)1518 4575 y Fq(\).)43 b FB(Ther)-5 b(e)35 b(exist)f(c)-5 b(onstants)34 b FF(")2542 4589 y FC(0)2616 4575 y FB(and)g FF(h)2845 4589 y FC(0)2884 4575 y FB(,)g(dep)-5 b(ending)35 b(only)f(on)118 4688 y FF(f)10 b FB(,)31 b FF(T)45 b FB(and)33 b FF(d)p FB(,)f(such)h(that)f (for)g FG(0)26 b FF(<)f(")h Fn(6)f FF(")1519 4702 y FC(0)1559 4688 y FB(,)32 b FF(h)25 b(<)g(h)1844 4702 y FC(0)1884 4688 y FF(\034)42 b FB(and)32 b FF(\034)k Fn(6)25 b FF(t)g Fn(6)g FF(T)13 b FB(,)643 4925 y Fo(P)704 4888 y Fy(\034)t(;)t FC(~)-39 b Fy(x)p FC(\()p Fy(\034)8 b FC(\))897 4852 y Fr(\010)950 4925 y FF(\034)990 4944 y FD(A)1046 4925 y Fj(\034)1085 4944 y FC(\()p Fy(h)p FC(\))1210 4925 y FF(<)25 b(t)1339 4852 y Fr(\011)1417 4925 y Fn(6)g FF(C)1585 4888 y Fy(\034)1627 4925 y FG(\()p FF(t;)15 b(")p FG(\))g(exp)1968 4824 y Fr(n)2028 4925 y FE(\000)2109 4864 y FG(1)p 2109 4904 V 2109 4988 a(2)2176 4864 y FF(h)2228 4831 y FC(2)p 2174 4904 95 4 v 2174 4988 a FF(\033)2229 4961 y FC(2)2279 4824 y Fr(h)2322 4925 y FG(1)20 b FE(\000)g(O)s FG(\()p FF(")p FG(\))i FE(\000)d(O)2852 4824 y Fr(\020)2917 4864 y FF(h)p 2917 4904 53 4 v 2918 4988 a(\034)2979 4824 y Fr(\021io)3430 4925 y FH(\(2.50\))118 5133 y FB(wher)-5 b(e)950 5279 y FF(C)1022 5242 y Fy(\034)1064 5279 y FG(\()p FF(t;)15 b(")p FG(\))27 b(=)1382 5218 y FE(j)p FF(\013)1465 5185 y Fy(\034)1509 5218 y FG(\()p FF(t;)15 b(\034)10 b FG(\))p FE(j)p 1382 5258 347 4 v 1514 5342 a FF(")1556 5315 y FC(2)1758 5279 y FG(+)20 b(2)26 b Fn(6)2044 5218 y FG(1)p 2025 5258 82 4 v 2025 5342 a FF(")2067 5315 y FC(2)2117 5148 y Fr(\014)2117 5202 y(\014)2117 5257 y(\014)2117 5311 y(\014)2148 5156 y(Z)2238 5182 y Fy(t)2198 5314 y FD(p)p 2257 5314 33 3 v 48 x Fy(")2309 5279 y FF(a)2357 5242 y Fy(?)2396 5279 y FG(\()p FF(s)p FG(\))15 b(d)p FF(s)2618 5148 y Fr(\014)2618 5202 y(\014)2618 5257 y(\014)2618 5311 y(\014)2668 5279 y FG(+)20 b(2)p FF(:)601 b FH(\(2.51\))259 5509 y(The)31 b(pro)s(of)g(is)e(giv)m(en)h (in)g(Section)h(4.4.)42 b(This)29 b(result)h(is)g(useful)f(for)h FF(\033)f FE(\034)d FF(h)g FE(\034)g FF(\034)10 b FH(,)31 b(and)g(sho)m(ws)g(that)118 5622 y(the)g(t)m(ypical)e(spreading)h(of)g (paths)h(around)g FF(x)1685 5578 y FC(det)q Fy(;\034)1685 5646 y(t)1876 5622 y FH(is)e(of)h(order)h FF(\033)s(=)2406 5548 y FE(p)p 2483 5548 33 4 v 2483 5622 a FF(t)o FH(,)g(see)f(Fig.)24 b(2.)1845 5871 y(12)p eop %%Page: 13 13 13 12 bop 118 328 a Fp(2.4)112 b(Discussion)118 499 y FH(Let)33 b(us)g(no)m(w)h(examine)d(some)h(of)g(the)h(consequences)h (of)e(these)h(results.)47 b(First)31 b(of)i(all,)e(they)i(allo)m(w)f (to)118 612 y(c)m(haracterize)j(the)g(in\035uence)f(of)f(additiv)m(e)g (noise)g(on)h(the)h(bifurcation)e(dela)m(y)-8 b(.)51 b(In)33 b(the)i(deterministic)118 725 y(case,)42 b(this)d(dela)m(y)g (is)g(de\034ned)h(as)g(the)g(\034rst)g(exit)e(time)h(from)f(a)i(strip)f (of)g(width)h FF(")g FH(around)g FF(x)h FG(=)g(0)p FH(,)118 838 y(see)32 b(Theorem)g(2.7.)46 b(A)31 b(p)s(ossible)f(de\034nition)h (of)g(the)i(dela)m(y)e(in)g(the)i(sto)s(c)m(hastic)e(case)h(is)e(th)m (us)j(the)f(\034rst)118 951 y(exit)j(time)f FF(\034)563 918 y FC(dela)n(y)764 951 y FH(from)h(a)h(similar)c(strip.)56 b(An)35 b(appropriate)i(c)m(hoice)f(for)f(the)h(width)g(of)f(the)h (strip)f(is)124 1064 y FG(~)-51 b FF(x)p FG(\()205 998 y FE(p)p 281 998 43 4 v 66 x FF(")q FG(\))25 b(=)g FE(O)s FG(\()p FF(")632 1031 y FC(1)p Fy(=)p FC(4)743 1064 y FG(\))p FH(,)30 b(since)f(suc)m(h)h(a)g(strip)e(will)g(con)m(tain)i FE(B)s FG(\()p FF(h)p FG(\))g FH(for)f(ev)m(ery)h(admissible)c FF(h)p FH(,)31 b(and)f(the)g(part)118 1177 y(of)h(the)h(strip)f(with)g FF(t)c Fn(>)957 1111 y FE(p)p 1032 1111 V 1032 1177 a FF(")32 b FH(will)d(b)s(e)i(con)m(tained)i(in)d FE(D)s FH(.)44 b(Theorems)31 b(2.8)h(and)g(2.9)g(then)g(imply)d(that)j(if)118 1290 y FF(t)25 b Fn(>)272 1224 y FE(p)p 348 1224 V 66 x FF(")p FH(,)208 1504 y Fo(P)269 1467 y FD(\000)p Fy(T)6 b(;x)431 1476 y Fw(0)469 1430 y Fr(\010)522 1504 y FF(\034)572 1467 y FC(dela)n(y)763 1504 y FF(<)859 1434 y FE(p)p 935 1434 V 70 x FF(")977 1430 y Fr(\011)1056 1504 y Fn(6)25 b FF(C)7 b FG(\()1259 1434 y FE(p)p 1334 1434 V 1334 1504 a FF(";)15 b(")p FG(\))g(e)1550 1467 y FD(\000O)r FC(\()p Fy("=\033)1800 1443 y Fw(2)1836 1467 y FC(\))3430 1504 y FH(\(2.52\))293 1720 y Fo(P)354 1682 y FD(\000)p Fy(T)6 b(;x)516 1691 y Fw(0)554 1646 y Fr(\010)607 1720 y FF(\034)657 1682 y FC(dela)n(y)848 1720 y Fn(>)25 b FF(t)977 1646 y Fr(\011)1056 1720 y Fn(6)g FF(C)1217 1734 y FC(0)1277 1720 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))1426 1637 y Fr(p)p 1517 1637 152 4 v 83 x FF(a)p FG(\()p FF(t)p FG(\))1689 1658 y FE(j)p FG(log)17 b FF(\033)s FE(j)p 1689 1699 239 4 v 1781 1782 a FF(\033)1937 1591 y Fr(\022)2004 1720 y FG(1)k(+)2171 1658 y FF(\013)p FG(\()p FF(t;)2337 1593 y FE(p)p 2414 1593 43 4 v 2414 1658 a FF(")p FG(\))p 2171 1699 321 4 v 2310 1782 a FF(")2501 1591 y Fr(\023)2720 1658 y FG(e)2760 1625 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;)2973 1577 y FD(p)p 3032 1577 33 3 v 48 x Fy(")p FC(\))p Fy(=")p 2578 1699 728 4 v 2578 1717 a Fr(p)p 2669 1717 637 4 v 91 x FG(1)g FE(\000)f FG(e)2866 1782 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;)3114 1734 y FD(p)p 3174 1734 33 3 v 3174 1782 a Fy(")o FC(\))p Fy(=")3316 1720 y FF(:)89 b FH(\(2.53\))118 1995 y(If)32 b(w)m(e)h(c)m(ho)s(ose)g FF(t)f FH(in)g(suc)m(h)h(a)g(w)m (a)m(y)g(that)g FF(\013)p FG(\()p FF(t;)1641 1929 y FE(p)p 1718 1929 43 4 v 1718 1995 a FF(")p FG(\))d(=)e FF(c)8 b(")p FE(j)p FG(log)17 b FF(\033)s FE(j)33 b FH(for)g(some)e FF(c)f(>)e FG(0)p FH(,)34 b(the)f(last)e(expression)118 2108 y(reduces)g(to)1119 2220 y Fo(P)1180 2183 y FD(\000)p Fy(T)6 b(;x)1342 2192 y Fw(0)1381 2147 y Fr(\010)1434 2220 y FF(\034)1484 2183 y FC(dela)n(y)1675 2220 y Fn(>)25 b FF(t)1804 2147 y Fr(\011)1882 2220 y FG(=)g FE(O)2053 2147 y Fr(\000)2095 2220 y FF(\033)2150 2183 y Fy(\024c)p FD(\000)p FC(1)2315 2220 y FE(j)p FG(log)17 b FF(\033)s FE(j)2553 2183 y FC(2)2593 2147 y Fr(\001)2635 2220 y FF(;)770 b FH(\(2.54\))118 2387 y(whic)m(h)37 b(b)s(ecomes)e(small)f (as)i(so)s(on)g(as)g FF(c)g(>)f FG(1)p FF(=\024)p FH(.)60 b(The)37 b(bifurcation)e(dela)m(y)i(will)c(th)m(us)k(lie)e(with)h(o)m (v)m(er-)118 2500 y(whelming)29 b(probabilit)m(y)g(in)g(the)i(in)m (terv)-5 b(al)1487 2631 y Fr(\002)1525 2635 y FE(p)p 1601 2635 V 69 x FF(")q(;)15 b FE(O)1759 2631 y Fr(\000)1800 2622 y(p)p 1891 2622 281 4 v 82 x FF(")p FE(j)p FG(log)j FF(\033)s FE(j)2187 2631 y Fr(\001\003)2267 2704 y FF(:)1138 b FH(\(2.55\))118 2921 y(Theorem)35 b(2.10)h(implies)31 b(that)36 b(for)f(times)e(larger)i(than)g FE(O)s FG(\()2187 2843 y Fr(p)p 2278 2843 V 78 x FF(")p FE(j)p FG(log)18 b FF(\033)s FE(j)d FG(\))p FH(,)37 b(the)e(paths)g(are)g(unlik)m(ely)e (to)118 3034 y(return)g(to)g(zero)g(in)e(a)i(time)e(of)h(order)h FG(1)p FH(.)47 b(The)33 b(wildest)d(b)s(eha)m(viour)j(of)f(the)h(paths) f(is)f(to)i(b)s(e)f(exp)s(ected)118 3147 y(in)e(the)g(in)m(terv)-5 b(al)30 b(\(2.55\))q(,)g(b)s(ecause)h(a)f(region)g(of)g(instabilit)m(y) d(is)i(crossed,)i(where)g FF(@)2941 3161 y Fy(x)2985 3147 y FF(f)k(>)25 b FG(0)p FH(.)259 3260 y(Our)41 b(results)e(on)i (the)f(pitc)m(hfork)h(bifurcation)e(require)h FF(\033)45 b FE(\034)2461 3194 y(p)p 2537 3194 43 4 v 66 x FF(")p FH(,)e(while)c(the)h(estimate)f(\(2.55\))118 3373 y(is)34 b(useful)f(as)i(long)f(as)h FF(\033)j FH(is)33 b(not)j(exp)s(onen)m (tially)d(small.)51 b(W)-8 b(e)35 b(can)h(th)m(us)f(distinguish)e (three)i(regimes,)118 3485 y(dep)s(ending)c(on)f(the)h(noise)e(in)m (tensit)m(y:)181 3622 y FE(\017)47 b FF(\033)29 b Fn(>)450 3557 y FE(p)p 525 3557 V 525 3622 a FF(")q FH(:)37 b(A)23 b(mo)s(di\034cation)f(of)i(Theorem)g(2.8)g(sho)m(ws)g(that)g(for)g FF(t)h(<)g FE(\000)p FF(\033)s FH(,)g(the)f(t)m(ypical)f(spreading)h (of)273 3735 y(paths)h(is)e(of)h(order)h FF(\033)s(=)1026 3657 y Fr(p)p 1117 3657 84 4 v 78 x FE(j)p FF(t)p FE(j)q FH(.)38 b(Near)24 b(the)h(bifurcation)f(p)s(oin)m(t,)h(the)g(pro)s (cess)f(is)e(dominated)i(b)m(y)h(noise,)273 3848 y(b)s(ecause)37 b(the)f(drift)g(term)f FF(f)45 b FE(\030)35 b(\000)p FF(x)1525 3815 y FC(3)1600 3848 y FH(is)g(to)s(o)h(w)m(eak)h(to)f(coun) m(teract)i(the)f(di\033usion.)56 b(Dep)s(ending)273 3961 y(on)35 b(the)g(global)e(structure)i(of)f FF(f)10 b FH(,)35 b(an)f(appreciable)h(fraction)f(of)g(the)g(paths)h(migh)m(t)f(escap)s (e)g(quite)273 4074 y(early)f(from)f(a)i(neigh)m(b)s(ourho)s(o)s(d)g (of)f(the)h(bifurcation)e(p)s(oin)m(t.)50 b(In)33 b(that)h(situation,)f (the)h(notion)g(of)273 4187 y(bifurcation)c(dela)m(y)g(b)s(ecomes)f (meaningless.)181 4299 y FE(\017)47 b FG(e)314 4266 y FD(\000)p FC(1)p Fy(=")472 4243 y Fj(p)537 4299 y Fn(6)25 b FF(\033)k FE(\034)830 4234 y(p)p 906 4234 43 4 v 65 x FF(")f FH(for)h(some)e FF(p)e(<)g FG(1)p FH(:)40 b(The)29 b(bifurcation)f(dela)m(y)g(lies)e(in)i(the)h(in)m(terv)-5 b(al)27 b(\(2.55\))j(with)273 4412 y(high)g(probabilit)m(y)-8 b(,)30 b(where)1215 4335 y Fr(p)p 1306 4335 281 4 v 77 x FF(")p FE(j)p FG(log)17 b FF(\033)s FE(j)26 b Fn(6)f FF(")1750 4379 y FC(\(1)p FD(\000)p Fy(p)p FC(\))p Fy(=)p FC(2)2036 4412 y FH(is)k(still)e(\020microscopic\021.)181 4525 y FE(\017)47 b FF(\033)30 b Fn(6)d FG(e)493 4492 y FD(\000)p Fy(K)q(=")711 4525 y FH(for)k(some)g FF(K)i(>)27 b FG(0)p FH(:)42 b(The)32 b(noise)e(is)g(so)h(small)e(that)i(the)h (paths)f(remain)g(concen)m(trated)273 4638 y(around)g(the)e (deterministic)e(solution)h(for)h(a)g(time)f(in)m(terv)-5 b(al)28 b(of)h(order)h FG(1)p FH(.)40 b(The)30 b(t)m(ypical)e (spreading)273 4751 y(is)g(of)g(order)h FF(\033)753 4673 y Fr(p)p 844 4673 151 4 v 78 x FF(\020)7 b FG(\()p FF(t)p FG(\))p FH(,)29 b(whic)m(h)g(b)s(eha)m(v)m(es)g(lik)m(e)e FF(\033)19 b FG(e)1915 4718 y Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")2128 4751 y FF(=")2215 4718 y FC(1)p Fy(=)p FC(4)2354 4751 y FH(for)29 b FF(t)c Fn(>)2645 4686 y FE(p)p 2721 4686 43 4 v 65 x FF(")p FH(,)k(see)f(Lemma)g(4.2.)41 b(Th)m(us)273 4864 y(the)28 b(paths)f(remain)f(close)g(to)i(the)f (origin)f(un)m(til)g FF(\013)p FG(\()p FF(t)p FG(\))g FE(')f FF(")p FE(j)p FG(log)17 b FF(\033)s FE(j)26 b Fn(>)f FF(K)7 b FH(.)39 b(If)26 b FF(")p FE(j)p FG(log)17 b FF(\033)s FE(j)26 b FF(>)f(\013)p FG(\(\005\()p FF(t)3454 4878 y FC(0)3495 4864 y FG(\)\))h(=)273 4977 y FE(j)p FF(\013)p FG(\()p FF(t)424 4991 y FC(0)465 4977 y FG(\))p FE(j)p FH(,)32 b(they)f(follo)m(w)g(the)h(deterministic)d(solution)h (whic)m(h)h(mak)m(es)g(a)h(quic)m(k)e(transition)h(to)g FF(x)3518 4944 y Fy(?)3558 4977 y FG(\()p FF(t)p FG(\))273 5090 y FH(at)g FF(t)25 b FG(=)g(\005\()p FF(t)674 5104 y FC(0)714 5090 y FG(\))p FH(.)118 5226 y(The)33 b(expression)f (\(2.55\))i(c)m(haracterizing)g(the)f(dela)m(y)f(is)g(in)g(accordance)i (with)f(exp)s(erimen)m(tal)e(results)118 5339 y([TM,)k(SMC)q(],)h(and)g (with)f(the)g(appro)m(ximate)g(calculation)f(of)h(the)h(last)e (crossing)g(of)h(zero)h([JL)q(].)55 b(The)118 5452 y(n)m(umerical)23 b(results)g(in)g([Ga],)i(whic)m(h)g(are)f(\034tted,)i(at)e FF(")i FG(=)f(0)p FF(:)p FG(01)p FH(,)h(to)e FF(\034)2413 5419 y FC(dela)n(y)2605 5452 y FE(')h FF(\033)2756 5419 y FC(0)p Fy(:)p FC(105)2944 5452 y FH(for)f(w)m(eak)h(noise)e(and)118 5565 y FF(\034)168 5532 y FC(dela)n(y)360 5565 y FE(')j FG(e)497 5532 y FD(\000)p FC(851)13 b Fy(\033)747 5565 y FH(for)31 b(strong)g(noise,)f(seem)g(rather)i(m)m(ysterious.)40 b(Finally)-8 b(,)29 b(the)i(results)f(in)g([Ku)q(],)h(who)1845 5871 y(13)p eop %%Page: 14 14 14 13 bop 118 328 a FH(appro)m(ximates)27 b(the)g(probabilit)m(y)e (densit)m(y)i(b)m(y)g(a)f(Gaussian)g(cen)m(tered)j(at)e(the)g (deterministic)d(solution,)118 441 y(can)31 b(ob)m(viously)e(only)g (apply)h(to)g(the)h(regime)e(of)h(exp)s(onen)m(tially)f(small)e(noise.) 259 553 y(Another)40 b(in)m(teresting)f(question)f(is)g(ho)m(w)i(fast)f (the)h(paths)f(concen)m(trate)j(near)d(the)h(equilibrium)118 666 y(branc)m(hes)d FE(\006)p FF(x)621 633 y Fy(?)661 666 y FG(\()p FF(t)p FG(\))p FH(.)57 b(The)37 b(deterministic)c (solutions,)j(starting)g(at)42 b FG(~)-51 b FF(x)p FG(\()p FF(t)2575 680 y FC(0)2615 666 y FG(\))36 b FH(at)g(some)f(time)g FF(t)3279 680 y FC(0)3353 666 y FF(>)f FG(0)p FH(,)k(all)118 779 y(trac)m(k)29 b FF(x)399 746 y Fy(?)438 779 y FG(\()p FF(t)p FG(\))f FH(at)g(a)g(distance)f(whic)m(h)i(is)d(asymptotically)f (of)i(order)i FF("=t)2504 746 y FC(3)p Fy(=)p FC(2)2614 779 y FH(.)40 b(Therefore,)29 b(w)m(e)g(can)f(c)m(ho)s(ose)118 892 y(one)j(of)f(them,)g(sa)m(y)h FF(x)852 845 y FC(det)q Fy(;)970 797 y FD(p)p 1028 797 33 3 v 1028 845 a Fy(")852 916 y(t)1065 892 y FH(,)f(and)h(measure)g(the)f(distance)h(of)f FF(x)2312 906 y Fy(t)2372 892 y FH(from)f(that)i(deterministic)d (solution.)118 1005 y(W)-8 b(e)35 b(restrict)f(our)h(atten)m(tion)h(to) e(those)h(paths)g(whic)m(h)g(are)g(still)c(in)j(a)g(neigh)m(b)s(ourho)s (o)s(d)h(of)f(the)h(origin)118 1118 y(at)30 b(time)433 1053 y FE(p)p 509 1053 43 4 v 65 x FF(")p FH(,)g(as)f(most)g(paths)h (are.)40 b(W)-8 b(e)30 b(w)m(an)m(t)h(to)f(sho)m(w)g(that)g(for)g (suitably)d(c)m(hosen)k FF(t)3074 1132 y FC(1)3138 1118 y FE(2)25 b FG(\()3259 1053 y FE(p)p 3335 1053 V 65 x FF(")q(;)15 b(t)p FG(\))30 b FH(and)118 1231 y FG(\001)25 b FE(2)g FG(\(0)p FF(;)15 b(t)p FG(\))p FH(,)30 b(most)d(paths)h(will)d (lea)m(v)m(e)k FE(D)h FH(un)m(til)d(time)g FF(t)1949 1245 y FC(1)2016 1231 y FH(and)h(reac)m(h)h(a)f FF(\016)s FH(-neigh)m(b)s(ourho)s(o)s(d)i(of)d FF(x)3340 1184 y FC(det)q Fy(;)3458 1136 y FD(p)p 3517 1136 33 3 v 48 x Fy(")3340 1255 y(t)3581 1231 y FH(at)118 1344 y(time)i FF(\034)364 1358 y FD(D)445 1344 y FG(+)20 b(\001)p FH(.)40 b(Let)31 b(us)f(estimate)208 1592 y Fo(P)269 1504 y FD(p)p 329 1504 V 329 1552 a Fy(")o(;x)421 1535 y Fk(p)p 470 1535 30 3 v 470 1570 a Fj(")508 1463 y Fr(\032\022)643 1592 y FF(\034)683 1606 y FD(D)769 1592 y FF(<)25 b(t)898 1606 y FC(1)937 1592 y FF(;)158 b FG(sup)1007 1674 y Fy(s)p FD(2)p FC([)p Fy(\034)1138 1685 y Fk(D)1192 1674 y FC(+\001)p Fy(;t)p FC(])1370 1487 y Fr(\014)1370 1542 y(\014)1370 1596 y(\014)1400 1592 y FE(j)p FF(x)1477 1606 y Fy(s)1514 1592 y FE(j)21 b(\000)f FF(x)1703 1554 y FC(det)q Fy(;)1821 1506 y FD(p)p 1879 1506 33 3 v 1879 1554 a Fy(")1703 1614 y(s)1916 1487 y Fr(\014)1916 1542 y(\014)1916 1596 y(\014)1971 1592 y FF(<)25 b(\016)2110 1463 y Fr(\023)2178 1486 y FC(c)2214 1463 y Fr(\033)3430 1592 y FH(\(2.56\))324 1865 y Fn(6)g Fo(P)481 1778 y FD(p)p 540 1778 V 48 x Fy(";x)633 1808 y Fk(p)p 682 1808 30 3 v 682 1844 a Fj(")719 1791 y Fr(\010)772 1865 y FF(\034)812 1879 y FD(D)898 1865 y Fn(>)g FF(t)1027 1879 y FC(1)1066 1791 y Fr(\011)1140 1865 y FG(+)19 b Fo(E)1291 1778 y FD(p)p 1356 1778 33 3 v 1356 1826 a Fy(")o(;x)1448 1808 y Fk(p)p 1497 1808 30 3 v 1497 1844 a Fj(")1535 1737 y Fr(\032)1603 1865 y FG(1)1648 1884 y FD(f)p Fy(\034)1714 1895 y Fk(D)1769 1884 y Fy()f FF(\016)3409 1737 y Fr(\033)q(\033)3546 1865 y FF(:)118 2160 y FH(The)49 b(\034rst)g(term)f(decreases)i(roughly)e (lik)m(e)f FF(\033)1759 2127 y FD(\000)p FC(1)1869 2160 y FG(e)1909 2127 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t)2102 2136 y Fw(1)2138 2127 y Fy(;)2158 2079 y FD(p)p 2216 2079 V 2216 2127 a Fy(")p FC(\))p Fy(=")2397 2160 y FH(and)i(b)s (ecomes)f(small)e(as)i(so)s(on)h(as)118 2273 y FF(\013)p FG(\()p FF(t)244 2287 y FC(1)284 2273 y FF(;)324 2207 y FE(p)p 400 2207 43 4 v 66 x FF(")q FG(\))25 b FE(\035)h FF(")p FE(j)p FG(log)17 b FF(\033)s FE(j)p FH(.)41 b(The)30 b(second)h(summand)e(is)g(b)s(ounded)i(ab)s(o)m(v)m(e)h(b)m(y)409 2532 y FB(c)-5 b(onst)35 b Fo(E)704 2445 y FD(p)p 769 2445 33 3 v 769 2493 a Fy(";x)862 2475 y Fk(p)p 911 2475 30 3 v 911 2511 a Fj(")948 2431 y Fr(n)1009 2532 y FG(1)1054 2551 y FD(f)p Fy(\034)1120 2562 y Fk(D)1175 2551 y Fy()g FG(0)p FH(.)40 b(Without)29 b(loss)e(of)118 3440 y(generalit)m(y)-8 b(,)31 b(w)m(e)g(ma)m(y)e(assume)h(that)h FF(x)1442 3454 y FC(0)1506 3440 y FF(>)25 b FG(0)p FH(.)41 b(The)31 b(symmetry)d(of)i FF(f)40 b FH(implies)877 3677 y Fo(P)938 3640 y Fy(t)963 3649 y Fw(0)999 3640 y Fy(;x)1059 3649 y Fw(0)1097 3604 y Fr(\010)1150 3677 y FF(x)1202 3691 y Fy(t)1257 3677 y Fn(>)25 b FG(0)1398 3604 y Fr(\011)1477 3677 y FG(=)g(1)20 b FE(\000)1739 3616 y FG(1)p 1739 3656 46 4 v 1739 3740 a(2)1795 3677 y Fo(P)1856 3640 y Fy(t)1881 3649 y Fw(0)1916 3640 y Fy(;x)1976 3649 y Fw(0)2014 3604 y Fr(\010)2067 3677 y FE(9)15 b FF(s)25 b FE(2)g FG([)p FF(t)2345 3691 y FC(0)2384 3677 y FF(;)15 b(t)p FG(\))26 b(:)f FF(x)2620 3691 y Fy(s)2682 3677 y FG(=)g(0)2823 3604 y Fr(\011)2877 3677 y FF(;)528 b FH(\(2.58\))118 3902 y(and)37 b(therefore)g(it)f(is)f(su\036cien)m(t)i (to)g(estimate)e(the)i(probabilit)m(y)e(for)i FF(x)2577 3916 y Fy(s)2650 3902 y FH(to)f(reac)m(h)i(zero)f(b)s(efore)g(time)118 4015 y(zero,)31 b(for)f(instance.)40 b(W)-8 b(e)31 b(linearize)d(the)j (SDE)f(\(2.1\))h(and)g(use)f(the)g(fact)h(that)f(the)h(solution)e FF(x)3364 3982 y FC(0)3364 4037 y Fy(s)3433 4015 y FH(of)h(the)118 4128 y(linearized)f(equation)1051 4274 y FG(d)p FF(x)1154 4236 y FC(0)1154 4296 y Fy(s)1218 4274 y FG(=)1324 4212 y(1)p 1324 4253 V 1326 4336 a FF(")1380 4274 y(a)p FG(\()p FF(s)p FG(\))p FF(x)1593 4236 y FC(0)1593 4296 y Fy(s)1648 4274 y FG(d)o FF(s)20 b FG(+)1893 4212 y FF(\033)p 1862 4253 119 4 v 1862 4271 a FE(p)p 1938 4271 43 4 v 66 x FF(")2005 4274 y FG(d)p FF(W)2142 4288 y Fy(s)2178 4274 y FF(;)196 b(x)2451 4236 y FC(0)2451 4296 y Fy(t)2476 4305 y Fw(0)2541 4274 y FG(=)25 b FF(x)2689 4288 y FC(0)3430 4274 y FH(\(2.59\))118 4496 y(satis\034es)k FF(x)503 4510 y Fy(s)565 4496 y Fn(6)c FF(x)713 4463 y FC(0)713 4519 y Fy(s)783 4496 y FH(as)30 b(long)g(as)g FF(x)1253 4510 y Fy(s)1319 4496 y FH(do)s(es)g(not)h(reac)m(h)h(zero.)41 b(F)-8 b(or)31 b(the)f(Gaussian)g(pro)s(cess)f FF(x)3221 4463 y FC(0)3221 4519 y Fy(s)3291 4496 y FH(w)m(e)i(kno)m(w)166 4766 y Fo(P)227 4728 y Fy(t)252 4737 y Fw(0)288 4728 y Fy(;x)348 4737 y Fw(0)386 4692 y Fr(\010)439 4766 y FE(9)15 b FF(s)24 b FE(2)h FG([)p FF(t)716 4780 y FC(0)756 4766 y FF(;)15 b(t)p FG(\))26 b(:)f FF(x)992 4728 y FC(0)992 4788 y Fy(s)1057 4766 y FG(=)g(0)1198 4692 y Fr(\011)1276 4766 y FG(=)g(2)1417 4665 y Fr(\020)1472 4766 y FG(1)c FE(\000)f Fo(P)1690 4728 y Fy(t)1715 4737 y Fw(0)1750 4728 y Fy(;x)1810 4737 y Fw(0)1848 4692 y Fr(\010)1901 4766 y FF(x)1953 4728 y FC(0)1953 4788 y Fy(t)2018 4766 y Fn(>)25 b FG(0)2159 4692 y Fr(\011)2212 4665 y(\021)2292 4766 y FG(=)g(1)c FE(\000)2620 4704 y FG(1)p 2554 4745 177 4 v 2554 4763 a FE(p)p 2630 4763 101 4 v 75 x FG(2)p FF(\031)2756 4642 y Fr(Z)2847 4668 y Fy(u)p FC(\()p Fy(t)p FC(\))2806 4848 y FD(\000)p Fy(u)p FC(\()p Fy(t)p FC(\))3002 4766 y FG(e)3042 4728 y FD(\000)p Fy(y)3134 4705 y Fw(2)3169 4728 y Fy(=)p FC(2)3259 4766 y FG(d)o FF(y)s(;)48 b FH(\(2.60\))118 5050 y(where)41 b FF(u)p FG(\()p FF(t)p FG(\))h(=)g FF(x)751 5064 y FC(0)805 5050 y FG(e)846 5017 y Fy(\013)p FC(\()p Fy(t;t)988 5026 y Fw(0)1024 5017 y FC(\))p Fy(=")1138 5050 y FF(=)1183 4972 y Fr(p)p 1275 4972 264 4 v 1275 5050 a FF(v)s FG(\()p FF(t;)15 b(t)1463 5064 y FC(0)1503 5050 y FG(\))40 b FH(and)h FF(v)s FG(\()p FF(t;)15 b(t)1952 5064 y FC(0)1992 5050 y FG(\))40 b FH(denotes)h(the)f(v)-5 b(ariance)40 b(of)g FF(x)3098 5017 y FC(0)3098 5072 y Fy(t)3137 5050 y FH(.)70 b(F)-8 b(or)41 b FF(t)g FG(=)h(0)p FH(,)118 5174 y FF(u)p FG(\(0\))d FH(is)e(of)g(order)i FF(x)829 5188 y FC(0)868 5174 y FF(")910 5141 y FC(1)p Fy(=)p FC(4)1021 5174 y FF(\033)1076 5141 y FD(\000)p FC(1)1185 5174 y FG(e)1226 5141 y FD(\000)p Fi(c)l(onst)26 b Fy(t)1490 5118 y Fw(2)1490 5162 y(0)1525 5141 y Fy(=")1597 5174 y FH(,)40 b(see)d(Lemma)h(4.2.)63 b(Th)m(us)39 b(the)f(probabilit) m(y)f(in)g(\(2.60\))i(is)118 5287 y(exp)s(onen)m(tially)26 b(close)h(to)h(one)g(for)f(small)e FF(")p FH(,)k(and)f(w)m(e)g (conclude)g(that)g(the)g(probabilit)m(y)f(for)g FF(x)3288 5301 y Fy(t)3345 5287 y FH(to)h(reac)m(h)118 5400 y(the)j(p)s(ositiv)m (e)d(branc)m(h)k(rather)f(than)g(the)g(negativ)m(e)g(one)f(is)f(exp)s (onen)m(tially)g(close)g(to)i FG(1)p FF(=)p FG(2)p FH(.)1845 5871 y(14)p eop %%Page: 15 15 15 14 bop 118 328 a FI(3)131 b(The)44 b(motion)g(near)g(non)l (bifurcating)h(equilibria)118 531 y FH(In)30 b(this)f(section)h(w)m(e)h (consider)f(the)h(nonlinear)f(SDE)1326 750 y FG(d)p FF(x)1429 764 y Fy(t)1483 750 y FG(=)1589 689 y(1)p 1589 729 46 4 v 1591 813 a FF(")1645 750 y(f)10 b FG(\()p FF(x)1787 764 y Fy(t)1816 750 y FF(;)15 b(t)p FG(\))g(d)p FF(t)20 b FG(+)2176 689 y FF(\033)p 2144 729 119 4 v 2144 748 a FE(p)p 2220 748 43 4 v 65 x FF(")2287 750 y FG(d)p FF(W)2424 764 y Fy(t)3476 750 y FH(\(3.1\))118 997 y(under)31 b(the)g(assumptions)181 1134 y FE(\017)47 b FF(t)25 b FE(2)g FF(I)33 b FG(=)25 b([0)p FF(;)15 b(T)e FG(])31 b FH(or)f FG([0)p FF(;)15 b FE(1)p FG(\))p FH(;)181 1246 y FE(\017)47 b FH(there)31 b(exists)e(an)i FB(e)-5 b(quilibrium)33 b(curve)f FF(x)1639 1213 y Fy(?)1704 1246 y FG(:)25 b FF(I)33 b FE(!)25 b Fo(R)54 b FH(suc)m(h)31 b(that)1509 1451 y FF(f)10 b FG(\()p FF(x)1651 1413 y Fy(?)1690 1451 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))26 b(=)f(0)91 b FE(8)p FF(t)25 b FE(2)g FF(I)7 b FG(;)1050 b FH(\(3.2\))181 1655 y FE(\017)47 b FH(there)33 b(is)d(a)h(constan)m(t)i FF(d)28 b(>)f FG(0)32 b FH(suc)m(h)g(that)g FF(f)41 b FH(is)30 b(t)m(wice)i(con)m(tin)m(uously)f(di\033eren)m(tiable)g(with)g (resp)s(ect)273 1768 y(to)25 b FF(x)f FH(and)i FF(t)e FH(for)g FE(j)p FF(x)9 b FE(\000)g FF(x)1032 1735 y Fy(?)1071 1768 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FF(d)g FH(and)g FF(t)g FE(2)g FF(I)7 b FH(,)26 b(with)e FE(j)p FF(@)2078 1782 y Fy(xx)2162 1768 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))p FE(j)25 b FH(uniformly)d(b)s(ounded)j(b)m(y)g FG(2)p FF(M)36 b(>)25 b FG(0)273 1881 y FH(in)30 b(that)h(domain;)181 1994 y FE(\017)47 b FH(there)31 b(is)e(a)i(constan)m(t)g FF(a)1083 2008 y FC(0)1148 1994 y FF(>)25 b FG(0)30 b FH(suc)m(h)h(that)g FF(a)p FG(\()p FF(t)p FG(\))26 b(=)f FF(@)2040 2008 y Fy(x)2084 1994 y FF(f)10 b FG(\()p FF(x)2226 1961 y Fy(?)2265 1994 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))32 b FH(satis\034es)1583 2198 y FE(j)p FF(a)p FG(\()p FF(t)p FG(\))p FE(j)27 b Fn(>)e FF(a)1955 2212 y FC(0)2085 2198 y FE(8)p FF(t)f FE(2)h FF(I)7 b(:)1125 b FH(\(3.3\))118 2402 y(W)-8 b(e)22 b(do)g(not)g(need)g(an)m(y)g (assumptions)e(on)i FF(\033)29 b(>)c FG(0)p FH(,)e(but)f(our)g(results) f(are)h(of)f(in)m(terest)h(only)f(for)g FF(\033)29 b FG(=)c FD(O)3483 2416 y Fy(")3520 2402 y FG(\(1\))p FH(.)259 2515 y(In)38 b(Section)h(3.1)g(w)m(e)g(consider)g(the)f(stable)g(case,) j(corresp)s(onding)e(to)g FF(a)p FG(\()p FF(t)p FG(\))g Fn(6)g FE(\000)p FF(a)3148 2529 y FC(0)3226 2515 y FF(<)g FG(0)g FH(for)f(all)118 2628 y FF(t)g FE(2)f FF(I)7 b FH(.)63 b(W)-8 b(e)38 b(\034rst)g(analyse)f(the)h(linearization)e(of)44 b(\(3.1\))38 b(around)h(a)f(giv)m(en)f(deterministic)e(solution.)118 2741 y(Prop)s(osition)25 b(3.3)i(sho)m(ws)f(that)h(the)f(solutions)e (of)i(the)h(linearized)d(equation)i(are)g(lik)m(ely)e(to)i(remain)f(in) g(a)118 2854 y(strip)c(of)h(width)g FF(h)716 2776 y Fr(p)p 807 2776 151 4 v 78 x FF(\020)7 b FG(\()p FF(t)p FG(\))22 b FH(around)i(the)e(deterministic)e(solution.)36 b(Here)23 b FF(\020)7 b FG(\()p FF(t)p FG(\))22 b FH(is)e(related)j(to)f(the)g(v) -5 b(ariance)118 2967 y(and)32 b(will)c(b)s(e)j(analyzed)g(in)g(Lemma)f (3.1.)43 b(Prop)s(osition)31 b(3.6)g(allo)m(ws)f(to)i(compare)f(the)h (tra)5 b(jectories)31 b(of)118 3080 y(the)g(linear)e(and)i(the)f (nonlinear)g(equation,)g(and)h(th)m(us)g(completes)e(the)i(pro)s(of)f (of)g(Theorem)h(2.3.)259 3193 y(In)j(Section)g(3.2,)i(w)m(e)e(consider) g(the)h(unstable)f(case,)h(i.)15 b(e.)51 b FF(a)p FG(\()p FF(t)p FG(\))32 b Fn(>)f FF(a)2626 3207 y FC(0)2697 3193 y FF(>)h FG(0)i FH(for)g(all)e FF(t)g FE(2)f FF(I)7 b FH(.)51 b(Theo-)118 3305 y(rem)27 b(2.5)h(is)e(equiv)-5 b(alen)m(t)26 b(to)i(Prop)s(osition)f(3.9,)h(whic)m(h)g(is)e(again)h (based)h(on)g(a)f(comparison)g(of)g(solutions)118 3418 y(of)i(the)g(nonlinear)g(equation)g(\(3.1\))h(and)f(its)f (linearization)f(around)k(a)e(giv)m(en)g(deterministic)d(solution.)118 3662 y Fp(3.1)112 b(Stable)37 b(case)118 3833 y FH(W)-8 b(e)33 b(\034rst)f(consider)g(the)g(case)h(of)e(a)i(stable)e (equilibrium,)e(that)k(is,)e(w)m(e)i(assume)e(that)i FF(a)p FG(\()p FF(t)p FG(\))c Fn(6)f FE(\000)p FF(a)3482 3847 y FC(0)3553 3833 y FH(for)118 3946 y(all)34 b FF(t)g FE(2)g FF(I)7 b FH(.)56 b(W)-8 b(e)36 b(will)d(assume)h(that)j(the)f (sto)s(c)m(hastic)f(pro)s(cess)g FF(x)2351 3960 y Fy(t)2381 3946 y FH(,)h(giv)m(en)g(b)m(y)g(the)g(SDE)g(\(3.1\))q(,)g(starts)118 4059 y(at)g(time)f FF(t)g FG(=)g(0)i FH(in)e FF(x)865 4073 y FC(0)904 4059 y FH(.)58 b(By)36 b(Theorem)h(2.1,)h(there)f (exists)d(a)i FF(c)2313 4073 y FC(0)2388 4059 y FF(>)f FG(0)i FH(suc)m(h)f(that)h(the)g(deterministic)118 4172 y(solution)29 b FF(x)512 4139 y FC(det)644 4172 y FH(of)37 b(\(2.6\))31 b(with)f(initial)d(condition)j FF(x)1886 4139 y FC(det)1886 4196 y(0)2013 4172 y FG(=)25 b FF(x)2161 4186 y FC(0)2231 4172 y FH(satis\034es)816 4376 y FE(j)p FF(x)893 4339 y FC(det)893 4399 y Fy(t)1015 4376 y FE(\000)20 b FF(x)1158 4339 y Fy(?)1198 4376 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FG(2)p FF(c)1532 4390 y FC(1)1572 4376 y FF(")20 b FG(+)g FE(j)p FF(x)1802 4390 y FC(0)1862 4376 y FE(\000)g FF(x)2005 4339 y Fy(?)2044 4376 y FG(\(0\))p FE(j)15 b FG(e)2241 4339 y FD(\000)p Fy(a)2333 4348 y Fw(0)2368 4339 y Fy(t=)p FC(2)p Fy(")2697 4376 y FE(8)p FF(t)25 b FE(2)g FF(I)7 b(;)512 b FH(\(3.4\))118 4581 y(pro)m(vided)39 b FE(j)p FF(x)575 4595 y FC(0)640 4581 y FE(\000)25 b FF(x)788 4548 y Fy(?)828 4581 y FG(\(0\))p FE(j)40 b Fn(6)e FF(c)1156 4595 y FC(0)1196 4581 y FH(.)64 b(W)-8 b(e)39 b(are)g(in)m(terested)f(in)g(the)h(sto)s(c)m(hastic)f (pro)s(cess)f FF(y)3101 4595 y Fy(t)3169 4581 y FG(=)i FF(x)3331 4595 y Fy(t)3386 4581 y FE(\000)25 b FF(x)3534 4548 y FC(det)3534 4603 y Fy(t)3636 4581 y FH(,)118 4693 y(whic)m(h)34 b(describ)s(es)e(the)i(deviation)f(due)h(to)g(noise)e (from)h(the)h(deterministic)d(solution)i FF(x)3162 4660 y FC(det)3264 4693 y FH(.)50 b(It)33 b(ob)s(eys)118 4806 y(an)e(SDE)g(of)f(the)g(form)886 5026 y FG(d)p FF(y)982 5040 y Fy(t)1036 5026 y FG(=)1142 4965 y(1)p 1142 5005 46 4 v 1144 5088 a FF(")1198 4952 y Fr(\002)1237 5026 y FG(\026)-47 b FF(a)p FG(\()p FF(t)p FG(\))p FF(y)1431 5040 y Fy(t)1481 5026 y FG(+)1569 5002 y(\026)1572 5026 y FF(b)p FG(\()p FF(y)1691 5040 y Fy(t)1721 5026 y FF(;)15 b(t)p FG(\))1829 4952 y Fr(\003)1883 5026 y FG(d)o FF(t)20 b FG(+)2118 4965 y FF(\033)p 2087 5005 119 4 v 2087 5023 a FE(p)p 2163 5023 43 4 v 66 x FF(")2230 5026 y FG(d)p FF(W)2367 5040 y Fy(t)2396 5026 y FF(;)196 b(y)2662 5040 y FC(0)2727 5026 y FG(=)25 b(0)p FF(;)583 b FH(\(3.5\))118 5273 y(where)31 b(w)m(e)g(ha)m(v)m(e)h(in)m(tro)s(duced)f(the)f (notations)945 5456 y FG(\026)-47 b FF(a)p FG(\()p FF(t)p FG(\))26 b(=)g(\026)-46 b FF(a)1264 5470 y Fy(")1301 5456 y FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1574 5470 y Fy(x)1618 5456 y FF(f)10 b FG(\()p FF(x)1760 5419 y FC(det)1760 5479 y Fy(t)1862 5456 y FF(;)15 b(t)p FG(\))861 5583 y(\026)864 5607 y FF(b)p FG(\()p FF(y)s(;)g(t)p FG(\))26 b(=)1213 5583 y(\026)1216 5607 y FF(b)1255 5621 y Fy(")1292 5607 y FG(\()p FF(y)s(;)15 b(t)p FG(\))26 b(=)f FF(f)10 b FG(\()p FF(x)1747 5569 y FC(det)1747 5629 y Fy(t)1869 5607 y FG(+)20 b FF(y)s(;)15 b(t)p FG(\))20 b FE(\000)g FF(f)10 b FG(\()p FF(x)2369 5569 y FC(det)2369 5629 y Fy(t)2471 5607 y FF(;)15 b(t)p FG(\))21 b FE(\000)g FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))p FF(y)s(:)3476 5530 y FH(\(3.6\))1845 5871 y(15)p eop %%Page: 16 16 16 15 bop 118 328 a FH(T)-8 b(aking)35 b FF(")f FH(and)h FE(j)p FF(x)757 342 y FC(0)820 328 y FE(\000)23 b FF(x)966 295 y Fy(?)1005 328 y FG(\(0\))p FE(j)36 b FH(su\036cien)m(tly)e (small,)e(w)m(e)k(ma)m(y)e(assume)f(that)i(there)h(exists)c(a)j (constan)m(t)134 417 y FG(\026)118 441 y FF(d)29 b(>)f FG(0)33 b FH(suc)m(h)g(that)f FE(j)p FF(x)851 408 y FC(det)851 463 y Fy(t)975 441 y FG(+)21 b FF(y)k FE(\000)c FF(x)1281 408 y Fy(?)1320 441 y FG(\()p FF(t)p FG(\))p FE(j)30 b Fn(6)e FF(d)k FH(whenev)m(er)i FE(j)p FF(y)s FE(j)29 b Fn(6)2297 417 y FG(\026)2281 441 y FF(d)p FH(.)47 b(It)31 b(follo)m(ws)g(from)h(T)-8 b(a)m(ylor's)32 b(form)m(ula)118 553 y(that)f(for)f(all)f FG(\()p FF(y)s(;)15 b(t)p FG(\))26 b FE(2)f FG([)p FE(\000)993 529 y FG(\026)977 553 y FF(d;)1080 529 y FG(\026)1064 553 y FF(d)11 b FG(])20 b FE(\002)g FF(I)7 b FH(,)1163 758 y FE(j)1185 734 y FG(\026)1188 758 y FF(b)p FG(\()p FF(y)s(;)15 b(t)p FG(\))p FE(j)26 b Fn(6)f FF(M)10 b(y)1711 720 y FC(2)3476 758 y FH(\(3.7\))979 912 y FE(j)q FG(\026)-46 b FF(a)q FG(\()p FF(t)p FG(\))20 b FE(\000)g FF(a)p FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FF(M)1663 838 y Fr(\000)1705 912 y FG(2)p FF(c)1789 926 y FC(1)1829 912 y FF(")c FG(+)f FE(j)p FF(x)2060 926 y FC(0)2120 912 y FE(\000)g FF(x)2263 875 y Fy(?)2302 912 y FG(\(0\))p FE(j)15 b FG(e)2499 875 y FD(\000)p Fy(a)2591 884 y Fw(0)2626 875 y Fy(t=)p FC(2)p Fy(")2759 838 y Fr(\001)3476 912 y FH(\(3.8\))118 1116 y(By)26 b(again)h(taking)f FF(")h FH(and)g FE(j)p FF(x)1087 1130 y FC(0)1140 1116 y FE(\000)13 b FF(x)1276 1083 y Fy(?)1315 1116 y FG(\(0\))p FE(j)28 b FH(su\036cien)m(tly)e(small,)e(w)m(e)k(ma)m (y)e(further)h(assume)f(that)h(there)h(are)118 1229 y(constan)m(ts)33 b FG(\026)-47 b FF(a)566 1243 y FC(+)651 1229 y Fn(>)26 b FG(\026)-46 b FF(a)795 1243 y FD(\000)879 1229 y FF(>)25 b(a)1023 1243 y FC(0)1062 1229 y FF(=)p FG(4)32 b FH(suc)m(h)e(that) 1291 1434 y FE(\000)q FG(\026)-46 b FF(a)1410 1448 y FC(+)1494 1434 y Fn(6)26 b FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))26 b Fn(6)f FE(\000)q FG(\026)-46 b FF(a)1982 1448 y FD(\000)2222 1434 y FE(8)p FF(t)24 b FE(2)h FF(I)7 b(:)988 b FH(\(3.9\))118 1638 y(Finally)-8 b(,)30 b(the)h(relation)h FG(\026)-46 b FF(a)982 1605 y FD(0)1005 1638 y FG(\()p FF(t)p FG(\))27 b(=)g FF(@)1281 1652 y Fy(xt)1351 1638 y FF(f)10 b FG(\()p FF(x)1493 1605 y FC(det)1493 1660 y Fy(t)1594 1638 y FF(;)15 b(t)p FG(\))22 b(+)e FF(@)1863 1652 y Fy(xx)1947 1638 y FF(f)10 b FG(\()p FF(x)2089 1605 y FC(det)2089 1660 y Fy(t)2191 1638 y FF(;)15 b(t)p FG(\))2309 1602 y FC(1)p 2309 1617 36 4 v 2310 1669 a Fy(")2355 1638 y FF(f)10 b FG(\()p FF(x)2497 1605 y FC(det)2497 1660 y Fy(t)2599 1638 y FF(;)15 b(t)p FG(\))31 b FH(implies)d(the)k (existence)f(of)118 1751 y(a)f(constan)m(t)i FF(c)597 1765 y FC(2)662 1751 y FF(>)25 b FG(0)31 b FH(suc)m(h)g(that)1142 2008 y FE(j)q FG(\026)-46 b FF(a)1215 1971 y FD(0)1238 2008 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)f FF(c)1527 2022 y FC(2)1567 1907 y Fr(\020)1621 2008 y FG(1)c(+)f FE(j)p FF(x)1855 2022 y FC(0)1915 2008 y FE(\000)g FF(x)2058 1971 y Fy(?)2097 2008 y FG(\(0\))p FE(j)2247 1947 y FG(e)2289 1914 y FD(\000)p Fy(a)2381 1923 y Fw(0)2416 1914 y Fy(t=)p FC(2)p Fy(")p 2247 1987 301 4 v 2377 2071 a FF(")2558 1907 y Fr(\021)2613 2008 y FF(:)792 b FH(\(3.10\))259 2238 y(Our)28 b(analysis)e(will)f(b)s(e)i(based)h(on)g(a)f(comparison)g (b)s(et)m(w)m(een)j(solutions)c(of)33 b(\(3.5\))c(and)f(those)g(of)f (the)118 2351 y(linearized)i(equation)1094 2497 y FG(d)o FF(y)1192 2460 y FC(0)1189 2520 y Fy(t)1257 2497 y FG(=)1363 2436 y(1)p 1363 2476 46 4 v 1365 2560 a FF(")1419 2497 y FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))p FF(y)1617 2460 y FC(0)1614 2520 y Fy(t)1672 2497 y FG(d)o FF(t)20 b FG(+)1908 2436 y FF(\033)p 1876 2476 119 4 v 1876 2495 a FE(p)p 1952 2495 43 4 v 65 x FF(")2020 2497 y FG(d)o FF(W)2156 2511 y Fy(t)2186 2497 y FF(;)196 b(y)2455 2460 y FC(0)2452 2520 y(0)2519 2497 y FG(=)25 b(0)p FF(:)745 b FH(\(3.11\))118 2707 y(Its)30 b(solution)f(is)g(giv)m(en)h(b)m(y)883 2962 y FF(y)931 2925 y FC(0)928 2985 y Fy(t)996 2962 y FG(=)1133 2901 y FF(\033)p 1102 2941 119 4 v 1102 2960 a FE(p)p 1177 2960 43 4 v 1177 3025 a FF(")1245 2839 y Fr(Z)1336 2865 y Fy(t)1295 3045 y FC(0)1381 2962 y FG(e)p 1428 2890 35 3 v -37 x Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1686 2962 y FG(d)p FF(W)1823 2976 y Fy(s)1859 2962 y FF(;)p 2091 2912 44 4 v 197 w(\013)p FG(\()p FF(t;)15 b(s)p FG(\))26 b(=)2447 2839 y Fr(Z)2538 2865 y Fy(t)2497 3045 y(s)2584 2962 y FG(\026)-47 b FF(a)q FG(\()p FF(u)p FG(\))15 b(d)p FF(u:)534 b FH(\(3.12\))118 3222 y(W)-8 b(e)32 b(will)d(write)p 690 3172 V 32 w FF(\013)p FG(\()p FF(t;)15 b FG(0\))29 b(=)p 1063 3172 V 27 w FF(\013)p FG(\()p FF(t)p FG(\))k FH(for)e(brevit)m(y)-8 b(.)45 b(The)32 b(Gaussian)f(random)h(v)-5 b(ariable)30 b FF(y)3025 3189 y FC(0)3022 3245 y Fy(t)3096 3222 y FH(has)h(mean)h(zero)118 3335 y(and)f(v)-5 b(ariance)1391 3482 y FF(v)s FG(\()p FF(t)p FG(\))27 b(=)1673 3420 y FF(\033)1728 3387 y FC(2)p 1673 3461 95 4 v 1700 3544 a FF(")1793 3358 y Fr(Z)1884 3384 y Fy(t)1844 3564 y FC(0)1929 3482 y FG(e)1969 3444 y FC(2)p 2011 3409 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2270 3482 y FG(d)o FF(s:)1042 b FH(\(3.13\))118 3695 y(Note)30 b(that)h(\(3.9\))g(implies)c(that)p 1260 3645 44 4 v 30 w FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))27 b Fn(6)e FE(\000)q FG(\026)-46 b FF(a)1736 3709 y FD(\000)1795 3695 y FG(\()p FF(t)20 b FE(\000)f FF(s)p FG(\))30 b FH(whenev)m(er)i FF(t)25 b Fn(>)g FF(s)p FH(,)30 b(whic)m(h)g(implies)d (in)i(partic-)118 3808 y(ular,)j(that)g FF(v)s FG(\()p FF(t)p FG(\))g FH(is)e(not)i(larger)g(than)g FF(\033)1489 3775 y FC(2)1529 3808 y FF(=)p FG(2)q(\026)-46 b FF(a)1667 3822 y FD(\000)1727 3808 y FH(.)44 b(W)-8 b(e)32 b(can,)g(ho)m(w)m(ev)m (er,)i(deriv)m(e)e(a)f(more)h(precise)f(b)s(ound,)118 3921 y(whic)m(h)g(is)e(useful)g(when)i FF(")f FH(and)h FG(e)1252 3888 y FD(\000)p Fy(a)1344 3897 y Fw(0)1379 3888 y Fy(t=)p FC(2)p Fy(")1541 3921 y FH(are)g(small.)38 b(T)-8 b(o)31 b(do)f(so,)h(w)m(e)g(in)m(tro)s(duce)f(the)h(function)582 4176 y FF(\020)7 b FG(\()p FF(t)p FG(\))26 b(=)971 4115 y(1)p 864 4155 261 4 v 864 4238 a(2)p FE(j)q FG(\026)-46 b FF(a)p FG(\(0\))p FE(j)1149 4176 y FG(e)1189 4139 y FC(2)p 1231 4104 35 3 v Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1438 4176 y FG(+)1519 4115 y(1)p 1519 4155 46 4 v 1521 4238 a FF(")1589 4052 y Fr(Z)1680 4079 y Fy(t)1639 4259 y FC(0)1724 4176 y FG(e)1765 4139 y FC(2)p 1807 4104 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2065 4176 y FG(d)p FF(s;)196 b FH(where)p 2651 4126 44 4 v 31 w FF(\013)p FG(\()p FF(t)p FG(\))26 b(=)p 2934 4126 V 25 w FF(\013)p FG(\()p FF(t;)15 b FG(0\))p FF(:)235 b FH(\(3.14\))118 4437 y(Note)26 b(that)g FF(v)s FG(\()p FF(t)p FG(\))g Fn(6)f FF(\033)850 4404 y FC(2)890 4437 y FF(\020)7 b FG(\()p FF(t)p FG(\))p FH(,)27 b(and)f(that)g(b)s(oth)g (functions)f(di\033er)g(b)m(y)h(a)g(term)g(whic)m(h)g(b)s(ecomes)f (negligible)118 4550 y(as)30 b(so)s(on)g(as)g FF(t)25 b(>)g FE(O)s FG(\()p FF(")p FE(j)p FG(log)18 b FF(")p FE(j)p FG(\))p FH(.)41 b(The)31 b(b)s(eha)m(viour)f(of)g FF(\020)7 b FG(\()p FF(t)p FG(\))30 b FH(is)f(c)m(haracterized)j(in)d (the)i(follo)m(wing)e(lemma.)118 4737 y Fq(Lemma)k(3.1.)41 b FB(The)33 b(function)f FF(\020)7 b FG(\()p FF(t)p FG(\))32 b FB(satis\034es)g(the)h(fol)5 b(lowing)31 b(r)-5 b(elations)33 b(for)f(al)5 b(l)32 b FF(t)25 b FE(2)g FF(I)7 b FB(.)948 4980 y FF(\020)g FG(\()p FF(t)p FG(\))26 b(=)1330 4918 y(1)p 1229 4959 248 4 v 1229 5042 a(2)p FE(j)q FG(\026)-46 b FF(a)q FG(\()p FF(t)p FG(\))p FE(j)1507 4980 y FG(+)20 b FE(O)s FG(\()p FF(")p FG(\))h(+)f FE(O)1972 4906 y Fr(\000)2014 4980 y FE(j)p FF(x)2091 4994 y FC(0)2151 4980 y FE(\000)f FF(x)2293 4942 y Fy(?)2333 4980 y FG(\(0\))p FE(j)c FG(e)2530 4942 y FD(\000)p Fy(a)2622 4951 y Fw(0)2657 4942 y Fy(t=)p FC(2)p Fy(")2790 4906 y Fr(\001)3430 4980 y FH(\(3.15\))1585 5157 y FG(1)p 1531 5197 153 4 v 1531 5280 a(2)q(\026)-46 b FF(a)1624 5294 y FC(+)1719 5218 y Fn(6)25 b FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)2149 5157 y FG(1)p 2096 5197 V 2096 5280 a(2)q(\026)-46 b FF(a)2189 5294 y FD(\000)3430 5218 y FH(\(3.16\))1710 5454 y FF(\020)1757 5417 y FD(0)1780 5454 y FG(\()p FF(t)p FG(\))25 b Fn(6)2014 5393 y FG(1)p 2014 5433 46 4 v 2016 5517 a FF(")3430 5454 y FH(\(3.17\))1845 5871 y(16)p eop %%Page: 17 17 17 16 bop 118 328 a Fh(Pr)m(oof:)47 b FH(By)29 b(in)m(tegration)i(b)m (y)f(parts,)h(w)m(e)g(obtain)f(that)1099 583 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b(=)1491 522 y(1)p 1380 562 268 4 v 1380 646 a FE(\000)p FG(2)q(\026)-46 b FF(a)q FG(\()p FF(t)p FG(\))1678 583 y FE(\000)1779 522 y FG(1)p 1779 562 46 4 v 1779 646 a(2)1850 460 y Fr(Z)1941 486 y Fy(t)1900 666 y FC(0)2005 522 y FG(\026)f FF(a)2051 489 y FD(0)2075 522 y FG(\()p FF(s)p FG(\))p 1995 562 201 4 v 1996 646 a(\026)h FF(a)p FG(\()p FF(s)p FG(\))2156 619 y FC(2)2221 583 y FG(e)2262 546 y FC(2)p 2304 511 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2562 583 y FG(d)p FF(s:)749 b FH(\(3.18\))118 836 y(Using)30 b(\(3.9\))h(and)g(\(3.10\))g(w)m(e)g(get)462 992 y Fr(\014)462 1047 y(\014)462 1101 y(\014)492 973 y(Z)583 999 y Fy(t)543 1179 y FC(0)647 1035 y FG(\026)-46 b FF(a)694 1002 y FD(0)717 1035 y FG(\()p FF(s)p FG(\))p 638 1076 201 4 v 639 1159 a(\026)g FF(a)p FG(\()p FF(s)p FG(\))799 1133 y FC(2)864 1097 y FG(e)904 1059 y FC(2)p 946 1024 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1205 1097 y FG(d)o FF(s)1298 992 y Fr(\014)1298 1047 y(\014)1298 1101 y(\014)668 1353 y Fn(6)788 1292 y FF(c)827 1306 y FC(2)p 774 1332 108 4 v 775 1417 a FG(\026)g FF(a)822 1386 y FC(2)822 1441 y FD(\000)906 1230 y Fr(Z)997 1256 y Fy(t)957 1436 y FC(0)1042 1353 y FG(e)1082 1316 y FD(\000)p FC(2)q(\026)-36 b Fy(a)1209 1325 y Fk(\000)1262 1316 y FC(\()p Fy(t)p FD(\000)p Fy(s)p FC(\))p Fy(=")1517 1353 y FG(d)p FF(s)19 b FG(+)1746 1292 y FF(c)1785 1306 y FC(2)p 1731 1332 V 1732 1417 a FG(\026)-46 b FF(a)1779 1386 y FC(2)1779 1441 y FD(\000)1858 1292 y FE(j)p FF(x)1935 1306 y FC(0)1995 1292 y FE(\000)20 b FF(x)2138 1259 y Fy(?)2178 1292 y FG(\(0\))p FE(j)p 1858 1332 461 4 v 2067 1416 a FF(")2344 1230 y Fr(Z)2435 1256 y Fy(t)2395 1436 y FC(0)2480 1353 y FG(e)2520 1316 y FC([)p FD(\000)p FC(2)q(\026)-36 b Fy(a)2667 1325 y Fk(\000)2720 1316 y FC(\()p Fy(t)p FD(\000)p Fy(s)p FC(\))p FD(\000)p Fy(a)2979 1325 y Fw(0)3014 1316 y Fy(s=)p FC(2])p Fy(=")3224 1353 y FG(d)p FF(s)668 1611 y Fn(6)811 1549 y FF(c)850 1563 y FC(2)p 774 1590 153 4 v 774 1675 a FG(2)q(\026)-46 b FF(a)867 1643 y FC(3)867 1698 y FD(\000)937 1611 y FF(")20 b FG(+)1114 1549 y FF(c)1153 1563 y FC(2)p 1100 1590 108 4 v 1101 1675 a FG(\026)-46 b FF(a)1148 1643 y FC(2)1148 1698 y FD(\000)1227 1549 y FE(j)p FF(x)1304 1563 y FC(0)1364 1549 y FE(\000)20 b FF(x)1507 1516 y Fy(?)1546 1549 y FG(\(0\))p FE(j)p 1227 1590 461 4 v 1236 1673 a FG(2)q(\026)-46 b FF(a)1329 1687 y FD(\000)1409 1673 y FE(\000)20 b FF(a)1548 1687 y FC(0)1587 1673 y FF(=)p FG(2)1713 1611 y(e)1753 1573 y FD(\000)p Fy(a)1845 1582 y Fw(0)1880 1573 y Fy(t=)p FC(2)p Fy(")2013 1611 y FF(;)1392 b FH(\(3.19\))118 1873 y(whic)m(h)31 b(pro)m(v)m(es)g (\(3.15\))q(.)40 b(W)-8 b(e)31 b(no)m(w)g(observ)m(e)g(that)g FF(\020)7 b FG(\()p FF(t)p FG(\))30 b FH(is)f(a)h(solution)g(of)f(the)i (linear)e(ODE)1101 2058 y FG(d)p FF(\020)p 1101 2099 97 4 v 1108 2182 a FG(d)o FF(t)1233 2120 y FG(=)1339 2058 y(1)p 1339 2099 46 4 v 1341 2182 a FF(")1395 2046 y Fr(\000)1436 2120 y FG(2)q(\026)-46 b FF(a)q FG(\()p FF(t)p FG(\))p FF(\020)27 b FG(+)20 b(1)1836 2046 y Fr(\001)1878 2120 y FF(;)196 b(\020)7 b FG(\(0\))26 b(=)2500 2058 y(1)p 2393 2099 261 4 v 2393 2182 a(2)p FE(j)q FG(\026)-46 b FF(a)q FG(\(0\))p FE(j)2663 2120 y FF(:)742 b FH(\(3.20\))118 2374 y(Since)36 b FF(\020)7 b FG(\()p FF(t)p FG(\))35 b FF(>)g FG(0)h FH(and)i FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))35 b FF(<)g FG(0)p FH(,)j(w)m(e)f(ha)m(v)m(e)g FF(\020)1714 2341 y FD(0)1737 2374 y FG(\()p FF(t)p FG(\))e Fn(6)g FG(1)p FF(=")p FH(.)59 b(W)-8 b(e)37 b(also)e(see)h(that)g FF(\020)2946 2341 y FD(0)2969 2374 y FG(\()p FF(t)p FG(\))g Fn(>)e FG(0)j FH(whenev)m(er)118 2487 y FF(\020)7 b FG(\()p FF(t)p FG(\))38 b Fn(6)f FG(1)p FF(=)p FG(2)q(\026)-46 b FF(a)597 2501 y FC(+)695 2487 y FH(and)38 b FF(\020)925 2454 y FD(0)947 2487 y FG(\()p FF(t)p FG(\))g Fn(6)f FG(0)h FH(whenev)m(er)h FF(\020)7 b FG(\()p FF(t)p FG(\))38 b Fn(>)f FG(1)p FF(=)p FG(2)q(\026)-46 b FF(a)2162 2501 y FD(\000)2222 2487 y FH(.)62 b(Since)38 b FF(\020)7 b FG(\(0\))38 b FH(b)s(elongs)e(to)i(the)g(in)m(terv)-5 b(al)118 2600 y FG([1)p FF(=)p FG(2)q(\026)-46 b FF(a)326 2614 y FC(+)387 2600 y FF(;)15 b FG(1)p FF(=)p FG(2)q(\026)-46 b FF(a)610 2614 y FD(\000)671 2600 y FG(])p FH(,)30 b FF(\020)7 b FG(\()p FF(t)p FG(\))30 b FH(m)m(ust)g(remain)g(in)f(this)g (in)m(terv)-5 b(al)30 b(for)g(all)f FF(t)p FH(.)p 3596 2600 4 62 v 3600 2543 55 4 v 3600 2600 V 3653 2600 4 62 v 259 2788 a(As)e(w)m(e)h(ha)m(v)m(e)g(already)f(seen)h(in)e (\(2.14\))q(,)i(the)g(probabilit)m(y)e(of)h(\034nding)g FF(y)2679 2755 y FC(0)2676 2810 y Fy(t)2746 2788 y FH(outside)g(a)g (strip)f(of)h(width)118 2901 y(m)m(uc)m(h)g(larger)f(than)814 2823 y Fr(p)p 905 2823 197 4 v 78 x FG(2)p FF(v)s FG(\()p FF(t)p FG(\))i FH(is)d(v)m(ery)h(small.)37 b(By)25 b(Lemma)h(3.1,)i(w)m (e)f(no)m(w)g(kno)m(w)g(that)3041 2823 y Fr(p)p 3132 2823 V 78 x FG(2)p FF(v)s FG(\()p FF(t)p FG(\))h FH(b)s(eha)m(v)m(es) 118 3014 y(appro)m(ximately)j(lik)m(e)f FF(\033)s FE(j)p FF(a)p FG(\()p FF(t)p FG(\))p FE(j)1138 2981 y FD(\000)p FC(1)p Fy(=)p FC(2)1304 3014 y FH(.)45 b(One)32 b(of)g(the)g(k)m(ey)g (p)s(oin)m(ts)f(of)g(the)h(presen)m(t)h(w)m(ork)g(is)d(to)i(sho)m(w)h (that)118 3127 y(the)k FB(whole)i(p)-5 b(ath)36 b FE(f)p FF(y)824 3141 y Fy(s)861 3127 y FE(g)906 3141 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1150 3127 y FH(remains)f(in)h(a)h(strip)e(of) h(similar)d(width)k(with)f(high)g(probabilit)m(y)-8 b(.)58 b(The)118 3240 y(strip)28 b(will)e(b)s(e)j(de\034ned)g(with)g(the)g (help)f(of)g FF(\020)7 b FG(\()p FF(t)p FG(\))29 b FH(instead)f(of)h FF(v)s FG(\()p FF(t)p FG(\))p FH(,)g(b)s(ecause)g(w)m(e)h(need)f(the)g (width)g(to)g(b)s(e)118 3352 y(b)s(ounded)i(a)m(w)m(a)m(y)h(from)e (zero,)h(ev)m(en)g(for)f(small)d FF(t)p FH(.)259 3465 y(T)-8 b(o)32 b(in)m(v)m(estigate)f FF(y)892 3432 y FC(0)889 3488 y Fy(t)962 3465 y FH(w)m(e)h(need)g(to)f(estimate)f(the)h(sto)s(c) m(hastic)g(in)m(tegral)g(from)f(\(3.12\))q(.)43 b(Lemma)30 b(A.1)118 3578 y(in)g(the)g(app)s(endix)g(pro)m(vides)g(the)h(estimate) 853 3833 y Fo(P)908 3732 y Fr(n)1002 3833 y FG(sup)969 3910 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1187 3709 y Fr(Z)1278 3735 y Fy(s)1238 3915 y FC(0)1330 3833 y FF(')p FG(\()p FF(u)p FG(\))15 b(d)q FF(W)1664 3847 y Fy(u)1734 3833 y Fn(>)25 b FF(\016)1873 3732 y Fr(o)1960 3833 y Fn(6)g FG(exp)2195 3705 y Fr(\032)2263 3833 y FE(\000)2554 3771 y FF(\016)2597 3738 y FC(2)p 2344 3812 506 4 v 2344 3911 a FG(2)2404 3837 y Fr(R)2465 3864 y Fy(t)2447 3943 y FC(0)2509 3911 y FF(')p FG(\()p FF(u)p FG(\))2690 3884 y FC(2)2746 3911 y FG(d)p FF(u)2859 3705 y Fr(\033)3430 3833 y FH(\(3.21\))118 4110 y(for)j(Borel-measurable)f(deterministic)f (functions)i FF(')p FG(\()p FF(u)p FG(\))p FH(.)40 b(Unfortunately)-8 b(,)29 b(this)e(estimate)g(cannot)j(b)s(e)118 4223 y(applied)g (directly)-8 b(,)31 b(b)s(ecause)g(in)f(\(3.12\))q(,)h(the)h(in)m (tegrand)g(dep)s(ends)f(explicitly)d(on)j(the)g(upp)s(er)h(in)m(tegra-) 118 4336 y(tion)e(limit.)37 b(This)29 b(is)g(wh)m(y)i(w)m(e)g(in)m(tro) s(duce)g(a)f(partition)g(of)g(the)h(in)m(terv)-5 b(al)29 b FG([0)p FF(;)15 b(t)p FG(])p FH(.)118 4524 y Fq(Lemma)33 b(3.2.)41 b FB(L)-5 b(et)32 b FF(\032)26 b FG(:)f FF(I)32 b FE(!)26 b Fo(R)1230 4538 y FC(+)1327 4524 y FB(b)-5 b(e)34 b(a)e(me)-5 b(asur)g(able,)34 b(strictly)e(p)-5 b(ositive)33 b(function.)41 b(Fix)32 b FF(K)g FE(2)25 b Fo(N)d FB(,)38 b(and)118 4637 y(let)33 b FG(0)26 b(=)f FF(u)464 4651 y FC(0)528 4637 y Fn(6)g FF(u)676 4651 y FC(1)741 4637 y FF(<)g FE(\001)15 b(\001)g(\001)26 b FF(<)f(u)1116 4651 y Fy(K)1210 4637 y FG(=)g FF(t)32 b FB(b)-5 b(e)34 b(a)e(p)-5 b(artition)32 b(of)g(the)h(interval)g FG([0)p FF(;)15 b(t)p FG(])p FB(.)42 b(Then)1238 4927 y Fo(P)1299 4890 y FC(0)p Fy(;)p FC(0)1394 4826 y Fr(n)1487 4927 y FG(sup)1454 5004 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1694 4866 y FE(j)p FF(y)1767 4833 y FC(0)1764 4888 y Fy(s)1806 4866 y FE(j)p 1683 4906 161 4 v 1683 4990 a FF(\032)p FG(\()p FF(s)p FG(\))1878 4927 y Fn(>)25 b FF(h)2026 4826 y Fr(o)2112 4927 y Fn(6)g FG(2)2302 4814 y Fy(K)2269 4841 y Fr(X)2270 5038 y Fy(k)r FC(=1)2415 4927 y FF(P)2473 4942 y Fy(k)2516 4927 y FF(;)889 b FH(\(3.22\))118 5210 y FB(wher)-5 b(e)359 5447 y FF(P)417 5462 y Fy(k)486 5447 y FG(=)25 b(exp)720 5319 y Fr(\032)789 5447 y FE(\000)870 5385 y FG(1)p 870 5426 46 4 v 870 5509 a(2)936 5385 y FF(h)988 5352 y FC(2)p 935 5426 95 4 v 935 5509 a FF(\033)990 5483 y FC(2)1039 5346 y Fr(\020)1228 5447 y FG(inf)1094 5506 y Fy(u)1135 5518 y Fj(k)q Fk(\000)p Fw(1)1251 5506 y Fx(6)p Fy(s)p Fx(6)p Fy(u)1435 5518 y Fj(k)1487 5447 y FF(\032)p FG(\()p FF(s)p FG(\))1647 5409 y FC(2)1702 5447 y FG(e)1743 5409 y FC(2)p 1785 5374 35 3 v Fy(\013)p FC(\()p Fy(u)1891 5421 y Fj(k)1930 5409 y Fy(;s)p FC(\))p Fy(=")2081 5346 y Fr(\021)q(\020)2200 5385 y FG(1)p 2200 5426 46 4 v 2202 5509 a FF(")2271 5323 y Fr(Z)2362 5349 y Fy(u)2403 5361 y Fj(k)2321 5529 y FC(0)2460 5447 y FG(e)2500 5409 y FC(2)p 2542 5374 35 3 v Fy(\013)p FC(\()p Fy(u)2648 5421 y Fj(k)2687 5409 y Fy(;s)p FC(\))p Fy(=")2854 5447 y FG(d)p FF(s)2948 5346 y Fr(\021)3001 5368 y FD(\000)p FC(1)3096 5319 y Fr(\033)3164 5447 y FF(:)241 b FH(\(3.23\))1845 5871 y(17)p eop %%Page: 18 18 18 17 bop 118 328 a Fh(Pr)m(oof:)47 b FH(W)-8 b(e)30 b(ha)m(v)m(e)347 561 y Fo(P)408 524 y FC(0)p Fy(;)p FC(0)503 461 y Fr(n)612 561 y FG(sup)579 638 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)818 500 y FE(j)p FF(y)891 467 y FC(0)888 522 y Fy(s)931 500 y FE(j)p 807 541 161 4 v 807 624 a FF(\032)p FG(\()p FF(s)p FG(\))1003 561 y Fn(>)25 b FF(h)1151 461 y Fr(o)3430 561 y FH(\(3.24\))589 826 y FG(=)g Fo(P)746 788 y FC(0)p Fy(;)p FC(0)841 725 y Fr(n)934 826 y FG(sup)901 903 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1187 764 y FG(1)p 1130 805 V 1130 888 a FF(\032)p FG(\()p FF(s)p FG(\))1300 721 y Fr(\014)1300 776 y(\014)1300 830 y(\014)1330 702 y(Z)1421 728 y Fy(s)1381 908 y FC(0)1473 826 y FG(e)p 1520 753 35 3 v -38 x Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")1794 826 y FG(d)o FF(W)1930 840 y Fy(u)1975 721 y Fr(\014)1975 776 y(\014)1975 830 y(\014)2031 826 y Fn(>)2137 764 y FF(h)2189 699 y FE(p)p 2265 699 43 4 v 65 x FF(")p 2137 805 171 4 v 2194 888 a(\033)2317 725 y Fr(o)589 1090 y FG(=)g Fo(P)746 1052 y FC(0)p Fy(;)p FC(0)841 989 y Fr(n)901 1090 y FE(9)p FF(k)j FE(2)d(f)p FG(1)p FF(;)15 b(:)g(:)g(:)j(;)d(K)7 b FE(g)25 b FG(:)147 b(sup)1611 1167 y Fy(u)1652 1179 y Fj(k)q Fk(\000)p Fw(1)1768 1167 y Fx(6)p Fy(s)p Fx(6)p Fy(u)1952 1179 y Fj(k)2072 1028 y FG(1)p 2015 1069 161 4 v 2015 1152 a FF(\032)p FG(\()p FF(s)p FG(\))2185 985 y Fr(\014)2185 1040 y(\014)2185 1094 y(\014)2215 966 y(Z)2306 992 y Fy(s)2266 1172 y FC(0)2358 1090 y FG(e)p 2405 1017 35 3 v 2399 1052 a Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2679 1090 y FG(d)p FF(W)2816 1104 y Fy(u)2860 985 y Fr(\014)2860 1040 y(\014)2860 1094 y(\014)2916 1090 y Fn(>)3022 1028 y FF(h)3074 963 y FE(p)p 3150 963 43 4 v 65 x FF(")p 3022 1069 171 4 v 3080 1152 a(\033)3202 989 y Fr(o)589 1396 y Fn(6)25 b FG(2)779 1282 y Fy(K)746 1309 y Fr(X)747 1507 y Fy(k)r FC(=1)892 1396 y Fo(P)953 1358 y FC(0)p Fy(;)p FC(0)1048 1295 y Fr(n)1229 1396 y FG(sup)1108 1473 y Fy(u)1149 1485 y Fj(k)q Fk(\000)p Fw(1)1266 1473 y Fx(6)p Fy(s)p Fx(6)p Fy(u)1450 1485 y Fj(k)1502 1272 y Fr(Z)1593 1298 y Fy(s)1553 1478 y FC(0)1645 1396 y FG(e)1685 1358 y FD(\000)p 1747 1323 35 3 v Fy(\013)p FC(\()p Fy(u)p FC(\))p Fy(=")1969 1396 y FG(d)o FF(W)2105 1410 y Fy(u)2175 1396 y Fn(>)2281 1334 y FF(h)2333 1269 y FE(p)p 2409 1269 43 4 v 65 x FF(")p 2281 1375 171 4 v 2339 1458 a(\033)2611 1396 y FG(inf)2477 1455 y Fy(u)2518 1467 y Fj(k)q Fk(\000)p Fw(1)2634 1455 y Fx(6)p Fy(s)p Fx(6)p Fy(u)2818 1467 y Fj(k)2871 1396 y FF(\032)p FG(\()p FF(s)p FG(\))15 b(e)3087 1358 y FD(\000)p 3149 1323 35 3 v Fy(\013)p FC(\()p Fy(s)p FC(\))p Fy(=")3346 1295 y Fr(o)3407 1396 y FF(:)118 1681 y FH(Applying)29 b(Lemma)g(A.1)i(to)f (the)h(last)e(expression,)h(w)m(e)h(obtain)f(\(3.22\))q(.)p 3596 1681 4 62 v 3600 1623 55 4 v 3600 1681 V 3653 1681 4 62 v 259 1868 a(W)-8 b(e)33 b(are)g(no)m(w)g(ready)g(to)g(deriv)m(e)f (an)h(upp)s(er)g(b)s(ound)f(for)h(the)g(probabilit)m(y)e(that)i FF(y)3070 1835 y FC(0)3067 1890 y Fy(s)3141 1868 y FH(lea)m(v)m(es)g(a) f(strip)118 1981 y(of)27 b(appropriate)i(width)f FF(h\032)p FG(\()p FF(s)p FG(\))g FH(b)s(efore)f(time)g FF(t)p FH(.)39 b(T)-8 b(aking)28 b FF(\032)p FG(\()p FF(s)p FG(\))d(=)2346 1903 y Fr(p)p 2437 1903 160 4 v 78 x FF(\020)7 b FG(\()p FF(s)p FG(\))27 b FH(will)e(b)s(e)j(a)g(go)s(o)s(d)f(c)m(hoice)h(since) 118 2094 y(it)i(leads)f(to)i(appro)m(ximately)e(constan)m(t)i FF(P)1563 2109 y Fy(k)1636 2094 y FH(in)f(\(3.22\))q(.)118 2279 y Fq(Prop)s(osition)35 b(3.3.)41 b FB(Ther)-5 b(e)34 b(exists)e(an)g FF(r)c FG(=)d FF(r)s FG(\()q(\026)-46 b FF(a)1809 2293 y FC(+)1868 2279 y FF(;)17 b FG(\026)-47 b FF(a)1956 2293 y FD(\000)2016 2279 y FG(\))32 b FB(such)h(that)798 2535 y Fo(P)859 2498 y FC(0)p Fy(;)p FC(0)954 2434 y Fr(n)1048 2535 y FG(sup)1015 2612 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1299 2474 y FE(j)p FF(y)1372 2441 y FC(0)1369 2496 y Fy(s)1412 2474 y FE(j)p 1243 2514 251 4 v 1243 2533 a Fr(p)p 1334 2533 160 4 v 77 x FF(\020)7 b FG(\()p FF(s)p FG(\))1529 2535 y Fn(>)25 b FF(h)1677 2434 y Fr(o)1763 2535 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)2271 2434 y Fr(n)2331 2535 y FE(\000)2412 2474 y FG(1)p 2412 2514 46 4 v 2412 2598 a(2)2479 2474 y FF(h)2531 2441 y FC(2)p 2478 2514 95 4 v 2478 2598 a FF(\033)2533 2571 y FC(2)2582 2535 y FG(\(1)21 b FE(\000)f FF(r)s(")p FG(\))2895 2434 y Fr(o)2956 2535 y FF(;)449 b FH(\(3.25\))118 2797 y FB(wher)-5 b(e)1493 2935 y FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))26 b(=)1882 2874 y FE(j)p 1917 2824 44 4 v FF(\013)p FG(\()p FF(t)p FG(\))p FE(j)p 1882 2914 213 4 v 1947 2997 a FF(")1989 2971 y FC(2)2125 2935 y FG(+)20 b(2)p FF(:)1144 b FH(\(3.26\))118 3140 y Fh(Pr)m(oof:)47 b FH(Let)1605 3281 y FF(K)32 b FG(=)1810 3152 y Fr(\030)1873 3219 y FE(j)p 1908 3169 44 4 v FF(\013)p FG(\()p FF(t)p FG(\))p FE(j)p 1873 3260 213 4 v 1916 3343 a FG(2)p FF(")2003 3317 y FC(2)2096 3152 y Fr(\031)2149 3281 y FF(:)1256 b FH(\(3.27\))118 3491 y(F)-8 b(or)31 b FF(k)e FG(=)24 b(1)p FF(;)15 b(:)g(:)g(:)j(;)d(K)27 b FE(\000)20 b FG(1)p FH(,)30 b(w)m(e)i(de\034ne)f(the)f(partition)g(times)f FF(u)2215 3506 y Fy(k)2288 3491 y FH(b)m(y)h(the)h(relation)1591 3692 y FE(j)p 1626 3642 44 4 v FF(\013)p FG(\()p FF(u)1761 3707 y Fy(k)1804 3692 y FG(\))p FE(j)26 b FG(=)f(2)p FF(")2073 3654 y FC(2)2113 3692 y FF(k)s(;)1242 b FH(\(3.28\))118 3893 y(whic)m(h)28 b(is)e(p)s(ossible)g(since)p 1028 3843 V 26 w FF(\013)p FG(\()p FF(t)p FG(\))j FH(is)d(con)m(tin)m(uous)j (and)f(decreasing.)40 b(This)26 b(de\034nition)h(implies)d(in)j (partic-)118 4006 y(ular)32 b(that)p 514 3956 V 33 w FF(\013)p FG(\()p FF(u)649 4021 y Fy(k)692 4006 y FF(;)15 b(u)784 4021 y Fy(k)r FD(\000)p FC(1)918 4006 y FG(\))28 b(=)h FE(\000)p FG(2)p FF(")1239 3973 y FC(2)1311 4006 y FH(and,)k(therefore,)h FF(u)1973 4021 y Fy(k)2037 4006 y FE(\000)21 b FF(u)2181 4021 y Fy(k)r FD(\000)p FC(1)2342 4006 y Fn(6)29 b FG(2)p FF(")2529 3973 y FC(2)2569 4006 y FF(=)q FG(\026)-46 b FF(a)2662 4020 y FD(\000)2721 4006 y FH(.)47 b(Bounding)32 b(the)h(in)m(tegral)118 4119 y(in)d(\(3.23\))h(b)m(y)f FF(\020)7 b FG(\()p FF(u)744 4134 y Fy(k)787 4119 y FG(\))p FH(,)30 b(w)m(e)i(obtain)993 4375 y FF(P)1051 4390 y Fy(k)1119 4375 y Fn(6)25 b FG(exp)1354 4274 y Fr(n)1415 4375 y FE(\000)1496 4314 y FG(1)p 1496 4354 46 4 v 1496 4438 a(2)1562 4314 y FF(h)1614 4281 y FC(2)p 1561 4354 95 4 v 1561 4438 a FF(\033)1616 4411 y FC(2)1815 4375 y FG(inf)1681 4435 y Fy(u)1722 4447 y Fj(k)q Fk(\000)p Fw(1)1838 4435 y Fx(6)p Fy(s)p Fx(6)p Fy(u)2022 4447 y Fj(k)2111 4314 y FF(\020)7 b FG(\()p FF(s)p FG(\))p 2084 4354 212 4 v 2084 4438 a FF(\020)g FG(\()p FF(u)2218 4453 y Fy(k)2261 4438 y FG(\))2321 4375 y(e)2362 4338 y FC(2)p 2404 4303 35 3 v Fy(\013)p FC(\()p Fy(u)2510 4350 y Fj(k)2549 4338 y Fy(;s)p FC(\))p Fy(=")2701 4274 y Fr(o)2761 4375 y FF(:)644 b FH(\(3.29\))118 4642 y(W)-8 b(e)31 b(ha)m(v)m(e)g FG(e)523 4609 y FC(2)p 565 4574 V Fy(\013)p FC(\()p Fy(u)671 4621 y Fj(k)710 4609 y Fy(;s)p FC(\))p Fy(=")887 4642 y Fn(>)25 b FG(e)1023 4609 y FD(\000)p FC(4)p Fy(")1180 4642 y FH(and)1052 4877 y FF(\020)7 b FG(\()p FF(s)p FG(\))20 b FE(\000)g FF(\020)7 b FG(\()p FF(u)1457 4892 y Fy(k)1500 4877 y FG(\))25 b(=)g FE(\000)1742 4753 y Fr(Z)1833 4779 y Fy(u)1874 4791 y Fj(k)1793 4959 y Fy(s)1931 4877 y FF(\020)1978 4839 y FD(0)2001 4877 y FG(\()p FF(u)p FG(\))15 b(d)q FF(u)25 b Fn(>)g FE(\000)2444 4815 y FF(u)2496 4830 y Fy(k)2558 4815 y FE(\000)20 b FF(s)p 2444 4856 249 4 v 2547 4939 a(")2702 4877 y(:)703 b FH(\(3.30\))118 5124 y(Since)30 b FF(\020)7 b FG(\()p FF(u)488 5139 y Fy(k)531 5124 y FG(\))25 b Fn(>)g FG(1)p FF(=)p FG(2)q(\026)-46 b FF(a)870 5138 y FC(+)931 5124 y FH(,)30 b(this)f(implies)1179 5380 y FF(P)1237 5395 y Fy(k)1306 5380 y Fn(6)c FG(exp)1541 5279 y Fr(n)1601 5380 y FE(\000)1682 5319 y FG(1)p 1682 5359 46 4 v 1682 5442 a(2)1749 5319 y FF(h)1801 5286 y FC(2)p 1747 5359 95 4 v 1747 5442 a FF(\033)1802 5416 y FC(2)1852 5279 y Fr(\020)1906 5380 y FG(1)c FE(\000)f FG(4)2119 5319 y(\026)-46 b FF(a)2166 5333 y FC(+)p 2118 5359 108 4 v 2119 5442 a FG(\026)g FF(a)2166 5456 y FD(\000)2235 5380 y FF(")2277 5279 y Fr(\021)2347 5380 y FG(e)2387 5343 y FD(\000)p FC(4)p Fy(")2514 5279 y Fr(o)2575 5380 y FF(;)830 b FH(\(3.31\))118 5622 y(and)31 b(the)g(result)e(follo)m(ws) g(from)g(Lemma)h(3.2.)p 3596 5622 4 62 v 3600 5564 55 4 v 3600 5622 V 3653 5622 4 62 v 1845 5871 a(18)p eop %%Page: 19 19 19 18 bop 118 328 a Fq(Remark)31 b(3.4.)39 b FH(If)27 b(w)m(e)i(only)e(assume)g(that)j FG(\026)-47 b FF(a)28 b FH(is)f(Borel-measurable)g(with)i FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))25 b Fn(6)g FE(\000)q FG(\026)-46 b FF(a)3099 342 y FD(\000)3186 328 y FH(for)28 b(all)e FF(t)f FE(2)g FF(I)7 b FH(,)118 441 y(w)m(e)31 b(still)d(ha)m(v)m(e)793 586 y Fo(P)854 549 y FC(0)p Fy(;)p FC(0)949 485 y Fr(n)1043 586 y FG(sup)1010 663 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1213 586 y FE(j)p FF(y)1286 549 y FC(0)1283 609 y Fy(s)1325 586 y FE(j)e Fn(>)f FF(h=)1569 508 y Fr(p)p 1660 508 153 4 v 78 x FG(2)q(\026)-46 b FF(a)1753 600 y FD(\000)1813 485 y Fr(o)1899 586 y Fn(6)25 b FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)2407 485 y Fr(n)2467 586 y FE(\000)2548 525 y FG(1)p 2548 565 46 4 v 2548 649 a(2)2615 525 y FF(h)2667 492 y FC(2)p 2613 565 95 4 v 2613 649 a FF(\033)2668 622 y FC(2)2733 586 y FG(e)2774 549 y FD(\000)p FC(4)p Fy(")2900 485 y Fr(o)2961 586 y FF(:)444 b FH(\(3.32\))118 813 y(T)-8 b(o)31 b(pro)m(v)m(e)h(this,)e(w)m(e)h(c)m(ho)s(ose)g(the)g (same)f(partition)f(as)i(b)s(efore)f(and)h(b)s(ound)g(the)g(in)m (tegral)f(in)g(\(3.23\))h(b)m(y)118 925 y FF("=)p FG(2)q(\026)-46 b FF(a)298 939 y FD(\000)359 925 y FH(.)259 1112 y(W)-8 b(e)43 b(no)m(w)h(return)g(to)f(the)h(nonlinear)e(equation)h(\(3.5\),)j (the)e(solutions)d(of)i(whic)m(h)g(w)m(e)h(w)m(an)m(t)g(to)118 1225 y(compare)31 b(to)f(those)h(of)f(its)f(linearization)f(\(3.11\))q (.)40 b(T)-8 b(o)32 b(this)d(end,)i(w)m(e)g(in)m(tro)s(duce)g(the)f(ev) m(en)m(ts)1038 1450 y FG(\012)1104 1464 y Fy(t)1133 1450 y FG(\()p FF(h)p FG(\))d(=)1378 1349 y Fr(n)1438 1450 y FF(!)13 b FG(:)1564 1373 y Fr(\014)1564 1427 y(\014)1594 1450 y FF(y)1639 1464 y Fy(s)1676 1450 y FG(\()p FF(!)s FG(\))1806 1373 y Fr(\014)1806 1427 y(\014)1862 1450 y FF(<)25 b(h)2010 1368 y Fr(p)p 2101 1368 160 4 v 82 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b FE(8)p FF(s)f FE(2)h FG([0)p FF(;)15 b(t)p FG(])2658 1349 y Fr(o)3430 1450 y FH(\(3.33\))1028 1647 y FG(\012)1094 1609 y FC(0)1094 1669 y Fy(t)1133 1647 y FG(\()p FF(h)p FG(\))27 b(=)1378 1546 y Fr(n)1438 1647 y FF(!)13 b FG(:)1564 1570 y Fr(\014)1564 1624 y(\014)1594 1647 y FF(y)1642 1609 y FC(0)1639 1669 y Fy(s)1681 1647 y FG(\()p FF(!)s FG(\))1811 1570 y Fr(\014)1811 1624 y(\014)1868 1647 y FF(<)25 b(h)2016 1565 y Fr(p)p 2107 1565 V 82 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b FE(8)p FF(s)f FE(2)h FG([0)p FF(;)15 b(t)p FG(])2664 1546 y Fr(o)2726 1647 y FF(:)679 b FH(\(3.34\))118 1880 y(Prop)s(osition)33 b(3.3)h(giv)m(es)f(us)g(an)h(upp)s(er)g(b)s(ound)h (on)e(the)i(probabilit)m(y)d(of)h(the)i(complemen)m(t)e(of)g FG(\012)3474 1847 y FC(0)3474 1902 y Fy(t)3513 1880 y FG(\()p FF(h)p FG(\))p FH(.)118 1993 y(The)g(k)m(ey)g(p)s(oin)m(t)f(to) h(con)m(trol)h(the)f(nonlinear)f(case)h(is)e(a)i(relation)f(b)s(et)m(w) m(een)i(the)f(sets)g FG(\012)3141 2007 y Fy(t)3202 1993 y FH(and)h FG(\012)3447 1960 y FC(0)3447 2015 y Fy(t)3518 1993 y FH(\(for)118 2106 y(sligh)m(tly)28 b(di\033eren)m(t)j(v)-5 b(alues)29 b(of)h FF(h)p FH(\).)41 b(This)29 b(is)g(done)i(in)e(Prop)s (osition)h(3.6)g(b)s(elo)m(w.)118 2315 y Fq(Notation)46 b(3.5.)g FB(F)-7 b(or)40 b(two)h(events)g FG(\012)1474 2329 y FC(1)1553 2315 y FB(and)f FG(\012)1802 2329 y FC(2)1841 2315 y FB(,)i(we)e(write)g FG(\012)2355 2329 y FC(1)2434 2248 y(a)p Fy(:)p FC(s)p Fy(:)2450 2315 y FE(\032)55 b FG(\012)2642 2329 y FC(2)2721 2315 y FB(if)k Fo(P)p FB(-almost)41 b(al)5 b(l)40 b FF(!)i FE(2)d FG(\012)3622 2329 y FC(1)118 2428 y FB(b)-5 b(elong)34 b(to)e FG(\012)564 2442 y FC(2)604 2428 y FB(.)118 2614 y Fq(Prop)s(osition)j(3.6.)41 b FB(L)-5 b(et)33 b FF(\015)d FG(=)25 b(2)1256 2549 y FE(p)p 1333 2549 153 4 v 1333 2614 a FG(2)q(\026)-46 b FF(a)1426 2628 y FC(+)1500 2614 y FF(M)5 b(=)q FG(\026)-46 b FF(a)1686 2581 y FC(2)1686 2637 y FD(\000)1778 2614 y FB(and)32 b(assume)h(that)g FF(h)25 b(<)2643 2590 y FG(\026)2627 2614 y FF(d)2675 2536 y Fr(p)p 2765 2536 199 4 v 78 x FG(\026)-46 b FF(a)2813 2628 y FD(\000)2873 2614 y FF(=)p FG(2)21 b FE(^)e FF(\015)3116 2581 y FD(\000)p FC(1)3211 2614 y FB(.)41 b(Then)1398 2846 y FG(\012)1464 2860 y Fy(t)1493 2846 y FG(\()p FF(h)p FG(\))1641 2779 y FC(a)p Fy(:)p FC(s)p Fy(:)1657 2846 y FE(\032)g FG(\012)1835 2809 y FC(0)1835 2869 y Fy(t)1874 2746 y Fr(\020)1928 2773 y(\002)1966 2846 y FG(1)21 b(+)2133 2785 y FF(\015)p 2133 2826 53 4 v 2136 2909 a FG(4)2195 2846 y FF(h)2247 2773 y Fr(\003)2285 2846 y FF(h)2337 2746 y Fr(\021)3430 2846 y FH(\(3.35\))1388 3047 y FG(\012)1454 3009 y FC(0)1454 3069 y Fy(t)1493 3047 y FG(\()p FF(h)p FG(\))1641 2980 y FC(a)p Fy(:)p FC(s)p Fy(:)1657 3047 y FE(\032)41 b FG(\012)1835 3061 y Fy(t)1864 2946 y Fr(\020)1918 2973 y(\002)1956 3047 y FG(1)21 b(+)f FF(\015)5 b(h)2217 2973 y Fr(\003)2255 3047 y FF(h)2307 2946 y Fr(\021)2362 3047 y FF(:)1043 b FH(\(3.36\))118 3341 y Fh(Pr)m(oof:)156 3478 y FH(1.)47 b(The)31 b(di\033erence)f FF(z)905 3492 y Fy(s)968 3478 y FG(=)25 b FF(y)1109 3492 y Fy(s)1165 3478 y FE(\000)20 b FF(y)1304 3445 y FC(0)1301 3500 y Fy(s)1374 3478 y FH(satis\034es)1361 3661 y FG(d)o FF(z)1453 3675 y Fy(s)p 1361 3702 130 4 v 1379 3785 a FG(d)o FF(s)1525 3723 y FG(=)1631 3661 y(1)p 1631 3702 46 4 v 1633 3785 a FF(")1687 3649 y Fr(\002)1726 3723 y FG(\026)-46 b FF(a)p FG(\()p FF(s)p FG(\))p FF(z)1928 3737 y Fy(s)1985 3723 y FG(+)2073 3699 y(\026)2076 3723 y FF(b)p FG(\()p FF(y)2198 3685 y FC(0)2195 3745 y Fy(s)2258 3723 y FG(+)20 b FF(z)2391 3737 y Fy(s)2428 3723 y FF(;)15 b(s)p FG(\))2546 3649 y Fr(\003)3430 3723 y FH(\(3.37\))273 3946 y(with)30 b FF(z)521 3960 y FC(0)586 3946 y FG(=)25 b(0)31 b Fo(P)p FH(-a.s.)41 b(No)m(w,)1279 4188 y FF(z)1321 4202 y Fy(s)1383 4188 y FG(=)1489 4126 y(1)p 1489 4167 V 1491 4250 a FF(")1559 4064 y Fr(Z)1650 4090 y Fy(s)1610 4270 y FC(0)1702 4188 y FG(e)p 1749 4115 35 3 v 1743 4150 a Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2020 4164 y FG(\026)2023 4188 y FF(b)p FG(\()p FF(y)2145 4150 y FC(0)2142 4210 y Fy(u)2207 4188 y FG(+)20 b FF(z)2340 4202 y Fy(u)2385 4188 y FF(;)15 b(u)p FG(\))g(d)q FF(u;)774 b FH(\(3.38\))273 4431 y(whic)m(h)31 b(implies)1347 4583 y FE(j)p FF(z)1414 4597 y Fy(s)1452 4583 y FE(j)25 b Fn(6)1608 4521 y FG(1)p 1608 4562 46 4 v 1610 4645 a FF(")1679 4459 y Fr(Z)1770 4485 y Fy(s)1729 4665 y FC(0)1822 4583 y FG(e)p 1869 4510 35 3 v -38 x Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2127 4583 y FE(j)2149 4559 y FG(\026)2152 4583 y FF(b)q FG(\()p FF(y)2272 4597 y Fy(u)2317 4583 y FF(;)15 b(u)p FG(\))p FE(j)g FG(d)p FF(u)843 b FH(\(3.39\))273 4795 y(for)30 b(all)f FF(s)c FE(2)g FG([0)p FF(;)15 b(t)p FG(])p FH(.)156 4908 y(2.)47 b(Let)31 b(us)f(assume)g(that)g FF(!)f FE(2)24 b FG(\012)1296 4922 y Fy(t)1326 4908 y FG(\()p FF(h)p FG(\))p FH(.)41 b(Then)31 b(w)m(e)g(ha)m(v)m(e)h(for)e(all)e FF(s)d FE(2)g FG([0)p FF(;)15 b(t)p FG(])1332 5166 y FE(j)p FF(y)1402 5180 y Fy(s)1439 5166 y FG(\()p FF(!)s FG(\))p FE(j)26 b Fn(6)f FF(h)1768 5083 y Fr(p)p 1859 5083 160 4 v 83 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b Fn(6)2238 5104 y FF(h)p 2150 5145 229 4 v 2150 5163 a FE(p)p 2226 5163 153 4 v 65 x FG(2)q(\026)-46 b FF(a)2319 5242 y FD(\000)2414 5166 y Fn(6)2536 5080 y FG(\026)2520 5104 y FF(d)p 2520 5145 48 4 v 2521 5228 a FG(2)2577 5166 y FF(;)828 b FH(\(3.40\))273 5417 y(and)31 b(th)m(us)g(b)m(y)h (\(3.39\))q(,)1328 5585 y FE(j)p FF(z)1395 5599 y Fy(s)1432 5585 y FG(\()p FF(!)s FG(\))p FE(j)27 b Fn(6)1719 5523 y FG(1)p 1719 5564 46 4 v 1721 5647 a FF(")1790 5461 y Fr(Z)1881 5487 y Fy(s)1841 5667 y FC(0)1933 5585 y FG(e)p 1980 5512 35 3 v -38 x Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2264 5523 y FF(M)10 b(h)2414 5490 y FC(2)p 2264 5564 190 4 v 2282 5647 a FG(2)q(\026)-46 b FF(a)2375 5661 y FD(\000)2479 5585 y FG(d)o FF(u:)824 b FH(\(3.41\))1845 5871 y(19)p eop %%Page: 20 20 20 19 bop 273 328 a FH(The)31 b(in)m(tegral)f(on)h(the)f(righ)m(t-hand) i(side)d(can)i(b)s(e)f(estimated)f(b)m(y)j(\(3.16\))q(,)e(yielding)1321 500 y FG(1)p 1321 540 46 4 v 1323 623 a FF(")1392 437 y Fr(Z)1483 464 y Fy(s)1442 644 y FC(0)1535 561 y FG(e)p 1582 489 35 3 v -37 x Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")1856 561 y FG(d)o FF(u)25 b Fn(6)g FG(2)p FF(\020)2164 575 y FC(2)p Fy(")2237 561 y FG(\()p FF(s)p FG(\))g Fn(6)2512 500 y FG(1)p 2481 540 108 4 v 2482 623 a(\026)-46 b FF(a)2529 637 y FD(\000)2598 561 y FF(:)807 b FH(\(3.42\))273 791 y(Therefore,)1263 955 y FE(j)p FF(z)1330 969 y Fy(s)1367 955 y FG(\()p FF(!)s FG(\))p FE(j)26 b Fn(6)1654 893 y FF(M)10 b(h)1804 860 y FC(2)p 1654 934 190 4 v 1673 1019 a FG(2)q(\026)-46 b FF(a)1766 987 y FC(2)1766 1042 y FD(\000)1879 955 y Fn(6)1985 893 y FF(M)2083 832 y FE(p)p 2159 832 108 4 v 2160 893 a FG(\026)g FF(a)2207 907 y FC(+)2281 893 y FF(h)p 1985 934 349 4 v 2045 952 a FE(p)p 2121 952 46 4 v 75 x FG(2)r(\026)f FF(a)2214 996 y FC(2)2214 1050 y FD(\000)2344 955 y FF(h)2396 872 y Fr(p)p 2487 872 160 4 v 83 x FF(\020)7 b FG(\()p FF(s)p FG(\))p FF(;)758 b FH(\(3.43\))273 1188 y(whic)m(h)31 b(pro)m(v)m(es)g(\(3.35\))g(b)s(ecause)g FE(j)p FF(y)1480 1155 y FC(0)1477 1210 y Fy(s)1519 1188 y FG(\()p FF(!)s FG(\))p FE(j)26 b Fn(6)f FE(j)p FF(y)1866 1202 y Fy(s)1903 1188 y FG(\()p FF(!)s FG(\))p FE(j)c FG(+)f FE(j)p FF(z)2237 1202 y Fy(s)2274 1188 y FG(\()p FF(!)s FG(\))p FE(j)p FH(.)156 1301 y(3.)47 b(Let)36 b(us)f(no)m(w)g(assume)f(that)i FF(!)g FE(2)c FG(\012)1523 1268 y FC(0)1523 1323 y Fy(t)1563 1301 y FG(\()p FF(h)p FG(\))p FH(.)55 b(Then)36 b(w)m(e)f(ha)m(v)m(e)h FE(j)p FF(y)2429 1268 y FC(0)2426 1323 y Fy(s)2469 1301 y FG(\()p FF(!)s FG(\))p FE(j)e Fn(6)2777 1277 y FG(\026)2761 1301 y FF(d)q(=)p FG(2)h FH(for)g(all)e FF(s)g FE(2)g FG([0)p FF(;)15 b(t)p FG(])36 b FH(as)273 1414 y(in)30 b(\(3.40\))q(.)40 b(F)-8 b(or)31 b FF(\016)e FG(=)c FF(\015)5 b(h)p FH(,)31 b(w)m(e)g(ha)m(v)m(e)g FF(\016)e(<)c FG(1)31 b FH(b)m(y)g(assumption.)39 b(W)-8 b(e)30 b(consider)h(the)f(\034rst)h (exit)e(time)999 1603 y FF(\034)35 b FG(=)25 b(inf)1281 1529 y Fr(\010)1334 1603 y FF(s)f FE(2)h FG([0)p FF(;)15 b(t)p FG(])10 b(:)32 b FE(j)p FF(z)1789 1617 y Fy(s)1826 1603 y FE(j)26 b Fn(>)f FF(\016)s(h)2068 1521 y Fr(p)p 2160 1521 V 2160 1603 a FF(\020)7 b FG(\()p FF(s)p FG(\))2320 1529 y Fr(\011)2398 1603 y FE(2)25 b FG([0)p FF(;)15 b(t)p FG(])21 b FE([)f(f1g)495 b FH(\(3.44\))273 1792 y(and)31 b(the)g(ev)m(en)m(t)1445 1905 y FF(A)25 b FG(=)g(\012)1700 1868 y FC(0)1700 1928 y Fy(t)1760 1905 y FE(\\)1840 1832 y Fr(\010)1893 1905 y FF(!)13 b FG(:)31 b FF(\034)10 b FG(\()p FF(!)s FG(\))26 b FF(<)f FE(1)2412 1832 y Fr(\011)2465 1905 y FF(:)940 b FH(\(3.45\))273 2077 y(If)30 b FF(!)f FE(2)d FF(A)p FH(,)31 b(then)h(for)e(all)f FF(s)d FE(2)g FG([0)p FF(;)15 b(\034)10 b FG(\()p FF(!)s FG(\)])p FH(,)33 b(w)m(e)f(ha)m(v)m(e)g FE(j)p FF(y)2074 2091 y Fy(s)2110 2077 y FG(\()p FF(!)s FG(\))p FE(j)27 b Fn(6)f FG(\(1)21 b(+)g FF(\016)s FG(\))p FF(h)2712 1999 y Fr(p)p 2804 1999 V 2804 2077 a FF(\020)7 b FG(\()p FF(s)p FG(\))26 b Fn(6)3103 2053 y FG(\026)3087 2077 y FF(d)p FH(,)31 b(and)h(th)m(us)f(b)m(y)273 2190 y(\(3.39\))g(and)g(\(3.42\))q(,)502 2435 y FE(j)p FF(z)569 2449 y Fy(s)607 2435 y FG(\()p FF(!)s FG(\))p FE(j)26 b Fn(6)894 2373 y FG(1)p 894 2414 46 4 v 896 2497 a FF(")964 2311 y Fr(Z)1055 2337 y Fy(s)1015 2517 y FC(0)1107 2435 y FG(e)p 1154 2362 35 3 v 1148 2397 a Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")1438 2373 y FF(M)10 b FG(\(1)21 b(+)f FF(\016)s FG(\))1806 2340 y FC(2)1847 2373 y FF(h)1899 2340 y FC(2)p 1438 2414 501 4 v 1612 2497 a FG(2)q(\026)-46 b FF(a)1705 2511 y FD(\000)1964 2435 y FG(d)o FF(u)25 b Fn(6)2197 2373 y FF(M)10 b FG(\(1)21 b(+)f FF(\016)s FG(\))2565 2340 y FC(2)2606 2373 y FF(h)2658 2340 y FC(2)p 2197 2414 V 2371 2499 a FG(2)q(\026)-46 b FF(a)2464 2467 y FC(2)2464 2522 y FD(\000)2733 2435 y FF(<)25 b(\016)s(h)2924 2352 y Fr(p)p 3016 2352 160 4 v 3016 2435 a FF(\020)7 b FG(\()p FF(s)p FG(\))p FF(:)229 b FH(\(3.46\))273 2699 y(Ho)m(w)m(ev)m(er,)36 b(b)m(y)e(the)g(de\034nition)f(of)g FF(\034)10 b FH(,)34 b(w)m(e)g(ha)m(v)m(e)h FE(j)p FF(z)1989 2717 y Fy(\034)8 b FC(\()p Fy(!)r FC(\))2133 2699 y FG(\()p FF(!)s FG(\))p FE(j)32 b FG(=)e FF(\016)s(h)2516 2621 y Fr(p)p 2608 2621 298 4 v 2608 2699 a FF(\020)7 b FG(\()p FF(\034)j FG(\()p FF(!)s FG(\)\))q FH(,)34 b(whic)m(h)f(con)m(tradicts) 273 2812 y(\(3.46\))d(for)e FF(s)d FG(=)g FF(\034)10 b FG(\()p FF(!)s FG(\))p FH(.)40 b(Therefore)30 b Fo(P)p FE(f)p FF(A)p FE(g)d FG(=)e(0)p FH(,)k(whic)m(h)g(implies)d(that)j(for) f(almost)g(all)e FF(!)j FE(2)24 b FG(\012)3465 2779 y FC(0)3465 2834 y Fy(t)3505 2812 y FH(,)k(w)m(e)273 2925 y(ha)m(v)m(e)k FE(j)p FF(z)549 2939 y Fy(s)586 2925 y FG(\()p FF(!)s FG(\))p FE(j)26 b FF(<)f(\016)s(h)958 2847 y Fr(p)p 1050 2847 160 4 v 1050 2925 a FF(\020)7 b FG(\()p FF(s)p FG(\))30 b FH(for)g(all)f FF(s)c FE(2)g FG([0)p FF(;)15 b(t)p FG(])p FH(,)31 b(and)g(hence)1257 3126 y FE(j)p FF(y)1327 3140 y Fy(s)1363 3126 y FG(\()p FF(!)s FG(\))p FE(j)26 b FF(<)f FG(\(1)c(+)f FF(\016)s FG(\))p FF(h)1962 3044 y Fr(p)p 2055 3044 V 2055 3126 a FF(\020)7 b FG(\()p FF(s)p FG(\))90 b FE(8)p FF(s)24 b FE(2)h FG([0)p FF(;)15 b(t)p FG(])753 b FH(\(3.47\))273 3316 y(for)30 b(these)h FF(!)s FH(,)f(whic)m(h)h(pro)m(v)m(es)g (\(3.36\))q(.)p 3596 3316 4 62 v 3600 3258 55 4 v 3600 3316 V 3653 3316 4 62 v 259 3501 a(W)-8 b(e)37 b(close)e(this)g (subsection)g(with)h(a)g(corollary)f(whic)m(h)h(is)f(Theorem)h(2.3,)i (restated)f(in)e(terms)h(of)118 3614 y(the)31 b(pro)s(cess)f FF(y)634 3628 y Fy(t)663 3614 y FH(.)118 3789 y Fq(Corollary)47 b(3.7.)f FB(Ther)-5 b(e)42 b(exist)f FF(h)1337 3803 y FC(0)1417 3789 y FB(and)f FF(")1642 3803 y FC(0)1682 3789 y FB(,)i(dep)-5 b(ending)42 b(only)e(on)g FF(f)10 b FB(,)41 b(such)g(that)g(for)f FF(")h(<)e(")3438 3803 y FC(0)3519 3789 y FB(and)118 3902 y FF(h)26 b(<)f(h)344 3916 y FC(0)384 3902 y FB(,)590 4142 y Fo(P)651 4104 y FC(0)p Fy(;)p FC(0)746 4041 y Fr(n)840 4142 y FG(sup)807 4218 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1094 4080 y FE(j)p FF(y)1164 4094 y Fy(s)1201 4080 y FE(j)p 1035 4121 251 4 v 1035 4139 a Fr(p)p 1126 4139 160 4 v 78 x FF(\020)7 b FG(\()p FF(s)p FG(\))1321 4142 y FF(>)25 b(h)1469 4041 y Fr(o)1555 4142 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)2063 4041 y Fr(n)2123 4142 y FE(\000)2204 4080 y FG(1)p 2204 4121 46 4 v 2204 4204 a(2)2271 4080 y FF(h)2323 4047 y FC(2)p 2269 4121 95 4 v 2269 4204 a FF(\033)2324 4178 y FC(2)2374 4068 y Fr(\002)2412 4142 y FG(1)21 b FE(\000)e(O)s FG(\()p FF(")p FG(\))j FE(\000)e(O)s FG(\()p FF(h)p FG(\))3065 4068 y Fr(\003)3103 4041 y(o)3164 4142 y FF(:)241 b FH(\(3.48\))118 4392 y Fh(Pr)m(oof:)47 b FH(By)29 b(Prop)s(osition)h(3.6)g(and)h(Prop)s(osition)f(3.3,)798 4631 y Fo(P)859 4594 y FC(0)p Fy(;)p FC(0)954 4530 y Fr(n)1048 4631 y FG(sup)1015 4708 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1302 4570 y FE(j)p FF(y)1372 4584 y Fy(s)1409 4570 y FE(j)p 1243 4610 251 4 v 1243 4629 a Fr(p)p 1334 4629 160 4 v 77 x FF(\020)7 b FG(\()p FF(s)p FG(\))1529 4631 y FF(>)25 b(h)1677 4530 y Fr(o)1763 4631 y Fn(6)g Fo(P)1920 4594 y FC(0)p Fy(;)p FC(0)2015 4530 y Fr(n)2109 4631 y FG(sup)2075 4708 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)2360 4570 y FE(j)p FF(y)2433 4537 y FC(0)2430 4592 y Fy(s)2473 4570 y FE(j)p 2304 4610 251 4 v 2304 4629 a Fr(p)p 2395 4629 160 4 v 77 x FF(\020)7 b FG(\()p FF(s)p FG(\))2590 4631 y FF(>)25 b(h)2738 4645 y FC(1)2777 4530 y Fr(o)1763 4905 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)2271 4804 y Fr(n)2331 4905 y FE(\000)2412 4843 y FG(1)p 2412 4884 46 4 v 2412 4967 a(2)2479 4843 y FF(h)2531 4810 y FC(2)2531 4867 y(1)p 2478 4884 95 4 v 2478 4967 a FF(\033)2533 4941 y FC(2)2582 4905 y FG(\(1)21 b FE(\000)f FF(r)s(")p FG(\))2895 4804 y Fr(o)2956 4905 y FF(;)3430 4754 y FH(\(3.49\))118 5120 y(where)31 b FF(h)26 b FG(=)f(\(1)c(+)f FF(\015)5 b(h)849 5134 y FC(1)889 5120 y FG(\))p FF(h)976 5134 y FC(1)1016 5120 y FH(,)30 b(whic)m(h)h(implies)1167 5347 y FF(h)1219 5361 y FC(1)1284 5347 y FG(=)1416 5286 y(1)p 1390 5326 98 4 v 1390 5410 a(2)p FF(\015)1498 5274 y Fr(\002)1535 5265 y(p)p 1626 5265 307 4 v 82 x FG(1)21 b(+)f(4)p FF(\015)5 b(h)21 b FE(\000)f FG(1)2089 5274 y Fr(\003)2153 5347 y Fn(>)25 b FF(h)p FG([1)c FE(\000)f FF(\015)5 b(h)p FG(])818 b FH(\(3.50\))118 5591 y(where)31 b(w)m(e)g(ha)m(v)m(e)h(used) e(the)h(relation)1414 5519 y FE(p)p 1490 5519 254 4 v 72 x FG(1)21 b(+)f(2)p FF(x)25 b Fn(>)g FG(1)c(+)f FF(x)g FE(\000)2195 5555 y FC(1)p 2195 5570 36 4 v 2195 5622 a(2)2240 5591 y FF(x)2292 5558 y FC(2)2332 5591 y FH(.)p 3596 5591 4 62 v 3600 5533 55 4 v 3600 5591 V 3653 5591 4 62 v 1845 5871 a(20)p eop %%Page: 21 21 21 20 bop 118 328 a Fp(3.2)112 b(Unstable)38 b(case)118 499 y FH(W)-8 b(e)35 b(no)m(w)h(consider)e(a)g(similar)d(situation)j (as)g(in)g(Section)g(3.1,)j(but)e(with)f(an)g(unstable)h(equilibrium,) 118 612 y(that)d(is,)f(w)m(e)i(assume)d(that)j FF(a)p FG(\()p FF(t)p FG(\))28 b Fn(>)f FF(a)1404 626 y FC(0)1471 612 y FF(>)g FG(0)32 b FH(for)g(all)e FF(t)d FE(2)g FF(I)7 b FH(.)45 b(Theorem)32 b(2.1)g(sho)m(ws)g(the)g(existence)f(of)g(a)118 725 y(particular)i(solution)i Fr(b)-54 b FF(x)943 692 y FC(det)935 748 y Fy(t)1078 725 y FH(of)33 b(the)h(deterministic)d (equation)i(\(2.6\))h(suc)m(h)g(that)g FE(j)s Fr(b)-54 b FF(x)2971 692 y FC(det)2963 748 y Fy(t)3096 725 y FE(\000)22 b FF(x)3241 692 y Fy(?)3280 725 y FG(\()p FF(t)p FG(\))p FE(j)31 b Fn(6)f FF(c)3579 739 y FC(1)3619 725 y FF(")118 838 y FH(for)k(all)f FF(t)f FE(2)f FF(I)7 b FH(.)53 b(W)-8 b(e)34 b(are)h(in)m(terested)g(in)e(the)i(sto)s(c)m(hastic)f(pro)s (cess)g FF(y)2460 852 y Fy(t)2521 838 y FG(=)d FF(x)2675 852 y Fy(t)2728 838 y FE(\000)26 b Fr(b)-55 b FF(x)2881 805 y FC(det)2873 860 y Fy(t)2983 838 y FH(,)36 b(whic)m(h)e(describ)s (es)118 951 y(the)d(deviation)e(due)i(to)f(noise)g(from)f(this)h (deterministic)d(solution)33 b Fr(b)-54 b FF(x)2511 918 y FC(det)2612 951 y FH(.)41 b(It)30 b(ob)s(eys)f(the)i(SDE)1122 1177 y FG(d)p FF(y)1218 1191 y Fy(t)1272 1177 y FG(=)1378 1116 y(1)p 1378 1156 46 4 v 1380 1240 a FF(")1433 1104 y Fr(\002)1473 1177 y FG(\026)-47 b FF(a)p FG(\()p FF(t)p FG(\))p FF(y)1667 1191 y Fy(t)1717 1177 y FG(+)1805 1153 y(\026)1808 1177 y FF(b)p FG(\()p FF(y)1927 1191 y Fy(t)1957 1177 y FF(;)15 b(t)p FG(\))2065 1104 y Fr(\003)2118 1177 y FG(d)p FF(t)20 b FG(+)2354 1116 y FF(\033)p 2323 1156 119 4 v 2323 1175 a FE(p)p 2399 1175 43 4 v 65 x FF(")2466 1177 y FG(d)p FF(W)2603 1191 y Fy(t)2632 1177 y FF(;)773 b FH(\(3.51\))118 1413 y(where)949 1585 y FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))26 b(=)g(\026)-46 b FF(a)1269 1599 y Fy(")1306 1585 y FG(\()p FF(t)p FG(\))25 b(=)g FF(@)1578 1599 y Fy(x)1623 1585 y FF(f)10 b FG(\()s Fr(b)-54 b FF(x)1773 1548 y FC(det)1765 1608 y Fy(t)1874 1585 y FF(;)15 b(t)p FG(\))866 1712 y(\026)869 1736 y FF(b)p FG(\()p FF(y)s(;)g(t)p FG(\))26 b(=)1218 1712 y(\026)1221 1736 y FF(b)1260 1750 y Fy(")1297 1736 y FG(\()p FF(y)s(;)15 b(t)p FG(\))26 b(=)f FF(f)10 b FG(\()s Fr(b)-54 b FF(x)1760 1698 y FC(det)1752 1758 y Fy(t)1881 1736 y FG(+)20 b FF(y)s(;)15 b(t)p FG(\))21 b FE(\000)f FF(f)10 b FG(\()s Fr(b)-54 b FF(x)2390 1698 y FC(det)2382 1758 y Fy(t)2492 1736 y FF(;)15 b(t)p FG(\))20 b FE(\000)h FG(\026)-46 b FF(a)p FG(\()p FF(t)p FG(\))p FF(y)3430 1659 y FH(\(3.52\))118 1934 y(are)37 b(the)h(analogs)e(of)i FG(\026)-46 b FF(a)37 b FH(and)1140 1910 y FG(\026)1143 1934 y FF(b)g FH(de\034ned)h(in)e (\(3.6\))q(.)61 b(T)-8 b(aking)37 b FF(")g FH(su\036cien)m(tly)f (small,)g(w)m(e)i(ma)m(y)e(assume)118 2047 y(that)30 b(there)g(exist)e(constan)m(ts)j FG(\026)-46 b FF(a)1203 2061 y FC(0)1243 2047 y FF(;)16 b FG(\026)-46 b FF(a)1331 2061 y FC(1)1370 2047 y FF(;)1427 2023 y FG(\026)1410 2047 y FF(d)26 b(>)f FG(0)p FH(,)30 b(suc)m(h)g(that)g(the)g(follo)m (wing)d(estimates)h(hold)h(for)g(all)e FF(t)e FE(2)g FF(I)118 2160 y FH(and)31 b(all)d FF(y)33 b FH(suc)m(h)e(that)g FE(j)p FF(y)s FE(j)26 b Fn(6)1133 2136 y FG(\026)1117 2160 y FF(d)p FH(:)931 2353 y FG(\026)-47 b FF(a)q FG(\()p FF(t)p FG(\))25 b Fn(6)g FE(\000)q FG(\026)-46 b FF(a)1321 2367 y FC(0)1360 2353 y FF(;)197 b FE(j)q FG(\026)-46 b FF(a)1655 2316 y FD(0)1678 2353 y FG(\()p FF(t)p FG(\))p FE(j)26 b Fn(6)h FG(\026)-47 b FF(a)1976 2367 y FC(1)2016 2353 y FF(;)196 b FE(j)2259 2329 y FG(\026)2262 2353 y FF(b)p FG(\()p FF(y)s(;)15 b(t)p FG(\))p FE(j)26 b Fn(6)f FF(M)10 b(y)2785 2316 y FC(2)2825 2353 y FF(:)580 b FH(\(3.53\))118 2546 y(The)34 b(b)s(ound)f(on)g FE(j)q FG(\026)-46 b FF(a)789 2513 y FD(0)813 2546 y FG(\()p FF(t)p FG(\))p FE(j)34 b FH(is)d(a)i(consequence)h(of)f(the)g(analog)g (of)39 b(\(3.10\))34 b(together)h(with)d(the)i(fact)f(that)118 2659 y FE(j)s Fr(b)-54 b FF(x)203 2626 y FC(det)195 2684 y(0)326 2659 y FE(\000)19 b FF(x)468 2626 y Fy(?)508 2659 y FG(\(0\))p FE(j)26 b FG(=)f FE(O)s FG(\()p FF(")p FG(\))p FH(.)259 2772 y(W)-8 b(e)31 b(\034rst)f(consider)g(the)h (linear)e(equation)1331 2998 y FG(d)p FF(y)1430 2961 y FC(0)1427 3021 y Fy(t)1494 2998 y FG(=)1600 2937 y(1)p 1600 2978 46 4 v 1602 3061 a FF(")1657 2998 y FG(\026)-47 b FF(a)q FG(\()p FF(t)p FG(\))p FF(y)1855 2961 y FC(0)1852 3021 y Fy(t)1909 2998 y FG(d)p FF(t)20 b FG(+)2145 2937 y FF(\033)p 2114 2978 119 4 v 2114 2996 a FE(p)p 2190 2996 43 4 v 65 x FF(")2257 2998 y FG(d)p FF(W)2394 3012 y Fy(t)2423 2998 y FF(:)982 b FH(\(3.54\))118 3247 y(Giv)m(en)36 b(the)g(initial)d(v)-5 b(alue)35 b FF(y)1101 3214 y FC(0)1098 3272 y(0)1141 3247 y FH(,)i(the)f(solution)f FF(y)1760 3214 y FC(0)1757 3270 y Fy(t)1835 3247 y FH(at)i(time)d FF(t)i FH(is)f(a)h(Gaussian)f(random)h(v)-5 b(ariable)35 b(with)118 3360 y(mean)30 b FF(y)407 3327 y FC(0)404 3385 y(0)462 3360 y FG(e)p 509 3292 35 3 v -33 x Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")730 3360 y FH(and)h(v)-5 b(ariance)1391 3612 y FF(v)s FG(\()p FF(t)p FG(\))27 b(=)1673 3550 y FF(\033)1728 3517 y FC(2)p 1673 3591 95 4 v 1700 3674 a FF(")1793 3488 y Fr(Z)1884 3514 y Fy(t)1844 3694 y FC(0)1929 3612 y FG(e)1969 3574 y FC(2)p 2011 3539 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2270 3612 y FG(d)o FF(s;)1042 b FH(\(3.55\))118 3870 y(where)p 392 3820 44 4 v 34 w FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))30 b(=)755 3797 y Fr(R)816 3823 y Fy(t)798 3902 y(s)862 3870 y FG(\026)-46 b FF(a)p FG(\()p FF(u)p FG(\))15 b(d)p FF(u)29 b Fn(>)h FG(\026)-46 b FF(a)1326 3884 y FC(0)1366 3870 y FG(\()p FF(t)22 b FE(\000)f FF(s)p FG(\))32 b FH(for)h FF(t)c Fn(>)g FF(s)p FH(.)46 b(The)33 b(v)-5 b(ariance)33 b(can)g(b)s(e)f(estimated)g(with)g(the)118 3983 y(help)e(of)g(the)h(follo)m(wing)d(lemma.)118 4162 y Fq(Lemma)33 b(3.8.)41 b FB(F)-7 b(or)33 b FG(0)25 b FF(<)g(")h(<)f FG(2)q(\026)-46 b FF(a)1281 4129 y FC(2)1281 4186 y(0)1321 4162 y FF(=)q FG(\026)g FF(a)1414 4176 y FC(1)1454 4162 y FB(,)32 b(one)g(has)938 4358 y FG(1)p 938 4398 46 4 v 940 4481 a FF(")1008 4295 y Fr(Z)1099 4322 y Fy(t)1059 4501 y FC(0)1144 4419 y FG(e)1184 4381 y FC(2)p 1226 4347 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1485 4419 y FG(d)o FF(s)25 b FG(=)1699 4318 y Fr(h)1752 4358 y FG(e)1792 4325 y FC(2)p 1834 4290 V Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")p 1752 4398 274 4 v 1784 4481 a FG(2)q(\026)-46 b FF(a)p FG(\(0\))2055 4419 y FE(\000)2232 4358 y FG(1)p 2156 4398 198 4 v 2156 4481 a(2)q(\026)g FF(a)q FG(\()p FF(t)p FG(\))2363 4318 y Fr(i)2406 4345 y(\002)2444 4419 y FG(1)21 b(+)f FE(O)s FG(\()p FF(")p FG(\))2788 4345 y Fr(\003)2827 4419 y FF(:)578 b FH(\(3.56\))118 4655 y Fh(Pr)m(oof:)47 b FH(By)29 b(in)m(tegration)i(b)m(y)f(parts,)h(w)m(e)g(obtain)f(that)623 4776 y Fr(Z)714 4802 y Fy(t)673 4982 y FC(0)758 4900 y FG(e)799 4862 y FC(2)p 841 4828 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1099 4900 y FG(d)p FF(s)25 b FG(=)1407 4838 y FF(")p 1323 4879 210 4 v 1323 4962 a FG(2)q(\026)-46 b FF(a)q FG(\(0\))1558 4900 y(e)1599 4862 y FC(2)p 1641 4828 35 3 v Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1847 4900 y FE(\000)2005 4838 y FF(")p 1928 4879 198 4 v 1928 4962 a FG(2)q(\026)g FF(a)p FG(\()p FF(t)p FG(\))2155 4900 y FE(\000)2257 4838 y FF(")p 2256 4879 46 4 v 2256 4962 a FG(2)2326 4776 y Fr(Z)2417 4802 y Fy(t)2377 4982 y FC(0)2481 4838 y FG(\026)g FF(a)2528 4805 y FD(0)2551 4838 y FG(\()p FF(s)p FG(\))p 2472 4879 201 4 v 2473 4962 a(\026)g FF(a)p FG(\()p FF(s)p FG(\))2633 4936 y FC(2)2698 4900 y FG(e)2738 4862 y FC(2)p 2780 4828 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")3038 4900 y FG(d)p FF(s;)273 b FH(\(3.57\))118 5136 y(whic)m(h)31 b(implies)c(that)230 5280 y Fr(h)273 5381 y FG(1)21 b FE(\000)441 5319 y FF(")p 439 5360 46 4 v 439 5443 a FG(2)506 5319 y(\026)-46 b FF(a)553 5333 y FC(1)p 505 5360 88 4 v 506 5445 a FG(\026)g FF(a)553 5414 y FC(2)553 5471 y(0)602 5280 y Fr(i)660 5257 y(Z)751 5283 y Fy(t)711 5463 y FC(0)796 5381 y FG(e)836 5343 y FC(2)p 878 5308 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1137 5381 y FG(d)o FF(s)25 b Fn(6)1445 5319 y FF(")p 1361 5360 210 4 v 1361 5443 a FG(2)q(\026)-46 b FF(a)q FG(\(0\))1596 5381 y(e)1636 5343 y FC(2)p 1678 5308 35 3 v Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1884 5381 y FE(\000)2042 5319 y FF(")p 1965 5360 198 4 v 1965 5443 a FG(2)q(\026)g FF(a)q FG(\()p FF(t)p FG(\))2197 5381 y Fn(6)2293 5280 y Fr(h)2336 5381 y FG(1)21 b(+)2504 5319 y FF(")p 2503 5360 46 4 v 2503 5443 a FG(2)2569 5319 y(\026)-46 b FF(a)2616 5333 y FC(1)p 2568 5360 88 4 v 2569 5445 a FG(\026)g FF(a)2616 5414 y FC(2)2616 5471 y(0)2666 5280 y Fr(i)2724 5257 y(Z)2815 5283 y Fy(t)2774 5463 y FC(0)2859 5381 y FG(e)2900 5343 y FC(2)p 2942 5308 35 3 v Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")3200 5381 y FG(d)p FF(s:)111 b FH(\(3.58\))118 5622 y(By)30 b(our)h(h)m(yp)s(othesis)e(on)i FF(")p FH(,)f(the)h(\034rst)f(term)g(in)g(brac)m(k)m(ets)h(is)e(p)s (ositiv)m(e.)p 3596 5622 4 62 v 3600 5564 55 4 v 3600 5622 V 3653 5622 4 62 v 1845 5871 a(21)p eop %%Page: 22 22 22 21 bop 259 328 a FH(Unlik)m(e)30 b(in)h(the)h(stable)e(case,)i(the)g (v)-5 b(ariance)31 b(gro)m(ws)i(exp)s(onen)m(tially)c(fast)i(\(at)h (least)f(with)g FG(e)3396 295 y FC(2)q(\026)-36 b Fy(a)3468 304 y Fw(0)3504 295 y Fy(t=")3601 328 y FH(\).)118 441 y(If)30 b FF(\032)25 b Fn(>)g FE(j)p FF(y)450 408 y FC(0)447 465 y(0)489 441 y FE(j)p FH(,)31 b(w)m(e)g(ha)m(v)m(e)668 650 y Fo(P)729 612 y FC(0)p Fy(;y)821 589 y Fw(0)819 633 y(0)860 576 y Fr(\010)946 650 y FG(sup)913 727 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1116 650 y FE(j)p FF(y)1189 612 y FC(0)1186 672 y Fy(s)1229 650 y FE(j)25 b FF(<)g(\032)1422 576 y Fr(\011)1500 650 y Fn(6)g Fo(P)1657 612 y FC(0)p Fy(;y)1749 589 y Fw(0)1747 633 y(0)1789 576 y Fr(\010)1842 650 y FE(j)p FF(y)1915 612 y FC(0)1912 672 y Fy(t)1954 650 y FE(j)h FF(<)f(\032)2148 576 y Fr(\011)1500 938 y FG(=)1596 814 y Fr(Z)1687 840 y Fy(\032)p FD(\000)p Fy(y)1815 817 y Fw(0)1813 861 y(0)1862 840 y FC(e)p 1898 792 33 2 v -23 x Fj(\013)p Fw(\()p Fj(t)p Fw(\))p Fj(=")1647 1020 y FD(\000)p Fy(\032)p FD(\000)p Fy(y)1830 997 y Fw(0)1828 1042 y(0)1876 1020 y FC(e)p 1912 976 V 1908 1001 a Fj(\013)p Fw(\()p Fj(t)p Fw(\))p Fj(=")2114 876 y FG(e)2155 843 y FD(\000)p Fy(x)2250 820 y Fw(2)2284 843 y Fy(=)p FC(2)p Fy(v)r FC(\()p Fy(t)p FC(\))p 2114 917 362 4 v 2124 935 a Fr(p)p 2215 935 252 4 v 78 x FG(2)p FF(\031)s(v)s FG(\()p FF(t)p FG(\))2501 938 y(d)o FF(x)g Fn(6)2859 876 y FG(2)p FF(\032)p 2734 917 343 4 v 2734 935 a Fr(p)p 2825 935 252 4 v 78 x FG(2)p FF(\031)s(v)s FG(\()p FF(t)p FG(\))3087 938 y FF(;)3430 822 y FH(\(3.59\))118 1216 y(whic)m(h)35 b(go)s(es)f(to)h(zero)h(as)e FF(\032\033)1112 1183 y FD(\000)p FC(1)1222 1216 y FG(e)1262 1183 y FD(\000)p 1324 1148 35 3 v Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1549 1216 y FH(for)h FF(t)d FE(!)h(1)p FH(.)53 b(In)34 b(this)g(estimate,)h (ho)m(w)m(ev)m(er,)j(w)m(e)d(neglect)g(all)118 1329 y(tra)5 b(jectories)31 b(that)g(lea)m(v)m(e)g(the)h(in)m(terv)-5 b(al)30 b FG(\()p FE(\000)p FF(\032;)15 b(\032)p FG(\))31 b FH(b)s(efore)g FF(t)g FH(and)g(come)f(bac)m(k.)43 b(W)-8 b(e)32 b(will)c(deriv)m(e)j(a)g(more)118 1442 y(precise)f(estimate)f (for)h(the)h(general,)f(nonlinear)g(case)h(b)m(y)f(in)m(tro)s(ducing)g (a)h(partition)e(of)h FG([0)p FF(;)15 b(t)p FG(])p FH(.)259 1555 y(The)26 b(follo)m(wing)e(prop)s(osition,)h(whic)m(h)h(restates)f (Theorem)h(2.5)g(in)e(terms)h(of)g FF(y)2902 1569 y Fy(t)2931 1555 y FH(,)h(is)e(the)i(main)e(result)118 1668 y(of)30 b(this)f(subsection.)118 1849 y Fq(Prop)s(osition)34 b(3.9.)40 b FB(Ther)-5 b(e)32 b(exist)g(c)-5 b(onstants)31 b FF(")1784 1863 y FC(0)1824 1849 y FF(;)15 b(h)1916 1863 y FC(0)1981 1849 y FF(>)25 b FG(0)32 b FB(such)g(that)f(for)g(al)5 b(l)31 b FF(h)26 b Fn(6)f FF(\033)c FE(^)c FF(h)3189 1863 y FC(0)3229 1849 y FB(,)31 b(al)5 b(l)31 b FF(")26 b Fn(6)f FF(")3622 1863 y FC(0)118 1962 y FB(and)32 b(for)g(any)g (given)h FF(y)890 1976 y FC(0)962 1962 y FB(with)f FE(j)p FF(y)1228 1976 y FC(0)1267 1962 y FE(j)1292 1884 y Fr(p)p 1383 1884 210 4 v 78 x FG(2)q(\026)-46 b FF(a)q FG(\(0\))26 b FF(<)f(h)p FB(,)32 b(we)h(have)852 2218 y Fo(P)913 2181 y FC(0)p Fy(;y)1003 2190 y Fw(0)1042 2118 y Fr(n)1136 2218 y FG(sup)1103 2295 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1306 2218 y FE(j)p FF(y)1376 2232 y Fy(s)1413 2218 y FE(j)1438 2136 y Fr(p)p 1529 2136 207 4 v 82 x FG(2)q(\026)-46 b FF(a)p FG(\()p FF(s)p FG(\))26 b FF(<)f(h)1909 2118 y Fr(o)1995 2218 y Fn(6)2091 2149 y FE(p)p 2167 2149 41 4 v 69 x FG(e)15 b(exp)2361 2118 y Fr(n)2422 2218 y FE(\000)p FF(\024)2555 2157 y(\033)2610 2124 y FC(2)p 2555 2198 95 4 v 2556 2281 a FF(h)2608 2255 y FC(2)p 2679 2107 44 4 v 2669 2157 a FF(\013)p FG(\()p FF(t)p FG(\))p 2669 2198 163 4 v 2729 2281 a FF(")2841 2118 y Fr(o)2902 2218 y FF(;)503 b FH(\(3.60\))118 2484 y FB(wher)-5 b(e)34 b FF(\024)25 b FG(=)568 2448 y Fy(\031)p 556 2463 67 4 v 556 2516 a FC(2e)633 2411 y Fr(\000)675 2484 y FG(1)20 b FE(\000)g(O)s FG(\()p FF(h)p FG(\))h FE(\000)f(O)s FG(\()p FF(")p FG(\))1327 2411 y Fr(\001)1370 2484 y FB(.)118 2739 y Fh(Pr)m(oof:)156 2875 y FH(1.)47 b(Let)33 b FF(K)h FE(2)27 b Fo(N)59 b FH(and)32 b(let)f FG(0)d(=)f FF(u)1280 2889 y FC(0)1347 2875 y FF(<)g(u)1497 2889 y FC(1)1564 2875 y FF(<)g FE(\001)15 b(\001)g(\001)28 b FF(<)g(u)1946 2889 y Fy(K)2041 2875 y FG(=)f FF(t)32 b FH(b)s(e)f(an)m(y)h(partition)f(of)g(the)h(in)m(terv)-5 b(al)31 b FG([0)p FF(;)15 b(t)p FG(])p FH(.)273 2988 y(W)-8 b(e)31 b(de\034ne)g(the)g(ev)m(en)m(ts)1204 3184 y FF(A)1272 3199 y Fy(k)1340 3184 y FG(=)1436 3083 y Fr(n)1496 3184 y FF(!)13 b FG(:)166 b(sup)1637 3261 y Fy(u)1678 3273 y Fj(k)1716 3261 y Fx(6)p Fy(s)p Fx(6)p Fy(u)1900 3273 y Fj(k)q Fw(+1)2014 3184 y FE(j)p FF(y)2084 3198 y Fy(s)2121 3184 y FE(j)2146 3102 y Fr(p)p 2237 3102 207 4 v 82 x FG(2)q(\026)-46 b FF(a)q FG(\()p FF(s)p FG(\))25 b FF(<)g(h)2617 3083 y Fr(o)1203 3427 y FF(B)1272 3442 y Fy(k)1340 3427 y FG(=)1436 3326 y Fr(n)1496 3427 y FF(!)13 b FG(:)31 b FE(j)p FF(y)1692 3441 y Fy(u)1733 3453 y Fj(k)1775 3427 y FE(j)1800 3345 y Fr(p)p 1891 3345 259 4 v 82 x FG(2)q(\026)-46 b FF(a)q FG(\()p FF(u)2072 3442 y Fy(k)2115 3427 y FG(\))25 b FF(<)g(h)2323 3326 y Fr(o)2409 3427 y FE(\033)g FF(A)2573 3442 y Fy(k)r FD(\000)p FC(1)2706 3427 y FF(:)3430 3306 y FH(\(3.61\))273 3646 y(Let)36 b FF(q)481 3661 y Fy(k)558 3646 y FH(b)s(e)f(a)h (deterministic)c(upp)s(er)k(b)s(ound)g(on)f FF(P)2050 3661 y Fy(k)2127 3646 y FG(=)e Fo(P)2292 3610 y Fy(u)2333 3622 y Fj(k)2371 3610 y Fy(;y)2426 3618 y Fj(u)2463 3636 y(k)2509 3646 y FE(f)p FF(A)2622 3661 y Fy(k)2665 3646 y FE(g)p FH(,)k(v)-5 b(alid)33 b(on)j FF(B)3195 3661 y Fy(k)3238 3646 y FH(.)55 b(Then)35 b(w)m(e)273 3759 y(ha)m(v)m(e)d(b)m(y)e(the)h(Mark)m(o)m(v)g(prop)s(ert)m(y)491 3973 y Fo(P)552 3935 y FC(0)p Fy(;y)642 3944 y Fw(0)681 3872 y Fr(n)775 3973 y FG(sup)742 4049 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)945 3973 y FE(j)p FF(y)1015 3987 y Fy(s)1051 3973 y FE(j)1076 3890 y Fr(p)p 1167 3890 207 4 v 83 x FG(2)q(\026)-46 b FF(a)q FG(\()p FF(s)p FG(\))26 b FF(<)f(h)1548 3872 y Fr(o)697 4271 y FG(=)g Fo(P)854 4233 y FC(0)p Fy(;y)944 4242 y Fw(0)983 4170 y Fr(n)1044 4157 y Fy(K)5 b FD(\000)p FC(1)1070 4184 y Fr(\\)1056 4382 y Fy(k)r FC(=0)1213 4271 y FF(A)1281 4286 y Fy(k)1324 4170 y Fr(o)1410 4271 y FG(=)25 b Fo(E)1566 4233 y FC(0)p Fy(;y)1656 4242 y Fw(0)1701 4170 y Fr(n)1761 4271 y FG(1)1806 4253 y Fg(T)1866 4274 y Fj(K)s Fk(\000)p Fw(2)1866 4327 y Fj(k)q Fw(=0)2014 4303 y Fy(A)2067 4315 y Fj(k)2110 4271 y Fo(E)2170 4233 y FC(0)p Fy(;y)2260 4242 y Fw(0)2305 4197 y Fr(\010)2358 4271 y FG(1)2403 4285 y Fy(A)2456 4296 y Fj(K)2544 4193 y Fr(\014)2544 4248 y(\014)2600 4271 y FE(f)p FF(y)2690 4285 y Fy(s)2726 4271 y FE(g)2771 4285 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(u)2990 4296 y Fj(K)s Fk(\000)p Fw(1)3132 4197 y Fr(\011)3185 4170 y(o)697 4592 y FG(=)g Fo(E)854 4555 y FC(0)p Fy(;y)944 4564 y Fw(0)988 4491 y Fr(n)1049 4592 y FG(1)1094 4575 y Fg(T)1153 4595 y Fj(K)s Fk(\000)p Fw(2)1153 4649 y Fj(k)q Fw(=0)1302 4625 y Fy(A)1355 4637 y Fj(k)1397 4592 y FF(P)1455 4606 y Fy(K)5 b FD(\000)p FC(1)1614 4491 y Fr(o)1700 4592 y Fn(6)25 b FF(q)1837 4606 y Fy(K)5 b FD(\000)p FC(1)1995 4592 y Fo(P)2056 4555 y FC(0)p Fy(;y)2146 4564 y Fw(0)2185 4491 y Fr(n)2245 4479 y Fy(K)g FD(\000)p FC(2)2272 4506 y Fr(\\)2258 4704 y Fy(k)r FC(=0)2415 4592 y FF(A)2483 4607 y Fy(k)2526 4491 y Fr(o)2611 4592 y Fn(6)25 b FE(\001)15 b(\001)g(\001)27 b Fn(6)2935 4479 y Fy(K)5 b FD(\000)p FC(1)2954 4506 y Fr(Y)2948 4704 y Fy(k)r FC(=0)3104 4592 y FF(q)3145 4607 y Fy(k)3187 4592 y FF(:)218 b FH(\(3.62\))156 4867 y(2.)47 b(T)-8 b(o)31 b(de\034ne)h(the)e(partition,)g(w)m(e)h(set) 1633 5114 y FF(K)h FG(=)1838 5013 y Fr(l)1899 5052 y FG(1)p 1895 5093 53 4 v 1895 5176 a FF(\015)p 1978 5002 44 4 v 1968 5052 a(\013)p FG(\()p FF(t)p FG(\))p 1968 5093 163 4 v 2027 5176 a FF(")2149 5052 y(\033)2204 5019 y FC(2)p 2149 5093 95 4 v 2150 5176 a FF(h)2202 5150 y FC(2)2254 5013 y Fr(m)3430 5114 y FH(\(3.63\))273 5354 y(for)e(some)g FF(\015)g FE(2)25 b FG(\(0)p FF(;)15 b FG(1])32 b FH(to)f(b)s(e)f(c)m(hosen)h(later,)f(and)p 1135 5555 44 4 v 1125 5605 a FF(\013)p FG(\()p FF(u)1270 5620 y Fy(k)r FC(+1)1404 5605 y FF(;)15 b(u)1496 5620 y Fy(k)1539 5605 y FG(\))25 b(=)g FF(\015)5 b(")1800 5544 y(h)1852 5511 y FC(2)p 1799 5585 95 4 v 1799 5668 a FF(\033)1854 5642 y FC(2)1904 5605 y FF(;)196 b(k)29 b FG(=)c(0)p FF(;)15 b(:)g(:)g(:)i(;)e(K)27 b FE(\000)20 b FG(2)p FF(:)621 b FH(\(3.64\))1845 5871 y(22)p eop %%Page: 23 23 23 22 bop 273 328 a FH(Since)p 526 278 44 4 v 37 w FF(\013)p FG(\()p FF(u)661 343 y Fy(k)r FC(+1)795 328 y FF(;)15 b(u)887 343 y Fy(k)930 328 y FG(\))36 b Fn(>)i FG(\026)-47 b FF(a)1156 342 y FC(0)1196 328 y FG(\()p FF(u)1283 343 y Fy(k)r FC(+1)1441 328 y FE(\000)24 b FF(u)1588 343 y Fy(k)1631 328 y FG(\))p FH(,)38 b(w)m(e)g(ha)m(v)m(e)g FF(u)2136 343 y Fy(k)r FC(+1)2293 328 y FE(\000)25 b FF(u)2441 343 y Fy(k)2520 328 y Fn(6)2638 292 y Fy(h)2679 268 y Fw(2)p 2637 307 78 4 v 2637 361 a Fy(\033)2679 342 y Fw(2)2750 287 y Fy(\015)p 2734 307 73 4 v 2735 359 a FC(\026)-36 b Fy(a)2771 368 y Fw(0)2816 328 y FF(")p FH(,)39 b(and)e(using)f(T)-8 b(a)m(ylor's)273 441 y(form)m(ula,)30 b(w)m(e)h(\034nd)g(for)f(all)e FF(s)d FE(2)g FG([)p FF(u)1438 456 y Fy(k)1481 441 y FF(;)15 b(u)1573 456 y Fy(k)r FC(+1)1706 441 y FG(])31 b FH(and)g(all)d FF(k)h FG(=)c(0)p FF(;)15 b(:)g(:)g(:)i(;)e(K)27 b FE(\000)20 b FG(1)1244 700 y(1)g FE(\000)1412 639 y FF(h)1464 606 y FC(2)p 1410 679 95 4 v 1410 763 a FF(\033)1465 736 y FC(2)1526 639 y FG(\026)-46 b FF(a)1573 653 y FC(1)p 1525 679 88 4 v 1526 764 a FG(\026)g FF(a)1573 733 y FC(2)1573 790 y(0)1622 700 y FF(\015)5 b(")26 b Fn(6)1875 639 y FG(\026)-46 b FF(a)p FG(\()p FF(s)p FG(\))p 1848 679 214 4 v 1849 763 a(\026)g FF(a)p FG(\()p FF(u)1983 778 y Fy(k)2026 763 y FG(\))2097 700 y Fn(6)25 b FG(1)20 b(+)2361 639 y FF(h)2413 606 y FC(2)p 2359 679 95 4 v 2359 763 a FF(\033)2414 736 y FC(2)2475 639 y FG(\026)-46 b FF(a)2522 653 y FC(1)p 2474 679 88 4 v 2475 764 a FG(\026)g FF(a)2522 733 y FC(2)2522 790 y(0)2571 700 y FF(\015)5 b(";)740 b FH(\(3.65\))273 960 y(where)33 b FG(\026)-47 b FF(a)582 974 y FC(1)652 960 y FH(is)29 b(the)h(upp)s(er)h(b)s(ound)f(on)g FE(j)q FG(\026)-46 b FF(a)1633 927 y FD(0)1657 960 y FE(j)p FH(,)30 b(see)g(\(3.53\))q(.)41 b(In)30 b(order)g(to)h(estimate)e FF(P)3059 975 y Fy(k)3102 960 y FH(,)h(w)m(e)h(in)m(tro)s(duce)273 1089 y(linear)e(appro)m(ximations)h FG(\()p FF(y)1246 1041 y FC(\()p Fy(k)r FC(\))1243 1113 y Fy(t)1343 1089 y FG(\))1378 1107 y Fy(t)p FD(2)p FC([)p Fy(u)1511 1119 y Fj(k)1550 1107 y Fy(;u)1611 1119 y Fj(k)q Fw(+1)1726 1107 y FC(])1779 1089 y FH(for)h FF(k)d FE(2)d(f)p FG(0)p FF(;)15 b(:)g(:)g(:)i(;)e(K)28 b FE(\000)20 b FG(2)p FE(g)p FH(,)31 b(de\034ned)g(b)m(y)1052 1345 y FG(d)o FF(y)1150 1298 y FC(\()p Fy(k)r FC(\))1147 1370 y Fy(t)1273 1345 y FG(=)1379 1284 y(1)p 1379 1325 46 4 v 1381 1408 a FF(")1436 1345 y FG(\026)-47 b FF(a)p FG(\()p FF(t)p FG(\))p FF(y)1633 1298 y FC(\()p Fy(k)r FC(\))1630 1370 y Fy(t)1751 1345 y FG(+)1884 1284 y FF(\033)p 1852 1325 119 4 v 1852 1343 a FE(p)p 1928 1343 43 4 v 65 x FF(")1996 1345 y FG(d)o FF(W)2145 1298 y FC(\()p Fy(k)r FC(\))2132 1370 y Fy(t)2242 1345 y FF(;)196 b(y)2511 1308 y FC(\()p Fy(k)r FC(\))2508 1368 y Fy(u)2549 1380 y Fj(k)2634 1345 y FG(=)25 b FF(y)2775 1359 y Fy(u)2816 1371 y Fj(k)2858 1345 y FF(;)547 b FH(\(3.66\))273 1627 y(where)29 b FF(W)631 1579 y FC(\()p Fy(k)r FC(\))618 1651 y Fy(t)754 1627 y FG(=)c FF(W)936 1641 y Fy(t)981 1627 y FE(\000)16 b FF(W)1154 1641 y Fy(u)1195 1653 y Fj(k)1265 1627 y FH(is)27 b(a)i(Bro)m(wnian)g(motion)e(with)h FF(W)2441 1579 y FC(\()p Fy(k)r FC(\))2428 1639 y Fy(u)2469 1651 y Fj(k)2563 1627 y FG(=)d(0)k FH(whic)m(h)f(is)f(indep)s(enden)m(t)j(of)273 1740 y FE(f)p FF(W)404 1754 y Fy(s)451 1740 y FG(:)h(0)26 b Fn(6)f FF(s)g Fn(6)g FF(u)890 1755 y Fy(k)932 1740 y FE(g)p FH(.)41 b(If)29 b FF(!)f FE(2)d FF(A)1372 1755 y Fy(k)1415 1740 y FH(,)30 b(w)m(e)i(ha)m(v)m(e)f(for)f(all)f FF(s)c FE(2)f FG([)p FF(u)2306 1755 y Fy(k)2349 1740 y FF(;)15 b(u)2441 1755 y Fy(k)r FC(+1)2575 1740 y FG(])751 1989 y FE(j)p FF(y)821 2003 y Fy(s)858 1989 y FG(\()p FF(!)s FG(\))20 b FE(\000)g FF(y)1147 1951 y FC(\()p Fy(k)r FC(\))1144 2011 y Fy(s)1245 1989 y FG(\()p FF(!)s FG(\))p FE(j)26 b Fn(6)1532 1927 y FG(1)p 1532 1968 46 4 v 1534 2051 a FF(")1602 1865 y Fr(Z)1693 1891 y Fy(s)1653 2071 y(u)1694 2083 y Fj(k)1751 1989 y FG(e)p 1798 1916 35 3 v -38 x Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2057 1989 y FE(j)2079 1965 y FG(\026)2082 1989 y FF(b)p FG(\()p FF(y)2201 2003 y Fy(u)2246 1989 y FF(;)15 b(u)p FG(\))p FE(j)g FG(d)q FF(u)1426 2260 y Fn(6)1532 2199 y FF(M)10 b(h)1682 2166 y FC(2)p 1532 2239 190 4 v 1560 2323 a FG(2)q(\026)-46 b FF(a)1653 2337 y FC(0)1742 2199 y FG(e)p 1789 2131 35 3 v -33 x Fy(\013)p FC(\()p Fy(u)1895 2178 y Fj(k)q Fw(+1)2011 2166 y Fy(;u)2072 2178 y Fj(k)2109 2166 y FC(\))p Fy(=")p 1742 2239 468 4 v 1870 2323 a FG(\026)f FF(a)p FG(\()p FF(u)2003 2338 y Fy(k)2047 2323 y FG(\))2219 2187 y Fr(\002)2257 2260 y FG(1)20 b(+)g FE(O)s FG(\()p FF(")p FG(\))2600 2187 y Fr(\003)2664 2260 y Fn(6)25 b FF(r)2801 2274 y FC(0)2954 2199 y FF(h)3006 2166 y FC(2)p 2851 2239 298 4 v 2851 2258 a Fr(p)p 2942 2258 207 4 v 77 x FG(2)q(\026)-46 b FF(a)p FG(\()p FF(s)p FG(\))3158 2260 y FF(;)3430 2136 y FH(\(3.67\))273 2546 y(where)31 b FF(r)575 2560 y FC(0)640 2546 y FG(=)25 b FF(M)g FG(e)q(\(2)q(\026)-46 b FF(a)1018 2513 y FC(3)1018 2570 y(0)1058 2546 y FG(\))1093 2513 y FD(\000)p FC(1)p Fy(=)p FC(2)1278 2546 y FG(+)20 b FE(O)s FG(\()p FF(")p FG(\))p FH(.)42 b(This)29 b(sho)m(ws)i(that)g(on)f FF(A)2478 2561 y Fy(k)2521 2546 y FH(,)1012 2794 y FE(j)p FF(y)1085 2757 y FC(\()p Fy(k)r FC(\))1082 2817 y Fy(s)1182 2794 y FG(\()p FF(!)s FG(\))p FE(j)c Fn(6)1459 2720 y Fr(\002)1497 2794 y FG(1)21 b(+)f FF(r)1695 2808 y FC(0)1734 2794 y FF(h)1786 2720 y Fr(\003)1957 2733 y FF(h)p 1835 2773 298 4 v 1835 2791 a Fr(p)p 1925 2791 207 4 v 1925 2869 a FG(2)q(\026)-46 b FF(a)q FG(\()p FF(s)p FG(\))2323 2794 y FE(8)p FF(s)24 b FE(2)h FG([)p FF(u)2604 2809 y Fy(k)2647 2794 y FF(;)15 b(u)2739 2809 y Fy(k)r FC(+1)2872 2794 y FG(])p FF(:)508 b FH(\(3.68\))156 3065 y(3.)47 b(W)-8 b(e)31 b(are)g(no)m(w)g(ready)g(to)f(estimate)f FF(P)1547 3080 y Fy(k)1590 3065 y FH(.)40 b(\(3.68\))32 b(sho)m(ws)e(that)h(on)g FF(B)2567 3080 y Fy(k)2609 3065 y FH(,)954 3292 y FF(P)1012 3307 y Fy(k)1081 3292 y Fn(6)25 b Fo(P)1238 3254 y Fy(u)1279 3266 y Fj(k)1317 3254 y Fy(;y)1372 3262 y Fj(u)1409 3280 y(k)1455 3191 y Fr(n)1635 3292 y FG(sup)1515 3368 y Fy(u)1556 3380 y Fj(k)1594 3368 y Fx(6)p Fy(s)p Fx(6)p Fy(u)1778 3380 y Fj(k)q Fw(+1)1893 3292 y FE(j)p FF(y)1966 3254 y FC(\()p Fy(k)r FC(\))1963 3314 y Fy(s)2063 3292 y FE(j)2088 3209 y Fr(p)p 2179 3209 V 83 x FG(2)q(\026)-46 b FF(a)q FG(\()p FF(s)p FG(\))25 b FF(<)g(h)p FG(\(1)d(+)e FF(r)2793 3306 y FC(0)2832 3292 y FF(h)p FG(\))2919 3191 y Fr(o)1081 3519 y Fn(6)25 b Fo(P)1238 3481 y Fy(u)1279 3493 y Fj(k)1317 3481 y Fy(;y)1372 3489 y Fj(u)1409 3507 y(k)1455 3445 y Fr(\010)1508 3519 y FE(j)p FF(y)1581 3481 y FC(\()p Fy(k)r FC(\))1578 3541 y Fy(u)1619 3553 y Fj(k)q Fw(+1)1738 3519 y FE(j)1763 3437 y Fr(p)p 1854 3437 350 4 v 82 x FG(2)q(\026)-46 b FF(a)p FG(\()p FF(u)2034 3534 y Fy(k)r FC(+1)2167 3519 y FG(\))26 b FF(<)f(h)p FG(\(1)c(+)f FF(r)2609 3533 y FC(0)2649 3519 y FF(h)p FG(\))2736 3445 y Fr(\011)1081 3723 y Fn(6)1362 3661 y FG(1)p 1186 3702 396 4 v 1186 3720 a Fr(q)p 1277 3720 305 4 v 116 x FG(2)p FF(\031)s(v)1424 3788 y FC(\()p Fy(k)r FC(\))1421 3848 y Fy(u)1462 3860 y Fj(k)q Fw(+1)1602 3661 y FG(2)p FF(h)p FG(\(1)i(+)e FF(r)1933 3675 y FC(0)1972 3661 y FF(h)p FG(\))p 1602 3702 458 4 v 1611 3720 a Fr(p)p 1702 3720 350 4 v 78 x FG(2)q(\026)-46 b FF(a)p FG(\()p FF(u)1882 3813 y Fy(k)r FC(+1)2016 3798 y FG(\))2070 3723 y FF(;)3430 3557 y FH(\(3.69\))273 4077 y(where)31 b FF(v)581 4029 y FC(\()p Fy(k)r FC(\))578 4089 y Fy(u)619 4101 y Fj(k)q Fw(+1)769 4077 y FH(denotes)g(the)f(conditional)g(v)-5 b(ariance)30 b(of)g FF(y)2224 4029 y FC(\()p Fy(k)r FC(\))2221 4089 y Fy(u)2262 4101 y Fj(k)q Fw(+1)2380 4077 y FH(,)h(giv)m(en)f FF(y)2717 4091 y Fy(u)2758 4103 y Fj(k)2799 4077 y FH(.)41 b(As)29 b(in)h(\(3.56\))q(,)342 4352 y FF(v)389 4314 y FC(\()p Fy(k)r FC(\))386 4374 y Fy(u)427 4386 y Fj(k)q Fw(+1)571 4352 y FG(=)677 4290 y FF(\033)732 4257 y FC(2)p 677 4331 95 4 v 703 4414 a FF(")797 4228 y Fr(Z)888 4254 y Fy(u)929 4266 y Fj(k)q Fw(+1)847 4434 y Fy(u)888 4446 y Fj(k)1063 4352 y FG(e)1104 4314 y FC(2)p 1146 4279 35 3 v Fy(\013)p FC(\()p Fy(u)1252 4326 y Fj(k)q Fw(+1)1368 4314 y Fy(;s)p FC(\))p Fy(=")1534 4352 y FG(d)p FF(s)25 b FG(=)1759 4290 y FF(\033)1814 4257 y FC(2)p 1759 4331 95 4 v 1783 4414 a FG(2)1863 4224 y Fr(\024)1921 4290 y FG(e)1962 4257 y FC(2)p 2004 4222 35 3 v Fy(\013)p FC(\()p Fy(u)2110 4269 y Fj(k)q Fw(+1)2226 4257 y Fy(;u)2287 4269 y Fj(k)2324 4257 y FC(\))p Fy(=")p 1921 4331 503 4 v 2067 4414 a FG(\026)-46 b FF(a)p FG(\()p FF(u)2201 4429 y Fy(k)2244 4414 y FG(\))2454 4352 y FE(\000)2684 4290 y FG(1)p 2555 4331 304 4 v 2556 4414 a(\026)g FF(a)p FG(\()p FF(u)2690 4429 y Fy(k)r FC(+1)2823 4414 y FG(\))2868 4224 y Fr(\025)2916 4278 y(\002)2954 4352 y FG(1)21 b(+)f FE(O)s FG(\()p FF(")p FG(\))3298 4278 y Fr(\003)3337 4352 y FF(:)68 b FH(\(3.70\))273 4608 y(It)30 b(follo)m(ws)f(that)620 4845 y FG(\026)-46 b FF(a)p FG(\()p FF(u)754 4860 y Fy(k)r FC(+1)887 4845 y FG(\))p FF(v)969 4808 y FC(\()p Fy(k)r FC(\))966 4868 y Fy(u)1007 4880 y Fj(k)q Fw(+1)1152 4845 y Fn(>)1258 4784 y FF(\033)1313 4751 y FC(2)p 1258 4824 95 4 v 1283 4907 a FG(2)1363 4744 y Fr(h)1406 4845 y FG(e)1446 4808 y FC(2)p Fy(\015)t(h)1562 4784 y Fw(2)1597 4808 y Fy(=\033)1674 4784 y Fw(2)36 b FG(\026)-46 b FF(a)p FG(\()p FF(u)1874 4799 y Fy(k)r FC(+1)2007 4784 y FG(\))p 1739 4824 304 4 v 1785 4907 a(\026)g FF(a)p FG(\()p FF(u)1919 4922 y Fy(k)1962 4907 y FG(\))2073 4845 y FE(\000)20 b FG(1)2209 4744 y Fr(i)2252 4771 y(\002)2290 4845 y FG(1)g FE(\000)g(O)s FG(\()p FF(")p FG(\))2633 4771 y Fr(\003)1152 5101 y Fn(>)1258 5039 y FF(\033)1313 5006 y FC(2)p 1258 5080 95 4 v 1283 5163 a FG(2)1363 5000 y Fr(h\020)1460 5101 y FG(1)h(+)f(2)p FF(\015)1726 5039 y(h)1778 5006 y FC(2)p 1724 5080 V 1724 5163 a FF(\033)1779 5137 y FC(2)1829 5000 y Fr(\021\020)1937 5101 y FG(1)h FE(\000)2105 5039 y FG(\026)-46 b FF(a)2152 5053 y FC(1)p 2104 5080 88 4 v 2105 5165 a FG(\026)g FF(a)2152 5133 y FC(2)2152 5191 y(0)2213 5039 y FF(h)2265 5006 y FC(2)p 2211 5080 95 4 v 2211 5163 a FF(\033)2266 5137 y FC(2)2316 5101 y FF(\015)5 b(")2410 5000 y Fr(\021)2485 5101 y FE(\000)20 b FG(1)2621 5000 y Fr(i)2664 5027 y(\002)2702 5101 y FG(1)h FE(\000)f(O)s FG(\()p FF(")p FG(\))3046 5027 y Fr(\003)1152 5337 y Fn(>)25 b FF(\015)5 b(h)1352 5300 y FC(2)1392 5236 y Fr(h)1435 5337 y FG(1)21 b FE(\000)1626 5276 y FG(\026)-47 b FF(a)1672 5290 y FC(1)p 1602 5316 133 4 v 1602 5401 a FG(2)q(\026)h FF(a)1695 5370 y FC(2)1695 5427 y(0)1745 5264 y Fr(\000)1786 5337 y FG(1)21 b(+)f(2)p FF(\015)2040 5264 y Fr(\001)2082 5337 y FF(")2124 5236 y Fr(i)2167 5264 y(\002)2205 5337 y FG(1)h FE(\000)f(O)s FG(\()p FF(")p FG(\))2549 5264 y Fr(\003)1152 5541 y Fn(>)25 b FF(\015)5 b(h)1352 5503 y FC(2)1392 5467 y Fr(\002)1430 5541 y FG(1)21 b FE(\000)f(O)s FG(\()p FF(")p FG(\))1774 5467 y Fr(\003)1812 5541 y FF(:)3430 5164 y FH(\(3.71\))1845 5871 y(23)p eop %%Page: 24 24 24 23 bop 273 328 a FH(Inserting)30 b(this)f(in)m(to)i(\(3.69\))q(,)f (w)m(e)h(obtain)f(for)g(eac)m(h)i FF(k)c FG(=)d(0)p FF(;)15 b(:)g(:)g(:)i(;)e(K)28 b FE(\000)19 b FG(2)31 b FH(on)g FF(B)2939 343 y Fy(k)3011 328 y FH(the)g(estimate)520 580 y FF(P)578 595 y Fy(k)646 580 y Fn(6)752 518 y FG(2)p FF(h)p FG(\(1)22 b(+)e FF(r)1083 532 y FC(0)1122 518 y FF(h)p FG(\))p 752 559 458 4 v 893 577 a FE(p)p 968 577 101 4 v 968 652 a FG(2)p FF(\031)1347 518 y FG(1)p 1230 559 281 4 v 1230 577 a Fr(p)p 1321 577 190 4 v 81 x FG(2)p FF(\015)5 b(h)1470 631 y FC(2)1520 506 y Fr(\002)1558 580 y FG(1)20 b(+)g FE(O)s FG(\()p FF(")p FG(\))1901 506 y Fr(\003)1965 580 y FG(=)2140 518 y(1)p 2071 559 183 4 v 2071 585 a FE(p)p 2147 585 108 4 v 57 x FF(\031)s(\015)2264 506 y Fr(\002)2302 580 y FG(1)h(+)f FE(O)s FG(\()p FF(")p FG(\))h(+)f FE(O)s FG(\()p FF(h)p FG(\))2955 506 y Fr(\003)3009 580 y FG(=)3075 577 y(:)3115 580 y FF(q)s(:)246 b FH(\(3.72\))273 850 y(Note)34 b(that)g(for)g(an)m(y)g FF(\015)i FE(2)30 b FG(\(1)p FF(=\031)s(;)15 b FG(1])p FH(,)37 b(there)d(exist)f FF(h)2040 864 y FC(0)2110 850 y FF(>)e FG(0)j FH(and)g FF(")2512 864 y FC(0)2583 850 y FF(>)c FG(0)k FH(suc)m(h)g(that)g FF(q)g(<)d FG(1)j FH(for)f(all)273 963 y FF(h)26 b Fn(6)f FF(h)499 977 y FC(0)569 963 y FH(and)31 b(all)d FF(")e Fn(6)f FF(")1076 977 y FC(0)1116 963 y FH(.)40 b(Since)30 b FF(q)1458 977 y Fy(K)5 b FD(\000)p FC(1)1641 963 y FG(=)25 b(1)31 b FH(is)e(an)h(ob)m(vious)g(b)s(ound,)h(w)m(e)g(obtain)f (from)g(\(3.62\))367 1223 y Fo(P)428 1185 y FC(0)p Fy(;y)518 1194 y Fw(0)557 1122 y Fr(n)651 1223 y FG(sup)618 1300 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)821 1223 y FE(j)p FF(y)891 1237 y Fy(s)927 1223 y FE(j)952 1141 y Fr(p)p 1044 1141 207 4 v 1044 1223 a FG(2)q(\026)-46 b FF(a)p FG(\()p FF(s)p FG(\))26 b FF(<)f(h)1424 1122 y Fr(o)1510 1223 y Fn(6)g FF(q)1650 1185 y Fy(K)5 b FD(\000)p FC(1)1833 1223 y Fn(6)1939 1161 y FG(1)p 1939 1202 46 4 v 1940 1285 a FF(q)2010 1223 y FG(exp)2149 1122 y Fr(n)2209 1223 y FE(\000)p 2300 1111 44 4 v 2290 1161 a FF(\013)p FG(\()p FF(t)p FG(\))p 2290 1202 163 4 v 2350 1285 a FF(")2472 1161 y(\033)2527 1128 y FC(2)p 2472 1202 95 4 v 2473 1285 a FF(h)2525 1259 y FC(2)2654 1161 y FG(1)p 2587 1202 181 4 v 2587 1285 a(2)p FF(\015)g(q)2728 1259 y FC(2)2777 1223 y FF(q)2821 1185 y FC(2)2876 1223 y FG(log)2993 1149 y Fr(\000)3035 1223 y FG(1)p FF(=q)3169 1185 y FC(2)3209 1149 y Fr(\001)3251 1122 y(o)3311 1223 y FF(:)94 b FH(\(3.73\))273 1495 y(Cho)s(osing)26 b FF(\015)32 b FH(so)26 b(that)i FF(q)1087 1462 y FC(2)1151 1495 y FG(=)d(1)p FF(=)15 b FG(e)28 b FH(holds,)f(yields)d(almost)h(the)j (optimal)c(exp)s(onen)m(t,)29 b(and)e(w)m(e)g(obtain)930 1755 y Fo(P)991 1717 y FC(0)p Fy(;y)1081 1726 y Fw(0)1120 1654 y Fr(n)1214 1755 y FG(sup)1180 1831 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1384 1755 y FE(j)p FF(y)1454 1769 y Fy(s)1490 1755 y FE(j)1515 1672 y Fr(p)p 1606 1672 207 4 v 83 x FG(2)q(\026)-46 b FF(a)q FG(\()p FF(s)p FG(\))25 b FF(<)g(h)1986 1654 y Fr(o)2073 1755 y Fn(6)2169 1685 y FE(p)p 2244 1685 41 4 v 2244 1755 a FG(e)16 b(exp)2439 1654 y Fr(n)2499 1755 y FE(\000)p FF(\024)p 2642 1643 44 4 v 2632 1693 a(\013)p FG(\()p FF(t)p FG(\))p 2632 1734 163 4 v 2692 1817 a FF(")2814 1693 y(\033)2869 1660 y FC(2)p 2814 1734 95 4 v 2815 1817 a FF(h)2867 1791 y FC(2)2919 1654 y Fr(o)2980 1755 y FF(:)425 b FH(\(3.74\))p 3596 2012 4 62 v 3600 1954 55 4 v 3600 2012 V 3653 2012 4 62 v 118 2298 a FI(4)131 b(Pitc)l(hfork)46 b(bifurcation)118 2505 y Fp(4.1)112 b(Preliminaries)118 2676 y FH(W)-8 b(e)31 b(consider)f(the)h(nonlinear)e(SDE)1326 2896 y FG(d)p FF(x)1429 2910 y Fy(t)1483 2896 y FG(=)1589 2834 y(1)p 1589 2875 46 4 v 1591 2958 a FF(")1645 2896 y(f)10 b FG(\()p FF(x)1787 2910 y Fy(t)1816 2896 y FF(;)15 b(t)p FG(\))g(d)p FF(t)20 b FG(+)2176 2834 y FF(\033)p 2144 2875 119 4 v 2144 2893 a FE(p)p 2220 2893 43 4 v 66 x FF(")2287 2896 y FG(d)p FF(W)2424 2910 y Fy(t)3476 2896 y FH(\(4.1\))118 3156 y(in)30 b(the)g(region)g FE(M)c FG(=)f FE(f)p FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f Fo(R)1308 3123 y FC(2)1364 3156 y FG(:)30 b FE(j)p FF(x)p FE(j)c Fn(6)f FF(d;)46 b FE(j)p FF(t)p FE(j)25 b Fn(6)g FF(T)13 b FE(g)p FH(.)41 b(W)-8 b(e)30 b(assume)g(that)181 3292 y FE(\017)47 b FH(there)22 b(exists)e(a)h(constan)m(t)i FF(M)35 b(>)25 b FG(0)d FH(suc)m(h)f(that)h FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))21 b FH(is)f(three)i(times)e(con)m(tin)m (uously)h(di\033eren)m(tiable)273 3405 y(with)30 b(resp)s(ect)h(to)f FF(x)g FH(and)h FF(t)f FH(and)h FE(j)p FF(@)1469 3419 y Fy(xxx)1593 3405 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))p FE(j)25 b Fn(6)g FG(6)p FF(M)41 b FH(for)30 b(all)f FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f(M)p FH(;)181 3518 y FE(\017)47 b FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)f FE(\000)p FF(f)10 b FG(\()p FE(\000)p FF(x;)15 b(t)p FG(\))29 b FH(for)i(all)d FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f(M)p FH(;)181 3631 y FE(\017)47 b FF(f)40 b FH(exhibits)28 b(a)j(sup)s(ercritical)d(pitc)m(hfork)i (bifurcation)g(at)g(the)h(origin,)e(that)i(is)e(\(after)i(rescaling\),) 743 3835 y FF(@)791 3849 y Fy(x)835 3835 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))26 b(=)f(0)p FF(;)197 b(@)1527 3849 y Fy(tx)1597 3835 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))26 b(=)f(1)181 b FH(and)h FF(@)2575 3849 y Fy(xxx)2698 3835 y FF(f)10 b FG(\(0)p FF(;)15 b FG(0\))27 b(=)e FE(\000)p FG(6)284 b FH(\(4.2\))118 4040 y(Using)29 b(T)-8 b(a)m(ylor)31 b(series)e(and)i(the)g(symmetry)d(assumptions,)h(w)m(e)i(ma)m(y)f (write)g(for)g(all)e FG(\()p FF(x;)15 b(t)p FG(\))27 b FE(2)d(M)1080 4241 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)f FF(a)p FG(\()p FF(t)p FG(\))p FF(x)20 b FG(+)g FF(b)p FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)f FF(x)2174 4167 y Fr(\002)2212 4241 y FF(a)p FG(\()p FF(t)p FG(\))c(+)f FF(g)2518 4255 y FC(0)2558 4241 y FG(\()p FF(x;)15 b(t)p FG(\))2753 4167 y Fr(\003)988 4379 y FF(@)1036 4393 y Fy(x)1080 4379 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)f FF(a)p FG(\()p FF(t)p FG(\))20 b(+)g FF(g)1757 4393 y FC(1)1797 4379 y FG(\()p FF(x;)15 b(t)p FG(\))3476 4311 y FH(\(4.3\))118 4581 y(where)31 b FF(a)p FG(\()p FF(t)p FG(\))p FH(,)g FF(g)629 4595 y FC(0)669 4581 y FG(\()p FF(x;)15 b(t)p FG(\))p FH(,)31 b FF(g)963 4595 y FC(1)1003 4581 y FG(\()p FF(x;)15 b(t)p FG(\))31 b FH(are)g(t)m(wice)f(con)m(tin)m(uously)g(di\033eren)m(tiable)g (functions)g(satisfying)783 4785 y FF(a)p FG(\()p FF(t)p FG(\))c(=)f FF(@)1104 4799 y Fy(x)1148 4785 y FF(f)10 b FG(\(0)p FF(;)15 b(t)p FG(\))26 b(=)f FF(t)20 b FG(+)g FE(O)s FG(\()p FF(t)1800 4747 y FC(2)1839 4785 y FG(\))656 4932 y FF(g)699 4946 y FC(0)739 4932 y FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)1056 4859 y Fr(\002)1094 4932 y FE(\000)p FG(1)20 b(+)g FF(\015)1368 4946 y FC(0)1408 4932 y FG(\()p FF(x;)15 b(t)p FG(\))1603 4859 y Fr(\003)1641 4932 y FF(x)1693 4895 y FC(2)2413 4932 y FE(j)p FF(g)2481 4946 y FC(0)2521 4932 y FG(\()p FF(x;)g(t)p FG(\))p FE(j)26 b Fn(6)f FF(M)10 b(x)3013 4895 y FC(2)3476 4932 y FH(\(4.4\))656 5080 y FF(g)699 5094 y FC(1)739 5080 y FG(\()p FF(x;)15 b(t)p FG(\))26 b(=)1056 5006 y Fr(\002)1094 5080 y FE(\000)p FG(3)20 b(+)g FF(\015)1368 5094 y FC(1)1408 5080 y FG(\()p FF(x;)15 b(t)p FG(\))1603 5006 y Fr(\003)1641 5080 y FF(x)1693 5042 y FC(2)2413 5080 y FE(j)p FF(g)2481 5094 y FC(1)2521 5080 y FG(\()p FF(x;)g(t)p FG(\))p FE(j)26 b Fn(6)f FG(3)p FF(M)10 b(x)3058 5042 y FC(2)3098 5080 y FF(;)118 5284 y FH(with)39 b FF(\015)380 5298 y FC(0)419 5284 y FF(;)15 b(\015)506 5298 y FC(1)585 5284 y FH(some)39 b(con)m(tin)m(uous)h(functions)f(suc)m(h)h(that)f FF(\015)2145 5298 y FC(0)2185 5284 y FG(\(0)p FF(;)15 b FG(0\))42 b(=)e FF(\015)2585 5298 y FC(1)2625 5284 y FG(\(0)p FF(;)15 b FG(0\))42 b(=)e(0)p FH(.)67 b(The)40 b(follo)m(wing)118 5397 y(standard)26 b(result)f(from)f(bifurcation)h(theory)h(is)e (easily)f(obtained)j(b)m(y)f(applying)f(the)i(implicit)21 b(function)118 5510 y(theorem,)31 b(see)f([GH,)g(p.)g(150])h(or)g([IJ,) f(Section)h(I)s(I.4])e(for)h(instance.)40 b(W)-8 b(e)31 b(state)g(it)e(without)h(pro)s(of.)1845 5871 y(24)p eop %%Page: 25 25 25 24 bop 118 328 a Fq(Prop)s(osition)38 b(4.1.)43 b FB(If)34 b FF(T)47 b FB(and)35 b FF(d)g FB(ar)-5 b(e)35 b(su\036ciently)g(smal)5 b(l,)34 b(ther)-5 b(e)36 b(exist)g(twic)-5 b(e)35 b(c)-5 b(ontinuously)36 b(di\033er-)118 441 y(entiable)d (functions)f FF(x)896 408 y Fy(?)936 441 y FF(;)21 b FG(\026)-51 b FF(x)25 b FG(:)h(\(0)p FF(;)15 b(T)e FG(])26 b FE(!)f Fo(R)1532 455 y FC(+)1630 441 y FB(of)32 b(the)g(form)1394 648 y FF(x)1446 611 y Fy(?)1485 648 y FG(\()p FF(t)p FG(\))26 b(=)1710 570 y FE(p)p 1786 570 33 4 v 78 x FF(t)1818 575 y Fr(\002)1856 648 y FG(1)21 b(+)f FD(O)2071 662 y Fy(T)2126 648 y FG(\(1\))2241 575 y Fr(\003)1439 805 y FG(\026)-51 b FF(x)p FG(\()p FF(t)p FG(\))26 b(=)1710 723 y Fr(p)p 1801 723 124 4 v 82 x FF(t=)p FG(3)1924 731 y Fr(\002)1962 805 y FG(1)21 b(+)f FD(O)2177 819 y Fy(T)2232 805 y FG(\(1\))2347 731 y Fr(\003)3476 724 y FH(\(4.5\))118 1001 y FB(with)32 b(the)h(fol)5 b(lowing)32 b(pr)-5 b(op)g(erties:)181 1138 y FE(\017)48 b FB(the)41 b(only)f(solutions)h(of)f FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))41 b(=)f(0)h FB(in)f FE(M)g FB(ar)-5 b(e)42 b(either)f(of)g(the)g(form)f FG(\(0)p FF(;)15 b(t)p FG(\))p FB(,)44 b(or)c(of)h(the)g(form)274 1251 y FG(\()p FE(\006)p FF(x)432 1218 y Fy(?)471 1251 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))34 b FB(with)e FF(t)25 b(>)g FG(0)p FB(;)181 1363 y FE(\017)48 b FB(the)33 b(only)f(solutions)g(of)g FF(@)1152 1377 y Fy(x)1196 1363 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))25 b(=)g(0)33 b FB(in)f FE(M)g FB(ar)-5 b(e)33 b(of)f(the)h(form)f FG(\()p FE(\006)6 b FG(\026)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))34 b FB(with)d FF(t)26 b Fn(>)e FG(0)p FB(;)181 1476 y FE(\017)48 b FB(the)33 b(derivative)g(of)g FF(f)41 b FB(at)32 b FE(\006)p FF(x)1261 1443 y Fy(?)1301 1476 y FG(\()p FF(t)p FG(\))g FB(is)1190 1681 y FF(a)1238 1643 y Fy(?)1278 1681 y FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1551 1695 y Fy(x)1595 1681 y FF(f)10 b FG(\()p FF(x)1737 1643 y Fy(?)1776 1681 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))26 b(=)f FE(\000)p FG(2)p FF(t)2258 1607 y Fr(\002)2296 1681 y FG(1)c(+)f FD(O)2510 1695 y Fy(T)2566 1681 y FG(\(1\))2681 1607 y Fr(\003)2720 1681 y FF(:)731 b FH(\(4.6\))181 1885 y FE(\017)48 b FB(the)33 b(derivatives)g(of)f FF(x)1032 1852 y Fy(?)1072 1885 y FG(\()p FF(t)p FG(\))h FB(and)38 b FG(\026)-51 b FF(x)p FG(\()p FF(t)p FG(\))33 b FB(satisfy)956 2071 y FG(d)p FF(x)1059 2038 y Fy(?)p 956 2112 142 4 v 986 2195 a FG(d)o FF(t)1134 2133 y FG(=)1294 2071 y(1)p 1239 2112 155 4 v 1239 2204 a(2)1284 2130 y FE(p)p 1361 2130 33 4 v 1361 2204 a FF(t)1403 2133 y FG([1)21 b(+)f FD(O)1643 2147 y Fy(T)1698 2133 y FG(\(1\)])p FF(;)2074 2071 y FG(d)6 b(\026)-51 b FF(x)p 2074 2112 103 4 v 2084 2195 a FG(d)o FF(t)2212 2133 y FG(=)2395 2071 y(1)p 2318 2112 200 4 v 2318 2205 a(2)2363 2130 y FE(p)p 2439 2130 79 4 v 75 x FG(3)p FF(t)2527 2133 y FG([1)21 b(+)f FD(O)2767 2147 y Fy(T)2822 2133 y FG(\(1\)])p FF(:)489 b FH(\(4.7\))259 2380 y(As)34 b(already)f(p)s(oin)m(ted)i(out)f(in)g(Section)g(2.3,)h (there)g(is)e(no)h(restriction)g(in)f(assuming)f FF(T)47 b FH(and)35 b FF(d)f FH(to)118 2493 y(b)s(e)29 b(small.)37 b(Th)m(us)30 b(w)m(e)g(ma)m(y)f(assume)f(that)i(the)g(terms)e FD(O)2035 2507 y Fy(T)2090 2493 y FG(\(1\))i FH(are)g(su\036cien)m(tly) f(small)d(to)j(do)h(no)f(harm.)118 2606 y(F)-8 b(or)31 b(instance,)f(w)m(e)h(ma)m(y)f(and)h(will)d(alw)m(a)m(ys)i(assume)f (that)i FF(a)2178 2573 y Fy(?)2218 2606 y FG(\()p FF(t)p FG(\))26 b FF(<)e FG(0)p FH(.)259 2719 y(Equation)31 b(\(4.4\))g(also)e(implies)e(the)k(existence)e(of)h(constan)m(ts)i FF(a)2446 2733 y FC(+)2530 2719 y Fn(>)25 b FF(a)2674 2733 y FD(\000)2758 2719 y FF(>)g FG(0)31 b FH(suc)m(h)g(that)1165 2920 y FF(a)1213 2934 y FC(+)1272 2920 y FF(t)25 b Fn(6)g FF(a)p FG(\()p FF(t)p FG(\))h Fn(6)f FF(a)1747 2934 y FD(\000)1806 2920 y FF(t)180 b FH(for)31 b FE(\000)p FF(T)37 b Fn(6)25 b FF(t)g Fn(6)g FG(0)1165 3058 y FF(a)1213 3072 y FD(\000)1272 3058 y FF(t)g Fn(6)g FF(a)p FG(\()p FF(t)p FG(\))h Fn(6)f FF(a)1747 3072 y FC(+)1806 3058 y FF(t)180 b FH(for)31 b FG(0)25 b Fn(6)g FF(t)g Fn(6)g FF(T)13 b FH(.)3476 2990 y(\(4.8\))118 3279 y(The)31 b(function)f FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))26 b(=)1024 3206 y Fr(R)1085 3232 y Fy(t)1067 3311 y(s)1130 3279 y FF(a)p FG(\()p FF(u)p FG(\))15 b(d)p FF(u)30 b FH(th)m(us)h(satis\034es)607 3483 y FE(\000)688 3448 y FC(1)p 688 3463 36 4 v 688 3515 a(2)733 3483 y FF(a)781 3497 y FC(+)840 3483 y FG(\()p FF(s)918 3446 y FC(2)978 3483 y FE(\000)20 b FF(t)1102 3446 y FC(2)1141 3483 y FG(\))26 b Fn(6)e FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))27 b Fn(6)e FE(\000)1745 3448 y FC(1)p 1745 3463 V 1745 3515 a(2)1790 3483 y FF(a)1838 3497 y FD(\000)1897 3483 y FG(\()p FF(s)1975 3446 y FC(2)2034 3483 y FE(\000)20 b FF(t)2158 3446 y FC(2)2198 3483 y FG(\))489 b FH(if)29 b FF(s)c Fn(6)g FF(t)g Fn(6)g FG(0)596 3595 y FC(1)p 596 3610 V 596 3662 a(2)641 3631 y FF(a)689 3645 y FD(\000)749 3631 y FF(t)782 3593 y FC(2)841 3631 y FE(\000)942 3595 y FC(1)p 942 3610 V 942 3662 a(2)987 3631 y FF(a)1035 3645 y FC(+)1094 3631 y FF(s)1137 3593 y FC(2)1202 3631 y Fn(6)f FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))27 b Fn(6)1674 3595 y FC(1)p 1674 3610 V 1674 3662 a(2)1719 3631 y FF(a)1767 3645 y FC(+)1826 3631 y FF(t)1859 3593 y FC(2)1919 3631 y FE(\000)2019 3595 y FC(1)p 2019 3610 V 2019 3662 a(2)2065 3631 y FF(a)2113 3645 y FD(\000)2172 3631 y FF(s)2215 3593 y FC(2)2722 3631 y FH(if)i FF(s)c Fn(6)g FG(0)g Fn(6)g FF(t)308 b FH(\(4.9\))688 3742 y FC(1)p 688 3757 V 688 3810 a(2)733 3778 y FF(a)781 3792 y FD(\000)840 3778 y FG(\()p FF(t)908 3741 y FC(2)968 3778 y FE(\000)20 b FF(s)1102 3741 y FC(2)1141 3778 y FG(\))26 b Fn(6)e FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))27 b Fn(6)1674 3742 y FC(1)p 1674 3757 V 1674 3810 a(2)1719 3778 y FF(a)1767 3792 y FC(+)1826 3778 y FG(\()p FF(t)1894 3741 y FC(2)1954 3778 y FE(\000)20 b FF(s)2088 3741 y FC(2)2127 3778 y FG(\))560 b FH(if)29 b FG(0)d Fn(6)e FF(s)h Fn(6)g FF(t)p FH(.)259 3982 y(W)-8 b(e)26 b(are)g(going)g(to)g (analyse)e(the)j(dynamics)d(in)h(three)h(di\033eren)m(t)g(regions)f(of) h(the)g FG(\()p FF(t;)15 b(x)p FG(\))p FH(-plane:)39 b(near)118 4095 y FF(x)d FG(=)g(0)h FH(for)f FF(t)g Fn(6)715 4030 y FE(p)p 790 4030 43 4 v 790 4095 a FF(")q FH(,)i(near)f FF(x)f FG(=)g(0)h FH(for)f FF(t)g Fn(>)1700 4030 y FE(p)p 1776 4030 V 65 x FF(")p FH(,)i(and)f(near)h FF(x)e FG(=)f FF(x)2517 4062 y Fy(?)2557 4095 y FG(\()p FF(t)p FG(\))i FH(for)f FF(t)g Fn(>)3017 4030 y FE(p)p 3092 4030 V 3092 4095 a FF(")q FH(.)59 b(In)37 b(order)g(to)118 4208 y(delimit)27 b(the)k(last)e(t)m(w)m(o)j(regions,)e(w)m(e)h(in)m(tro)s(duce)g (\(somewhat)g(arbitrarily\))d(the)j(function)1575 4412 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))26 b(=)1846 4331 y FE(p)p 1922 4331 54 4 v 81 x FF(\025)15 b(x)2042 4375 y Fy(?)2081 4412 y FG(\()p FF(t)p FG(\))p FF(;)1221 b FH(\(4.10\))118 4617 y(set)1519 4730 y FG(~)-46 b FF(a)p FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1839 4744 y Fy(x)1883 4730 y FF(f)10 b FG(\()c(~)-51 b FF(x)o FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))p FF(;)1170 b FH(\(4.11\))118 4896 y(and)31 b(de\034ne)g(the)g(region)1160 5009 y FE(D)c FG(=)1353 4936 y Fr(\010)1406 5009 y FG(\()p FF(x;)15 b(t)p FG(\))10 b(:)1668 4939 y FE(p)p 1744 4939 43 4 v 70 x FF(")25 b Fn(6)g FF(t)g Fn(6)g FF(T)8 b(;)15 b FE(j)p FF(x)p FE(j)26 b FF(<)31 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))2541 4936 y Fr(\011)2595 5009 y FF(;)810 b FH(\(4.12\))118 5176 y(whic)m(h)31 b(has)f(the)h(follo)m(wing)d(prop) s(erties:)111 5313 y(\(a\))47 b(for)30 b(all)f FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f(D)33 b FH(with)d FF(x)25 b FE(6)p FG(=)g(0)p FH(,)31 b(one)f(has)1089 5494 y FG(1)p 1086 5534 52 4 v 1086 5618 a FF(x)1148 5555 y(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))25 b Fn(>)g FF(\024a)p FG(\()p FF(t)p FG(\))182 b FH(with)30 b FF(\024)c FG(=)f(1)20 b FE(\000)g FF(\025)g FE(\000)g FD(O)2662 5569 y Fy(T)2717 5555 y FG(\(1\))p FH(.)573 b(\(4.13\))1845 5871 y(25)p eop %%Page: 26 26 26 25 bop 106 328 a FH(\(b\))47 b(for)30 b(all)f FG(\()p FF(x;)15 b(t)p FG(\))26 b FE(2)f FG([)p FE(\000)p FF(d;)15 b(d)10 b FG(])22 b FE(\002)d FG([)1246 262 y FE(p)p 1322 262 43 4 v 66 x FF(")q(;)c(T)e FG(])20 b FE(n)h(D)s FH(,)876 502 y FF(@)924 516 y Fy(x)968 502 y FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))25 b Fn(6)h FG(~)-46 b FF(a)p FG(\()p FF(t)p FG(\))26 b Fn(6)f FE(\000)p FF(\021)s(a)p FG(\()p FF(t)p FG(\))181 b FH(with)30 b FF(\021)f FG(=)c(3)p FF(\025)c FE(\000)e FG(1)i FE(\000)f FD(O)2863 516 y Fy(T)2918 502 y FG(\(1\))p FH(.)372 b(\(4.14\))118 677 y(F)-8 b(or)31 b(our)h(results)d(to)i(b)s(e)g(of)f(in)m(terest,)h FF(\024)26 b(>)g FG(0)31 b FH(and)g FF(\021)e(>)d FG(0)31 b FH(are)g(necessary)-8 b(,)31 b(whic)m(h)g(requires)f FF(\025)c FE(2)g FG(\()3470 641 y FC(1)p 3470 656 36 4 v 3470 708 a(3)3515 677 y FF(;)15 b FG(1\))p FH(.)118 789 y(As)31 b(w)m(e)g(shall)f(see,)h(w)m(e)h(will)d(actually)h(need)i FF(\025)26 b FE(2)h FG(\()1846 754 y FC(1)p 1846 769 V 1846 821 a(3)1891 789 y FF(;)1942 754 y FC(1)p 1942 769 V 1942 821 a(2)1987 789 y FG(\))p FH(.)43 b(F)-8 b(urthermore,)33 b(in)d(Section)h(4.3,)h(w)m(e)g(need)g(to)118 912 y(assume)d(that)i FF(\033)s FE(j)p FG(log)17 b FF(\033)s FE(j)919 879 y FC(3)p Fy(=)p FC(2)1055 912 y FG(=)25 b FE(O)s FG(\()1261 846 y FE(p)p 1337 846 43 4 v 66 x FF(")p FG(\))p FH(.)259 1025 y(In)30 b(the)g(follo)m(wing)e (subsections,)i(w)m(e)g(in)m(v)m(estigate)g(the)g(three)g(di\033eren)m (t)h(regimes:)38 b(In)30 b(Section)g(4.2,)118 1138 y(w)m(e)37 b(analyse)e(the)h(b)s(eha)m(viour)g(for)g FF(t)f Fn(6)1485 1072 y FE(p)p 1561 1072 V 66 x FF(")p FH(.)58 b(Theorem)36 b(2.8)g(is)f(pro)m(v)m(ed)i(in)e(the)i(same)e(w)m(a)m(y)i(as)e(Theo-) 118 1251 y(rem)27 b(2.3,)i(the)g(main)d(di\033erence)i(lying)e(in)h (the)h(b)s(eha)m(viour)g(of)g(the)g(v)-5 b(ariance)28 b(whic)m(h)g(is)e(in)m(v)m(estigated)i(in)118 1364 y(Lemma)i(4.2.)259 1476 y(Section)g(4.3)f(is)f(dev)m(oted)i(to)f(the)h(rather)g(in)m(v)m (olv)m(ed)f(pro)s(of)g(of)g(Theorem)g(2.9.)41 b(W)-8 b(e)29 b(start)g(b)m(y)h(giving)118 1589 y(some)43 b(preparatory)i (results.)81 b(Prop)s(osition)43 b(4.7)h(estimates)e(the)j(probabilit)m (y)d(of)i(remaining)e(in)h(a)118 1702 y(smaller)28 b(strip)i FE(S)38 b FH(in)30 b(a)g(similar)e(w)m(a)m(y)j(as)g(Prop)s(osition)f (3.9.)42 b(W)-8 b(e)31 b(then)g(sho)m(w)h(in)d(Lemma)h(4.8)i(that)f (the)118 1815 y(paths)39 b(are)h(lik)m(ely)c(to)j(lea)m(v)m(e)h FE(D)h FH(as)e(w)m(ell,)h(unless)e(the)i(solution)e(of)g(a)h(suitably)f (c)m(hosen)i(linear)e(SDE)118 1928 y(returns)29 b(to)g(zero.)41 b(The)29 b(probabilit)m(y)e(of)h(suc)m(h)i(a)e(return)i(to)f(zero)g(is) e(studied)h(in)g(Lemma)g(4.9.)40 b(Finally)-8 b(,)118 2041 y(Theorem)31 b(2.9)f(is)f(pro)m(v)m(ed,)j(the)f(pro)s(of)f(b)s (eing)g(based)g(on)h(an)f(iterativ)m(e)g(sc)m(heme.)259 2154 y(The)k(last)e(subsection)h(analyses)f(the)h(motion)g(after)g FF(\034)2145 2168 y FD(D)2205 2154 y FH(.)49 b(Here,)34 b(the)g(main)e(di\036cult)m(y)h(is)e(to)j(con-)118 2267 y(trol)h(the)g(b)s(eha)m(viour)h(of)f(the)g(deterministic)e(solutions,) i(whic)m(h)h(are)f(sho)m(wn)i(to)e(approac)m(h)i FF(x)3364 2234 y Fy(?)3404 2267 y FG(\()p FF(t)p FG(\))p FH(,)f(cf.)118 2380 y(Prop)s(osition)27 b(4.11.)40 b(W)-8 b(e)27 b(then)i(pro)m(v)m(e) f(that)g(the)g(paths)g(of)f(the)h(random)f(pro)s(cess)g(are)h(lik)m (ely)d(to)j(sta)m(y)f(in)118 2493 y(a)k(neigh)m(b)s(ourho)s(o)s(d)h(of) e(the)i(deterministic)c(solutions.)41 b(The)32 b(pro)s(of)e(is)g (similar)d(to)32 b(the)f(corresp)s(onding)118 2606 y(pro)s(of)f(in)g (Section)g(3.1.)118 2844 y Fp(4.2)112 b(The)38 b(b)s(eha)m(viour)g(for) f Ff(t)28 b Fe(6)1465 2773 y Fd(p)p 1548 2773 46 4 v 71 x Ff(")118 3016 y FH(W)-8 b(e)31 b(\034rst)f(consider)g(the)h (linear)e(equation)1340 3223 y FG(d)o FF(x)1442 3186 y FC(0)1442 3246 y Fy(t)1507 3223 y FG(=)1613 3162 y(1)p 1613 3203 V 1615 3286 a FF(")1668 3223 y(a)p FG(\()p FF(t)p FG(\))p FF(x)1871 3186 y FC(0)1871 3246 y Fy(t)1926 3223 y FG(d)p FF(t)20 b FG(+)2162 3162 y FF(\033)p 2131 3203 119 4 v 2131 3221 a FE(p)p 2206 3221 43 4 v 2206 3286 a FF(")2274 3223 y FG(d)o FF(W)2410 3237 y Fy(t)3430 3223 y FH(\(4.15\))118 3454 y(with)30 b(initial)d(condition)j FF(x)1034 3421 y FC(0)1034 3476 y Fy(t)1059 3485 y Fw(0)1123 3454 y FG(=)25 b FF(x)1271 3468 y FC(0)1341 3454 y FH(at)30 b(time)f FF(t)1690 3468 y FC(0)1755 3454 y FE(2)c FG([)p FE(\000)p FF(T)8 b(;)15 b FG(0\))p FH(.)41 b(Let)1335 3693 y FF(v)s FG(\()p FF(t;)15 b(t)1523 3707 y FC(0)1563 3693 y FG(\))26 b(=)1730 3632 y FF(\033)1785 3599 y FC(2)p 1730 3673 95 4 v 1756 3756 a FF(")1850 3570 y Fr(Z)1940 3596 y Fy(t)1900 3776 y(t)1925 3785 y Fw(0)1985 3693 y FG(e)2026 3656 y FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2326 3693 y FG(d)p FF(s:)985 b FH(\(4.16\))118 3931 y(denote)29 b(the)g(v)-5 b(ariance)28 b(of)h FF(x)1067 3898 y FC(0)1067 3953 y Fy(t)1106 3931 y FH(.)40 b(As)27 b(b)s(efore,)i(w)m(e)g(no)m(w)h(in)m(tro)s(duce)e(a)h(function)f FF(\020)7 b FG(\()p FF(t)p FG(\))28 b FH(whic)m(h)h(will)d(allo)m(w)i (us)118 4044 y(to)k(de\034ne)h(a)g(strip)e(that)h(the)h(pro)s(cess)f FF(x)1508 4058 y Fy(t)1569 4044 y FH(is)f(unlik)m(ely)f(to)i(lea)m(v)m (e)g(b)s(efore)g(time)2821 3978 y FE(p)p 2897 3978 43 4 v 66 x FF(")p FH(,)h(see)f(Corollary)f(4.5)118 4157 y(b)s(elo)m(w.)40 b(Let)1036 4303 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b(=)1438 4242 y(1)p 1317 4282 287 4 v 1317 4366 a(2)p FE(j)p FF(a)p FG(\()p FF(t)1503 4380 y FC(0)1544 4366 y FG(\))p FE(j)1629 4303 y FG(e)1670 4266 y FC(2)p Fy(\013)p FC(\()p Fy(t;t)1847 4275 y Fw(0)1883 4266 y FC(\))p Fy(=")1997 4303 y FG(+)2078 4242 y(1)p 2078 4282 46 4 v 2080 4366 a FF(")2149 4179 y Fr(Z)2240 4206 y Fy(t)2199 4386 y(t)2224 4395 y Fw(0)2284 4303 y FG(e)2325 4266 y FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2625 4303 y FG(d)p FF(s:)686 b FH(\(4.17\))118 4510 y(The)31 b(follo)m(wing)e(lemma)f(describ)s(es)h(the)i(b)s(eha)m (viour)f(of)g FF(\020)7 b FG(\()p FF(t)p FG(\))p FH(.)118 4673 y Fq(Lemma)33 b(4.2.)42 b FB(Assuming)32 b FF(")26 b Fn(6)f FG(4)p FF(a)p FG(\()p FF(t)1430 4687 y FC(0)1471 4673 y FG(\))1506 4640 y FC(2)1566 4673 y FE(^)20 b FG(\()p FF(t)1715 4687 y FC(0)1754 4673 y FF(=)p FG(2\))1879 4640 y FC(2)1920 4673 y FB(,)32 b(ther)-5 b(e)34 b(exist)f(c)-5 b(onstants)33 b FF(c)2851 4687 y FD(\006)2936 4673 y FG(=)25 b FF(c)3071 4687 y FD(\006)3131 4673 y FG(\()p FF(a)3214 4687 y FC(+)3273 4673 y FF(;)15 b(a)3361 4687 y FD(\000)3421 4673 y FG(\))33 b FB(such)118 4786 y(that)960 4899 y FF(c)999 4913 y FD(\000)p 960 4940 99 4 v 968 5023 a FE(j)p FF(t)p FE(j)1094 4961 y Fn(6)25 b FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1471 4899 y FF(c)1510 4913 y FC(+)p 1471 4940 V 1479 5023 a FE(j)p FF(t)p FE(j)2411 4961 y FB(for)32 b FF(t)2588 4975 y FC(0)2653 4961 y Fn(6)25 b FF(t)g Fn(6)g FE(\000)2974 4891 y(p)p 3049 4891 43 4 v 3049 4961 a FF(")950 5118 y(c)989 5132 y FD(\000)p 940 5159 119 4 v 940 5177 a FE(p)p 1016 5177 43 4 v 65 x FF(")1094 5179 y Fn(6)g FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1481 5118 y FF(c)1520 5132 y FC(+)p 1471 5159 119 4 v 1471 5177 a FE(p)p 1547 5177 43 4 v 65 x FF(")2411 5179 y FB(for)32 b FE(\000)2626 5110 y(p)p 2702 5110 V 69 x FF(")26 b Fn(6)e FF(t)i Fn(6)3020 5110 y FE(p)p 3095 5110 V 3095 5179 a FF(")293 b FH(\(4.18\))662 5336 y FF(c)701 5350 y FD(\000)p 652 5377 119 4 v 652 5395 a FE(p)p 728 5395 43 4 v 66 x FF(")795 5398 y FG(e)835 5360 y FC(2)p Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1094 5398 y Fn(6)25 b FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1481 5336 y FF(c)1520 5350 y FC(+)p 1471 5377 119 4 v 1471 5395 a FE(p)p 1547 5395 43 4 v 66 x FF(")1614 5398 y FG(e)1655 5360 y FC(2)p Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")2411 5398 y FB(for)2555 5328 y FE(p)p 2631 5328 V 70 x FF(")h Fn(6)f FF(t)g Fn(6)g FF(T)13 b FB(.)118 5622 y(If,)31 b(mor)-5 b(e)g(over,)35 b FF(a)706 5589 y FD(0)729 5622 y FG(\()p FF(t)p FG(\))26 b FF(>)f FG(0)33 b FB(on)f FG([)p FF(t)1219 5636 y FC(0)1258 5622 y FF(;)15 b(t)p FG(])p FB(,)33 b(then)f FF(\020)7 b FG(\()p FF(t)p FG(\))32 b FB(is)g(incr)-5 b(e)g(asing)34 b(on)e FG([)p FF(t)2509 5636 y FC(0)2548 5622 y FF(;)15 b(t)p FG(])p FB(.)1845 5871 y FH(26)p eop %%Page: 27 27 27 26 bop 118 328 a Fh(Pr)m(oof:)47 b FH(The)27 b(upp)s(er)h(b)s(ounds) g(are)f(easy)g(to)h(obtain.)39 b(F)-8 b(or)28 b FF(t)2186 342 y FC(0)2251 328 y Fn(6)d FF(t)g Fn(6)g FE(\000)2572 262 y(p)p 2647 262 43 4 v 2647 328 a FF(")j FH(w)m(e)g(ha)m(v)m(e,)h (using)e FF(t)3345 295 y FC(2)3398 328 y FE(\000)14 b FF(s)3526 295 y FC(2)3591 328 y Fn(6)118 441 y FG(2)p FF(t)p FG(\()p FF(t)21 b FE(\000)f FF(s)p FG(\))p FH(,)768 610 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1049 548 y FG(1)p 1049 589 46 4 v 1051 672 a FF(")1119 486 y Fr(Z)1210 512 y Fy(t)1170 692 y(t)1195 701 y Fw(0)1255 610 y FG(e)1295 572 y Fy(a)1332 581 y Fk(\000)1385 572 y FC(\()p Fy(t)1437 549 y Fw(2)1473 572 y FD(\000)p Fy(s)1561 549 y Fw(2)1595 572 y FC(\))p Fy(=")1709 610 y FG(d)p FF(s)20 b FG(+)2044 548 y(1)p 1923 589 287 4 v 1923 672 a(2)p FE(j)p FF(a)p FG(\()p FF(t)2109 686 y FC(0)2150 672 y FG(\))p FE(j)2246 610 y Fn(6)2370 548 y FG(1)p 2352 589 84 4 v 2352 672 a FE(j)p FF(t)p FE(j)2445 509 y Fr(h)2551 548 y FG(1)p 2498 589 153 4 v 2498 672 a(2)p FF(a)2591 686 y FD(\000)2680 610 y FG(+)2835 548 y(1)p 2781 589 V 2781 672 a(2)p FF(a)2874 686 y FC(+)2944 509 y Fr(i)2987 610 y FF(:)418 b FH(\(4.19\))118 840 y(F)-8 b(or)31 b FE(\000)351 774 y(p)p 427 774 43 4 v 66 x FF(")25 b Fn(6)g FF(t)g Fn(6)g FG(0)p FH(,)31 b(the)g(h)m(yp)s(othesis)e FF(")d Fn(6)f FG(4)p FF(a)p FG(\()p FF(t)1771 854 y FC(0)1811 840 y FG(\))1846 807 y FC(2)1916 840 y FH(implies)344 1102 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)625 1041 y FG(1)p 625 1081 46 4 v 627 1165 a FF(")696 1102 y FG(e)736 1065 y FD(\000)p Fy(a)828 1074 y Fk(\000)900 978 y Fr(Z)991 1005 y FC(0)950 1185 y Fy(t)975 1194 y Fw(0)1045 1102 y FG(e)1086 1065 y FD(\000)p Fy(a)1178 1074 y Fk(\000)1230 1065 y Fy(s)1263 1041 y Fw(2)1298 1065 y Fy(=")1385 1102 y FG(d)o FF(s)20 b FG(+)1720 1041 y(1)p 1599 1081 287 4 v 1599 1165 a(2)p FE(j)p FF(a)p FG(\()p FF(t)1785 1179 y FC(0)1825 1165 y FG(\))p FE(j)1921 1102 y Fn(6)2063 1041 y FG(1)p 2027 1081 119 4 v 2027 1099 a FE(p)p 2103 1099 43 4 v 66 x FF(")2155 1001 y Fr(h)2198 1102 y FG(e)2239 1065 y FD(\000)p Fy(a)2331 1074 y Fk(\000)2402 978 y Fr(Z)2493 1005 y FC(0)2453 1185 y FD(\0001)2598 1102 y FG(e)2638 1065 y FD(\000)p Fy(a)2730 1074 y Fk(\000)2782 1065 y Fy(u)2823 1041 y Fw(2)2877 1102 y FG(d)p FF(u)g FG(+)g(1)3136 1001 y Fr(i)3179 1102 y FF(:)226 b FH(\(4.20\))118 1366 y(F)-8 b(or)38 b FG(0)g Fn(6)e FF(t)h Fn(6)655 1301 y FE(p)p 731 1301 V 65 x FF(")p FH(,)i(a)f(similar)c(estimate)i(is)g(obtained)h(b)m(y)h (splitting)d(the)j(in)m(tegrals)f(for)g FF(s)f Fn(6)h FG(0)h FH(and)118 1479 y FF(s)25 b Fn(>)g FG(0)p FH(.)41 b(F)-8 b(or)31 b FF(t)25 b Fn(>)709 1414 y FE(p)p 784 1414 V 784 1479 a FF(")q FH(,)30 b(w)m(e)h(ha)m(v)m(e)754 1741 y FG(e)795 1703 y FD(\000)p FC(2)p Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")1098 1741 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(6)1415 1679 y FG(1)p 1379 1720 119 4 v 1379 1738 a FE(p)p 1455 1738 43 4 v 66 x FF(")1507 1640 y Fr(h)1550 1617 y(Z)1641 1643 y FC(0)1601 1823 y FD(\0001)1745 1741 y FG(e)1786 1703 y FD(\000)p Fy(a)1878 1712 y Fk(\000)1930 1703 y Fy(u)1971 1680 y Fw(2)2025 1741 y FG(d)p FF(u)20 b FG(+)2239 1617 y Fr(Z)2330 1643 y FD(1)2289 1823 y FC(0)2419 1741 y FG(e)2460 1703 y FD(\000)p Fy(a)2552 1712 y Fw(+)2603 1703 y Fy(u)2644 1680 y Fw(2)2698 1741 y FG(d)o FF(u)h FG(+)e(1)2956 1640 y Fr(i)3000 1741 y FF(:)405 b FH(\(4.21\))118 2008 y(T)-8 b(o)35 b(obtain)e(the)i(lo)m(w)m(er)f(b)s(ound,)h(w)m(e)g(\034rst)f (consider)f(the)h(in)m(terv)-5 b(al)33 b FF(t)2450 2022 y FC(0)2521 2008 y Fn(6)d FF(t)h Fn(6)2798 1973 y FC(1)p 2798 1988 36 4 v 2798 2040 a(2)2843 2008 y FF(t)2876 2022 y FC(0)2915 2008 y FH(,)k(where)g(w)m(e)f(use)g(the)118 2121 y(estimate)29 b FF(t)513 2088 y FC(2)573 2121 y FE(\000)20 b FF(s)707 2088 y FC(2)771 2121 y Fn(>)25 b FG(2)p FF(t)945 2135 y FC(0)985 2121 y FG(\()p FF(t)20 b FE(\000)g FF(s)p FG(\))p FH(,)30 b(v)-5 b(alid)29 b(for)h(all)e FF(s)d FE(2)g FG([)p FF(t)1991 2135 y FC(0)2031 2121 y FF(;)15 b(t)p FG(])p FH(,)30 b(whic)m(h)h(yields)754 2388 y FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(>)1035 2326 y FG(1)p 1035 2367 46 4 v 1037 2450 a FF(")1105 2264 y Fr(Z)1196 2291 y Fy(t)1156 2470 y(t)1181 2479 y Fw(0)1241 2388 y FG(e)1281 2350 y FD(\000)p FC(2)p Fy(a)1408 2359 y Fw(+)1460 2350 y FD(j)p Fy(t)1505 2359 y Fw(0)1539 2350 y FD(j)p FC(\()p Fy(t)p FD(\000)p Fy(s)p FC(\))p Fy(=")1814 2388 y FG(d)p FF(s)19 b FG(+)2028 2326 y(e)2069 2293 y FD(\000)p FC(2)p Fy(a)2196 2302 y Fw(+)2247 2293 y FD(j)p Fy(t)2292 2302 y Fw(0)2327 2293 y FD(j)p FC(\()p Fy(t)p FD(\000)p Fy(t)2479 2302 y Fw(0)2514 2293 y FC(\))p Fy(=")p 2028 2367 586 4 v 2183 2450 a FG(2)p FF(a)2276 2464 y FC(+)2336 2450 y FE(j)p FF(t)2394 2464 y FC(0)2433 2450 y FE(j)2649 2388 y Fn(>)2850 2326 y FG(1)p 2755 2367 236 4 v 2755 2450 a(2)p FF(a)2848 2464 y FC(+)2907 2450 y FE(j)p FF(t)p FE(j)3001 2388 y FF(:)404 b FH(\(4.22\))118 2658 y(F)-8 b(or)290 2622 y FC(1)p 290 2637 36 4 v 290 2689 a(2)335 2658 y FF(t)368 2672 y FC(0)433 2658 y Fn(6)25 b FF(t)g Fn(6)g FE(\000)754 2592 y(p)p 829 2592 43 4 v 829 2658 a FF(")q FH(,)30 b(w)m(e)h(ha)m(v)m(e)g FF(t)1301 2625 y FC(2)1361 2658 y FE(\000)20 b FF(s)1495 2625 y FC(2)1559 2658 y Fn(>)25 b FG(3)p FF(t)p FG(\()p FF(t)c FE(\000)f FF(s)p FG(\))30 b FH(for)g(all)e FF(s)d FE(2)g FG([2)p FF(t;)15 b(t)p FG(])p FH(,)31 b(and)g(th)m(us)1057 2927 y FF(\020)7 b FG(\()p FF(t)p FG(\))26 b Fn(>)1338 2866 y FG(1)p 1338 2906 46 4 v 1340 2989 a FF(")1409 2803 y Fr(Z)1500 2830 y Fy(t)1460 3010 y FC(2)p Fy(t)1545 2927 y FG(e)1585 2890 y FD(\000)p FC(3)p Fy(a)1712 2899 y Fw(+)1764 2890 y FD(j)p Fy(t)p FD(j)p FC(\()p Fy(t)p FD(\000)p Fy(s)p FC(\))p Fy(=")2083 2927 y FG(d)p FF(s)f Fn(>)2308 2866 y FG(1)20 b FE(\000)g FG(e)2505 2833 y FD(\000)p FC(3)p Fy(a)2632 2842 y Fw(+)p 2308 2906 380 4 v 2379 2989 a FG(3)p FF(a)2472 3003 y FC(+)2532 2989 y FE(j)p FF(t)p FE(j)2697 2927 y FF(;)708 b FH(\(4.23\))118 3185 y(where)31 b(w)m(e)g(used)g(the)f(relation)g FF(t)1239 3199 y FC(0)1303 3185 y Fn(6)25 b FE(\000)p FG(2)1515 3119 y FE(p)p 1591 3119 43 4 v 66 x FF(")31 b FH(in)e(the)i(last)e (step.)41 b(By)30 b(the)g(same)g(relation,)f(w)m(e)j(obtain)944 3446 y FF(\020)7 b FG(\()p FF(t)p FG(\))26 b Fn(>)1262 3385 y FG(1)p 1226 3425 119 4 v 1226 3444 a FE(p)p 1301 3444 43 4 v 1301 3509 a FF(")1369 3323 y Fr(Z)1460 3349 y FD(\000)p FC(1)1419 3529 y FD(\000)p FC(2)1569 3446 y FG(e)1610 3409 y FD(\000)p Fy(a)1702 3418 y Fw(+)1753 3409 y Fy(u)1794 3385 y Fw(2)1848 3446 y FG(d)o FF(u)483 b FH(for)30 b FE(\000)2642 3376 y(p)p 2718 3376 V 70 x FF(")25 b Fn(6)g FF(t)g Fn(6)3035 3376 y FE(p)p 3111 3376 V 70 x FF(")p FH(,)252 b(\(4.24\))601 3709 y FG(e)641 3672 y FD(\000)p FC(2)p Fy(\013)p FC(\()p Fy(t)p FC(\))p Fy(=")944 3709 y FF(\020)7 b FG(\()p FF(t)p FG(\))26 b Fn(>)1262 3648 y FG(1)p 1226 3688 119 4 v 1226 3706 a FE(p)p 1301 3706 43 4 v 1301 3772 a FF(")1369 3585 y Fr(Z)1460 3612 y FC(1)1419 3792 y(0)1514 3709 y FG(e)1555 3672 y FD(\000)p Fy(a)1647 3681 y Fw(+)1698 3672 y Fy(u)1739 3648 y Fw(2)1793 3709 y FG(d)o FF(u)538 b FH(for)30 b FF(t)25 b Fn(>)2725 3639 y FE(p)p 2801 3639 V 70 x FF(")p FH(.)562 b(\(4.25\))118 3963 y(Finally)-8 b(,)36 b(assume)f(that)h FF(a)1018 3930 y FD(0)1042 3963 y FG(\()p FF(t)p FG(\))f FF(>)g FG(0)h FH(for)g(all)f FF(t)p FH(,)i(and)g(recall)d(that)j FF(\020)7 b FG(\()p FF(t)p FG(\))36 b FH(is)f(the)h(solution)f(of)h (the)g(initial)118 4076 y(v)-5 b(alue)30 b(problem)1129 4170 y FG(d)o FF(\020)p 1129 4211 97 4 v 1136 4294 a FG(d)o FF(t)1261 4232 y FG(=)1367 4170 y(2)p FF(a)p FG(\()p FF(t)p FG(\))p 1367 4211 198 4 v 1444 4294 a FF(")1574 4232 y(\020)c FG(+)1741 4170 y(1)p 1741 4211 46 4 v 1743 4294 a FF(")1797 4232 y(;)196 b(\020)7 b FG(\()p FF(t)2133 4246 y FC(0)2172 4232 y FG(\))26 b(=)2459 4170 y(1)p 2339 4211 287 4 v 2339 4294 a(2)p FE(j)p FF(a)p FG(\()p FF(t)2525 4308 y FC(0)2565 4294 y FG(\))p FE(j)2636 4232 y FF(:)769 b FH(\(4.26\))118 4449 y(Since)28 b FF(\020)7 b FG(\()p FF(t)p FG(\))25 b Fn(>)g FG(0)p FH(,)j FF(\020)768 4416 y FD(0)816 4449 y FF(>)d FG(0)j FH(for)g(all)e(p)s(ositiv)m(e)g FF(t)p FH(.)39 b(F)-8 b(or)29 b(negativ)m(e)f FF(t)p FH(,)g FF(\020)2320 4416 y FD(0)2370 4449 y FH(is)e(p)s(ositiv)m(e)g (whenev)m(er)j(the)f(function)118 4562 y FF(V)20 b FG(\()p FF(t)p FG(\))26 b(=)f FF(\020)7 b FG(\()p FF(t)p FG(\))20 b(+)g(1)p FF(=)p FG(2)p FF(a)p FG(\()p FF(t)p FG(\))32 b FH(is)d(negativ)m(e.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)g FF(V)20 b FG(\()p FF(t)1983 4576 y FC(0)2023 4562 y FG(\))25 b(=)g(0)31 b FH(and)1426 4754 y FG(d)o FF(V)p 1426 4795 124 4 v 1446 4878 a FG(d)p FF(t)1585 4816 y FG(=)1691 4754 y(2)p FF(a)p FG(\()p FF(t)p FG(\))p 1691 4795 198 4 v 1768 4878 a FF(")1898 4816 y(V)40 b FE(\000)2123 4754 y FF(a)2171 4721 y FD(0)2194 4754 y FG(\()p FF(t)p FG(\))p 2092 4795 237 4 v 2092 4878 a(2)p FF(a)p FG(\()p FF(t)p FG(\))2288 4852 y FC(2)2338 4816 y FF(:)1067 b FH(\(4.27\))118 5070 y(Since)30 b FF(V)427 5037 y FD(0)476 5070 y FF(<)25 b FG(0)30 b FH(whenev)m(er)i FF(V)46 b FG(=)25 b(0)p FH(,)30 b FF(V)51 b FH(can)30 b(nev)m(er)i(b)s(ecome)d(p) s(ositiv)m(e.)39 b(This)30 b(implies)c FF(\020)3103 5037 y FD(0)3151 5070 y Fn(>)f FG(0)p FH(.)p 3596 5070 4 62 v 3600 5012 55 4 v 3600 5070 V 3653 5070 4 62 v 259 5258 a(The)33 b(follo)m(wing)d(prop)s(osition)h(sho)m(ws)h(that)g(the)h (solution)d FF(x)2318 5225 y FC(0)2318 5280 y Fy(t)2389 5258 y FH(of)i(the)g(linearized)f(equation)g(\(4.15\))118 5371 y(is)e(lik)m(ely)f(to)i(trac)m(k)h(the)g(solution)e(of)h(the)h (corresp)s(onding)f(deterministic)e(equation.)1845 5871 y(27)p eop %%Page: 28 28 28 27 bop 118 328 a Fq(Prop)s(osition)35 b(4.3.)41 b FB(Assume)33 b(that)g FE(\000)p FF(T)k Fn(6)25 b FF(t)1694 342 y FC(0)1759 328 y FF(<)g(t)g Fn(6)2009 262 y FE(p)p 2084 262 43 4 v 2084 328 a FF(")q FB(.)41 b(F)-7 b(or)32 b(su\036ciently)h(smal)5 b(l)31 b FF(")p FB(,)384 593 y Fo(P)445 556 y Fy(t)470 565 y Fw(0)505 556 y Fy(;x)565 565 y Fw(0)603 492 y Fr(n)709 593 y FG(sup)664 670 y Fy(t)689 679 y Fw(0)724 670 y Fx(6)p Fy(s)p Fx(6)p Fy(t)917 532 y FE(j)p FF(x)994 499 y FC(0)994 554 y Fy(s)1054 532 y FE(\000)20 b FF(x)1197 546 y FC(0)1251 532 y FG(e)1292 499 y Fy(\013)p FC(\()p Fy(s;t)1442 508 y Fw(0)1477 499 y FC(\))p Fy(=")1576 532 y FE(j)p 917 572 685 4 v 1134 591 a Fr(p)p 1225 591 160 4 v 77 x FF(\020)7 b FG(\()p FF(s)p FG(\))1637 593 y FF(>)25 b(h)1785 492 y Fr(o)1871 593 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)2379 492 y Fr(n)2439 593 y FE(\000)2520 532 y FG(1)p 2520 572 46 4 v 2520 656 a(2)2587 532 y FF(h)2639 499 y FC(2)p 2585 572 95 4 v 2585 656 a FF(\033)2640 629 y FC(2)2690 520 y Fr(\002)2728 593 y FG(1)20 b FE(\000)g FF(r)s FG(\()p FF(")p FG(\))3040 520 y Fr(\003)3079 492 y(o)3140 593 y FF(;)265 b FH(\(4.28\))118 858 y FB(wher)-5 b(e)1193 1001 y FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))26 b(=)1582 939 y FE(j)p FF(\013)p FG(\()p FF(t;)15 b(t)1806 953 y FC(0)1847 939 y FG(\))p FE(j)p 1582 980 326 4 v 1704 1063 a FF(")1746 1037 y FC(2)1937 1001 y FG(+)2038 939 y FF(a)2086 953 y FC(+)2165 939 y FG(+)20 b(4)2301 874 y FE(p)p 2378 874 43 4 v 2378 939 a FF(")g FG(+)g(4)p 2038 980 539 4 v 2286 1063 a FF(")3430 1001 y FH(\(4.29\))118 1198 y FB(and)32 b(wher)-5 b(e)34 b FF(r)s FG(\()p FF(")p FG(\))26 b(=)f FE(O)s FG(\()p FF(")p FG(\))33 b FB(for)f FF(t)1223 1212 y FC(0)1288 1198 y Fn(6)25 b FF(t)g Fn(6)g FE(\000)1609 1133 y(p)p 1684 1133 43 4 v 1684 1198 a FF(")q FB(,)32 b(and)g FF(r)s FG(\()p FF(")p FG(\))26 b(=)f FE(O)s FG(\()2350 1133 y FE(p)p 2426 1133 V 65 x FF(")p FG(\))33 b FB(for)f FE(\000)2751 1133 y(p)p 2826 1133 V 2826 1198 a FF(")26 b Fn(6)f FF(t)g Fn(6)3144 1133 y FE(p)p 3220 1133 V 65 x FF(")p FB(.)118 1386 y Fh(Pr)m(oof:)47 b FH(Let)33 b FF(t)675 1400 y FC(0)743 1386 y FG(=)28 b FF(u)894 1400 y FC(0)963 1386 y FF(<)g FE(\001)15 b(\001)g(\001)30 b FF(<)f(u)1349 1400 y Fy(K)1446 1386 y FG(=)g FF(t)j FH(b)s(e)g(a)h(partition)f(of)g(the)h(in)m(terv)-5 b(al)31 b FG([)p FF(t)2842 1400 y FC(0)2882 1386 y FF(;)15 b(t)p FG(])p FH(.)47 b(By)32 b(Lemma)g(3.2,)118 1499 y(the)f(probabilit)m(y)e(in)g(\(4.28\))j(is)c(b)s(ounded)k(b)m(y)e FG(2)1748 1431 y Fr(P)1845 1457 y Fy(K)1845 1526 y(k)r FC(=1)1993 1499 y FF(P)2051 1514 y Fy(k)2094 1499 y FH(,)g(where)900 1763 y FF(P)958 1778 y Fy(k)1027 1763 y FG(=)24 b(exp)1261 1662 y Fr(n)1322 1763 y FE(\000)1403 1701 y FG(1)p 1403 1742 46 4 v 1403 1825 a(2)1469 1701 y FF(h)1521 1668 y FC(2)p 1468 1742 95 4 v 1468 1825 a FF(\033)1523 1799 y FC(2)1666 1701 y FG(1)p 1583 1742 212 4 v 1583 1825 a FF(\020)7 b FG(\()p FF(u)1717 1840 y Fy(k)1759 1825 y FG(\))1958 1763 y(inf)1820 1822 y Fy(u)1861 1834 y Fj(k)q Fk(\000)p Fw(1)1977 1822 y Fx(6)p Fy(u)p Fx(6)p Fy(u)2169 1834 y Fj(k)2222 1763 y FF(\020)g FG(\()p FF(u)p FG(\))15 b(e)2446 1725 y FC(2)p Fy(\013)p FC(\()p Fy(u)2594 1737 y Fj(k)2634 1725 y Fy(;u)p FC(\))p Fy(=")2793 1662 y Fr(o)2854 1763 y FF(:)551 b FH(\(4.30\))118 2022 y(If)30 b FF(t)25 b Fn(6)g FE(\000)434 1956 y(p)p 509 1956 43 4 v 509 2022 a FF(")p FH(,)31 b(w)m(e)g(de\034ne)g(the)g(partition)e(b) m(y)632 2275 y FF(K)j FG(=)836 2147 y Fr(\030)899 2214 y FE(\000)p FF(\013)p FG(\()p FF(t;)15 b(t)1169 2228 y FC(0)1210 2214 y FG(\))p 899 2254 346 4 v 1009 2337 a(2)p FF(")1096 2311 y FC(2)1255 2147 y Fr(\031)1308 2275 y FF(;)196 b FE(\000)p FF(\013)p FG(\()p FF(u)1745 2290 y Fy(k)1788 2275 y FF(;)15 b(t)1861 2289 y FC(0)1901 2275 y FG(\))26 b(=)f(2)p FF(")2145 2238 y FC(2)2185 2275 y FF(k)93 b FH(for)31 b FF(k)d FG(=)d(0)p FF(;)15 b(:)g(:)g(:)i(;)e(K)27 b FE(\000)20 b FG(1)q FF(:)282 b FH(\(4.31\))118 2524 y(Estimating)29 b FF(P)640 2539 y Fy(k)713 2524 y FH(as)h(in)g(the)g(pro)s(of)g(of)g(Prop)s(osition)g (3.3,)g(w)m(e)i(obtain)1174 2778 y FF(P)1232 2793 y Fy(k)1300 2778 y Fn(6)25 b FG(exp)1535 2677 y Fr(n)1596 2778 y FE(\000)1677 2717 y FG(1)p 1677 2757 46 4 v 1677 2841 a(2)1743 2717 y FF(h)1795 2684 y FC(2)p 1742 2757 95 4 v 1742 2841 a FF(\033)1797 2814 y FC(2)1847 2677 y Fr(\020)1901 2778 y FG(1)20 b FE(\000)2126 2717 y FG(2)p FF(")p 2067 2757 206 4 v 2067 2841 a(a)2115 2855 y FD(\000)2174 2841 y FF(c)2213 2855 y FD(\000)2283 2677 y Fr(\021)2352 2778 y FG(e)2393 2741 y FD(\000)p FC(4)p Fy(")2520 2677 y Fr(o)2580 2778 y FF(:)825 b FH(\(4.32\))118 3037 y(Therefore,)32 b(\(4.28\))f(holds)e(with)h FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))26 b(=)f FE(j)p FF(\013)p FG(\()p FF(t;)15 b(t)1859 3051 y FC(0)1900 3037 y FG(\))p FE(j)p FF(=")2047 3004 y FC(2)2108 3037 y FG(+)20 b(2)p FH(.)259 3150 y(F)-8 b(or)31 b FE(\000)492 3084 y(p)p 568 3084 43 4 v 66 x FF(")25 b Fn(6)g FF(t)g Fn(6)885 3084 y FE(p)p 961 3084 V 66 x FF(")p FH(,)31 b(w)m(e)g(de\034ne)g(the)g(partition)e (separately)h(in)g(t)m(w)m(o)i(di\033eren)m(t)e(regions.)40 b(Let)924 3405 y FF(K)1001 3419 y FC(0)1065 3405 y FG(=)1161 3277 y Fr(\030)1224 3344 y FE(\000)p FF(\013)p FG(\()p FE(\000)1459 3278 y(p)p 1535 3278 V 66 x FF(")q(;)15 b(t)1651 3358 y FC(0)1690 3344 y FG(\))p 1224 3384 502 4 v 1411 3468 a(2)p FF(")1498 3441 y FC(2)1736 3277 y Fr(\031)1789 3405 y FF(;)196 b(K)32 b FG(=)25 b FF(K)2292 3419 y FC(0)2352 3405 y FG(+)2443 3277 y Fr(\030)2506 3344 y FF(t)20 b FG(+)2650 3278 y FE(p)p 2725 3278 43 4 v 2725 3344 a FF(")p 2506 3384 263 4 v 2616 3468 a(")2778 3277 y Fr(\031)2831 3405 y FF(:)574 b FH(\(4.33\))118 3654 y(The)31 b(partition)f(times)e(are)j(de\034ned)g(via)622 3858 y FE(\000)p FF(\013)p FG(\()p FF(u)838 3873 y Fy(k)881 3858 y FF(;)15 b(t)954 3872 y FC(0)994 3858 y FG(\))25 b(=)g(2)p FF(")1237 3821 y FC(2)1278 3858 y FF(k)1020 b FH(for)30 b FG(0)c Fn(6)f FF(k)j Fn(6)d FF(K)2898 3872 y FC(0)2958 3858 y FE(\000)20 b FG(1)934 3996 y FF(u)986 4011 y Fy(k)1054 3996 y FG(=)25 b FE(\000)1221 3926 y(p)p 1297 3926 43 4 v 70 x FF(")20 b FG(+)g FF(")p FG(\()p FF(k)k FE(\000)c FF(K)1766 4010 y FC(0)1806 3996 y FG(\))504 b FH(for)30 b FF(K)2560 4010 y FC(0)2625 3996 y Fn(6)25 b FF(k)j Fn(6)d FF(K)i FE(\000)20 b FG(1)q FF(:)272 b FH(\(4.34\))118 4200 y(In)31 b(the)i(\034rst)e(case,)h(w)m(e)h (immediately)28 b(obtain)k(the)g(b)s(ound)g(\(4.32\))q(.)44 b(In)32 b(the)g(second)g(case,)g(estimating)118 4313 y FF(P)176 4328 y Fy(k)249 4313 y FH(in)e(the)g(usual)g(w)m(a)m(y)h (sho)m(ws)g(that)962 4568 y FF(P)1020 4583 y Fy(k)1089 4568 y Fn(6)25 b FG(exp)1323 4467 y Fr(n)1384 4568 y FE(\000)1465 4506 y FG(1)p 1465 4547 46 4 v 1465 4630 a(2)1532 4506 y FF(h)1584 4473 y FC(2)p 1530 4547 95 4 v 1530 4630 a FF(\033)1585 4604 y FC(2)1635 4467 y Fr(\020)1689 4568 y FG(1)c FE(\000)1856 4441 y(p)p 1931 4441 43 4 v 1931 4506 a FF(")p 1856 4547 119 4 v 1866 4630 a(c)1905 4644 y FD(\000)1984 4568 y FG([1)g(+)e(2)p FF(a)2258 4582 y FC(+)2318 4568 y FF(c)2357 4582 y FC(+)2416 4568 y FG(])2441 4467 y Fr(\021)2511 4568 y FG(e)2551 4530 y FD(\000)p Fy(a)2643 4539 y Fw(+)2695 4530 y Fy(")2731 4467 y Fr(o)2792 4568 y FF(:)613 b FH(\(4.35\))118 4823 y(Finally)-8 b(,)28 b(let)i(us)g(note)h(that,)g(for)f FE(\000)1321 4757 y(p)p 1396 4757 43 4 v 1396 4823 a FF(")c Fn(6)f FF(t)g Fn(6)1714 4757 y FE(p)p 1790 4757 V 66 x FF(")p FH(,)613 5079 y FG(2)p FF(K)33 b Fn(6)874 5017 y FE(j)p FF(\013)p FG(\()p FE(\000)1063 4952 y(p)p 1139 4952 V 65 x FF(")q(;)15 b(t)1255 5031 y FC(0)1294 5017 y FG(\))p FE(j)p 874 5058 482 4 v 1073 5141 a FF(")1115 5115 y FC(2)1385 5079 y FG(+)1486 5017 y(2)p 1486 5058 46 4 v 1488 5141 a FF(")1541 5079 y FG(\()p FF(t)21 b FG(+)1721 5009 y FE(p)p 1796 5009 43 4 v 1796 5079 a FF(")q FG(\))f(+)g(4)26 b Fn(6)2162 5017 y FE(j)p FF(\013)p FG(\()p FF(t;)15 b(t)2386 5031 y FC(0)2426 5017 y FG(\))p FE(j)p 2162 5058 326 4 v 2284 5141 a FF(")2326 5115 y FC(2)2517 5079 y FG(+)2618 5017 y FF(a)2666 5031 y FC(+)p 2618 5058 108 4 v 2650 5141 a FF(")2755 5079 y FG(+)2893 5017 y(4)p 2856 5058 119 4 v 2856 5076 a FE(p)p 2932 5076 43 4 v 65 x FF(")3005 5079 y FG(+)k(4)p FF(;)265 b FH(\(4.36\))118 5325 y(whic)m(h)31 b(concludes)f(the)h(pro)s(of)f(of) g(the)g(prop)s(osition.)p 3596 5325 4 62 v 3600 5268 55 4 v 3600 5325 V 3653 5325 4 62 v 1845 5871 a(28)p eop %%Page: 29 29 29 28 bop 259 328 a FH(Let)31 b(us)f(no)m(w)i(compare)e(solutions)f(of) h(the)g(t)m(w)m(o)i(SDEs)802 536 y FG(d)p FF(x)905 498 y FC(0)905 558 y Fy(t)969 536 y FG(=)1075 474 y(1)p 1075 515 46 4 v 1077 598 a FF(")1131 536 y(a)p FG(\()p FF(t)p FG(\))p FF(x)1334 498 y FC(0)1334 558 y Fy(t)1389 536 y FG(d)o FF(t)20 b FG(+)1625 474 y FF(\033)p 1593 515 119 4 v 1593 533 a FE(p)p 1669 533 43 4 v 66 x FF(")1736 536 y FG(d)p FF(W)1873 550 y Fy(t)2623 536 y FF(x)2675 498 y FC(0)2675 558 y Fy(t)2700 567 y Fw(0)2765 536 y FG(=)25 b FF(x)2913 550 y FC(0)3430 536 y FH(\(4.37\))812 774 y FG(d)o FF(x)914 788 y Fy(t)969 774 y FG(=)1075 712 y(1)p 1075 753 46 4 v 1077 836 a FF(")1131 774 y(f)10 b FG(\()p FF(x)1273 788 y Fy(t)1302 774 y FF(;)15 b(t)p FG(\))g(d)p FF(t)20 b FG(+)1661 712 y FF(\033)p 1630 753 119 4 v 1630 771 a FE(p)p 1706 771 43 4 v 65 x FF(")1773 774 y FG(d)p FF(W)1910 788 y Fy(t)2623 774 y FF(x)2675 788 y Fy(t)2700 797 y Fw(0)2765 774 y FG(=)25 b FF(x)2913 788 y FC(0)2952 774 y FF(;)453 b FH(\(4.38\))118 997 y(where)31 b FF(t)412 1011 y FC(0)477 997 y FE(2)25 b FG([)p FE(\000)p FF(T)8 b(;)15 b FG(0\))p FH(.)41 b(W)-8 b(e)30 b(de\034ne)i(the)e(ev)m(en)m(ts)743 1195 y FG(\012)809 1157 y FC(0)809 1217 y Fy(t)848 1195 y FG(\()p FF(h)p FG(\))c(=)1092 1094 y Fr(n)1153 1195 y FF(!)13 b FG(:)1278 1117 y Fr(\014)1278 1172 y(\014)1309 1195 y FF(x)1361 1157 y FC(0)1361 1217 y Fy(s)1400 1195 y FG(\()p FF(!)s FG(\))21 b FE(\000)f FF(x)1694 1209 y FC(0)1748 1195 y FG(e)1789 1157 y Fy(\013)p FC(\()p Fy(s;t)1939 1166 y Fw(0)1974 1157 y FC(\))p Fy(=")2073 1117 y Fr(\014)2073 1172 y(\014)2129 1195 y Fn(6)25 b FF(h)2277 1113 y Fr(p)p 2368 1113 160 4 v 82 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b FE(8)p FF(s)g FE(2)f FG([)p FF(t)2815 1209 y FC(0)2855 1195 y FF(;)15 b(t)p FG(])2953 1094 y Fr(o)3430 1195 y FH(\(4.39\))753 1392 y FG(\012)819 1406 y Fy(t)848 1392 y FG(\()p FF(h)p FG(\))26 b(=)1092 1291 y Fr(n)1153 1392 y FF(!)13 b FG(:)1278 1314 y Fr(\014)1278 1369 y(\014)1309 1392 y FF(x)1361 1406 y Fy(s)1397 1392 y FG(\()p FF(!)s FG(\))21 b FE(\000)f FF(x)1691 1406 y FC(0)1746 1392 y FG(e)1786 1354 y Fy(\013)p FC(\()p Fy(s;t)1936 1363 y Fw(0)1971 1354 y FC(\))p Fy(=")2071 1314 y Fr(\014)2071 1369 y(\014)2126 1392 y Fn(6)25 b FF(h)2274 1309 y Fr(p)p 2365 1309 V 83 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b FE(8)p FF(s)g FE(2)g FG([)p FF(t)2813 1406 y FC(0)2852 1392 y FF(;)15 b(t)p FG(])2950 1291 y Fr(o)3011 1392 y FF(:)394 b FH(\(4.40\))118 1597 y(Prop)s(osition)33 b(4.3)h(giv)m(es)f(us)g(an)h(upp)s(er)g(b)s(ound)h(on)e(the)i (probabilit)m(y)d(of)h(the)i(complemen)m(t)e(of)g FG(\012)3474 1564 y FC(0)3474 1619 y Fy(t)3513 1597 y FG(\()p FF(h)p FG(\))p FH(.)118 1710 y(W)-8 b(e)31 b(no)m(w)g(giv)m(e)f(relations)f(b) s(et)m(w)m(een)j(these)f(ev)m(en)m(ts.)118 1873 y Fq(Prop)s(osition)47 b(4.4.)g FB(L)-5 b(et)42 b FF(t)h FE(2)f FG([)p FF(t)1302 1887 y FC(0)1342 1873 y FF(;)1382 1808 y FE(p)p 1458 1808 43 4 v 65 x FF(")15 b FG(])42 b FB(and)g FE(j)p FF(x)1844 1887 y FC(0)1884 1873 y FE(j)h Fn(6)f FF(h=")2204 1840 y FC(1)p Fy(=)p FC(4)2315 1873 y FB(,)i(wher)-5 b(e)43 b(we)g(assume)f FF(h)3172 1840 y FC(2)3254 1873 y FF(<)h("=\015)48 b FB(for)118 1986 y FF(\015)30 b FG(=)25 b FF(M)10 b FG(\(1)22 b(+)d(2)626 1931 y FE(p)p 703 1931 99 4 v 703 1986 a FF(c)742 2000 y FC(+)801 1986 y FG(\))836 1953 y FC(3)876 1986 y FF(c)915 2000 y FC(+)974 1986 y FF(=)1019 1931 y FE(p)p 1095 1931 V 55 x FF(c)1134 2000 y FD(\000)1226 1986 y FB(and)32 b FF(h)1453 1953 y FC(2)1518 1986 y Fn(6)25 b FF(d)1661 1953 y FC(2)1701 1921 y FE(p)p 1777 1921 43 4 v 65 x FF("=)p FG(\(1)d(+)e(2)2102 1931 y FE(p)p 2178 1931 99 4 v 55 x FF(c)2217 2000 y FC(+)2276 1986 y FG(\))2311 1953 y FC(2)2351 1986 y FB(.)41 b(Then)1365 2226 y FG(\012)1431 2240 y Fy(t)1460 2226 y FG(\()p FF(h)p FG(\))1609 2159 y FC(a)p Fy(:)p FC(s)p Fy(:)1625 2226 y FE(\032)f FG(\012)1802 2188 y FC(0)1802 2248 y Fy(t)1841 2125 y Fr(\020)q(h)1939 2226 y FG(1)20 b(+)g FF(\015)2157 2164 y(h)2209 2131 y FC(2)p 2157 2205 92 4 v 2182 2288 a FF(")2259 2125 y Fr(i)2302 2226 y FF(h)2354 2125 y Fr(\021)3430 2226 y FH(\(4.41\))1355 2459 y FG(\012)1421 2421 y FC(0)1421 2481 y Fy(t)1460 2459 y FG(\()p FF(h)p FG(\))1609 2392 y FC(a)p Fy(:)p FC(s)p Fy(:)1625 2459 y FE(\032)40 b FG(\012)1802 2473 y Fy(t)1832 2358 y Fr(\020h)1929 2459 y FG(1)21 b(+)e FF(\015)2147 2397 y(h)2199 2364 y FC(2)p 2148 2438 V 2172 2521 a FF(")2249 2358 y Fr(i)2292 2459 y FF(h)2344 2358 y Fr(\021)2399 2459 y FF(:)1006 b FH(\(4.42\))118 2667 y Fh(Pr)m(oof:)47 b FH(Assume)29 b(\034rst)h(that)h FF(!)d FE(2)d FG(\012)1432 2634 y FC(0)1432 2690 y Fy(t)1471 2667 y FG(\()p FF(h)p FG(\))32 b FH(and)f(let)e FF(\016)h FG(=)25 b FF(\015)5 b(h)2201 2634 y FC(2)2241 2667 y FF(=")p FH(.)41 b(Then)31 b(w)m(e)h(ha)m(v)m(e)f FF(\016)e(<)d FG(1)k FH(b)m(y)h(assump-)118 2780 y(tion.)40 b(By)31 b(\(4.3\))q(,)f(the)h(di\033erence)f FF(z)1325 2794 y Fy(s)1388 2780 y FG(=)25 b FF(x)1536 2794 y Fy(s)1593 2780 y FE(\000)19 b FF(x)1735 2747 y FC(0)1735 2803 y Fy(s)1805 2780 y FH(satis\034es)1296 3000 y FF(z)1338 3014 y Fy(s)1401 3000 y FG(=)1507 2938 y(1)p 1507 2979 46 4 v 1509 3062 a FF(")1577 2876 y Fr(Z)1668 2902 y Fy(s)1628 3082 y(t)1653 3091 y Fw(0)1720 3000 y FG(e)1761 2962 y Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2041 3000 y FF(b)p FG(\()p FF(x)2167 3014 y Fy(u)2212 3000 y FF(;)c(u)p FG(\))g(d)q FF(u:)947 b FH(\(4.43\))118 3224 y(W)-8 b(e)31 b(consider)f(the)h(\034rst)f(exit)f(time)882 3399 y FF(\034)35 b FG(=)25 b(inf)1164 3326 y Fr(\010)1217 3399 y FF(s)f FE(2)h FG([)p FF(t)1428 3413 y FC(0)1468 3399 y FF(;)15 b(t)p FG(])10 b(:)31 b FE(j)p FF(z)1699 3413 y Fy(s)1736 3399 y FE(j)26 b Fn(>)f FF(\016)s(h)1978 3317 y Fr(p)p 2070 3317 160 4 v 2070 3399 a FF(\020)7 b FG(\()p FF(s)p FG(\))2230 3326 y Fr(\011)2308 3399 y FE(2)25 b FG([)p FF(t)2452 3413 y FC(0)2491 3399 y FF(;)15 b(t)p FG(])21 b FE([)f(f1g)p FF(:)533 b FH(\(4.44\))118 3574 y(F)-8 b(or)31 b(all)e FF(!)k FH(in)c(the)i(set)1306 3687 y FF(A)25 b FG(=)g(\012)1561 3650 y FC(0)1561 3710 y Fy(t)1600 3687 y FG(\()p FF(h)p FG(\))c FE(\\)1824 3614 y Fr(\010)1877 3687 y FF(!)13 b FG(:)31 b FF(\034)10 b FG(\()p FF(!)s FG(\))26 b FF(<)f FE(1)2396 3614 y Fr(\011)2449 3687 y FF(;)956 b FH(\(4.45\))118 3840 y(and)31 b FF(s)25 b FE(2)g FG([)p FF(t)506 3854 y FC(0)545 3840 y FF(;)15 b(\034)10 b FG(\()p FF(!)s FG(\)])p FH(,)32 b(w)m(e)f(ha)m(v)m(e)g(b)m (y)g(the)f(h)m(yp)s(otheses)h(on)f FF(h)h FH(and)g FF(x)2366 3854 y FC(0)2435 3840 y FH(together)h(with)e(Lemma)f(4.2)830 4058 y FE(j)p FF(x)907 4072 y Fy(s)944 4058 y FG(\()p FF(!)s FG(\))p FE(j)d Fn(6)f FE(j)p FF(x)1298 4072 y FC(0)1338 4058 y FE(j)20 b FG(+)g FF(h)1526 3975 y Fr(p)p 1617 3975 V 83 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b Fn(6)1898 3984 y Fr(\000)1940 4058 y FG(1)c(+)f(\(1)h(+)e FF(\016)s FG(\))2366 3998 y FE(p)p 2443 3998 99 4 v 2443 4058 a FF(c)2482 4072 y FC(+)2542 3984 y Fr(\001)2643 3996 y FF(h)p 2593 4037 153 4 v 2593 4124 a(")2635 4097 y FC(1)p Fy(=)p FC(4)2781 4058 y Fn(6)25 b FF(d:)481 b FH(\(4.46\))118 4262 y(Therefore,)32 b(\(4.4\))f(yields)875 4482 y FE(j)p FF(z)942 4496 y Fy(s)980 4482 y FE(j)25 b Fn(6)g FF(M)1224 4381 y Fr(h)1267 4408 y(\000)1309 4482 y FG(1)20 b(+)g(\(1)h(+)f FF(\016)s FG(\))1735 4422 y FE(p)p 1812 4422 99 4 v 1812 4482 a FF(c)1851 4496 y FC(+)1911 4408 y Fr(\001)2012 4420 y FF(h)p 1962 4461 153 4 v 1962 4548 a(")2004 4521 y FC(1)p Fy(=)p FC(4)2125 4381 y Fr(i)2168 4403 y FC(3)2242 4420 y FG(1)p 2242 4461 46 4 v 2244 4544 a FF(")2313 4358 y Fr(Z)2404 4384 y Fy(s)2363 4564 y(t)2388 4573 y Fw(0)2456 4482 y FG(e)2496 4444 y Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2776 4482 y FG(d)p FF(u:)526 b FH(\(4.47\))118 4712 y(The)39 b(in)m(tegral)g(is)e(b)s(ounded)j(b)m(y)f FG(2)p FF(\020)1346 4726 y FC(2)p Fy(")1418 4712 y FG(\()p FF(s)p FG(\))p FH(,)j(whic)m(h)d(can)g(b)s(e)g(estimated)f(b)m(y)h(Lemma)f(4.2)h(once) g(again.)118 4825 y(Thereb)m(y)-8 b(,)32 b(w)m(e)f(obtain)802 5050 y FE(j)p FF(z)869 5064 y Fy(s)906 5050 y FE(j)26 b Fn(6)f FF(M)1151 4977 y Fr(\000)1192 5050 y FG(1)c(+)f(\(1)h(+)f FF(\016)s FG(\))1619 4991 y FE(p)p 1696 4991 99 4 v 1696 5050 a FF(c)1735 5064 y FC(+)1794 4977 y Fr(\001)1836 4999 y FC(3)1923 4989 y FF(c)1962 5003 y FC(+)p 1885 5029 175 4 v 1885 5057 a FE(p)p 1961 5057 99 4 v 56 x FF(c)2000 5127 y FD(\000)2079 4989 y FF(h)2131 4956 y FC(2)p 2079 5029 92 4 v 2104 5113 a FF(")2181 5050 y(h)2233 4968 y Fr(p)p 2324 4968 160 4 v 82 x FF(\020)7 b FG(\()p FF(s)p FG(\))25 b FF(<)g(\016)s(h)2700 4968 y Fr(p)p 2793 4968 V 2793 5050 a FF(\020)7 b FG(\()p FF(s)p FG(\))o FF(;)453 b FH(\(4.48\))118 5283 y(whic)m(h)45 b(leads)e(to)h(a)h(con)m (tradiction)f(for)g FF(s)k FG(=)h FF(\034)10 b FG(\()p FF(!)s FG(\))p FH(.)82 b(W)-8 b(e)45 b(conclude)f(that)h Fo(P)p FG(\()p FF(A)p FG(\))50 b(=)e(0)p FH(,)g(and)d(th)m(us)118 5396 y FF(\034)10 b FG(\()p FF(!)s FG(\))45 b(=)f FE(1)d FH(for)h Fo(P)p FH(-almost)f(all)e FF(!)48 b FE(2)43 b FG(\012)1540 5363 y FC(0)1540 5419 y Fy(t)1580 5396 y FG(\()p FF(h)p FG(\))p FH(.)75 b(This)41 b(sho)m(ws)h(that)g FE(j)p FF(z)2565 5410 y Fy(s)2602 5396 y FG(\()p FF(!)s FG(\))p FE(j)j FF(<)f(\016)s(h)3012 5319 y Fr(p)p 3104 5319 V 3104 5396 a FF(\020)7 b FG(\()p FF(s)p FG(\))42 b FH(and)g(th)m(us)118 5509 y FE(j)p FF(x)195 5523 y Fy(s)232 5509 y FG(\()p FF(!)s FG(\))11 b FE(\000)g FF(x)507 5523 y FC(0)562 5509 y FG(e)603 5476 y Fy(\013)p FC(\()p Fy(s;t)753 5485 y Fw(0)788 5476 y FC(\))p Fy(=")887 5509 y FE(j)26 b FF(<)f FG(\(1)11 b(+)g FF(\016)s FG(\))p FF(h)1337 5432 y Fr(p)p 1430 5432 V 1430 5509 a FF(\020)c FG(\()p FF(s)p FG(\))25 b FH(for)h(all)e(these)i FF(!)j FH(and)d(all)e FF(s)h FE(2)g FG([)p FF(t)2687 5523 y FC(0)2726 5509 y FF(;)15 b(t)p FG(])p FH(,)27 b(whic)m(h)g(pro)m(v)m (es)f(\(4.42\))q(.)118 5622 y(The)31 b(pro)s(of)f(of)g(the)h(inclusion) d(\(4.41\))j(is)e(straigh)m(tforw)m(ard,)j(using)d(the)i(same)f (estimates.)p 3596 5622 4 62 v 3600 5564 55 4 v 3600 5622 V 3653 5622 4 62 v 1845 5871 a(29)p eop %%Page: 30 30 30 29 bop 259 328 a FH(The)36 b(t)m(w)m(o)h(preceding)f(prop)s (ositions)e(immediately)e(imply)g(the)k(main)e(result)h(on)g(the)h(b)s (eha)m(viour)118 441 y(of)g(the)h(solution)d(of)i(the)h(nonlinear)f (equation)g(\(4.38\))h(for)f FF(t)f Fn(6)2369 375 y FE(p)p 2445 375 43 4 v 66 x FF(")q FH(,)i(i.e.,)g(Theorem)g(2.8,)h(whic)m(h)e (w)m(e)118 553 y(restate)31 b(here)g(with)f(an)g(arbitrary)h(initial)c (time)i FF(t)1827 567 y FC(0)1891 553 y FE(2)c FG([)p FE(\000)p FF(T)8 b(;)2174 488 y FE(p)p 2250 488 V 65 x FF(")15 b FG(])p FH(.)118 741 y Fq(Corollary)41 b(4.5.)i FB(Assume)35 b(that)h FE(\000)p FF(T)43 b Fn(6)30 b FF(t)1618 755 y FC(0)1688 741 y FF(<)g(t)g Fn(6)1954 676 y FE(p)p 2029 676 V 2029 741 a FF(")q FB(.)50 b(Then)35 b(ther)-5 b(e)36 b(exists)g(an)f FF(h)3049 755 y FC(0)3119 741 y FF(>)c FG(0)k FB(such)h(that)118 854 y(for)h(al)5 b(l)38 b FF(h)d Fn(6)f FF(h)646 868 y FC(0)686 789 y FE(p)p 762 789 V 65 x FF(")k FB(and)f(al)5 b(l)37 b(initial)f(c)-5 b(onditions)38 b FF(x)1915 868 y FC(0)1992 854 y FB(with)f FE(j)p FF(x)2270 868 y FC(0)2309 854 y FE(j)e Fn(6)g FF(h=")2614 821 y FC(1)p Fy(=)p FC(4)2725 854 y FB(,)j(the)g(fol)5 b(lowing)37 b(estimate)118 967 y(holds:)167 1211 y Fo(P)228 1173 y Fy(t)253 1182 y Fw(0)288 1173 y Fy(;x)348 1182 y Fw(0)386 1110 y Fr(n)493 1211 y FG(sup)447 1288 y Fy(t)472 1297 y Fw(0)507 1288 y Fx(6)p Fy(s)p Fx(6)p Fy(t)700 1149 y FE(j)p FF(x)777 1163 y Fy(s)834 1149 y FE(\000)20 b FF(x)977 1163 y FC(0)1032 1149 y FG(e)1072 1116 y Fy(\013)p FC(\()p Fy(s;t)1222 1125 y Fw(0)1257 1116 y FC(\))p Fy(=")1357 1149 y FE(j)p 700 1190 682 4 v 916 1208 a Fr(p)p 1007 1208 160 4 v 78 x FF(\020)7 b FG(\()p FF(s)p FG(\))1417 1211 y FF(>)25 b(h)1565 1110 y Fr(o)1651 1211 y Fn(6)g FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))g(exp)2159 1110 y Fr(n)2220 1211 y FE(\000)2301 1149 y FG(1)p 2301 1190 46 4 v 2301 1273 a(2)2367 1149 y FF(h)2419 1116 y FC(2)p 2366 1190 95 4 v 2366 1273 a FF(\033)2421 1247 y FC(2)2470 1137 y Fr(\002)2508 1211 y FG(1)21 b FE(\000)f FF(r)s FG(\()p FF(")p FG(\))h FE(\000)f(O)s FG(\()p FF(h)3095 1173 y FC(2)3135 1211 y FF(=")p FG(\))3257 1137 y Fr(\003)3296 1110 y(o)3356 1211 y FF(;)49 b FH(\(4.49\))118 1481 y FB(wher)-5 b(e)34 b FF(C)7 b FG(\()p FF(t;)15 b(")p FG(\))33 b FB(and)f FF(r)s FG(\()p FF(")p FG(\))h FB(ar)-5 b(e)33 b(given)g(in)f(Pr)-5 b(op)g(osition)33 b(4.3.)118 1725 y Fp(4.3)112 b(Escap)s(e)38 b(from)f(the)g(origin)118 1896 y FH(W)-8 b(e)31 b(no)m(w)g(consider)f(the)h(SDE)g(\(4.1\))q(,)f (written)g(in)g(the)g(form)1111 2139 y FG(d)p FF(x)1214 2153 y Fy(t)1268 2139 y FG(=)1374 2077 y(1)p 1374 2118 46 4 v 1376 2201 a FF(")1430 2065 y Fr(\002)1468 2139 y FF(a)p FG(\()p FF(t)p FG(\))p FF(x)1671 2153 y Fy(t)1721 2139 y FG(+)20 b FF(b)p FG(\()p FF(x)1938 2153 y Fy(t)1968 2139 y FF(;)15 b(t)p FG(\))2076 2065 y Fr(\003)2130 2139 y FG(d)o FF(t)20 b FG(+)2365 2077 y FF(\033)p 2334 2118 119 4 v 2334 2136 a FE(p)p 2410 2136 43 4 v 66 x FF(")2477 2139 y FG(d)p FF(W)2614 2153 y Fy(t)2643 2139 y FF(;)762 b FH(\(4.50\))118 2396 y(for)31 b FF(t)26 b Fn(>)g FF(t)446 2410 y FC(0)512 2396 y Fn(>)609 2330 y FE(p)p 685 2330 V 66 x FF(")p FH(,)31 b(where)h(w)m(e)g(assume)e(that)i FE(j)p FF(x)1766 2410 y Fy(t)1791 2419 y Fw(0)1830 2396 y FE(j)27 b Fn(6)32 b FG(~)-51 b FF(x)p FG(\()p FF(t)2099 2410 y FC(0)2138 2396 y FG(\))p FH(.)43 b(Our)32 b(aim)d(is)h(to)h (estimate)f(the)h(\034rst)g(exit)118 2509 y(time)j FF(\034)369 2523 y FD(D)464 2509 y FH(of)h FF(x)624 2523 y Fy(t)689 2509 y FH(from)f FE(D)k FH(de\034ned)e(in)f(\(4.12\))q(.)55 b(W)-8 b(e)35 b(recall)f(that)i FF(a)p FG(\()p FF(t)p FG(\))24 b(+)2642 2473 y FC(1)p 2640 2488 40 4 v 2640 2540 a Fy(x)2689 2509 y FF(b)p FG(\()p FF(x;)15 b(t)p FG(\))35 b Fn(>)e FF(\024a)p FG(\()p FF(t)p FG(\))j FH(in)e FE(D)s FH(,)i(see)118 2622 y(\(4.13\))q(.)k(Moreo)m(v)m(er,)32 b(w)m(e)f(ha)m(v)m(e)h FF(a)1226 2636 y FD(\000)1285 2622 y FF(t)25 b Fn(6)g FF(a)p FG(\()p FF(t)p FG(\))h Fn(6)f FF(a)1760 2636 y FC(+)1819 2622 y FF(t)p FH(,)30 b FG(0)c Fn(6)f FF(a)2122 2589 y FD(0)2145 2622 y FG(\()p FF(t)p FG(\))h Fn(6)f FF(a)2418 2636 y FC(1)2457 2622 y FH(,)30 b(and)h FE(j)p FF(b)p FG(\()p FF(x;)15 b(t)p FG(\))p FE(j)27 b Fn(6)e FF(M)10 b FE(j)p FF(x)p FE(j)3295 2589 y FC(3)3365 2622 y FH(in)29 b FE(D)s FH(.)259 2734 y(W)-8 b(e)31 b(\034rst)f(state)h(a)f(result)g(allo)m(wing)f(to)h (estimate)f(the)i(v)-5 b(ariance)30 b(of)g(the)h(linearization)d(of)37 b(\(4.50\))q(.)118 2922 y Fq(Lemma)31 b(4.6.)39 b FB(L)-5 b(et)31 b FF(a)p FG(\()p FF(t)p FG(\))g FB(b)-5 b(e)32 b(any)e(c)-5 b(ontinuously)31 b(di\033er)-5 b(entiable,)33 b(strictly)d(p)-5 b(ositive,)31 b(incr)-5 b(e)g(asing)32 b(func-)118 3035 y(tion,)g(and)g(set)g FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))27 b(=)1015 2962 y Fr(R)1075 2988 y Fy(t)1057 3067 y(s)1120 3035 y FF(a)p FG(\()p FF(u)p FG(\))15 b(d)p FF(u)p FB(.)42 b(Then)32 b(the)h(inte)-5 b(gr)g(al)1354 3305 y FF(v)s FG(\()p FF(t;)15 b(s)p FG(\))26 b(=)1719 3243 y FF(\033)1774 3210 y FC(2)p 1719 3284 95 4 v 1745 3367 a FF(")1839 3181 y Fr(Z)1929 3207 y Fy(t)1889 3387 y(s)1974 3305 y FG(e)2015 3267 y FC(2)p Fy(\013)p FC(\()p Fy(t;u)p FC(\))p Fy(=")2323 3305 y FG(d)p FF(u)1004 b FH(\(4.51\))118 3550 y FB(satis\034es)32 b(the)h(ine)-5 b(qualities)1034 3743 y FF(\033)1089 3710 y FC(2)p 983 3783 198 4 v 983 3866 a FG(2)p FF(a)p FG(\()p FF(t)p FG(\))1190 3731 y Fr(\002)1228 3804 y FG(e)1268 3767 y FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1568 3804 y FE(\000)p FG(1)1684 3731 y Fr(\003)1748 3804 y Fn(6)25 b FF(v)s FG(\()p FF(t;)15 b(s)p FG(\))26 b Fn(6)2265 3743 y FF(\033)2320 3710 y FC(2)p 2209 3783 207 4 v 2209 3866 a FG(2)p FF(a)p FG(\()p FF(s)p FG(\))2441 3804 y(e)2481 3767 y FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2781 3804 y FF(:)624 b FH(\(4.52\))118 4052 y Fh(Pr)m(oof:)47 b FH(Using)29 b(in)m(tegration)h(b)m(y)h(parts,)f(w)m(e)h(ha)m(v)m(e) 300 4307 y FG(e)341 4270 y FD(\000)p FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")696 4307 y FF(v)s FG(\()p FF(t;)15 b(s)p FG(\))26 b(=)f FF(\033)1106 4270 y FC(2)1146 4206 y Fr(h)1279 4246 y FG(1)p 1198 4286 V 1198 4369 a(2)p FF(a)p FG(\()p FF(s)p FG(\))1435 4307 y FE(\000)1612 4246 y FG(1)p 1536 4286 198 4 v 1536 4369 a(2)p FF(a)p FG(\()p FF(t)p FG(\))1759 4307 y(e)1799 4270 y FD(\000)p FC(2)p Fy(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2154 4307 y FE(\000)2240 4183 y Fr(Z)2331 4210 y Fy(t)2291 4390 y(s)2417 4246 y FF(a)2465 4213 y FD(0)2488 4246 y FG(\()p FF(u)p FG(\))p 2386 4286 256 4 v 2386 4369 a(2)p FF(a)p FG(\()p FF(u)p FG(\))2601 4343 y FC(2)2667 4307 y FG(e)2707 4270 y FD(\000)p FC(2)p Fy(\013)p FC(\()p Fy(u;s)p FC(\))p Fy(=")3078 4307 y FG(d)o FF(u)3180 4206 y Fr(i)3223 4307 y FF(:)182 b FH(\(4.53\))118 4554 y(The)33 b(upp)s(er)g(b)s(ound)g(follo)m(ws)f(immediately)-8 b(,)30 b(and)j(the)g(lo)m(w)m(er)h(b)s(ound)f(is)e(obtained)i(b)m(y)g(b)s (ounding)g(the)118 4667 y(exp)s(onen)m(tial)d(in)f(the)i(last)e(in)m (tegral)h(b)m(y)h FG(1)p FH(.)p 3596 4667 4 62 v 3600 4610 55 4 v 3600 4667 V 3653 4667 4 62 v 259 4855 a(Our)d(\034rst)f (step)g(to)m(w)m(ards)h(estimating)e FF(\034)1626 4869 y FD(D)1713 4855 y FH(is)f(to)j(estimate)d(the)j(\034rst)f(exit)f(time) f FF(\034)3019 4869 y FD(S)3098 4855 y FH(from)h(a)h(smaller)118 4968 y(strip)j FE(S)7 b FH(,)30 b(de\034ned)h(as)1091 5126 y FE(S)i FG(=)1275 4997 y Fr(\032)1343 5126 y FG(\()p FF(x;)15 b(t)p FG(\))10 b(:)1604 5056 y FE(p)p 1680 5056 43 4 v 70 x FF(")26 b Fn(6)f FF(t)g Fn(6)g FF(T)8 b(;)15 b FE(j)p FF(x)p FE(j)25 b FF(<)2432 5064 y(h)p 2332 5105 253 4 v 2332 5123 a Fr(p)p 2423 5123 162 4 v 78 x FF(a)p FG(\()p FF(s)p FG(\))2595 4997 y Fr(\033)2663 5126 y FF(;)742 b FH(\(4.54\))118 5353 y(where)31 b(w)m(e)g(will)d(c)m(ho)s (ose)1575 5466 y FF(h)e FG(=)f(2)p FF(\033)1849 5384 y Fr(p)p 1941 5384 239 4 v 1941 5466 a FE(j)p FG(log)16 b FF(\033)s FE(j)q FF(:)1226 b FH(\(4.55\))1845 5871 y(30)p eop %%Page: 31 31 31 30 bop 118 328 a Fq(Prop)s(osition)35 b(4.7.)41 b FB(L)-5 b(et)33 b FF(t)1071 342 y FC(0)1136 328 y Fn(>)1232 262 y FE(p)p 1307 262 43 4 v 1307 328 a FF(")g FB(and)f FE(j)p FF(x)1634 342 y FC(0)1674 328 y FE(j)25 b Fn(6)g FF(h=)1917 250 y Fr(p)p 2009 250 192 4 v 2009 328 a FF(a)p FG(\()p FF(t)2125 342 y FC(0)2165 328 y FG(\))p FB(.)41 b(Then,)32 b(for)g(any)g FF(\026)25 b(>)g FG(0)p FB(,)32 b(we)h(have)441 587 y Fo(P)502 549 y Fy(t)527 558 y Fw(0)562 549 y Fy(;x)622 558 y Fw(0)661 513 y Fr(\010)714 587 y FF(\034)754 601 y FD(S)830 587 y Fn(>)25 b FF(t)959 513 y Fr(\011)1037 587 y Fn(6)1133 486 y Fr(\020)1199 525 y FF(h)p 1198 566 56 4 v 1198 649 a(\033)1263 486 y Fr(\021)1317 509 y Fy(\026)1379 587 y FG(exp)1518 459 y Fr(\032)1586 587 y FE(\000)1745 525 y FF(\026)p 1667 566 212 4 v 1667 649 a FG(1)20 b(+)g FF(\026)1898 525 y(\013)p FG(\()p FF(t;)15 b(t)2097 539 y FC(0)2137 525 y FG(\))p 1898 566 275 4 v 2014 649 a FF(")2182 486 y Fr(h)2225 587 y FG(1)21 b FE(\000)f(O)2457 486 y Fr(\020)2704 525 y FG(1)p 2521 566 412 4 v 2521 649 a FF(\026)15 b FG(log)q(\()p FF(h=\033)s FG(\))2942 486 y Fr(\021i)3039 459 y(\033)3430 587 y FH(\(4.56\))118 835 y FB(under)33 b(the)g(c)-5 b(ondition)1403 880 y Fr(\020)1469 920 y FF(h)p 1468 960 56 4 v 1468 1043 a(\033)1533 880 y Fr(\021)1587 903 y FC(3+)p Fy(\026)1724 981 y FE(O)1799 880 y Fr(\020)1853 981 y FG(log)1997 920 y FF(h)p 1996 960 V 1996 1043 a(\033)2061 880 y Fr(\021)2140 981 y Fn(6)2257 920 y FF(t)2290 887 y FC(2)2290 944 y(0)p 2246 960 95 4 v 2246 1043 a FF(\033)2301 1017 y FC(2)2351 981 y FF(:)1054 b FH(\(4.57\))118 1264 y Fh(Pr)m(oof:)156 1400 y FH(1.)47 b(F)-8 b(or)22 b FF(K)32 b FE(2)25 b Fo(N)d FH(,)29 b(w)m(e)22 b(in)m(tro)s(duce)g(a)g (partition)e FF(t)1734 1414 y FC(0)1799 1400 y FG(=)25 b FF(u)1947 1414 y FC(0)2012 1400 y FF(<)f FE(\001)15 b(\001)g(\001)27 b FF(<)e(u)2387 1414 y Fy(K)2480 1400 y FG(=)g FF(t)c FH(of)g(the)h(in)m(terv)-5 b(al)21 b FG([)p FF(t)3247 1414 y FC(0)3286 1400 y FF(;)15 b(t)p FG(])p FH(,)24 b(whic)m(h)273 1529 y(will)30 b(b)s(e)j(c)m(hosen)h (later,)f(and)g(for)f(eac)m(h)i FF(k)s FH(,)f(w)m(e)h(de\034ne)f(a)g (linear)f(appro)m(ximation)g FG(\()p FF(x)3158 1481 y FC(\()p Fy(k)r FC(\))3158 1553 y Fy(t)3256 1529 y FG(\))3291 1548 y Fy(t)p FD(2)p FC([)p Fy(u)3424 1560 y Fj(k)3462 1548 y Fy(;u)3523 1560 y Fj(k)q Fw(+1)3638 1548 y FC(])273 1642 y FH(b)m(y)1013 1788 y FG(d)o FF(x)1115 1740 y FC(\()p Fy(k)r FC(\))1115 1812 y Fy(t)1238 1788 y FG(=)1344 1727 y(1)p 1344 1767 46 4 v 1346 1850 a FF(")1400 1788 y(a)p FG(\()p FF(t)p FG(\))p FF(x)1603 1740 y FC(\()p Fy(k)r FC(\))1603 1812 y Fy(t)1716 1788 y FG(d)o FF(t)20 b FG(+)1952 1727 y FF(\033)p 1920 1767 119 4 v 1920 1785 a FE(p)p 1996 1785 43 4 v 66 x FF(")2063 1788 y FG(d)p FF(W)2213 1740 y FC(\()p Fy(k)r FC(\))2200 1812 y Fy(t)2491 1788 y FF(x)2543 1750 y FC(\()p Fy(k)r FC(\))2543 1811 y Fy(u)2584 1823 y Fj(k)2666 1788 y FG(=)k FF(x)2813 1802 y Fy(u)2854 1814 y Fj(k)2897 1788 y FF(;)508 b FH(\(4.58\))273 2033 y(where)33 b FF(W)635 1985 y FC(\()p Fy(k)r FC(\))622 2057 y Fy(t)760 2033 y FG(=)27 b FF(W)944 2047 y Fy(t)995 2033 y FE(\000)21 b FF(W)1173 2047 y Fy(u)1214 2059 y Fj(k)1256 2033 y FH(.)45 b(Assume)31 b(that)h FE(j)p FF(x)1937 2047 y Fy(s)1974 2033 y FE(j)1999 1955 y Fr(p)p 2090 1955 162 4 v 78 x FF(a)p FG(\()p FF(s)p FG(\))c Fn(6)g FF(h)k FH(for)g(all)e FF(s)d FE(2)h FG([)p FF(u)2965 2048 y Fy(k)3008 2033 y FF(;)15 b(u)3100 2048 y Fy(k)r FC(+1)3233 2033 y FG(])p FH(.)45 b(Then)33 b(b)m(y)273 2145 y(Lemma)d(4.6)931 2371 y FE(j)p FF(x)1008 2385 y Fy(s)1066 2371 y FE(\000)20 b FF(x)1209 2334 y FC(\()p Fy(k)r FC(\))1209 2394 y Fy(s)1306 2371 y FE(j)26 b Fn(6)1463 2310 y FG(1)p 1463 2350 46 4 v 1465 2434 a FF(")1533 2247 y Fr(Z)1624 2274 y Fy(s)1584 2454 y(u)1625 2466 y Fj(k)1667 2371 y FE(j)p FF(b)p FG(\()p FF(x)1818 2385 y Fy(u)1863 2371 y FF(;)15 b(u)p FG(\))p FE(j)g FG(e)2072 2334 y Fy(\013)p FC(\()p Fy(s;u)p FC(\))p Fy(=")2352 2371 y FG(d)p FF(u)1357 2636 y Fn(6)25 b FF(M)1676 2574 y(h)1728 2541 y FC(3)p 1561 2615 324 4 v 1561 2702 a FF(a)p FG(\()p FF(u)1696 2717 y Fy(k)1739 2702 y FG(\))1774 2676 y FC(3)p Fy(=)p FC(2)1988 2574 y FG(1)p 1904 2615 214 4 v 1904 2698 a FF(a)p FG(\()p FF(u)2039 2713 y Fy(k)2082 2698 y FG(\))2143 2636 y(e)2183 2598 y Fy(\013)p FC(\()p Fy(u)2296 2610 y Fj(k)q Fw(+1)2412 2598 y Fy(;u)2473 2610 y Fj(k)2510 2598 y FC(\))p Fy(=")2635 2636 y Fn(6)2841 2574 y FF(h)p 2741 2615 253 4 v 2741 2633 a Fr(p)p 2832 2633 162 4 v 78 x FF(a)p FG(\()p FF(s)p FG(\))3430 2515 y FH(\(4.59\))273 2906 y(for)30 b FF(s)25 b FE(2)g FG([)p FF(u)642 2921 y Fy(k)685 2906 y FF(;)15 b(u)777 2921 y Fy(k)r FC(+1)910 2906 y FG(])p FH(,)31 b(pro)m(vided)g(the)f (partition)g(is)f(c)m(hosen)i(in)f(suc)m(h)h(a)f(w)m(a)m(y)h(that)g (for)f(all)f FF(k)1265 3200 y(h)1317 3162 y FC(2)1382 3200 y Fn(6)1488 3135 y FF(a)1536 3102 y FC(2)1536 3157 y FD(\000)p 1488 3179 108 4 v 1492 3262 a FF(M)1605 3036 y Fr(s)p 1696 3036 324 4 v 1751 3138 a FF(a)p FG(\()p FF(u)1886 3153 y Fy(k)1929 3138 y FG(\))p 1706 3179 304 4 v 1706 3262 a FF(a)p FG(\()p FF(u)1841 3277 y Fy(k)r FC(+1)1974 3262 y FG(\))2035 3200 y(e)2075 3162 y FD(\000)p Fy(\013)p FC(\()p Fy(u)2243 3174 y Fj(k)q Fw(+1)2359 3162 y Fy(;u)2420 3174 y Fj(k)2457 3162 y FC(\))p Fy(=")2572 3200 y FF(t)2605 3162 y FC(2)2605 3222 y(0)2644 3200 y FF(:)761 b FH(\(4.60\))156 3490 y(2.)47 b(If)30 b FE(j)p FF(x)441 3504 y Fy(u)482 3516 y Fj(k)524 3490 y FE(j)549 3412 y Fr(p)p 640 3412 214 4 v 78 x FF(a)p FG(\()p FF(u)775 3505 y Fy(k)818 3490 y FG(\))c Fn(6)f FF(h)p FH(,)30 b(then)h(w)m(e)g(ha)m(v)m(e)593 3722 y Fo(P)654 3684 y Fy(u)695 3696 y Fj(k)734 3684 y Fy(;x)794 3692 y Fj(u)831 3710 y(k)876 3621 y Fr(n)1057 3722 y FG(sup)937 3798 y Fy(u)978 3810 y Fj(k)1016 3798 y Fx(6)p Fy(s)p Fx(6)p Fy(u)1200 3810 y Fj(k)q Fw(+1)1314 3722 y FE(j)p FF(x)1391 3736 y Fy(s)1428 3722 y FE(j)1453 3639 y Fr(p)p 1544 3639 162 4 v 83 x FF(a)p FG(\()p FF(s)p FG(\))26 b Fn(6)f FF(h)1879 3621 y Fr(o)1965 3722 y Fn(6)g Fo(P)2122 3684 y Fy(u)2163 3696 y Fj(k)2202 3684 y Fy(;x)2262 3692 y Fj(u)2299 3710 y(k)2344 3621 y Fr(n)2405 3722 y FE(j)p FF(x)2482 3684 y FC(\()p Fy(k)r FC(\))2482 3744 y Fy(u)2523 3756 y Fj(k)q Fw(+1)2642 3722 y FE(j)2667 3639 y Fr(p)p 2758 3639 304 4 v 83 x FF(a)p FG(\()p FF(u)2893 3737 y Fy(k)r FC(+1)3026 3722 y FG(\))h Fn(6)f FG(2)p FF(h)3280 3621 y Fr(o)1965 3984 y Fn(6)2372 3923 y FG(4)p FF(h)p 2071 3963 700 4 v 2071 3981 a Fr(q)p 2162 3981 609 4 v 116 x FG(2)p FF(\031)s(v)2309 4049 y FC(\()p Fy(k)r FC(\))2306 4109 y Fy(u)2347 4121 y Fj(k)q Fw(+1)2467 4097 y FF(a)p FG(\()p FF(u)2602 4112 y Fy(k)r FC(+1)2735 4097 y FG(\))2780 3984 y FF(;)625 b FH(\(4.61\))273 4304 y(where)31 b(the)g(v)-5 b(ariance)1324 4449 y FF(v)1371 4412 y FC(\()p Fy(k)r FC(\))1368 4472 y Fy(u)1409 4484 y Fj(k)q Fw(+1)1554 4449 y FG(=)1660 4388 y FF(\033)1715 4355 y FC(2)p 1660 4428 95 4 v 1686 4511 a FF(")1780 4325 y Fr(Z)1871 4352 y Fy(u)1912 4364 y Fj(k)q Fw(+1)1830 4532 y Fy(u)1871 4544 y Fj(k)2046 4449 y FG(e)2086 4412 y FC(2)p Fy(\013)p FC(\()p Fy(u)2234 4424 y Fj(k)q Fw(+1)2350 4412 y Fy(;s)p FC(\))p Fy(=")2517 4449 y FG(d)p FF(s)819 b FH(\(4.62\))273 4668 y(can)31 b(b)s(e)f(estimated)f(b)m(y)i(Lemma)f (4.6.)40 b(W)-8 b(e)31 b(th)m(us)g(ha)m(v)m(e)g(b)m(y)g(the)g(Mark)m(o) m(v)f(prop)s(ert)m(y)491 4954 y FF(P)38 b FG(=)25 b Fo(P)744 4916 y Fy(t)769 4925 y Fw(0)804 4916 y Fy(;x)864 4925 y Fw(0)903 4853 y Fr(n)1009 4954 y FG(sup)963 5030 y Fy(t)988 5039 y Fw(0)1023 5030 y Fx(6)p Fy(s)p Fx(6)p Fy(t)1191 4954 y FE(j)p FF(x)1268 4968 y Fy(s)1305 4954 y FE(j)1330 4871 y Fr(p)p 1421 4871 162 4 v 83 x FF(a)p FG(\()p FF(s)p FG(\))h Fn(6)f FF(h)1756 4853 y Fr(o)1842 4954 y Fn(6)1938 4840 y Fy(K)5 b FD(\000)p FC(1)1957 4867 y Fr(Y)1951 5065 y Fy(k)r FC(=0)2092 4825 y Fr(\022)2235 4892 y FG(4)p 2169 4933 177 4 v 2169 4951 a FE(p)p 2245 4951 101 4 v 75 x FG(2)p FF(\031)2639 4892 y(h)p 2366 4933 599 4 v 2366 4951 a Fr(q)p 2456 4951 508 4 v 2456 5067 a FF(v)2503 5019 y FC(\()p Fy(k)r FC(\))2500 5079 y Fy(u)2541 5091 y Fj(k)q Fw(+1)2661 5067 y FF(a)p FG(\()p FF(u)2796 5082 y Fy(k)r FC(+1)2929 5067 y FG(\))2994 4954 y FE(^)20 b FG(1)3120 4825 y Fr(\023)3188 4954 y FF(:)217 b FH(\(4.63\))156 5308 y(3.)47 b(W)-8 b(e)41 b(no)m(w)g(c)m(ho)s(ose)g(the)g FF(u)1152 5323 y Fy(k)1234 5308 y FH(in)f(suc)m(h)h(a)f(w)m(a)m(y)h(that)g FF(v)2095 5260 y FC(\()p Fy(k)r FC(\))2092 5320 y Fy(u)2133 5332 y Fj(k)q Fw(+1)2252 5308 y FF(a)p FG(\()p FF(u)2387 5323 y Fy(k)r FC(+1)2520 5308 y FG(\))g FH(is)e(appro)m(ximately)g(constan)m (t.)273 5421 y(Giv)m(en)30 b FF(\026)25 b(>)g FG(0)p FH(,)31 b(let)1670 5584 y FF(`)25 b FG(=)1843 5523 y(8)p 1839 5563 56 4 v 1839 5647 a FF(\031)1904 5584 y(h)1956 5547 y FC(2)1995 5483 y Fr(\020)2061 5523 y FF(h)2113 5490 y FC(2)p 2060 5563 95 4 v 2060 5647 a FF(\033)2115 5620 y FC(2)2164 5483 y Fr(\021)2219 5506 y Fy(\026)3430 5584 y FH(\(4.64\))1845 5871 y(31)p eop %%Page: 32 32 32 31 bop 273 328 a FH(\(Observ)m(e)31 b(that)g FF(`)25 b Fn(>)g FG(8)p FF(h)1107 295 y FC(2)1148 328 y FF(=\031)j(>)d(\033) 1424 295 y FC(2)1464 328 y FF(=)p FG(2)p FH(.\))41 b(Cho)s(osing)30 b FF(K)37 b FH(as)30 b(the)h(smallest)c(in)m(teger)k(satisfying)1608 580 y FF(K)h Fn(>)1897 518 y FG(2)p FF(\013)p FG(\()p FF(t;)15 b(t)2141 532 y FC(0)2182 518 y FG(\))p 1823 559 470 4 v 1823 642 a FF(")g FG(log)r(\(2)p FF(`=\033)2216 616 y FC(2)2256 642 y FG(\))2302 580 y FF(;)1103 b FH(\(4.65\))273 827 y(w)m(e)31 b(de\034ne)h(the)e(partition)g(b)m(y)g(the)h(relations) 1065 1069 y FF(\013)p FG(\()p FF(u)1210 1084 y Fy(k)r FC(+1)1344 1069 y FF(;)15 b(u)1436 1084 y Fy(k)1479 1069 y FG(\))26 b(=)1647 1007 y FF(")p 1645 1048 46 4 v 1645 1131 a FG(2)1716 1069 y(log)1864 1007 y(2)p FF(`)p 1859 1048 95 4 v 1859 1131 a(\033)1914 1105 y FC(2)1963 1069 y FF(;)196 b FH(for)31 b FF(k)d FE(2)d(f)p FG(0)p FF(;)15 b(:)g(:)g(:)i(;)e(K)28 b FE(\000)20 b FG(2)p FE(g)p FH(,)343 b(\(4.66\))847 1289 y FG(0)26 b FF(<)f(\013)p FG(\()p FF(u)1159 1303 y Fy(K)1228 1289 y FF(;)15 b(u)1320 1303 y Fy(K)5 b FD(\000)p FC(1)1479 1289 y FG(\))26 b Fn(6)1647 1228 y FF(")p 1645 1268 46 4 v 1645 1351 a FG(2)1716 1289 y(log)1864 1228 y(2)p FF(`)p 1859 1268 95 4 v 1859 1351 a(\033)1914 1325 y FC(2)1963 1289 y FF(:)1442 b FH(\(4.67\))273 1514 y(Then)31 b(w)m(e)g(ha)m(v)m(e)346 1774 y FF(P)38 b Fn(6)538 1646 y Fr(\022)658 1712 y FG(4)p 615 1753 131 4 v 615 1771 a FE(p)p 691 1771 56 4 v 66 x FF(\031)767 1712 y(h)p 766 1753 V 766 1836 a(\033)1054 1712 y FG(1)p 841 1753 471 4 v 841 1771 a Fr(p)p 932 1771 380 4 v 78 x FG(2)p FF(`=\033)1115 1823 y FC(2)1176 1849 y FE(\000)20 b FG(1)1322 1646 y Fr(\023)1389 1668 y Fy(K)5 b FD(\000)p FC(1)1573 1774 y Fn(6)1669 1673 y Fr(\020)1734 1712 y FF(h)p 1733 1753 56 4 v 1733 1836 a(\033)1798 1673 y Fr(\021)1852 1696 y Fy(\026)1914 1774 y FG(exp)2053 1646 y Fr(\032)2121 1774 y FE(\000)2202 1712 y FF(\013)p FG(\()p FF(t;)15 b(t)2401 1726 y FC(0)2441 1712 y FG(\))p 2202 1753 275 4 v 2318 1836 a FF(")2516 1705 y FG(log)2633 1631 y Fr(\002\000)2724 1669 y Fy(h)2765 1645 y Fw(2)p 2723 1684 78 4 v 2723 1738 a Fy(\033)2765 1720 y Fw(2)2810 1631 y Fr(\001)2852 1654 y Fy(\026)2918 1705 y FE(\000)3033 1669 y Fy(\031)p 3019 1684 71 4 v 3019 1736 a FC(16)3110 1669 y Fy(\033)3152 1645 y Fw(2)p 3110 1684 78 4 v 3111 1738 a Fy(h)3152 1720 y Fw(2)3197 1631 y Fr(\003)p 2497 1753 758 4 v 2497 1858 a FG(log)2614 1784 y Fr(\002)2662 1822 y FC(16)p 2662 1837 71 4 v 2676 1889 a Fy(\031)2742 1784 y Fr(\000)2795 1822 y Fy(h)2836 1803 y Fw(2)p 2794 1837 78 4 v 2794 1891 a Fy(\033)2836 1872 y Fw(2)2881 1784 y Fr(\001)2923 1807 y FC(1+)p Fy(\026)3080 1858 y FE(\000)20 b FG(1)3216 1784 y Fr(\003)3264 1646 y(\033)3332 1774 y FF(;)73 b FH(\(4.68\))273 2059 y(whic)m(h)31 b(pro)m(v)m(es)g(\(4.56\))q(.)156 2172 y(4.)47 b(It)30 b(remains)f(to)i(sho)m(w)g(that)g(condition)e(\(4.60\))i(is)e (satis\034ed.)40 b(Since)570 2370 y FF(a)p FG(\()p FF(u)705 2385 y Fy(k)r FC(+1)838 2370 y FG(\))p 570 2411 304 4 v 615 2494 a FF(a)p FG(\()p FF(u)750 2509 y Fy(k)793 2494 y FG(\))909 2432 y Fn(6)25 b FG(1)20 b(+)1234 2370 y FF(a)1282 2384 y FC(1)p 1171 2411 214 4 v 1171 2494 a FF(a)p FG(\()p FF(u)1306 2509 y Fy(k)1349 2494 y FG(\))1395 2432 y(\()p FF(u)1482 2447 y Fy(k)r FC(+1)1635 2432 y FE(\000)g FF(u)1778 2447 y Fy(k)1821 2432 y FG(\))25 b Fn(6)g FG(1)c(+)2191 2370 y FF(a)2239 2384 y FC(1)2279 2370 y FF(")p 2144 2411 225 4 v 2144 2496 a FG(2)p FF(a)2237 2465 y FC(2)2237 2519 y FD(\000)2296 2496 y FF(t)2329 2465 y FC(2)2329 2522 y(0)2394 2432 y FG(log)2511 2304 y Fr(\032)2589 2370 y FG(16)p 2589 2411 91 4 v 2607 2494 a FF(\031)2690 2331 y Fr(\020)2756 2370 y FF(h)2808 2337 y FC(2)p 2755 2411 95 4 v 2755 2494 a FF(\033)2810 2468 y FC(2)2859 2331 y Fr(\021)2913 2353 y FC(1+)p Fy(\026)3050 2304 y Fr(\033)3118 2432 y FF(;)287 b FH(\(4.69\))273 2688 y(the)31 b(condition)f(reduces)h(to)916 2828 y Fr(\020)981 2867 y FF(h)p 980 2908 56 4 v 980 2991 a(\033)1045 2828 y Fr(\021)1099 2850 y FC(3+)p Fy(\026)1236 2800 y Fr(\022)1303 2929 y FG(1)21 b(+)1502 2867 y FF(a)1550 2881 y FC(1)p 1470 2908 153 4 v 1470 2993 a FG(4)p FF(a)1563 2961 y FC(2)1563 3016 y FD(\000)1657 2867 y FF(")p 1642 2908 73 4 v 1642 2993 a(t)1675 2961 y FC(2)1675 3019 y(0)1739 2929 y FG(log)1857 2828 y Fr(h)1910 2867 y FG(16)p 1910 2908 91 4 v 1928 2991 a FF(\031)2011 2828 y Fr(\020)2076 2867 y FF(h)2128 2834 y FC(2)p 2075 2908 95 4 v 2075 2991 a FF(\033)2130 2965 y FC(2)2179 2828 y Fr(\021)2234 2850 y FC(1+)p Fy(\026)2370 2828 y Fr(i)2413 2800 y(\023)2506 2929 y Fn(6)2611 2864 y FF(a)2659 2831 y FC(2)2659 2886 y FD(\000)p 2611 2908 108 4 v 2615 2991 a FF(M)2739 2802 y FE(p)p 2814 2802 56 4 v 2814 2867 a FF(\031)p 2739 2908 131 4 v 2781 2991 a FG(4)2900 2867 y FF(t)2933 2834 y FC(2)2933 2892 y(0)p 2889 2908 95 4 v 2889 2991 a FF(\033)2944 2965 y FC(2)2994 2929 y FF(;)411 b FH(\(4.70\))273 3191 y(whic)m(h)31 b(is)e(satis\034ed)g(whenev)m(er)j(condition)e(\(4.57\))h (is)e(satis\034ed.)p 3596 3191 4 62 v 3600 3133 55 4 v 3600 3191 V 3653 3191 4 62 v 259 3378 a(W)-8 b(e)40 b(w)m(an)m(t)i(to)e(c)m(ho)s(ose)g FF(\026)f FH(in)g(suc)m(h)h(a)g(w)m (a)m(y)h(that)f Fo(P)2037 3345 y Fy(t)2062 3354 y Fw(0)2098 3345 y Fy(;x)2158 3354 y Fw(0)2196 3378 y FE(f)p FF(\034)2281 3392 y FD(S)2374 3378 y Fn(>)g FF(t)p FE(g)i Fn(6)f FG(\()p FF(h=\033)s FG(\))2939 3345 y Fy(\026)3002 3378 y FG(e)3042 3345 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;t)3280 3354 y Fw(0)3316 3345 y FC(\))p Fy(=")3455 3378 y FH(holds)118 3491 y(with)i(the)h(same)f FF(\024)h FH(as)f(in)g(\(4.13\))q(.)80 b(W)-8 b(e)44 b(opt)g(for)f FF(\026)48 b FG(=)f(2)p FH(,)g(b)s(ecause)d (this)f(c)m(hoice)h(guaran)m(tees)h(the)118 3604 y(ab)s(o)m(v)m(e)33 b(estimate)e(for)h(all)e(p)s(ossible)g FF(\024)i FH(without)g(c)m(ho)s (osing)g(a)g FF(\024)p FH(-dep)s(enden)m(t)i FF(\026)p FH(.)45 b(F)-8 b(or)33 b FF(h)c FG(=)e(2)p FF(\033)3306 3527 y Fr(p)p 3398 3527 239 4 v 3398 3604 a FE(j)p FG(log)17 b FF(\033)s FE(j)p FH(,)118 3717 y(Condition)30 b(\(4.57\))h(b)s (ecomes)f(a)g(consequence)h(of)f(the)h(follo)m(wing)e(sligh)m(tly)f (stronger)j(condition)1483 3921 y FF(\033)s FE(j)p FG(log)17 b FF(\033)s FE(j)1776 3884 y FC(3)p Fy(=)p FC(2)1912 3921 y FG(=)25 b FE(O)s FG(\()2118 3852 y FE(p)p 2194 3852 43 4 v 69 x FF(")p FG(\))p FF(;)1134 b FH(\(4.71\))118 4126 y(whic)m(h)31 b(w)m(e)g(will)d(assume)h(to)h(b)s(e)h(satis\034ed)e (from)h(no)m(w)h(on)f(for)h(the)f(rest)h(of)e(this)h(subsection.)259 4239 y(The)e(second)f(step)g(is)e(to)i(con)m(trol)g(the)h(probabilit)m (y)d(that)j FF(x)2268 4253 y Fy(t)2324 4239 y FH(returns)f(to)g(zero)h (after)f(it)e(has)i(left)f(the)118 4351 y(strip)k FE(S)7 b FH(.)40 b(T)-8 b(o)31 b(do)f(so,)g(w)m(e)i(will)27 b(compare)k(solutions)e(of)36 b(\(4.50\))31 b(with)f(those)h(of)f(the)g (linear)g(equation)1307 4594 y FG(d)p FF(x)1410 4556 y FC(0)1410 4616 y Fy(t)1474 4594 y FG(=)1580 4532 y(1)p 1580 4573 46 4 v 1582 4656 a FF(")1636 4594 y(a)1684 4608 y FC(0)1723 4594 y FG(\()p FF(t)p FG(\))p FF(x)1878 4556 y FC(0)1878 4616 y Fy(t)1933 4594 y FG(d)p FF(t)20 b FG(+)2169 4532 y FF(\033)p 2138 4573 119 4 v 2138 4591 a FE(p)p 2213 4591 43 4 v 2213 4657 a FF(")2281 4594 y FG(d)p FF(W)2418 4608 y Fy(t)2447 4594 y FF(;)958 b FH(\(4.72\))118 4846 y(where)32 b FF(a)428 4860 y FC(0)467 4846 y FG(\()p FF(t)p FG(\))27 b(=)e FF(\024a)p FG(\()p FF(t)p FG(\))32 b FH(satis\034es)e FF(a)1310 4860 y FC(0)1349 4846 y FG(\()p FF(t)p FG(\))d Fn(6)e FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))p FF(=x)31 b FH(in)f FE(D)s FH(.)42 b(The)31 b(follo)m(wing)e(lemma)g(sho)m(ws)i(that)g(this)118 4959 y(c)m(hoice)e(of)f FF(a)534 4973 y FC(0)574 4959 y FG(\()p FF(s)p FG(\))g FH(implies)d(that)30 b FE(j)p FF(x)1292 4973 y Fy(s)1329 4959 y FE(j)25 b Fn(>)g FE(j)p FF(x)1552 4926 y FC(0)1552 4982 y Fy(s)1592 4959 y FE(j)j FH(holds)g(as)g(long)g(as)h FF(x)2345 4973 y Fy(s)2410 4959 y FH(do)s(es)f(not)h(return)g(to)g(zero)g(\(Fig.)23 b(3\).)118 5072 y(This)33 b(implies)d(that)k(if)e FF(x)975 5039 y FC(0)975 5095 y Fy(s)1048 5072 y FH(do)s(es)h(not)h(return)g(to) g(zero)g(b)s(efore)f(time)f FF(t)p FH(,)i(then)g FF(x)2842 5086 y Fy(s)2912 5072 y FH(is)f(lik)m(ely)d(to)k(lea)m(v)m(e)g FE(D)118 5185 y FH(b)s(efore)c(time)f FF(t)h FH(without)h(returning)f (to)h(zero.)118 5373 y Fq(Lemma)i(4.8.)41 b FB(L)-5 b(et)33 b FF(t)878 5387 y FC(0)942 5373 y Fn(>)1038 5307 y FE(p)p 1114 5307 V 66 x FF(")g FB(and)f(assume)h(that)f FG(0)26 b FF(<)f(x)2083 5387 y FC(0)2148 5373 y FF(<)30 b FG(~)-51 b FF(x)p FG(\()p FF(t)2363 5387 y FC(0)2403 5373 y FG(\))p FB(.)42 b(W)-7 b(e)32 b(de\034ne)949 5577 y FE(D)1022 5539 y FC(+)1080 5577 y FG(\()p FF(t)p FG(\))26 b(=)1305 5503 y Fr(\010)1358 5577 y FG(\()p FF(x;)15 b(s)p FG(\))10 b(:)1629 5507 y FE(p)p 1705 5507 V 70 x FF(")26 b Fn(6)f FF(s)g Fn(6)g FF(t)32 b FB(and)g FG(0)26 b FF(<)e(x)i(<)k FG(~)-50 b FF(x)p FG(\()p FF(s)p FG(\))2778 5503 y Fr(\011)3430 5577 y FH(\(4.73\))1845 5871 y(32)p eop %%Page: 33 33 33 32 bop 883 236 a 15887066 11188078 10261954 24733941 28943974 37824512 startTexFig doclip 883 236 a %%BeginDocument: fig3.ps %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: escape.dvi %%Pages: 1 %%PageOrder: Ascend %%BoundingBox: 156 376 440 575 %%EndComments %DVIPSCommandLine: dvips escape -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.04:2026 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup 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 /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore 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 /IE 256 array N 2 string 0 1 255{IE S dup 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (escape.dvi) @start %DVIPSBitmapFont: Fa ecrm1095 10.95 1 /Fa 1 50 df49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb ecrm1000 10 27 /Fb 27 123 df66 D68 D<913A01FFC001C0021F13F891B5EAFE03010390 38C07F07903A07FE001FCFD91FF8EB07EFD93FE0EB01FF49487F4948147F4890C8123F48 5A4848151F49150F120F491507121F491503123FA2485A1701A3484892C7FCAB93B6FCA2 6C7E9338007FE0EF3FC0A26C7EA2121F7F120F7F12077F6C7E6C7E6C6D147F6D6C14FF6D 7ED91FF81303D907FEEB0FEF903A03FFE03FC7010090B51203021FEBFC01020101E0C7FC 383B7CB941>71 D78 D97 DI<90380FFF80013F13F09038FF03F8D803FC13FCEA07F0120FEA1FE013C012 3F90388001F8007F90C7FCA290C8FC5AAA6C7EA3003F140E6D131E121F6D133C6C6C137C 6C6C13F83903FC01F03900FF0FE090383FFF8090380FFE001F247DA325>III<14FF0107 13C090381FE3E090383F8FF0EB7F0F137E13FE13FC12019038F807E091C7FCACB512FCA3 D801F8C7FCB3AB487E387FFFF8A31C3A7FB919>IIII107 DI<2703F01FF8EB3F F000FFD97FFEEBFFFC903BF1F87F83F0FF903BF3E01F87C03F3D0FF7800FCF001F802603 FF0013DE4902FC14C0496D48130FA2495CA2495CB3A3486C496CEB1FE0B500C1B50083B5 FCA340247EA345>I<3903F01FF800FFEB7FFE9039F1F87F809038F3E01F3A0FF7800FC0 3803FF004980491307A25BA25BB3A3486C497EB500C1B51280A329247EA32E>II<3903 F03FF039FFF1FFFC9038F7F0FF9039FFC03FC000079038001FE06C48EB0FF049EB07F85B 49EB03FCA2ED01FEA316FF81A85D16FEA3ED03FC7FED07F86DEB0FF06D14E06DEB1FC091 38807F809039F7F0FF009038F1FFFC9038F07FE091C8FCAC487EB512C0A328347EA32E> I<3907E07F8039FFE1FFC09038E3E7E09038E78FF0000F130FEA03EEA213FCEC07E09038 F803C091C7FCA25BB3A2487EB512F0A31C247EA321>114 D<3801FFC7000F13FFEA1F81 383E007F48131F1278487FA2807E7E6C90C7FC6C7E13F8387FFFC06C13F06C13FC6C7F00 037FC66C138013039038007FC000E0131F140F6C1307A214037EA26CEB0780A26CEB0F00 6C6C5AEBE0FE38F3FFF800E013E01A247DA321>I<1338A51378A413F8A21201A2120312 07001FB5FCB6FCA2D801F8C7FCB2EC01C0A8EBFC03000014801407137E90387F9F00EB1F FEEB07F81A337FB220>IIII121 D<003FB512FCA2EBC003 90380007F8003C14F0140F0038EB1FE0007814C0143F0070EB7F80ECFF00A2495A495A00 005B1307495A495AA2495A495AEC000E5B485A485AA24848131E485A49131C121F484813 3C49137C007F14FC38FF000790B5FCA21F247EA325>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc eccc1000 10 8 /Fc 8 118 df<123E127FEAFF80A5EA7F00123E090977881B>46 D51 D70 D101 D<91387FF007903903FFFC0F01 0F13FF90393FF83FDF90397FC007FF4848C67E48481300D807F880120F49804848805B00 3F81A2485A82A248C8FC93C7FCA892387FFFF8A26C7E03001380EE7F006C7EA26C7EA26C 7E7F6C7E6C6C5C6C6C5B39007FC00390393FF81FDF010FB5128F010314079026007FF8C7 FC2D2D7BAB35>103 D105 D114 D117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmmi7 7 1 /Fd 1 117 df<133CA2137CA413FC5BA31201B512E0A33803F0005BA312075BA3120F5B A3121FEB00E0130114C0EA3F03383E0780381F0F00131EEA0FFCEA07F013247EA319> 116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr7 7 1 /Fe 1 49 df48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy7 7 1 /Ff 1 84 df<913803FF80021F13F0027F13F849B5FC903903F81FFC903807E00390380F C001148049C712F849130116E06EC7FCA28080EB1FF814FE6D6C7E6D13E0010113F89038 007FFEEC1FFF0207138014019138007FC0D80780133F001F141F48C7FC007E140F127C12 FC1680151F6C15006C5C6D137E9038E001FC397FFC0FF86CB512E06C1480000749C7FCC6 13E0262A7DA829>83 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi10 10 4 /Fg 4 121 df<017FB6FC48B7FC5A5A4815FE261FC01CC7FCEB003C123E5A0078137C48 1378A2C7FC14F8A25C1301A313035CA31307A25C130FA3131F5CA591C8FC28247EA324> 28 D34 D116 D<90391FE00FF090397FF83FFC3A01F8FC7C7E3903E07EF83A07C03FF0FE9038801FE1D8 0F0013C1121E158148133FED80F8003891C7FCA2C75AA2147EA214FEA25CA21301A24A13 38A2D83E03147812FF4A13F01307ED01E0010FEB03C039FE1FF8073AFC3EFC0F8090397C 7E3F00397FF83FFC391FE00FF027247DA32F>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmsy10 10 3 /Fh 3 113 df<0203B512F0023F14FF49B712E0010F16F84916FE90277FC3F80F7F2601 FE0701001480D803F0031F13C0D807E0030713E0D80FC01501D81F806F13F0003F4AEC7F F80100163F48171F007E010F16FC00F8170F00E01707C7FC5D1803A2141F5DA219F8143F A24B140719F0A2147F92C8EA0FE0A219C002FE151F1980F03F0049485D187E6049484A5A 4D5A4D5A49484A5A4D5A057FC7FC494814FCEE03F84AEB0FE0011FEC3FC04BB4C8FC9039 3F803FFC49B512F048B612C0484AC9FC4814E002FCCAFC3E397FB840>68 D83 D112 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 707 1619 a 18665439 13052758 20063436 8288501 43876433 24931287 startTexFig doclip 707 1619 a %%BeginDocument: escape.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: escape.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta2 %%CreationDate: Fri Aug 4 20:25:46 2000 %%For: berglund@hilbert.wias-berlin.de (Nils Berglund,0518,,) %%Orientation: Portrait %%BoundingBox: 305 126 667 379 %%Pages: 0 %%BeginSetup %%IncludeFeature: *PageSize Letter %%EndSetup %%Magnification: 1.00 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -35.0 577.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n 0 9641 m 0 0 l 16841 0 l 16841 9641 l cp clip 0.06000 0.06000 sc % Polyline n 600 9600 m 16800 9600 l 16800 1200 l 600 1200 l cp gs col7 1.00 shd ef gr % Polyline n 7200 6593 m 6900 6375 l 6900 4425 l 7200 4208 l 7673 3968 l 8318 3728 l 9083 3533 l 9705 3413 l 10298 3353 l 10800 3315 l 10800 7485 l 10073 7433 l 9428 7335 l 8723 7193 l 8100 7005 l 7545 6773 l cp gs col4 0.60 tnt ef gr % Polyline n 6905 4580 m 7212 4775 l 7572 4940 l 7910 5023 l 8502 5083 l 9260 5120 l 10115 5150 l 10790 5165 l 10790 5615 l 9672 5660 l 8637 5705 l 8007 5765 l 7655 5825 l 7392 5930 l 7152 6050 l 6987 6163 l 6897 6223 l cp gs col6 0.50 tnt ef gr % Polyline 15.000 slw n 6900 6375 m 6900 4425 l gs col0 s gr % Polyline n 6900 4575 m 6901 4576 l 6903 4577 l 6907 4580 l 6914 4584 l 6923 4591 l 6936 4599 l 6951 4609 l 6970 4622 l 6993 4636 l 7019 4653 l 7047 4671 l 7079 4690 l 7113 4711 l 7149 4733 l 7188 4755 l 7227 4777 l 7268 4800 l 7310 4822 l 7353 4844 l 7396 4864 l 7440 4884 l 7485 4903 l 7530 4921 l 7576 4938 l 7624 4954 l 7672 4968 l 7723 4981 l 7775 4994 l 7831 5005 l 7889 5015 l 7950 5025 l 7988 5030 l 8028 5036 l 8069 5041 l 8113 5046 l 8158 5050 l 8206 5055 l 8256 5060 l 8309 5064 l 8364 5068 l 8422 5072 l 8482 5076 l 8545 5081 l 8611 5084 l 8680 5088 l 8751 5092 l 8824 5096 l 8901 5100 l 8979 5104 l 9059 5107 l 9142 5111 l 9226 5114 l 9312 5118 l 9399 5122 l 9487 5125 l 9575 5128 l 9664 5132 l 9753 5135 l 9840 5138 l 9927 5141 l 10013 5144 l 10096 5147 l 10177 5150 l 10256 5153 l 10331 5155 l 10403 5158 l 10471 5160 l 10535 5162 l 10594 5164 l 10649 5166 l 10699 5167 l 10744 5169 l 10784 5170 l 10819 5171 l 10849 5172 l 10875 5173 l 10896 5173 l 10913 5174 l 10926 5174 l 10936 5175 l 10943 5175 l 10947 5175 l 10949 5175 l 10950 5175 l gs col0 s gr % Ellipse n 11400 5400 5100 2100 0 360 DrawEllipse gs col0 s gr % Polyline n 6900 6225 m 6901 6224 l 6903 6223 l 6907 6220 l 6914 6216 l 6923 6209 l 6936 6201 l 6951 6191 l 6970 6178 l 6993 6164 l 7019 6147 l 7047 6129 l 7079 6110 l 7113 6089 l 7149 6067 l 7188 6045 l 7227 6023 l 7268 6000 l 7310 5978 l 7353 5956 l 7396 5936 l 7440 5916 l 7485 5897 l 7530 5879 l 7576 5862 l 7624 5846 l 7672 5832 l 7723 5819 l 7775 5806 l 7831 5795 l 7889 5785 l 7950 5775 l 7988 5770 l 8028 5764 l 8069 5759 l 8113 5754 l 8158 5750 l 8206 5745 l 8256 5740 l 8309 5736 l 8364 5732 l 8422 5728 l 8482 5724 l 8545 5719 l 8611 5716 l 8680 5712 l 8751 5708 l 8824 5704 l 8901 5700 l 8979 5696 l 9059 5693 l 9142 5689 l 9226 5686 l 9312 5682 l 9399 5678 l 9487 5675 l 9575 5672 l 9664 5668 l 9753 5665 l 9840 5662 l 9927 5659 l 10013 5656 l 10096 5653 l 10177 5650 l 10256 5647 l 10331 5645 l 10403 5642 l 10471 5640 l 10535 5638 l 10594 5636 l 10649 5634 l 10699 5633 l 10744 5631 l 10784 5630 l 10819 5629 l 10849 5628 l 10875 5627 l 10896 5627 l 10913 5626 l 10926 5626 l 10936 5625 l 10943 5625 l 10947 5625 l 10949 5625 l 10950 5625 l gs col0 s gr % Polyline 0.000 slw n 10875 9000 m 16800 9000 l 16800 1800 l 10875 1800 l cp gs col7 1.00 shd ef gr % Polyline 15.000 slw gs clippath 6240 3585 m 6300 3344 l 6360 3585 l 6360 3270 l 6240 3270 l cp clip n 6300 7500 m 6300 3300 l gs col0 s gr gr % arrowhead n 6240 3585 m 6300 3344 l 6360 3585 l 6300 3545 l 6240 3585 l cp gs col7 1.00 shd ef gr col0 s % Polyline gs clippath 11406 5350 m 11646 5400 l 11406 5450 l 11730 5450 l 11730 5350 l cp clip n 5700 5400 m 11700 5400 l gs col0 s gr gr % arrowhead n 11406 5350 m 11646 5400 l 11406 5450 l 11446 5400 l 11406 5350 l cp gs col7 1.00 shd ef gr col0 s % Polyline 2 slj 7.500 slw n 8774 5400 m 8834 5620 l 8879 5485 l 8954 5785 l 8989 5605 l 9024 5720 l 9074 5485 l 9104 5630 l 9134 5535 l 9224 5895 l 9259 5770 l 9309 5945 l gs col1 s gr % Polyline n 7899 5015 m 7969 4840 l 8024 4970 l 8069 4745 l 8124 5200 l 8164 4980 l 8224 5285 l 8304 4870 l 8339 5030 l 8374 4875 l 8434 5230 l 8469 5130 l 8509 5330 l 8554 5020 l 8589 5220 l 8624 5100 l 8675 5340 l 8720 5290 l 8769 5400 l gs col1 s gr % Polyline n 7895 5023 m 7962 4685 l 8009 4830 l 8045 4453 l 8084 4675 l 8120 4513 l 8210 4933 l 8262 4573 l 8292 4768 l 8367 4490 l 8435 4978 l 8480 4715 l 8529 4890 l 8559 4790 l 8604 4910 l 8607 4625 l 8712 5225 l 8772 4963 l 8810 5203 l 8854 5040 l 8885 5240 l 8929 4500 l 8989 4720 l 9029 4350 l 9084 4640 l 9155 4415 l gs col0 s gr % Polyline n 6905 5000 m 6950 5248 l 6980 4903 l 7025 5143 l 7062 4895 l 7137 5593 l 7160 5435 l 7220 5870 l 7250 5638 l 7295 5788 l 7332 5473 l 7385 5638 l 7430 5263 l 7467 5480 l 7475 5308 l 7527 5533 l 7587 5038 l 7625 5263 l 7655 5120 l 7737 5645 l 7760 5240 l 7790 5360 l 7805 5090 l 7842 5240 l 7887 5015 l gs col0 s gr 15.000 slw % Ellipse n 8775 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 7875 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 6900 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 6300 5400 50 50 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr $F2psEnd rs %%EndDocument endTexFig 1046 2475 a Fh(p)p 1115 2475 39 4 v 60 x Fg(")388 b(\034)1578 2547 y Ff(S)1825 2535 y Fg(\034)1870 2505 y Fe(0)2865 2535 y Fg(t)2062 2015 y(x)2109 2027 y Fd(t)2133 2724 y Fg(x)2180 2693 y Fe(0)2180 2744 y Fd(t)857 1826 y Fg(x)2558 1779 y Fh(D)1329 2334 y(S)268 4124 y Fc(Figure)32 b(3.)40 b Fb(Nils)28 b(Berglund)f(and)g(Barbara)e(Gen)n(tz)268 4224 y(Dynamic)i(pitc)n(hfork)g(bifurcations)g(with)h(additiv)n(e)f (noise)1867 5871 y Fa(1)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF %%EndDocument endTexFig 268 1802 a FA(Figure)31 b(3.)41 b FQ(Assume)26 b(the)i(path)e FP(x)1385 1814 y FM(t)1442 1802 y FQ(exits)h(the)g(region)e FO(S)34 b FQ(at)26 b(time)h FP(\034)2436 1814 y Fs(S)2486 1802 y FQ(,)g(sa)n(y)f(b)n(y)g(passing)g(through)g(the)268 1902 y(upp)r(er)h(b)r(oundary)g(of)h FO(S)6 b FQ(.)38 b(W)-7 b(e)28 b(in)n(tro)r(duce)g(a)f(pro)r(cess)g FP(x)2010 1872 y FN(0)2010 1922 y FM(t)2047 1902 y FQ(,)h(starting)f(on)h(the)g (same)f(b)r(oundary)g(at)h(time)268 2001 y FP(\034)304 2013 y Fs(S)353 2001 y FQ(,)34 b(whic)n(h)e(ob)r(eys)g(the)g(linear)g (SDE)i(\(4.72\))o(.)51 b(Let)32 b FP(\034)1954 1971 y FN(0)2025 2001 y FQ(b)r(e)h(the)f(time)h(of)f(\034rst)h(return)e(to)i (zero)e(of)h FP(x)3451 1971 y FN(0)3451 2022 y FM(t)3489 2001 y FQ(.)268 2101 y(Then)25 b FP(x)529 2113 y FM(t)583 2101 y FQ(lies)g(ab)r(o)n(v)n(e)f FP(x)1004 2071 y FN(0)1004 2122 y FM(t)1067 2101 y FQ(for)g FP(\034)1227 2113 y Fs(S)1300 2101 y FP(<)e(t)h Ft(6)g FP(\034)1573 2071 y FN(0)1611 2101 y FQ(.)36 b(In)25 b(case)f FP(x)1991 2113 y FM(t)2046 2101 y FQ(also)g(b)r(ecomes)h(negativ)n(e,)f(the)h(t)n (w)n(o)g(pro)r(cesses)268 2201 y(ma)n(y)j(cross)f(eac)n(h)h(other.)40 b(The)29 b(probabilit)n(y)f(of)h FP(x)1836 2171 y FN(0)1836 2221 y FM(t)1902 2201 y FQ(ev)n(er)f(returning)g(to)h(zero)f(is)g(b)r (ounded)i(b)n(y)e FP(\033)3316 2171 y FN(4)p FM(\024)3393 2201 y FQ(.)41 b(If)268 2300 y FP(x)315 2270 y FN(0)315 2321 y FM(t)380 2300 y FQ(do)r(es)27 b(not)h(return)f(to)g(zero,)g FP(x)1316 2312 y FM(t)1373 2300 y FQ(is)h(lik)n(ely)f(to)g(lea)n(v)n(e) f FO(D)r FQ(.)118 2578 y FB(and)34 b(denote)h(by)g FF(\034)745 2598 y FD(D)802 2579 y Fw(+)890 2578 y FB(the)g(\034rst)e(exit)i(time)f (of)g FF(x)1773 2592 y Fy(s)1844 2578 y FB(fr)-5 b(om)34 b FE(D)2132 2545 y FC(+)2191 2578 y FG(\()p FF(t)p FG(\))p FB(.)47 b(L)-5 b(et)35 b FF(\034)2577 2545 y FC(0)2650 2578 y FB(b)-5 b(e)36 b(the)f(time)f(of)g(\034rst)f(r)-5 b(eturn)118 2691 y(to)34 b(zer)-5 b(o)34 b(of)g FF(x)579 2658 y FC(0)579 2714 y Fy(s)652 2691 y FB(in)f FG([)p FF(t)822 2705 y FC(0)861 2691 y FF(;)15 b(t)p FG(])p FB(,)34 b(wher)-5 b(e)35 b(we)f(set)g FF(\034)1604 2658 y FC(0)1671 2691 y FG(=)28 b FE(1)33 b FB(if)g FF(x)2035 2658 y FC(0)2035 2714 y Fy(s)2102 2691 y FF(>)28 b FG(0)34 b FB(for)f(al)5 b(l)33 b FF(t)28 b FE(2)f FG([)p FF(t)2762 2705 y FC(0)2802 2691 y FF(;)15 b(t)p FG(])p FB(.)45 b(Then)34 b FF(x)3262 2705 y Fy(s)3326 2691 y Fn(>)28 b FF(x)3477 2658 y FC(0)3477 2714 y Fy(s)3550 2691 y FB(for)118 2804 y(al)5 b(l)32 b FF(s)25 b Fn(6)g FF(\034)451 2823 y FD(D)508 2804 y Fw(+)i FE(^)20 b FF(t)32 b FB(and)233 3028 y Fo(P)294 2990 y Fy(t)319 2999 y Fw(0)355 2990 y Fy(;x)415 2999 y Fw(0)453 2927 y Fr(n)513 3028 y FG(0)26 b FF(<)f(x)732 3042 y Fy(s)794 3028 y FF(<)31 b FG(~)-51 b FF(x)p FG(\()p FF(s)p FG(\))25 b FE(8)p FF(s)g FE(2)g FG([)p FF(t)1343 3042 y FC(0)1382 3028 y FF(;)15 b(t)p FG(])p FF(;)g(\034)1570 2990 y FC(0)1636 3028 y FG(=)25 b FE(1)1823 2927 y Fr(o)1909 3028 y Fn(6)g Fo(P)2066 2990 y Fy(t)2091 2999 y Fw(0)2126 2990 y Fy(;x)2186 2999 y Fw(0)2224 2927 y Fr(n)2285 3028 y FG(0)g FF(<)g(x)2503 2990 y FC(0)2503 3050 y Fy(s)2568 3028 y FF(<)31 b FG(~)-51 b FF(x)p FG(\()p FF(s)p FG(\))26 b FE(8)p FF(s)e FE(2)h FG([)p FF(t)3117 3042 y FC(0)3156 3028 y FF(;)15 b(t)p FG(])3254 2927 y Fr(o)1909 3268 y Fn(6)2020 3205 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))2169 3127 y Fr(p)p 2261 3127 192 4 v 2261 3205 a FF(a)2309 3219 y FC(0)2348 3205 y FG(\()p FF(t)p FG(\))p 2014 3247 438 4 v 2140 3265 a FE(p)p 2216 3265 56 4 v 66 x FF(\031)s(\033)2613 3207 y FG(e)2654 3174 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;t)2892 3183 y Fw(0)2927 3174 y FC(\))p Fy(=")p 2472 3247 697 4 v 2472 3265 a Fr(p)p 2563 3265 606 4 v 90 x FG(1)20 b FE(\000)g FG(e)2760 3328 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3033 3337 y Fw(0)3068 3328 y FC(\))p Fy(=")3178 3268 y FF(:)3430 3170 y FH(\(4.74\))118 3608 y Fh(Pr)m(oof:)156 3744 y FH(1.)47 b(Let)31 b FF(g)s FG(\()p FF(x;)15 b(s)p FG(\))27 b(=)e FF(f)10 b FG(\()p FF(x;)15 b(s)p FG(\))20 b FE(\000)g FF(a)1228 3758 y FC(0)1268 3744 y FG(\()p FF(s)p FG(\))p FF(x)p FH(.)40 b(By)30 b(assumption,)g FF(g)s FG(\()p FF(x;)15 b(s)p FG(\))31 b FH(is)e(non-negativ)m(e)i(for) g FG(\()p FF(x;)15 b(s)p FG(\))26 b FE(2)f(D)3578 3711 y FC(+)3636 3744 y FH(.)273 3857 y(The)31 b(di\033erence)f FF(z)905 3871 y Fy(s)968 3857 y FG(=)25 b FF(x)1116 3871 y Fy(s)1173 3857 y FE(\000)20 b FF(x)1316 3824 y FC(0)1316 3880 y Fy(s)1385 3857 y FH(satis\034es)29 b(the)i(equation)1200 4105 y FF(z)1242 4119 y Fy(s)1304 4105 y FG(=)25 b FF(z)1442 4119 y Fy(t)1467 4128 y Fw(0)1527 4105 y FG(+)1628 4044 y(1)p 1628 4084 46 4 v 1630 4168 a FF(")1698 3982 y Fr(Z)1789 4008 y Fy(s)1749 4188 y(t)1774 4197 y Fw(0)1826 4032 y Fr(\002)1864 4105 y FF(g)s FG(\()p FF(x)1997 4119 y Fy(u)2043 4105 y FF(;)15 b(u)p FG(\))21 b(+)f FF(a)2330 4119 y FC(0)2369 4105 y FG(\()p FF(u)p FG(\))p FF(z)2533 4119 y Fy(u)2579 4032 y Fr(\003)2632 4105 y FG(d)p FF(u)695 b FH(\(4.75\))273 4365 y(with)30 b FF(z)521 4379 y Fy(t)546 4388 y Fw(0)611 4365 y FG(=)25 b(0)p FH(.)41 b(Since)30 b FF(g)s FG(\()p FF(x)1187 4379 y Fy(s)1224 4365 y FF(;)15 b(s)p FG(\))26 b Fn(>)f FG(0)31 b FH(for)f FF(t)1711 4379 y FC(0)1775 4365 y Fn(6)25 b FF(s)g Fn(6)g FF(\034)2075 4384 y FD(D)2132 4365 y Fw(+)i FE(^)20 b FF(t)p FH(,)1426 4613 y FF(z)1468 4627 y Fy(s)1531 4613 y Fn(>)25 b FF(z)1669 4627 y Fy(t)1694 4636 y Fw(0)1753 4613 y FG(+)1854 4552 y(1)p 1854 4592 V 1856 4676 a FF(")1925 4490 y Fr(Z)2016 4516 y Fy(s)1975 4696 y FC(0)2068 4613 y FF(a)2116 4627 y FC(0)2155 4613 y FG(\()p FF(u)p FG(\))p FF(z)2319 4627 y Fy(u)2380 4613 y FG(d)p FF(u;)922 b FH(\(4.76\))273 4858 y(follo)m(ws)29 b(for)h(all)f(suc)m(h)i FF(s)e FH(and,)i (therefore,)h(Gron)m(w)m(all's)e(lemma)e(yields)1004 5062 y FF(z)1046 5076 y Fy(s)1109 5062 y Fn(>)d FF(z)1247 5076 y Fy(t)1272 5085 y Fw(0)1326 5062 y FG(e)1367 5025 y Fy(\024\013)p FC(\()p Fy(s;t)1558 5034 y Fw(0)1592 5025 y FC(\))p Fy(=")1717 5062 y FG(=)g(0)181 b FH(for)31 b(all)d FF(s)d FE(2)g FG([)p FF(t)2515 5076 y FC(0)2554 5062 y FF(;)15 b(\034)2634 5082 y FD(D)2691 5063 y Fw(+)2766 5062 y FE(^)20 b FF(t)p FG(])p FF(:)500 b FH(\(4.77\))273 5267 y(This)36 b(sho)m(ws)h FF(x)804 5281 y Fy(s)876 5267 y Fn(>)e FF(x)1034 5234 y FC(0)1034 5289 y Fy(s)1110 5267 y FH(for)h(those)h FF(s)p FH(.)58 b(No)m(w)37 b(assume)f FF(\034)2194 5286 y FD(D)2251 5267 y Fw(+)42 b FG(=)36 b FE(1)g FH(and)h FF(\034)2806 5234 y FC(0)2881 5267 y FG(=)e FE(1)p FH(.)59 b(Then,)38 b(\(4.77\))273 5379 y(implies)22 b(that)27 b FG(0)e FF(<)g(x)985 5346 y FC(0)985 5402 y Fy(s)1050 5379 y Fn(6)g FF(x)1198 5393 y Fy(s)1260 5379 y FF(<)30 b FG(~)-50 b FF(x)p FG(\()p FF(s)p FG(\))25 b FH(for)h(all)e FF(s)g Fn(6)h FF(t)p FH(,)i(whic)m(h)f(sho)m(ws)f(the) h(\034rst)g(inequalit)m(y)e(in)g(\(4.74\))q(.)1845 5871 y(33)p eop %%Page: 34 34 34 33 bop 156 328 a FH(2.)47 b FF(x)325 295 y FC(0)325 350 y Fy(s)395 328 y FH(b)s(eing)29 b(distributed)h(according)h(to)f(a) h(normal)e(la)m(w,)h(w)m(e)h(ha)m(v)m(e)826 494 y Fo(P)887 457 y Fy(t)912 466 y Fw(0)948 457 y Fy(;x)1008 466 y Fw(0)1046 421 y Fr(\010)1099 494 y FG(0)26 b FF(<)f(x)1318 457 y FC(0)1318 517 y Fy(s)1382 494 y FF(<)31 b FG(\026)-51 b FF(x)p FG(\()p FF(s)p FG(\))31 b FE(8)p FF(s)24 b FE(2)h FG([)p FF(t)1936 508 y FC(0)1975 494 y FF(;)15 b(t)p FG(])2073 421 y Fr(\011)2152 494 y Fn(6)25 b Fo(P)2309 457 y Fy(t)2334 466 y Fw(0)2370 457 y Fy(;x)2430 466 y Fw(0)2468 421 y Fr(\010)2521 494 y FG(0)g FF(<)g(x)2739 457 y FC(0)2739 517 y Fy(t)2804 494 y FF(<)31 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))3055 421 y Fr(\011)2152 691 y Fn(6)2432 630 y FG(~)g FF(x)p FG(\()p FF(t)p FG(\))p 2258 670 492 4 v 2258 688 a Fr(p)p 2349 688 401 4 v 78 x FG(2)p FF(\031)s(v)2493 780 y FC(0)2533 766 y FG(\()p FF(t;)15 b(t)2674 780 y FC(0)2714 766 y FG(\))2759 691 y FF(;)3430 627 y FH(\(4.78\))273 926 y(where)35 b(the)f(v)-5 b(ariance)34 b FF(v)1099 940 y FC(0)1139 926 y FG(\()p FF(t;)15 b(t)1280 940 y FC(0)1319 926 y FG(\))35 b FH(can)f(b)s(e)g (estimated)e(b)m(y)j(Lemma)e(4.6.)52 b(This)33 b(pro)m(v)m(es)h(the)h (second)273 1039 y(inequalit)m(y)29 b(in)g(\(4.74\))q(.)p 3596 1039 4 62 v 3600 981 55 4 v 3600 1039 V 3653 1039 4 62 v 259 1221 a(The)d(previous)f(lemma)f(is)g(useful)g(only)h(if)f(w) m(e)i(can)g(con)m(trol)g(the)g(probabilit)m(y)f(that)h(the)g(solution)e FF(x)3622 1188 y FC(0)3622 1243 y Fy(t)118 1333 y FH(of)33 b(the)g(linearized)e(equation)i(returns)g(to)g(zero.)49 b(The)33 b(follo)m(wing)e(result)h(estimates)g(this)f(probabilit)m(y) 118 1446 y(and)g(its)e(densit)m(y)-8 b(.)118 1604 y Fq(Lemma)35 b(4.9.)41 b FB(L)-5 b(et)34 b FF(t)881 1618 y FC(0)948 1604 y Fn(>)1046 1539 y FE(p)p 1122 1539 43 4 v 65 x FF(")f FB(and)h(assume)g(that)f FF(x)1928 1571 y FC(0)1928 1627 y Fy(t)1953 1636 y Fw(0)2020 1604 y FG(=)27 b FF(\032)g(>)g(\033)s (=)2390 1526 y Fr(p)p 2482 1526 231 4 v 2482 1604 a FF(a)2530 1618 y FC(0)2569 1604 y FG(\()p FF(t)2637 1618 y FC(0)2677 1604 y FG(\))p FB(.)45 b(Denote)34 b(by)g FF(\034)3265 1571 y FC(0)3338 1604 y FB(the)g(time)118 1717 y(of)e(the)h(\034rst)f (r)-5 b(eturn)33 b(of)f FF(x)994 1684 y FC(0)994 1740 y Fy(t)1066 1717 y FB(to)g(zer)-5 b(o.)42 b(Then)32 b(we)h(have)309 1897 y Fo(P)370 1859 y Fy(t)395 1868 y Fw(0)431 1859 y Fy(;\032)490 1897 y FE(f)p FF(\034)585 1859 y FC(0)651 1897 y FF(<)25 b(t)p FE(g)g Fn(6)g Fo(P)1007 1859 y Fy(t)1032 1868 y Fw(0)1068 1859 y Fy(;\032)1127 1897 y FE(f)p FF(\034)1222 1859 y FC(0)1288 1897 y FF(<)g FE(1g)g Fn(6)g FG(e)1682 1859 y FD(\000)p Fy(a)1774 1868 y Fw(0)1809 1859 y FC(\()p Fy(t)1861 1868 y Fw(0)1896 1859 y FC(\))p Fy(\032)1959 1836 y Fw(2)1994 1859 y Fy(=\033)2071 1836 y Fw(2)3430 1897 y FH(\(4.79\))240 2046 y FG(d)p 216 2087 84 4 v 216 2170 a(d)o FF(t)309 2108 y Fo(P)370 2070 y Fy(t)395 2079 y Fw(0)431 2070 y Fy(;\032)490 2108 y FE(f)p FF(\034)585 2070 y FC(0)651 2108 y FF(<)g(t)p FE(g)g Fn(6)999 2046 y FG(2)p 956 2087 131 4 v 956 2105 a FE(p)p 1032 2105 56 4 v 66 x FF(\031)1097 2026 y Fr(p)p 1188 2026 231 4 v 82 x FF(a)1236 2122 y FC(0)1275 2108 y FG(\()p FF(t)1343 2122 y FC(0)1383 2108 y FG(\))1447 2046 y FF(\032)p 1443 2087 56 4 v 1443 2170 a(\033)1524 2108 y FG(e)1564 2070 y FD(\000)p Fy(a)1656 2079 y Fw(0)1691 2070 y FC(\()p Fy(t)1743 2079 y Fw(0)1778 2070 y FC(\))p Fy(\032)1841 2047 y Fw(2)1877 2070 y Fy(=\033)1954 2047 y Fw(2)2019 2046 y FG(1)p 2019 2087 46 4 v 2021 2170 a FF(")2074 2026 y Fr(p)p 2165 2026 422 4 v 82 x FF(a)2213 2122 y FC(0)2252 2108 y FG(\()p FF(t)p FG(\))p FF(a)2403 2122 y FC(0)2443 2108 y FG(\()p FF(t)2511 2122 y FC(0)2551 2108 y FG(\))2735 2046 y(e)2776 2013 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3049 2022 y Fw(0)3084 2013 y FC(\))p Fy(=")p 2611 2087 697 4 v 2611 2105 a Fr(p)p 2702 2105 606 4 v 90 x FG(1)c FE(\000)f FG(e)2899 2168 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3172 2177 y Fw(0)3208 2168 y FC(\))p Fy(=")3318 2108 y FF(:)87 b FH(\(4.80\))118 2406 y Fh(Pr)m(oof:)156 2543 y FH(1.)47 b(Since)36 b(b)m(y)g(symmetry)-8 b(,)36 b Fo(P)1154 2510 y Fy(\034)1193 2486 y Fw(0)1228 2510 y Fy(;)p FC(0)1287 2543 y FE(f)p FF(x)1384 2510 y FC(0)1384 2565 y Fy(t)1459 2543 y Fn(>)f FG(0)p FE(g)h FG(=)1806 2507 y FC(1)p 1806 2522 36 4 v 1806 2574 a(2)1887 2543 y FH(on)g FE(f)p FF(\034)2113 2510 y FC(0)2188 2543 y FF(<)f(t)p FE(g)p FH(,)j(w)m(e)e(ha)m(v)m(e)h(b)m(y)g(the)f(strong)h(Mark)m(o)m(v)273 2655 y(prop)s(ert)m(y)1462 2802 y Fo(P)1523 2764 y Fy(t)1548 2773 y Fw(0)1583 2764 y Fy(;\032)1643 2802 y FE(f)p FF(x)1740 2764 y FC(0)1740 2824 y Fy(t)1805 2802 y Fn(>)25 b FG(0)p FE(j)p FF(\034)2021 2764 y FC(0)2087 2802 y FF(<)g(t)p FE(g)g FG(=)2392 2740 y(1)p 2392 2781 46 4 v 2392 2864 a(2)2448 2802 y FF(:)957 b FH(\(4.81\))273 2971 y(W)-8 b(e)31 b(no)m(w)g(observ)m(e)g(that)555 3115 y Fo(P)616 3078 y Fy(t)641 3087 y Fw(0)677 3078 y Fy(;\032)737 3115 y FE(f)p FF(x)834 3078 y FC(0)834 3138 y Fy(t)899 3115 y Fn(>)25 b FG(0)p FE(g)h FG(=)f Fo(P)1268 3078 y Fy(t)1293 3087 y Fw(0)1328 3078 y Fy(;\032)1388 3115 y FE(f)p FF(x)1485 3078 y FC(0)1485 3138 y Fy(t)1550 3115 y Fn(>)g FG(0)p FF(;)15 b(\034)1781 3078 y FC(0)1847 3115 y Fn(>)25 b FF(t)p FE(g)20 b FG(+)g Fo(P)2193 3078 y Fy(t)2218 3087 y Fw(0)2254 3078 y Fy(;\032)2313 3115 y FE(f)p FF(x)2410 3078 y FC(0)2410 3138 y Fy(t)2476 3115 y Fn(>)25 b FG(0)p FF(;)15 b(\034)2707 3078 y FC(0)2772 3115 y FF(<)25 b(t)p FE(g)1111 3263 y FG(=)g Fo(P)1268 3225 y Fy(t)1293 3234 y Fw(0)1328 3225 y Fy(;\032)1388 3263 y FE(f)p FF(\034)1483 3225 y FC(0)1548 3263 y Fn(>)g FF(t)p FE(g)c FG(+)f Fo(P)1895 3225 y Fy(t)1920 3234 y Fw(0)1955 3225 y Fy(;\032)2015 3263 y FE(f)p FF(x)2112 3225 y FC(0)2112 3285 y Fy(t)2177 3263 y Fn(>)25 b FG(0)p FE(j)p FF(\034)2393 3225 y FC(0)2459 3263 y FF(<)f(t)p FE(g)p Fo(P)2693 3225 y Fy(t)2718 3234 y Fw(0)2754 3225 y Fy(;\032)2814 3263 y FE(f)p FF(\034)2909 3225 y FC(0)2974 3263 y FF(<)h(t)p FE(g)1111 3410 y FG(=)g(1)20 b FE(\000)g Fo(P)1424 3372 y Fy(t)1449 3381 y Fw(0)1485 3372 y Fy(;\032)1545 3410 y FE(f)p FF(\034)1640 3372 y FC(0)1705 3410 y FF(<)25 b(t)p FE(g)20 b FG(+)2000 3374 y FC(1)p 2000 3389 36 4 v 2000 3441 a(2)2045 3410 y Fo(P)2106 3372 y Fy(t)2131 3381 y Fw(0)2167 3372 y Fy(;\032)2227 3410 y FE(f)p FF(\034)2322 3372 y FC(0)2387 3410 y FF(<)25 b(t)p FE(g)1111 3557 y FG(=)g(1)20 b FE(\000)1373 3522 y FC(1)p 1373 3537 V 1373 3589 a(2)1418 3557 y Fo(P)1479 3520 y Fy(t)1504 3529 y Fw(0)1540 3520 y Fy(;\032)1600 3557 y FE(f)p FF(\034)1695 3520 y FC(0)1760 3557 y FF(<)25 b(t)p FE(g)p FF(;)3430 3336 y FH(\(4.82\))273 3718 y(whic)m(h)31 b(implies)884 3886 y Fo(P)945 3849 y Fy(t)970 3858 y Fw(0)1005 3849 y Fy(;\032)1065 3886 y FE(f)p FF(\034)1160 3849 y FC(0)1225 3886 y FF(<)25 b(t)p FE(g)h FG(=)f(2)1566 3813 y Fr(\002)1604 3886 y FG(1)c FE(\000)f Fo(P)1822 3849 y Fy(t)1847 3858 y Fw(0)1882 3849 y Fy(;\032)1942 3886 y FE(f)p FF(x)2039 3849 y FC(0)2039 3909 y Fy(t)2104 3886 y Fn(>)25 b FG(0)p FE(g)2290 3813 y Fr(\003)2354 3886 y FG(=)g(2)p Fo(P)2556 3849 y Fy(t)2581 3858 y Fw(0)2617 3849 y Fy(;\032)2677 3886 y FE(f)p FF(x)2774 3849 y FC(0)2774 3909 y Fy(t)2839 3886 y FF(<)g FG(0)p FE(g)p FF(:)380 b FH(\(4.83\))156 4065 y(2.)47 b(Next,)25 b(w)m(e)g(use)f(that)h FF(x)1033 4032 y FC(0)1033 4087 y Fy(t)1096 4065 y FH(is)d(a)i (Gaussian)f(random)i(v)-5 b(ariable)22 b(with)i(mean)g FF(\032)15 b FG(e)2826 4032 y Fy(\024\013)p FC(\()p Fy(t;t)3009 4041 y Fw(0)3045 4032 y FC(\))p Fy(=")3168 4065 y FH(and)24 b(v)-5 b(ariance)1374 4290 y FF(v)1418 4304 y FC(0)1458 4290 y FG(\()p FF(t;)15 b(t)1599 4304 y FC(0)1639 4290 y FG(\))25 b(=)1805 4228 y FF(\033)1860 4195 y FC(2)p 1805 4269 95 4 v 1831 4352 a FF(")1925 4166 y Fr(Z)2016 4192 y Fy(t)1975 4372 y(t)2000 4381 y Fw(0)2061 4290 y FG(e)2101 4252 y FC(2)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2442 4290 y FG(d)p FF(s:)869 b FH(\(4.84\))273 4508 y(By)30 b(Lemma)g(4.6,)1373 4678 y FG(\004)25 b(=)1565 4616 y FF(\032)1612 4583 y FC(2)1666 4616 y FG(e)1707 4583 y FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)1925 4592 y Fw(0)1961 4583 y FC(\))p Fy(=")p 1565 4657 496 4 v 1640 4740 a FG(2)p FF(v)1729 4754 y FC(0)1769 4740 y FG(\()p FF(t;)15 b(t)1910 4754 y FC(0)1950 4740 y FG(\))2095 4678 y Fn(>)25 b FF(a)2239 4692 y FC(0)2279 4678 y FG(\()p FF(t)2347 4692 y FC(0)2386 4678 y FG(\))2435 4616 y FF(\032)2482 4583 y FC(2)p 2431 4657 95 4 v 2431 4740 a FF(\033)2486 4714 y FC(2)2536 4678 y FF(;)869 b FH(\(4.85\))273 4870 y(and)31 b(w)m(e)g(th)m(us)g(ha)m(v)m(e)700 5078 y Fo(P)761 5041 y Fy(t)786 5050 y Fw(0)821 5041 y Fy(;\032)881 5078 y FE(f)p FF(x)978 5041 y FC(0)978 5101 y Fy(t)1043 5078 y FF(<)25 b FG(0)p FE(g)h FG(=)1584 5017 y(1)p 1361 5057 492 4 v 1361 5076 a Fr(p)p 1452 5076 401 4 v 77 x FG(2)p FF(\031)s(v)1596 5167 y FC(0)1636 5153 y FG(\()p FF(t;)15 b(t)1777 5167 y FC(0)1817 5153 y FG(\))1878 4955 y Fr(Z)1968 4981 y FC(0)1928 5161 y FD(\0001)2073 5078 y FG(exp)2212 4977 y Fr(n)2272 5078 y FE(\000)2353 5017 y FG(\()p FF(x)20 b FE(\000)g FF(\032)15 b FG(e)2654 4984 y Fy(\024\013)p FC(\()p Fy(t;t)2837 4993 y Fw(0)2873 4984 y FC(\))p Fy(=")2972 5017 y FG(\))3007 4984 y FC(2)p 2353 5057 694 4 v 2527 5141 a FG(2)p FF(v)2616 5155 y FC(0)2656 5141 y FG(\()p FF(t;)g(t)2797 5155 y FC(0)2837 5141 y FG(\))3057 4977 y Fr(o)3133 5078 y FG(d)o FF(x)1255 5401 y FG(=)1427 5339 y(1)p 1361 5380 177 4 v 1361 5398 a FE(p)p 1437 5398 101 4 v 75 x FG(2)p FF(\031)1563 5277 y Fr(Z)1654 5303 y FD(\000)1719 5272 y Fj(\032)10 b Fw(e)1788 5252 y Fj(\024\013)p Fw(\()p Fj(t;t)1955 5267 y Fw(0)1989 5252 y(\))p Fj(=")p 1718 5288 361 3 v 1762 5297 a FD(p)p 1821 5297 215 3 v 45 x Fj(v)1852 5357 y Fw(0)1886 5342 y(\()p Fj(t;t)1977 5357 y Fw(0)2011 5342 y(\))1613 5483 y FD(\0001)2108 5401 y FG(e)2148 5363 y FD(\000)p Fy(y)2240 5340 y Fw(2)2275 5363 y Fy(=)p FC(2)2365 5401 y FG(d)o FF(y)28 b Fn(6)2594 5339 y FG(1)p 2594 5380 46 4 v 2594 5463 a(2)2665 5401 y(e)2705 5363 y FD(\000)p FC(\004)2811 5401 y FF(;)594 b FH(\(4.86\))273 5622 y(whic)m(h)31 b(pro)m(v)m(es)g(\(4.79\))q(,)f(using)h(\(4.83\))g(and)g(\(4.85\))q(.) 1845 5871 y(34)p eop %%Page: 35 35 35 34 bop 156 328 a FH(3.)47 b(In)30 b(order)h(to)g(compute)f(the)h (deriv)-5 b(ativ)m(e)29 b(of)h Fo(P)1833 295 y Fy(t)1858 304 y Fw(0)1894 295 y Fy(;\032)1954 328 y FE(f)p FF(x)2051 295 y FC(0)2051 350 y Fy(t)2116 328 y FF(<)25 b FG(0)p FE(g)p FH(,)31 b(w)m(e)g(\034rst)g(note)f(that)1336 479 y FG(d)p 1312 519 84 4 v 1312 603 a(d)o FF(t)1405 540 y(v)1449 554 y FC(0)1489 540 y FG(\()p FF(t;)15 b(t)1630 554 y FC(0)1669 540 y FG(\))26 b(=)1836 479 y FF(\033)1891 446 y FC(2)p 1836 519 95 4 v 1862 603 a FF(")1961 540 y FG(+)2062 479 y(2)p FF(a)2155 493 y FC(0)2195 479 y FG(\()p FF(t)p FG(\))p 2062 519 237 4 v 2159 603 a FF(")2308 540 y(v)2352 554 y FC(0)2392 540 y FG(\()p FF(t;)15 b(t)2533 554 y FC(0)2572 540 y FG(\))p FF(:)798 b FH(\(4.87\))273 723 y(Di\033eren)m(tiating)30 b(the)g(second)h(line)e(of)37 b(\(4.86\))q(,)30 b(w)m(e)h(get)380 882 y FG(d)p 356 922 84 4 v 356 1005 a(d)p FF(t)449 943 y Fo(P)510 906 y Fy(t)535 915 y Fw(0)571 906 y Fy(;\032)631 943 y FE(f)p FF(x)728 906 y FC(0)728 966 y Fy(t)793 943 y FF(<)25 b FG(0)p FE(g)h FG(=)1176 882 y(1)p 1111 922 177 4 v 1111 940 a FE(p)p 1187 940 101 4 v 76 x FG(2)p FF(\031)1312 943 y FG(exp)1451 815 y Fr(\032)1519 943 y FE(\000)1600 882 y FF(\032)1647 849 y FC(2)1702 882 y FG(e)1742 849 y FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)1960 858 y Fw(0)1996 849 y FC(\))p Fy(=")p 1600 922 496 4 v 1675 1005 a FG(2)p FF(v)1764 1019 y FC(0)1804 1005 y FG(\()p FF(t;)15 b(t)1945 1019 y FC(0)1985 1005 y FG(\))2105 815 y Fr(\033)2207 882 y FG(d)p 2184 922 84 4 v 2184 1005 a(d)o FF(t)2277 815 y Fr(\024)2325 943 y FE(\000)2406 882 y FF(\032)g FG(e)2508 849 y Fy(\024\013)p FC(\()p Fy(t;t)2691 858 y Fw(0)2727 849 y FC(\))p Fy(=")p 2406 922 421 4 v 2420 940 a Fr(p)p 2511 940 300 4 v 78 x FF(v)2555 1032 y FC(0)2595 1018 y FG(\()p FF(t;)g(t)2736 1032 y FC(0)2776 1018 y FG(\))2836 815 y Fr(\025)1005 1223 y FG(=)1176 1162 y(1)p 1111 1202 177 4 v 1111 1221 a FE(p)p 1187 1221 101 4 v 75 x FG(2)p FF(\031)1312 1223 y FG(e)1353 1186 y FD(\000)p FC(\004)1484 1162 y FF(\032)p 1484 1202 48 4 v 1485 1286 a FG(2)1551 1162 y FF(\033)1606 1129 y FC(2)p 1551 1202 95 4 v 1577 1286 a FF(")1691 1162 y FG(e)1731 1129 y Fy(\024\013)p FC(\()p Fy(t;t)1914 1138 y Fw(0)1950 1129 y FC(\))p Fy(=")p 1665 1202 410 4 v 1665 1289 a FF(v)1709 1303 y FC(0)1749 1289 y FG(\()p FF(t;)g(t)1890 1303 y FC(0)1930 1289 y FG(\))1965 1263 y FC(3)p Fy(=)p FC(2)1005 1490 y FG(=)1176 1428 y(1)p 1111 1469 177 4 v 1111 1487 a FE(p)p 1187 1487 101 4 v 75 x FG(2)p FF(\031)1308 1428 y FG(1)p 1307 1469 48 4 v 1307 1552 a FF(\032)1374 1428 y(\033)1429 1395 y FC(2)p 1374 1469 95 4 v 1400 1552 a FF(")1489 1428 y FG(e)1529 1395 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;t)1767 1404 y Fw(0)1803 1395 y FC(\))p Fy(=")p 1489 1469 414 4 v 1500 1487 a Fr(p)p 1591 1487 300 4 v 78 x FF(v)1635 1579 y FC(0)1674 1565 y FG(\()p FF(t;)g(t)1815 1579 y FC(0)1855 1565 y FG(\))1912 1490 y(\004)g(e)2028 1452 y FD(\000)p FC(\004)3430 1490 y FH(\(4.88\))1005 1770 y Fn(6)1176 1708 y FG(1)p 1111 1749 177 4 v 1111 1767 a FE(p)p 1187 1767 101 4 v 75 x FG(2)p FF(\031)1297 1688 y Fr(p)p 1388 1688 231 4 v 82 x FF(a)1436 1784 y FC(0)1476 1770 y FG(\()p FF(t)1544 1784 y FC(0)1583 1770 y FG(\))1648 1708 y FF(\032)p 1644 1749 56 4 v 1644 1832 a(\033)1724 1770 y FG(e)1764 1732 y FD(\000)p Fy(a)1856 1741 y Fw(0)1891 1732 y FC(\()p Fy(t)1943 1741 y Fw(0)1979 1732 y FC(\))p Fy(\032)2042 1709 y Fw(2)2077 1732 y Fy(=\033)2154 1709 y Fw(2)2219 1708 y FG(1)p 2219 1749 46 4 v 2221 1832 a FF(")2274 1688 y Fr(p)p 2365 1688 467 4 v 82 x FG(2)p FF(a)2458 1784 y FC(0)2498 1770 y FG(\()p FF(t)p FG(\))p FF(a)2649 1784 y FC(0)2689 1770 y FG(\()p FF(t)2757 1784 y FC(0)2797 1770 y FG(\))2981 1708 y(e)3021 1675 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3294 1684 y Fw(0)3330 1675 y FC(\))p Fy(=")p 2857 1749 697 4 v 2857 1767 a Fr(p)p 2948 1767 606 4 v 90 x FG(1)21 b FE(\000)f FG(e)3145 1830 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3418 1839 y Fw(0)3454 1830 y FC(\))p Fy(=")3563 1770 y FF(;)273 2004 y FH(where)34 b(w)m(e)g(ha)m(v)m(e)h(used)e(the)h (facts)f(that)h FG(\004)c FF(>)g(a)1909 2018 y FC(0)1948 2004 y FG(\()p FF(t)2016 2018 y FC(0)2056 2004 y FG(\))p FF(\032)2138 1971 y FC(2)2177 2004 y FF(=\033)2277 1971 y FC(2)2348 2004 y FF(>)g FG(1)j FH(and)h(that)g FG(\004)15 b(e)3021 1971 y FD(\000)p FC(\004)3160 2004 y FH(is)32 b(decreasing)273 2117 y(for)e FG(\004)25 b FF(>)g FG(1)p FH(.)41 b(No)m(w,)31 b(\(4.80\))g(follo)m(ws)e(from)g(\(4.83\))q(.)p 3596 2117 4 62 v 3600 2059 55 4 v 3600 2117 V 3653 2117 4 62 v 259 2296 a(Assume)c(for)i(the)f(momen)m(t)g(that)h FF(x)1465 2263 y FC(0)1465 2319 y Fy(t)1531 2296 y FH(starts)f(\020on)h (the)g(b)s(order\021)34 b(of)26 b FE(S)7 b FH(,)27 b(i.e.)39 b(in)26 b FF(\032)p FG(\()p FF(t)2990 2310 y FC(0)3029 2296 y FG(\))g(=)f FF(h=)3283 2219 y Fr(p)p 3374 2219 192 4 v 77 x FF(a)p FG(\()p FF(t)3490 2310 y FC(0)3530 2296 y FG(\))h(=)118 2344 y FE(p)p 194 2344 53 4 v 65 x FF(\024h=)343 2332 y Fr(p)p 435 2332 231 4 v 435 2409 a FF(a)483 2423 y FC(0)522 2409 y FG(\()p FF(t)590 2423 y FC(0)630 2409 y FG(\))p FH(.)84 b(Then,)50 b(b)m(y)45 b(our)g(c)m(hoice)g FF(h)50 b FG(=)f(2)p FF(\033)1974 2332 y Fr(p)p 2066 2332 239 4 v 2066 2409 a FE(j)p FG(log)17 b FF(\033)s FE(j)p FH(,)49 b(Estimate)44 b(\(4.79\))h(sho)m(ws)g(that)h (the)118 2534 y(probabilit)m(y)29 b(for)h FF(x)770 2501 y FC(0)770 2556 y Fy(t)840 2534 y FH(to)g(return)i(to)e(zero)h(cannot)g (exceed)g FG(e)2153 2501 y FD(\000)p Fy(a)2245 2510 y Fw(0)2280 2501 y FC(\()p Fy(t)2332 2510 y Fw(0)2368 2501 y FC(\))p Fy(\032)2431 2477 y Fw(2)2466 2501 y Fy(=\033)2543 2477 y Fw(2)2608 2534 y FG(=)25 b FF(\033)2759 2501 y FC(4)p Fy(\024)2839 2534 y FH(.)259 2647 y(W)-8 b(e)35 b(are)h(no)m(w)g(ready)f(to)g(pro)m(v)m(e)h(the)f(main)f(estimate)f(on) i(the)h(\034rst)f(exit)e(time)h FF(\034)3056 2661 y FD(D)3116 2647 y FH(,)i(whic)m(h)f(is)f(the)118 2760 y(most)c(imp)s(ortan)m(t)h (of)g(our)g(results.)42 b(Since)31 b(the)g(pro)s(of)g(is)f(rather)i(in) m(v)m(olv)m(ed,)f(w)m(e)h(restate)f(Theorem)h(2.9)118 2873 y(here)f(for)f(con)m(v)m(enience.)118 3021 y Fq(Prop)s(osition)35 b(4.10)g(\()p FH(Theorem)c(2.9)p Fq(\).)42 b FB(L)-5 b(et)33 b FF(t)1736 3035 y FC(0)1800 3021 y Fn(>)1896 2956 y FE(p)p 1972 2956 43 4 v 65 x FF(")g FB(and)f FE(j)p FF(x)2299 3035 y FC(0)2338 3021 y FE(j)26 b Fn(6)31 b FG(~)-51 b FF(x)p FG(\()p FF(t)2605 3035 y FC(0)2644 3021 y FG(\))p FB(.)42 b(Then)367 3241 y Fo(P)428 3203 y Fy(t)453 3212 y Fw(0)489 3203 y Fy(;x)549 3212 y Fw(0)587 3167 y Fr(\010)640 3241 y FF(\034)680 3255 y FD(D)766 3241 y Fn(>)25 b FF(t)895 3167 y Fr(\011)973 3241 y Fn(6)g FF(C)1134 3255 y FC(0)1194 3241 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))1343 3159 y Fr(p)p 1435 3159 152 4 v 1435 3241 a FF(a)p FG(\()p FF(t)p FG(\))1606 3179 y FE(j)p FG(log)17 b FF(\033)s FE(j)p 1606 3220 239 4 v 1698 3303 a FF(\033)1855 3113 y Fr(\022)1922 3241 y FG(1)j(+)2088 3179 y FF(\013)p FG(\()p FF(t;)15 b(t)2287 3193 y FC(0)2327 3179 y FG(\))p 2088 3220 275 4 v 2204 3303 a FF(")2373 3113 y Fr(\023)2591 3179 y FG(e)2631 3146 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;t)2869 3155 y Fw(0)2905 3146 y FC(\))p Fy(=")p 2450 3220 697 4 v 2450 3238 a Fr(p)p 2541 3238 606 4 v 89 x FG(1)20 b FE(\000)g FG(e)2738 3301 y FD(\000)p FC(2)p Fy(\024\013)p FC(\()p Fy(t;t)3011 3310 y Fw(0)3046 3301 y FC(\))p Fy(=")3156 3241 y FF(;)249 b FH(\(4.89\))118 3464 y FB(wher)-5 b(e)34 b FF(C)438 3478 y FC(0)502 3464 y FF(>)25 b FG(0)33 b FB(is)f(a)g(\(numeric)-5 b(al\))34 b(c)-5 b(onstant.)259 3613 y FH(The)32 b(strategy)f(of)g(the) g(pro)s(of)g(can)g(b)s(e)g(summarized)e(as)i(follo)m(ws.)41 b(The)32 b(paths)f(are)h(lik)m(ely)c(to)j(lea)m(v)m(e)118 3726 y FE(S)37 b FH(after)31 b(a)f(short)h(time.)40 b(Then)31 b(there)g(are)g(t)m(w)m(o)h(p)s(ossibilities.)k(Either)31 b(the)f(solution)g FF(x)3112 3693 y FC(0)3112 3748 y Fy(t)3181 3726 y FH(of)h(the)f(linear)118 3838 y(equation)24 b(\(4.72\))h(do)s(es)e(not)i(return)g(to)f(zero,)i(and)e(Lemma)f(4.8)i (sho)m(ws)f(that)h FF(x)2793 3852 y Fy(t)2846 3838 y FH(is)e(lik)m(ely)e(to)j(lea)m(v)m(e)g FE(D)j FH(as)118 3951 y(w)m(ell.)40 b(Or)30 b FF(x)524 3918 y FC(0)524 3974 y Fy(t)593 3951 y FH(do)s(es)g(return)h(to)f(zero.)41 b(Using)29 b(the)i(\(strong\))g(Mark)m(o)m(v)f(prop)s(ert)m(y)h(and)g (in)m(tegrating)f(o)m(v)m(er)118 4064 y(the)g(distribution)d(of)i(the)g (time)f(of)h(suc)m(h)g(a)h(\(\034rst\))f(return)h(to)f(zero,)h(w)m(e)g (obtain)f(an)h(in)m(tegral)e(equation)118 4177 y(for)g(an)h(upp)s(er)f (b)s(ound)h(on)f(the)h(probabilit)m(y)e(of)h(remaining)e(in)i FE(D)s FH(.)39 b(Finally)-8 b(,)27 b(this)g(in)m(tegral)h(equation)g (is)118 4290 y(solv)m(ed)i(b)m(y)g(iterations.)118 4470 y Fh(Pr)m(oof)35 b(of)g(Pr)m(oposition)h(4.10.)156 4606 y FH(1.)47 b(W)-8 b(e)31 b(\034rst)f(in)m(tro)s(duce)h(some)f (notations.)40 b(Let)992 4789 y FG(\010)1058 4803 y Fy(t)1087 4789 y FG(\()p FF(s;)15 b(x)p FG(\))26 b(=)f Fo(P)1475 4751 y Fy(s;x)1572 4715 y Fr(\010)1625 4789 y FF(\034)1665 4803 y FD(D)1750 4789 y Fn(>)g FF(t)1879 4715 y Fr(\011)1958 4789 y FG(=)f Fo(P)2114 4751 y Fy(s;x)2211 4688 y Fr(n)2308 4789 y FG(sup)2272 4866 y Fy(s)p Fx(6)p Fy(u)p Fx(6)p Fy(t)2519 4727 y FE(j)p FF(x)2596 4741 y Fy(u)2641 4727 y FE(j)p 2506 4768 175 4 v 2512 4851 a FG(~)-51 b FF(x)o FG(\()p FF(u)p FG(\))2715 4789 y FF(<)25 b FG(1)2856 4688 y Fr(o)2917 4789 y FF(;)488 b FH(\(4.90\))273 5023 y(and)35 b(de\034ne)h FF(\032)p FG(\()p FF(t)p FG(\))d(=)f FF(h=)1102 4945 y Fr(p)p 1194 4945 152 4 v 1194 5023 a FF(a)p FG(\()p FF(t)p FG(\))p FH(.)54 b(W)-8 b(e)35 b(ma)m(y)f(assume)g(that)h FF(\032)p FG(\()p FF(t)p FG(\))e Fn(6)38 b FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))35 b FH(for)f(all)f FF(t)i FH(\(otherwise)f(w)m(e)273 5136 y(replace)i FG(~)-50 b FF(x)30 b FH(b)m(y)g(its)f(maxim)m(um)f(with)i FF(\032)p FH(\).)41 b(F)-8 b(or)31 b FF(t)25 b Fn(>)g FF(s)f Fn(>)2168 5071 y FE(p)p 2244 5071 43 4 v 65 x FF(")31 b FH(w)m(e)g(de\034ne)g(the)g(quan)m(tities)1440 5293 y FF(q)1481 5307 y Fy(t)1510 5293 y FG(\()p FF(s)p FG(\))26 b(=)85 b(sup)1745 5376 y FD(j)p Fy(x)p FD(j)p Fx(6)p Fy(\032)p FC(\()p Fy(s)p FC(\))2018 5293 y FG(\010)2084 5307 y Fy(t)2113 5293 y FG(\()p FF(s;)15 b(x)p FG(\))p FF(;)1087 b FH(\(4.91\))1409 5514 y FF(Q)1481 5528 y Fy(t)1510 5514 y FG(\()p FF(s)p FG(\))26 b(=)176 b(sup)1745 5596 y Fy(\032)p FC(\()p Fy(s)p FC(\))p Fx(6)p FD(j)p Fy(x)p FD(j)p Fx(6)t FC(~)-39 b Fy(x)o FC(\()p Fy(s)p FC(\))2200 5514 y FG(\010)2266 5528 y Fy(t)2295 5514 y FG(\()p FF(s;)15 b(x)p FG(\))p FF(:)905 b FH(\(4.92\))1845 5871 y(35)p eop %%Page: 36 36 36 35 bop 156 328 a FH(2.)47 b(Let)40 b(us)f(\034rst)h(consider)f(the)h (case)f FE(j)p FF(x)p FE(j)i Fn(6)f FF(\032)p FG(\()p FF(s)p FG(\))p FH(.)68 b(Recall)37 b(that)j FE(S)47 b FG(=)40 b FE(f)p FG(\()p FF(x;)15 b(t)p FG(\))10 b(:)35 b FE(j)p FF(x)p FE(j)41 b FF(<)f(\032)p FG(\()p FF(t)p FG(\))p FE(g)p FH(.)69 b(By)273 441 y(Prop)s(osition)30 b(4.7)g(and)h(the)g(strong)g(Mark)m(o)m(v)f(prop)s(ert)m(y)-8 b(,)32 b(w)m(e)f(ha)m(v)m(e)h(the)e(estimate)605 688 y FG(\010)671 702 y Fy(t)700 688 y FG(\()p FF(s;)15 b(x)p FG(\))26 b(=)f Fo(P)1088 650 y Fy(s;x)1185 614 y Fr(\010)1238 688 y FF(\034)1278 702 y FD(S)1354 688 y Fn(>)g FF(t)1483 614 y Fr(\011)1556 688 y FG(+)20 b Fo(P)1708 650 y Fy(s;x)1805 587 y Fr(n)1866 688 y FF(\034)1906 702 y FD(S)1982 688 y FF(<)25 b(t;)74 b FG(sup)2151 764 y Fy(\034)2182 775 y Fk(S)2228 764 y Fx(6)p Fy(u)p Fx(6)p Fy(t)2443 626 y FE(j)p FF(x)2520 640 y Fy(u)2565 626 y FE(j)p 2430 667 175 4 v 2436 750 a FG(~)-51 b FF(x)p FG(\()p FF(u)p FG(\))2640 688 y FF(<)24 b FG(1)2780 587 y Fr(o)931 948 y Fn(6)1027 847 y Fr(\020)1093 886 y FF(h)p 1091 927 56 4 v 1091 1010 a(\033)1157 847 y Fr(\021)1211 869 y FC(2)1265 948 y FG(e)1306 910 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1666 948 y FG(+)p Fo(E)1798 910 y Fy(s;x)1900 847 y Fr(n)1960 948 y FG(1)2005 966 y FD(f)p Fy(\034)2071 977 y Fk(S)2118 966 y Fy()f FG(1)p FF(=)p FG(2)p FH(.)273 1848 y(Using)33 b(the)h(trivial)d(b)s(ound)p 1244 1774 72 4 v 34 w FF(Q)1316 1869 y Fy(t)1345 1848 y FG(\()p FF(u)p FG(\))h(=)e(1)k FH(in)f(\(4.107\))q(,)i(w)m(e)f (\034nd)g(that)g(\(4.108\))h(holds)e(with)g FF(a)3450 1862 y FC(1)3521 1848 y FG(=)d FF(c)273 1960 y FH(and)h FF(b)488 1974 y FC(1)553 1960 y FG(=)25 b(3)p FF(c=\024)p FH(.)42 b(Inserting)31 b(\(4.108\))g(in)m(to)f(\(4.107\))i(again,)e(w)m (e)h(get)398 2165 y FF(Q)470 2179 y Fy(t)500 2165 y FG(\()p FF(s)p FG(\))25 b Fn(6)g FF(C)7 b(g)s FG(\()p FF(t;)15 b(s)p FG(\))21 b(+)f FF(c)15 b FG(e)1245 2127 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")755 2370 y FG(+)k FF(c)899 2246 y Fr(Z)991 2273 y Fy(t)950 2452 y(s)1020 2269 y Fr(h)1063 2370 y FF(C)7 b(g)s FG(\()p FF(t;)15 b(u)p FG(\))22 b(+)e FF(a)1537 2384 y Fy(n)1599 2370 y FG(e)1639 2333 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;u)p FC(\))p Fy(=")2008 2370 y FG(+)p FF(b)2118 2384 y Fy(n)2165 2269 y Fr(i)2218 2309 y FF(a)p FG(\()p FF(u)p FG(\))p 2218 2349 171 4 v 2282 2432 a FF(")2414 2370 y FG(e)2454 2333 y FD(\000)p Fy(\024\013)p FC(\()p Fy(u;s)p FC(\))p Fy(=")2815 2296 y Fr(\002)2853 2370 y FG(1)g(+)g FF(g)s FG(\()p FF(u;)15 b(s)p FG(\))3260 2296 y Fr(\003)3315 2370 y FG(d)o FF(u)638 2618 y Fn(6)25 b FF(C)7 b(g)s FG(\()p FF(t;)15 b(s)p FG(\))21 b(+)f FF(c)1189 2489 y Fr(\024)1237 2618 y FG(1)h(+)f FF(C)1466 2517 y Fr(\020)1530 2556 y FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))p 1530 2597 246 4 v 1631 2680 a FF(")1805 2618 y FG(+)1909 2556 y(3)p 1906 2597 53 4 v 1906 2680 a FF(\024)1968 2517 y Fr(\021)2043 2618 y FG(+)20 b FF(a)2182 2632 y Fy(n)2229 2517 y Fr(\020)2293 2556 y FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))p 2293 2597 246 4 v 2394 2680 a FF(")2568 2618 y FG(+)2672 2556 y(2)p 2669 2597 53 4 v 2669 2680 a FF(\024)2731 2517 y Fr(\021)2786 2489 y(\025)2849 2618 y FG(e)2889 2580 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")3250 2618 y FG(+)3331 2556 y(3)p FF(c)p 3331 2597 85 4 v 3347 2680 a(\024)3425 2618 y(b)3464 2632 y Fy(n)3511 2618 y FF(:)273 2866 y FH(By)30 b(induction,)g(w)m(e)h (\034nd)486 3149 y FF(a)534 3163 y Fy(n)p FC(+1)696 3149 y FG(=)25 b FF(c)831 3048 y Fr(h)875 3149 y FG(1)20 b(+)g FF(C)1103 3048 y Fr(\020)1167 3087 y FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))p 1167 3128 246 4 v 1268 3211 a FF(")1442 3149 y FG(+)1546 3087 y(3)p 1543 3128 53 4 v 1543 3211 a FF(\024)1605 3048 y Fr(\021)q(i)1718 3035 y Fy(n)p FD(\000)p FC(1)1718 3063 y Fr(X)1723 3258 y Fy(j)t FC(=0)1851 3048 y Fr(h)1894 3149 y FF(c)1933 3048 y Fr(\020)1997 3087 y FF(\013)p FG(\()p FF(t;)g(s)p FG(\))p 1997 3128 246 4 v 2098 3211 a FF(")2272 3149 y FG(+)2377 3087 y(2)p 2373 3128 53 4 v 2373 3211 a FF(\024)2436 3048 y Fr(\021i)2533 3071 y Fy(j)2589 3149 y FG(+)20 b FF(c)2719 3048 y Fr(h)2763 3149 y FF(c)2802 3048 y Fr(\020)2866 3087 y FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))p 2866 3128 246 4 v 2968 3211 a FF(")3141 3149 y FG(+)3246 3087 y(2)p 3242 3128 53 4 v 3242 3211 a FF(\024)3305 3048 y Fr(\021i)3402 3071 y Fy(n)696 3443 y Fn(6)792 3342 y Fr(h)835 3443 y FG(1)21 b(+)f FF(C)1064 3342 y Fr(\020)1127 3381 y FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))p 1127 3422 246 4 v 1229 3505 a FF(")1403 3443 y FG(+)1507 3381 y(3)p 1504 3422 53 4 v 1504 3505 a FF(\024)1566 3342 y Fr(\021i)1978 3381 y FF(c)p 1673 3422 649 4 v 1673 3528 a FG(1)21 b FE(\000)f FF(c)1869 3454 y Fr(\000)1921 3483 y Fy(\013)p FC(\()p Fy(t;s)p FC(\))p 1921 3507 178 4 v 1993 3559 a Fy(")2129 3528 y FG(+)2232 3492 y FC(2)p 2230 3507 41 4 v 2230 3559 a Fy(\024)2280 3454 y Fr(\001)3385 3443 y FH(\(4.111\))495 3714 y FF(b)534 3728 y Fy(n)p FC(+1)696 3714 y FG(=)792 3613 y Fr(\020)856 3652 y FG(3)p FF(c)p 856 3693 85 4 v 873 3776 a(\024)951 3613 y Fr(\021)1006 3635 y Fy(n)p FC(+1)3385 3714 y FH(\(4.112\))273 3954 y(as)34 b(a)h(p)s(ossible)d(c)m(hoice,)k(where)f(w)m(e)g(ha)m(v)m(e)h (used)f(the)f(fact)h(that)g FF(c)p FG(\()p FF(\013)p FG(\()p FF(t;)15 b(s)p FG(\))p FF(=")25 b FG(+)e(2)p FF(=\024)p FG(\))34 b Fn(6)3327 3918 y FC(1)p 3327 3933 36 4 v 3327 3986 a(2)3406 3954 y FH(b)m(y)h(the)273 4067 y(h)m(yp)s(othesis)30 b(\(4.109\))q(.)40 b(T)-8 b(aking)31 b(the)f(limit)d FF(n)e FE(!)g(1)p FH(,)30 b(and)h(using)f FF(c)25 b Fn(6)2658 4031 y Fy(\024)p 2658 4046 41 4 v 2661 4098 a FC(4)2734 4067 y Fn(6)2840 4031 y FC(1)p 2840 4046 36 4 v 2840 4098 a(4)2885 4067 y FH(,)31 b(w)m(e)g(obtain)932 4318 y FF(Q)1004 4332 y Fy(t)1033 4318 y FG(\()p FF(s)p FG(\))26 b Fn(6)f FF(C)7 b(g)s FG(\()p FF(t;)15 b(s)p FG(\))20 b(+)1693 4257 y(1)p 1693 4297 46 4 v 1693 4381 a(2)1749 4245 y Fr(\000)1790 4318 y FG(1)h(+)f(3)p FF(C)2064 4245 y Fr(\001)2121 4318 y FG(e)2161 4281 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")2532 4318 y Fn(6)25 b FG(3)p FF(C)7 b(g)s FG(\()p FF(t;)15 b(s)p FG(\))p FF(:)383 b FH(\(4.113\))273 4548 y(In)39 b(order)i(to)e(obtain) h(also)e(a)i(b)s(ound)f(on)h FF(q)1789 4562 y Fy(t)1818 4548 y FG(\()p FF(s)p FG(\))p FH(,)i(w)m(e)f(insert)d(the)i(ab)s(o)m(v) m(e)h(b)s(ound)e(on)h FF(Q)3326 4562 y Fy(t)3356 4548 y FG(\()p FF(s)p FG(\))f FH(in)m(to)273 4661 y(\(4.95\))q(,)31 b(whic)m(h)f(yields)674 4922 y FF(q)715 4936 y Fy(t)744 4922 y FG(\()p FF(s)p FG(\))c Fn(6)f FG(2)1024 4821 y Fr(\020)1090 4860 y FF(h)p 1089 4901 56 4 v 1089 4984 a(\033)1154 4821 y Fr(\021)1208 4844 y FC(2)1263 4922 y FG(e)1303 4884 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")1664 4922 y FG(+3)p FF(\024C)1904 4821 y Fr(\020)1970 4860 y FF(h)p 1968 4901 V 1968 4984 a(\033)2033 4821 y Fr(\021)2088 4844 y FC(2)2142 4798 y Fr(Z)2233 4824 y Fy(t)2193 5004 y(s)2288 4860 y FF(a)p FG(\()p FF(u)p FG(\))p 2288 4901 171 4 v 2352 4984 a FF(")2484 4922 y FG(e)2524 4884 y FD(\000)p Fy(\024\013)p FC(\()p Fy(u;s)p FC(\))p Fy(=")2900 4922 y FF(g)s FG(\()p FF(t;)15 b(u)p FG(\))g(d)q FF(u)883 5167 y Fn(6)979 5066 y Fr(h)1022 5167 y FG(2)21 b(+)e(3)p FF(\024C)1347 5066 y Fr(\020)1416 5106 y FG(1)p 1412 5146 53 4 v 1412 5230 a FF(\024)1495 5167 y FG(+)1595 5106 y FF(\013)p FG(\()p FF(t;)c(s)p FG(\))p 1595 5146 246 4 v 1697 5230 a FF(")1850 5066 y Fr(\021)q(i\020)2013 5106 y FF(h)p 2012 5146 56 4 v 2012 5230 a(\033)2077 5066 y Fr(\021)2131 5089 y FC(2)2186 5167 y FG(e)2226 5130 y FD(\000)p Fy(\024\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")3385 5167 y FH(\(4.114\))273 5397 y(b)m(y)28 b(\(4.103\))q(.)39 b(This)25 b(pro)m(v)m(es)i(the)g(prop)s (osition,)f(and)g(therefore)h(Theorem)g(2.9,)g(b)m(y)f(taking)g(the)g (sum)273 5510 y(of)k(the)h(ab)s(o)m(v)m(e)g(estimates)e(on)h FF(q)1352 5524 y Fy(t)1382 5510 y FG(\()p FF(s)p FG(\))g FH(and)h FF(Q)1773 5524 y Fy(t)1802 5510 y FG(\()p FF(s)p FG(\))p FH(.)p 3596 5510 4 62 v 3600 5453 55 4 v 3600 5510 V 3653 5510 4 62 v 1845 5871 a(38)p eop %%Page: 39 39 39 38 bop 118 328 a Fp(4.4)112 b(Approac)m(h)38 b(to)f Ff(x)1077 291 y Fy(?)1117 328 y Fc(\()p Ff(t)p Fc(\))118 499 y FH(W)-8 b(e)35 b(\034nally)d(turn)j(to)f(the)h(b)s(eha)m(viour)f (after)g(the)h(time)e FF(\034)41 b FG(=)32 b FF(\034)2268 513 y FD(D)2360 499 y FF(>)2462 434 y FE(p)p 2538 434 43 4 v 65 x FF(")q FH(,)j(when)f FF(x)2932 513 y Fy(t)2996 499 y FH(lea)m(v)m(es)g(the)h(set)f FE(D)s FH(.)118 612 y(By)28 b(symmetry)-8 b(,)27 b(w)m(e)j(can)e(restrict)g(the)h(analysis) d(to)j(the)g(case)f FF(x)2297 626 y Fy(\034)2365 612 y FG(=)j(~)-51 b FF(x)p FG(\()p FF(\034)10 b FG(\))p FH(.)41 b(Our)29 b(aim)d(is)h(to)i(pro)m(v)m(e)h(that)118 725 y(with)g(high)g(probabilit)m(y)-8 b(,)30 b FF(x)1057 739 y Fy(t)1116 725 y FH(so)s(on)g(reac)m(hes)i(a)e(neigh)m(b)s(ourho)s (o)s(d)h(of)f FF(x)2488 692 y Fy(?)2527 725 y FG(\()p FF(t)p FG(\))p FH(.)259 846 y(W)-8 b(e)31 b(start)f(b)m(y)h(analysing)e (the)h(solution)f FF(x)1706 802 y FC(det)q Fy(;\034)1706 871 y(t)1897 846 y FH(of)h(the)h(deterministic)d(equation)1622 1095 y FF(")1674 1033 y FG(d)p FF(x)p 1674 1074 103 4 v 1684 1157 a FG(d)o FF(t)1812 1095 y FG(=)d FF(f)10 b FG(\()p FF(x;)15 b(t)p FG(\))1227 b FH(\(4.115\))118 1347 y(with)30 b(initial)d(condition)j FF(x)1034 1303 y FC(det)q Fy(;\034)1034 1359 y(\034)1220 1347 y FG(=)h(~)-51 b FF(x)p FG(\()p FF(\034)10 b FG(\))p FH(.)118 1535 y Fq(Prop)s(osition)35 b(4.11.)41 b FB(F)-7 b(or)33 b(su\036ciently)f (smal)5 b(l)32 b FF(")g FB(and)h FF(T)13 b FB(,)780 1739 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))26 b Fn(6)f FF(x)1103 1695 y FC(det)q Fy(;\034)1103 1763 y(t)1289 1739 y Fn(6)g FF(x)1437 1701 y Fy(?)1476 1739 y FG(\()p FF(t)p FG(\))1806 b FH(\(4.116\))774 1938 y FG(0)26 b Fn(6)f FF(x)993 1900 y Fy(?)1032 1938 y FG(\()p FF(t)p FG(\))c FE(\000)f FF(x)1299 1894 y FC(det)q Fy(;\034)1299 1962 y(t)1485 1938 y Fn(6)25 b FF(C)1653 1810 y Fr(\024)1760 1876 y FF(")p 1710 1917 143 4 v 1710 2004 a(t)1743 1978 y FC(3)p Fy(=)p FC(2)1883 1938 y FG(+)1974 1864 y Fr(\000)2016 1938 y FF(x)2068 1900 y Fy(?)2107 1938 y FG(\()p FF(\034)10 b FG(\))21 b FE(\000)26 b FG(~)-51 b FF(x)p FG(\()p FF(\034)10 b FG(\))2511 1864 y Fr(\001)2568 1938 y FG(e)2609 1900 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)e FC(\))p Fy(=")2958 1810 y Fr(\025)3385 1938 y FH(\(4.117\))774 2158 y FG(0)26 b Fn(6)f FF(x)993 2111 y FC(det)q Fy(;)1111 2063 y FD(p)p 1169 2063 33 3 v 1169 2111 a Fy(")993 2182 y(t)1226 2158 y FE(\000)20 b FF(x)1369 2114 y FC(det)q Fy(;\034)1369 2182 y(t)1555 2158 y Fn(6)1651 2085 y Fr(\000)1692 2158 y FF(x)1744 2121 y FC(det)q Fy(;)1862 2073 y FD(p)p 1921 2073 V 48 x Fy(")1744 2181 y(\034)1978 2158 y FE(\000)25 b FG(~)-50 b FF(x)p FG(\()p FF(\034)10 b FG(\))2241 2085 y Fr(\001)2298 2158 y FG(e)2339 2121 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)e FC(\))p Fy(=")3385 2158 y FH(\(4.118\))118 2362 y FB(for)32 b(al)5 b(l)32 b FF(t)25 b FE(2)g FG([)p FF(\034)5 b(;)15 b(T)e FG(])33 b FB(and)f(al)5 b(l)32 b FF(\034)k FE(2)25 b FG([)1260 2297 y FE(p)p 1336 2297 43 4 v 65 x FF(";)15 b(T)e FG(])p FB(,)32 b(wher)-5 b(e)34 b FF(C)e(>)25 b FG(0)32 b FB(is)g(a)h(c)-5 b(onstant)32 b(dep)-5 b(ending)34 b(only)e(on)g FF(f)10 b FB(.)118 2625 y Fh(Pr)m(oof:)156 2761 y FH(1.)47 b(Whenev)m(er)32 b FF(x)750 2717 y FC(det)q Fy(;\034)750 2785 y(t)936 2761 y FG(=)25 b FF(x)1084 2728 y Fy(?)1123 2761 y FG(\()p FF(t)p FG(\))p FH(,)31 b(w)m(e)g(ha)m(v)m(e)678 3015 y FF(")755 2953 y FG(d)p 730 2994 84 4 v 730 3077 a(d)p FF(t)824 2941 y Fr(\000)866 3015 y FF(x)918 2977 y Fy(?)957 3015 y FG(\()p FF(t)p FG(\))21 b FE(\000)f FF(x)1224 2970 y FC(det)q Fy(;\034)1224 3039 y(t)1384 2941 y Fr(\001)1451 3015 y FG(=)25 b FF(")1599 2953 y FG(d)p FF(x)1702 2920 y Fy(?)1742 2953 y FG(\()p FF(t)p FG(\))p 1599 2994 246 4 v 1681 3077 a(d)o FF(t)1875 3015 y FE(\000)20 b FF(f)10 b FG(\()p FF(x)2108 2977 y Fy(?)2147 3015 y FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))27 b(=)d FF(")2532 2953 y FG(d)p FF(x)2635 2920 y Fy(?)2675 2953 y FG(\()p FF(t)p FG(\))p 2532 2994 V 2614 3077 a(d)o FF(t)2813 3015 y Fn(>)h FG(0)p FF(;)406 b FH(\(4.119\))273 3267 y(whic)m(h)46 b(sho)m(ws)g(that)h FF(x)1086 3223 y FC(det)q Fy(;\034)1086 3291 y(t)1292 3267 y FH(can)f(nev)m(er)h(b)s(ecome)e(larger)h(than)g FF(x)2619 3234 y Fy(?)2659 3267 y FG(\()p FF(t)p FG(\))p FH(.)87 b(Similarly)-8 b(,)46 b(whenev)m(er)273 3390 y FF(x)325 3346 y FC(det)q Fy(;\034)325 3414 y(t)511 3390 y FG(=)31 b(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FH(,)31 b(w)m(e)g(get)422 3643 y FF(")499 3582 y FG(d)p 474 3622 84 4 v 474 3706 a(d)p FF(t)568 3570 y Fr(\000)610 3643 y FF(x)662 3599 y FC(det)q Fy(;\034)662 3667 y(t)843 3643 y FE(\000)25 b FG(~)-50 b FF(x)p FG(\()p FF(t)p FG(\))1089 3570 y Fr(\001)1156 3643 y FG(=)25 b FF(f)10 b FG(\()c(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FF(;)15 b(t)p FG(\))21 b FE(\000)f FF(")1769 3582 y FG(d)5 b(~)-50 b FF(x)p FG(\()p FF(t)p FG(\))p 1769 3622 206 4 v 1831 3706 a(d)o FF(t)1156 3885 y FG(=)1252 3803 y FE(p)p 1328 3803 54 4 v 82 x FF(\025)15 b FG(\(1)21 b FE(\000)f FF(\025)p FG(\))p FF(t)1709 3847 y FC(3)p Fy(=)p FC(2)1819 3811 y Fr(\002)1857 3885 y FG(1)h(+)f FD(O)2071 3899 y Fy(T)2126 3885 y FG(\(1\))2241 3811 y Fr(\003)2301 3885 y FE(\000)g FF(")2457 3746 y FE(p)p 2532 3746 V 2532 3823 a FF(\025)p 2444 3864 155 4 v 2444 3956 a FG(2)2489 3882 y FE(p)p 2565 3882 33 4 v 74 x FF(t)2608 3811 y Fr(\002)2646 3885 y FG(1)g(+)g FD(O)2860 3899 y Fy(T)2915 3885 y FG(\(1\))3030 3811 y Fr(\003)3095 3885 y FF(>)25 b FG(0)3385 3765 y FH(\(4.120\))273 4159 y(pro)m(vided)39 b FF(\025)f(<)863 4124 y FC(1)p 863 4139 36 4 v 863 4191 a(2)908 4159 y FG([1)26 b FE(\000)f FD(O)1158 4173 y Fy(T)1213 4159 y FG(\(1\)])p FH(,)41 b(whic)m(h)e(sho)m(ws)f(that)h FF(x)2209 4115 y FC(det)q Fy(;\034)2209 4184 y(t)2407 4159 y FH(can)g(nev)m(er)f(b)s(ecome)g(smaller)e(than)279 4272 y FG(~)-51 b FF(x)p FG(\()p FF(t)p FG(\))p FH(.)41 b(This)29 b(completes)h(the)h(pro)s(of)f(of)36 b(\(4.116\))q(.)156 4394 y(2.)47 b(W)-8 b(e)36 b(no)m(w)g(in)m(tro)s(duce)g(the)f (di\033erence)g FF(y)1651 4350 y FC(det)q Fy(;\034)1648 4418 y(t)1845 4394 y FG(=)f FF(x)2002 4361 y Fy(?)2041 4394 y FG(\()p FF(t)p FG(\))24 b FE(\000)f FF(x)2314 4350 y FC(det)q Fy(;\034)2314 4418 y(t)2475 4394 y FH(.)55 b(Using)34 b(T)-8 b(a)m(ylor's)35 b(form)m(ula,)h(one)273 4517 y(immediately)27 b(obtains)j(that)h FF(y)1349 4472 y FC(det)q Fy(;\034)1346 4541 y(t)1540 4517 y FH(satis\034es)e(the)h (ODE)1333 4765 y FF(")1385 4703 y FG(d)o FF(y)p 1385 4744 99 4 v 1393 4827 a FG(d)o FF(t)1518 4765 y FG(=)25 b FF(a)1662 4727 y Fy(?)1702 4765 y FG(\()p FF(t)p FG(\))p FF(y)e FG(+)d FF(b)2003 4727 y Fy(?)2043 4765 y FG(\()p FF(y)s(;)15 b(t)p FG(\))21 b(+)f FF("x)2440 4727 y Fy(?)p FD(0)2499 4765 y FG(\()p FF(t)p FG(\))783 b FH(\(4.121\))273 4989 y(where)1760 5168 y FF(a)1808 5131 y Fy(?)1847 5168 y FG(\()p FF(t)p FG(\))26 b Fn(6)f FE(\000)p FF(a)2191 5131 y Fy(?)2191 5191 y FC(0)2230 5168 y FF(t)1514 5321 y FG(0)h Fn(6)f FF(b)1720 5283 y Fy(?)1759 5321 y FG(\()p FF(y)s(;)15 b(t)p FG(\))26 b Fn(6)f FF(M)2170 5283 y Fy(?)2210 5242 y FE(p)p 2285 5242 33 4 v 2285 5321 a FF(t)15 b(y)2381 5283 y FC(2)1737 5513 y FF(x)1789 5476 y Fy(?)p FD(0)1847 5513 y FG(\()p FF(t)p FG(\))26 b Fn(6)2082 5452 y FF(K)2166 5419 y Fy(?)p 2082 5492 124 4 v 2089 5511 a FE(p)p 2165 5511 33 4 v 73 x FF(t)2215 5513 y(;)3385 5367 y FH(\(4.122\))1845 5871 y(39)p eop %%Page: 40 40 40 39 bop 273 328 a FH(with)31 b FF(a)528 295 y Fy(?)528 352 y FC(0)595 328 y FG(=)c(2[1)22 b(+)e FD(O)979 342 y Fy(T)1034 328 y FG(\(1\)])p FH(,)33 b FF(M)1330 295 y Fy(?)1397 328 y FG(=)27 b(3[1)22 b(+)e FD(O)1781 342 y Fy(T)1836 328 y FG(\(1\)])33 b FH(and)f FF(K)2270 295 y Fy(?)2336 328 y FG(=)2444 292 y FC(1)p 2444 307 36 4 v 2444 359 a(2)2489 328 y FG([1)22 b(+)e FD(O)2730 342 y Fy(T)2785 328 y FG(\(1\)])p FH(.)45 b(W)-8 b(e)32 b(\034rst)f(consider)273 441 y(the)k(particular)g(solution)i Fr(b)-55 b FF(y)1257 408 y FC(det)1246 463 y Fy(t)1393 441 y FH(of)41 b(\(4.121\))36 b(starting)e(at)h(time)e FG(4)2529 375 y FE(p)p 2605 375 43 4 v 66 x FF(")i FH(in)j Fr(b)-55 b FF(y)2848 408 y FC(det)2837 476 y(4)2872 428 y FD(p)p 2931 428 33 3 v 48 x Fy(")2993 441 y FG(=)25 b(0)p FH(.)54 b(By)35 b(\(4.119\))q(,)273 553 y(w)m(e)c(kno)m(w)g(that) k Fr(b)-55 b FF(y)894 520 y FC(det)883 576 y Fy(t)1021 553 y Fn(>)25 b FG(0)31 b FH(for)f(all)f FF(t)c Fn(>)g FG(4)1656 488 y FE(p)p 1732 488 43 4 v 65 x FF(")p FH(.)41 b(W)-8 b(e)30 b(will)e(use)i(the)h(fact)f(that)651 690 y Fr(Z)742 716 y Fy(t)702 896 y(\034)834 752 y FG(1)p 797 793 119 4 v 797 811 a FE(p)p 873 811 43 4 v 66 x FF(s)941 814 y FG(e)981 776 y FD(\000)p Fy(a)1073 753 y Fj(?)1073 797 y Fw(0)1109 776 y FC(\()p Fy(t)1161 753 y Fw(2)1197 776 y FD(\000)p Fy(s)1285 753 y Fw(2)1319 776 y FC(\))p Fy(=)p FC(4)p Fy(")1469 814 y FG(d)o FF(s)25 b Fn(6)1683 690 y Fr(Z)1774 716 y Fy(t)1734 896 y(\034)1865 752 y FG(1)p 1829 793 119 4 v 1829 811 a FE(p)p 1905 811 43 4 v 66 x FF(s)1972 814 y FG(e)2013 776 y FD(\000)p Fy(a)2105 753 y Fj(?)2105 797 y Fw(0)2141 776 y Fy(t)p FC(\()p Fy(t)p FD(\000)p Fy(s)p FC(\))p Fy(=)p FC(4)p Fy(")2457 814 y FG(d)o FF(s)1587 1075 y Fn(6)1764 1014 y FG(4)p FF(")p 1693 1054 231 4 v 1693 1141 a(a)1741 1110 y Fy(?)1741 1167 y FC(0)1781 1141 y FF(t)1814 1115 y FC(3)p Fy(=)p FC(2)1949 951 y Fr(Z)2040 978 y Fy(\030)1999 1158 y FC(0)2227 1014 y FG(e)2267 981 y FD(\000)p Fy(u)p 2102 1054 389 4 v 2102 1072 a Fr(p)p 2193 1072 299 4 v 78 x FG(1)c FE(\000)f FF(u=\030)2516 1075 y FG(d)p FF(u)25 b(<)g(c)2779 1089 y FC(0)2879 1014 y FF(")p 2829 1054 143 4 v 2829 1141 a(t)2862 1115 y FC(3)p Fy(=)p FC(2)2982 1075 y FF(;)3385 950 y FH(\(4.123\))273 1346 y(where)41 b FF(c)583 1360 y FC(0)665 1346 y FG(=)g(8)p FF(=a)915 1313 y Fy(?)915 1370 y FC(0)956 1346 y FH(.)70 b(W)-8 b(e)40 b(ha)m(v)m(e)h(used)g(the)f(transformation)g FF(s)h FG(=)g FF(t)27 b FE(\000)g FG(4)p FF("u=)p FG(\()p FF(a)3063 1313 y Fy(?)3063 1370 y FC(0)3104 1346 y FF(t)p FG(\))p FH(,)43 b(in)m(tro)s(duced)273 1458 y FF(\030)31 b FG(=)c FF(a)490 1425 y Fy(?)490 1483 y FC(0)530 1458 y FF(t)563 1425 y FC(2)602 1458 y FF(=)p FG(4)p FF(")32 b FH(and)g(b)s(ounded)g(the)g(last)e(in)m(tegral)i(b)m(y)f FG(2)p FH(.)44 b(W)-8 b(e)32 b(no)m(w)g(in)m(tro)s(duce)g(the)g (\034rst)f(exit)g(time)278 1571 y FG(^)-50 b FF(\034)36 b FG(=)25 b(inf)5 b FE(f)p FF(t)26 b Fn(>)f FG(4)800 1506 y FE(p)p 876 1506 43 4 v 65 x FF(")10 b FG(:)35 b Fr(b)-55 b FF(y)1040 1538 y FC(det)1029 1594 y Fy(t)1167 1571 y Fn(>)25 b FF(c)1302 1585 y FC(0)1342 1571 y FF("t)1417 1538 y FD(\000)p FC(3)p Fy(=)p FC(2)1582 1571 y FE(g)p FH(.)41 b(F)-8 b(or)31 b FG(4)1900 1506 y FE(p)p 1976 1506 V 65 x FF(")25 b Fn(6)g FF(t)g Fn(6)30 b FG(^)-50 b FF(\034)10 b FH(,)31 b(w)m(e)g(ha)m(v)m(e)527 1819 y FF(a)575 1781 y Fy(?)614 1819 y FG(\()p FF(t)p FG(\))p FF(y)24 b FG(+)c FF(b)916 1781 y Fy(?)955 1819 y FG(\()p FF(y)s(;)15 b(t)p FG(\))26 b Fn(6)1268 1718 y Fr(\020)1323 1819 y FE(\000)p FF(a)1442 1781 y Fy(?)1442 1842 y FC(0)1496 1819 y FF(t)20 b FG(+)g FF(M)1738 1781 y Fy(?)1777 1741 y FE(p)p 1853 1741 33 4 v 78 x FF(t)15 b(c)1940 1833 y FC(0)2040 1758 y FF(")p 1990 1798 143 4 v 1990 1885 a(t)2023 1859 y FC(3)p Fy(=)p FC(2)2143 1718 y Fr(\021)2197 1819 y FF(y)28 b Fn(6)d FE(\000)p FF(a)2485 1781 y Fy(?)2485 1842 y FC(0)2524 1718 y Fr(\020)2579 1819 y FG(1)20 b FE(\000)2745 1758 y FF(c)2784 1772 y FC(0)2824 1758 y FF(M)2922 1725 y Fy(?)p 2745 1798 217 4 v 2764 1881 a FG(16)p FF(a)2902 1850 y Fy(?)2902 1907 y FC(0)2971 1718 y Fr(\021)3026 1819 y FF(ty)s(:)253 b FH(\(4.124\))273 2086 y(Since)30 b FF(M)607 2053 y Fy(?)647 2086 y FF(=)p FG(\()p FF(a)775 2053 y Fy(?)775 2110 y FC(0)815 2086 y FG(\))850 2053 y FC(2)915 2086 y FG(=)1021 2050 y FC(3)p 1021 2065 36 4 v 1021 2117 a(4)1067 2086 y FG([1)21 b(+)f FD(O)6 b FG(\(1\)])p FH(,)31 b(the)g(term)f(in)g(brac)m(k)m(ets)h(can)g (b)s(e)f(assumed)g(to)g(b)s(e)g(larger)h(than)283 2163 y FC(1)p 283 2178 V 283 2230 a(2)328 2198 y FH(.)41 b(Hence)30 b(\(4.121\))i(sho)m(ws)e(that)1422 2470 y FF(")1474 2408 y FG(d)t Fr(b)-55 b FF(y)1581 2375 y FC(det)p 1474 2449 209 4 v 1537 2532 a FG(d)p FF(t)1718 2470 y Fn(6)25 b FE(\000)1895 2408 y FF(a)1943 2375 y Fy(?)1943 2433 y FC(0)p 1895 2449 88 4 v 1916 2532 a FG(2)1995 2470 y FF(t)9 b Fr(b)-55 b FF(y)2089 2432 y FC(det)2211 2470 y FG(+)20 b FF(")2354 2408 y(K)2438 2375 y Fy(?)p 2354 2449 124 4 v 2361 2467 a FE(p)p 2437 2467 33 4 v 74 x FF(t)2487 2470 y(;)873 b FH(\(4.125\))273 2717 y(whic)m(h)31 b(implies)1118 2893 y Fr(b)-54 b FF(y)1171 2855 y FC(det)1160 2915 y Fy(t)1298 2893 y Fn(6)25 b FF(K)1478 2855 y Fy(?)1532 2769 y Fr(Z)1623 2795 y Fy(t)1582 2975 y(\034)1678 2831 y FG(e)1718 2798 y FD(\000)p Fy(a)1810 2775 y Fj(?)1810 2819 y Fw(0)1846 2798 y FC(\()p Fy(t)1898 2775 y Fw(2)1934 2798 y FD(\000)p Fy(s)2022 2775 y Fw(2)2056 2798 y FC(\))p Fy(=)p FC(4)p Fy(")p 1678 2872 514 4 v 1875 2890 a FE(p)p 1951 2890 43 4 v 65 x FF(s)2216 2893 y FG(d)o FF(s)g(<)g(K)2514 2855 y Fy(?)2553 2893 y FF(c)2592 2907 y FC(0)2692 2831 y FF(")p 2642 2872 143 4 v 2642 2959 a(t)2675 2932 y FC(3)p Fy(=)p FC(2)2795 2893 y FF(:)565 b FH(\(4.126\))273 3122 y(Since)30 b FF(K)593 3089 y Fy(?)658 3122 y FG(=)764 3086 y FC(1)p 764 3101 36 4 v 764 3154 a(2)809 3122 y FG([1)21 b(+)f FD(O)6 b FG(\(1\)])p FH(,)31 b(w)m(e)g(obtain)j Fr(b)-54 b FF(y)1720 3089 y FC(det)1709 3145 y Fy(t)1847 3122 y FF(<)25 b(c)1982 3136 y FC(0)2022 3122 y FF("t)2097 3089 y FD(\000)p FC(3)p Fy(=)p FC(2)2262 3122 y FH(,)30 b(and)h(th)m(us)k FG(^)-49 b FF(\034)35 b FG(=)25 b FE(1)p FH(.)40 b(This)29 b(sho)m(ws)1138 3354 y FG(0)d Fn(6)i Fr(b)-55 b FF(y)1360 3316 y FC(det)1349 3376 y Fy(t)1488 3354 y Fn(6)25 b FF(K)1668 3316 y Fy(?)1707 3354 y FF(c)1746 3368 y FC(0)1846 3292 y FF(")p 1795 3333 143 4 v 1795 3420 a(t)1828 3394 y FC(3)p Fy(=)p FC(2)2129 3354 y FH(for)30 b FG(4)2312 3284 y FE(p)p 2388 3284 43 4 v 70 x FF(")c Fn(6)f FF(t)g Fn(6)g FF(T)13 b FH(.)588 b(\(4.127\))156 3617 y(3.)47 b(Let)30 b FF(\034)35 b Fn(>)605 3552 y FE(p)p 681 3552 V 65 x FF(")29 b FH(and)g FG(0)d Fn(6)f FF(y)1138 3631 y FC(1)1202 3617 y FF(<)g(y)1343 3631 y FC(2)1407 3617 y Fn(6)g FF(x)1555 3584 y Fy(?)1595 3617 y FG(\()p FF(\034)10 b FG(\))18 b FE(\000)k FG(~)-50 b FF(x)p FG(\()p FF(\034)10 b FG(\))29 b FH(b)s(e)g(giv)m(en.)39 b(Let)30 b FF(y)2623 3569 y FC(\(1\))2620 3641 y Fy(t)2746 3617 y FH(and)f FF(y)2968 3569 y FC(\(2\))2965 3641 y Fy(t)3091 3617 y FH(b)s(e)f(solutions)g(of)273 3747 y(\(4.121\))33 b(with)e(initial)d(conditions)i FF(y)1531 3699 y FC(\(1\))1528 3759 y Fy(\034)1652 3747 y FG(=)d FF(y)1795 3761 y FC(1)1865 3747 y FH(and)32 b FF(y)2090 3699 y FC(\(2\))2087 3759 y Fy(\034)2211 3747 y FG(=)27 b FF(y)2354 3761 y FC(2)2393 3747 y FH(,)k(resp)s(ectiv)m(ely)-8 b(.)43 b(Then)32 b(there)g(exists)273 3876 y(a)f FF(\022)c FE(2)e FG([0)p FF(;)15 b FG(1])32 b FH(suc)m(h)f(that)g(the)f(di\033erence)h FF(z)1719 3890 y Fy(t)1774 3876 y FG(=)25 b FF(y)1918 3828 y FC(\(2\))1915 3900 y Fy(t)2032 3876 y FE(\000)20 b FF(y)2171 3828 y FC(\(1\))2168 3900 y Fy(t)2295 3876 y FH(satis\034es)1090 4124 y FF(")1142 4063 y FG(d)o FF(z)p 1142 4103 97 4 v 1149 4187 a FG(d)o FF(t)1274 4124 y FG(=)25 b FE(\000)p FF(@)1489 4138 y Fy(x)1533 4124 y FF(f)10 b FG(\()p FF(x)1675 4087 y Fy(?)1714 4124 y FG(\()p FF(t)p FG(\))20 b FE(\000)g FF(y)1976 4076 y FC(\(1\))1973 4148 y Fy(t)2091 4124 y FE(\000)g FF(\022)s(z)t(;)15 b(t)p FG(\))25 b Fn(6)g FE(\000)p FF(\021)s(a)p FG(\()p FF(t)p FG(\))p FF(z)t(;)541 b FH(\(4.128\))273 4354 y(where)31 b(w)m(e)h(ha)m(v)m(e)f(used)f(\(4.116\))i(and)e(\(4.14\))r(.)40 b(It)30 b(follo)m(ws)f(that)1236 4574 y FG(0)d Fn(6)f FF(y)1451 4527 y FC(\(2\))1448 4599 y Fy(t)1565 4574 y FE(\000)20 b FF(y)1704 4527 y FC(\(1\))1701 4599 y Fy(t)1823 4574 y Fn(6)25 b FG(\()p FF(y)1999 4588 y FC(2)2059 4574 y FE(\000)20 b FF(y)2195 4588 y FC(1)2234 4574 y FG(\))15 b(e)2325 4537 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")2674 4574 y FF(;)686 b FH(\(4.129\))273 4788 y(whic)m(h)28 b(pro)m(v)m(es)f(\(4.118\))h(in)f(particular.)38 b(If)26 b FF(\034)36 b Fn(>)25 b FG(4)1964 4723 y FE(p)p 2040 4723 43 4 v 65 x FF(")p FH(,)j(w)m(e)g(can)f(use)g(the)g(relation) f FF(x)3112 4755 y Fy(?)3152 4788 y FG(\()p FF(t)p FG(\))13 b FE(\000)g FF(x)3404 4744 y FC(det)r Fy(;\034)3404 4813 y(t)3591 4788 y FG(=)277 4911 y Fr(b)-55 b FF(y)329 4878 y FC(det)318 4934 y Fy(t)451 4911 y FG(+)20 b(\()p FF(y)625 4867 y FC(det)q Fy(;\034)622 4935 y(t)806 4911 y FE(\000)k Fr(b)-55 b FF(y)953 4878 y FC(det)942 4934 y Fy(t)1055 4911 y FG(\))31 b FH(to)f(sho)m(w)h(that)905 5135 y FF(x)957 5098 y Fy(?)996 5135 y FG(\()p FF(t)p FG(\))21 b FE(\000)f FF(x)1263 5091 y FC(det)q Fy(;\034)1263 5160 y(t)1449 5135 y Fn(6)25 b FF(K)1629 5098 y Fy(?)1668 5135 y FF(c)1707 5149 y FC(0)1807 5074 y FF(")p 1757 5115 143 4 v 1757 5202 a(t)1790 5175 y FC(3)p Fy(=)p FC(2)1930 5135 y FG(+)2021 5062 y Fr(\000)2062 5135 y FF(x)2114 5098 y Fy(?)2154 5135 y FG(\()p FF(\034)10 b FG(\))21 b FE(\000)k FG(~)-50 b FF(x)p FG(\()p FF(\034)10 b FG(\))2558 5062 y Fr(\001)2615 5135 y FG(e)2656 5098 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)e FC(\))p Fy(=")3005 5135 y FF(;)355 b FH(\(4.130\))273 5374 y(whic)m(h)39 b(pro)m(v)m(es)h(\(4.117\))g(for)f FF(\034)49 b Fn(>)40 b FG(4)1535 5308 y FE(p)p 1611 5308 43 4 v 66 x FF(")p FH(.)66 b(Finally)-8 b(,)39 b(if)2171 5308 y FE(p)p 2247 5308 V 66 x FF(")h Fn(6)f FF(\034)50 b Fn(6)39 b FG(4)2684 5308 y FE(p)p 2760 5308 V 66 x FF(")q FH(,)h(w)m(e)g(can)f(use)g(the)g(fact)273 5506 y(that)g FF(x)529 5473 y Fy(?)569 5506 y FG(\()p FF(t)p FG(\))26 b FE(\000)f FF(x)846 5461 y FC(det)q Fy(;\034)846 5530 y(t)1046 5506 y Fn(6)39 b FF(x)1208 5473 y Fy(?)1247 5506 y FG(\()p FF(t)p FG(\))27 b FE(\000)e FF(x)1525 5459 y FC(det)q Fy(;)p FC(4)1678 5411 y FD(p)p 1737 5411 33 3 v 48 x Fy(")1525 5530 y(t)1812 5506 y FH(to)39 b(pro)m(v)m(e)g (that)g(\(4.117\))h(holds)e(for)g(some)g(constan)m(t)273 5619 y FF(C)32 b(>)25 b FG(0)p FH(.)p 3596 5619 4 62 v 3600 5561 55 4 v 3600 5619 V 3653 5619 4 62 v 1845 5871 a(40)p eop %%Page: 41 41 41 40 bop 259 328 a FH(Let)37 b(us)e(no)m(w)i(consider)e(the)h(pro)s (cess)f FF(y)1625 342 y Fy(t)1689 328 y FG(=)f FF(y)1842 295 y Fy(\034)1839 350 y(t)1919 328 y FG(=)g FF(x)2076 342 y Fy(t)2129 328 y FE(\000)24 b FF(x)2276 283 y FC(det)q Fy(;\034)2276 352 y(t)2436 328 y FH(,)37 b(starting)f(at)f(time)g FF(\034)45 b FH(in)35 b FF(y)3409 342 y Fy(\034)3486 328 y FG(=)f(0)p FH(,)118 450 y(whic)m(h)23 b(describ)s(es)f(the)h (deviation)f(due)i(to)f(noise)f(from)g(the)h(deterministic)e(solution)g FF(x)3044 406 y FC(det)q Fy(;\034)3044 474 y(t)3205 450 y FH(.)38 b(It)23 b(satis\034es)118 563 y(the)31 b(SDE)1092 692 y FG(d)p FF(y)1188 706 y Fy(t)1242 692 y FG(=)1348 630 y(1)p 1348 671 46 4 v 1350 754 a FF(")1404 618 y Fr(\002)1441 692 y FF(a)1489 654 y Fy(\034)1533 692 y FG(\()p FF(t)p FG(\))p FF(y)23 b FG(+)d FF(b)1834 654 y Fy(\034)1877 692 y FG(\()p FF(y)1957 706 y Fy(t)1987 692 y FF(;)15 b(t)p FG(\))2095 618 y Fr(\003)2148 692 y FG(d)p FF(t)20 b FG(+)2384 630 y FF(\033)p 2353 671 119 4 v 2353 689 a FE(p)p 2429 689 43 4 v 65 x FF(")2496 692 y FG(d)p FF(W)2633 706 y Fy(t)2662 692 y FF(;)698 b FH(\(4.131\))118 901 y(where)31 b(w)m(e)g(ha)m(v)m(e)h(in)m(tro)s (duced)1073 1091 y FF(a)1121 1054 y Fy(\034)1164 1091 y FG(\()p FF(t)p FG(\))26 b(=)f FF(@)1437 1105 y Fy(x)1481 1091 y FF(f)10 b FG(\()p FF(x)1623 1047 y FC(det)p Fy(;\034)1623 1115 y(t)1783 1091 y FF(;)15 b(t)p FG(\))993 1249 y FF(b)1032 1211 y Fy(\034)1076 1249 y FG(\()p FF(y)s(;)g(t)p FG(\))26 b(=)f FF(f)10 b FG(\()p FF(x)1531 1204 y FC(det)p Fy(;\034)1531 1273 y(t)1711 1249 y FG(+)20 b FF(y)s(;)15 b(t)p FG(\))21 b FE(\000)f FF(f)10 b FG(\()p FF(x)2212 1204 y FC(det)p Fy(;\034)2212 1273 y(t)2372 1249 y FG(\))20 b FE(\000)g FF(a)2566 1211 y Fy(\034)2610 1249 y FG(\()p FF(t)p FG(\))p FF(y)s(:)3385 1165 y FH(\(4.132\))118 1445 y(The)31 b(follo)m(wing)e(b) s(ounds)h(are)h(direct)f(consequences)h(of)f(T)-8 b(a)m(ylor's)30 b(form)m(ula)g(and)g(Prop)s(osition)g(4.11:)705 1649 y FF(a)753 1611 y Fy(?)792 1649 y FG(\()p FF(t)p FG(\))c Fn(6)f FF(a)1065 1611 y Fy(\034)1108 1649 y FG(\()p FF(t)p FG(\))h Fn(6)g FG(~)-46 b FF(a)p FG(\()p FF(t)p FG(\))1901 b FH(\(4.133\))705 1820 y FF(a)753 1783 y Fy(\034)796 1820 y FG(\()p FF(t)p FG(\))26 b(=)f FF(a)1069 1783 y Fy(?)1108 1820 y FG(\()p FF(t)p FG(\))c(+)f FE(O)1398 1720 y Fr(\020)1462 1759 y FF(")p 1462 1800 V 1467 1883 a(t)1514 1720 y Fr(\021)1589 1820 y FG(+)g FE(O)s FG(\()p FF(t)15 b FG(e)1878 1783 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")2227 1820 y FG(\))1123 b FH(\(4.134\))705 2053 y FG(\()p FF(a)788 2016 y Fy(\034)831 2053 y FG(\))866 2016 y FD(0)890 2053 y FG(\()p FF(t)p FG(\))26 b(=)f FE(O)1190 1952 y Fr(\020)1244 2053 y FG(1)20 b(+)1410 1992 y FF(t)1443 1959 y FC(2)p 1410 2032 73 4 v 1425 2116 a FF(")1508 2053 y FG(e)1548 2016 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")1897 1952 y Fr(\021)3385 2053 y FH(\(4.135\))705 2229 y FE(j)p FF(b)769 2192 y Fy(\034)812 2229 y FG(\()p FF(y)s(;)15 b(t)p FG(\))p FE(j)26 b Fn(6)f FG(3)p FF(M)10 b(y)1341 2192 y FC(2)1381 2156 y Fr(\000)1423 2229 y FF(x)1475 2192 y Fy(?)1514 2229 y FG(\()p FF(t)p FG(\))21 b(+)f FE(j)p FF(y)s FE(j)1827 2156 y Fr(\001)1869 2229 y FF(;)196 b FH(v)-5 b(alid)28 b(for)j FF(x)2499 2192 y Fy(?)2538 2229 y FG(\()p FF(t)p FG(\))21 b(+)f FE(j)p FF(y)s FE(j)25 b Fn(6)g FF(d:)341 b FH(\(4.136\))259 2433 y(F)-8 b(or)31 b(comparison,)f(w)m(e)h(will)d(also)h(consider)h(the)h(linear)e(SDE) 1310 2671 y FG(d)o FF(y)1408 2633 y FC(0)1405 2693 y Fy(t)1473 2671 y FG(=)1578 2609 y(1)p 1578 2650 46 4 v 1580 2733 a FF(")1634 2671 y(a)1682 2633 y Fy(\034)1725 2671 y FG(\()p FF(t)p FG(\))p FF(y)1876 2633 y FC(0)1873 2693 y Fy(t)1931 2671 y FG(d)p FF(t)20 b FG(+)2167 2609 y FF(\033)p 2135 2650 119 4 v 2135 2668 a FE(p)p 2211 2668 43 4 v 66 x FF(")2279 2671 y FG(d)o FF(W)2415 2685 y Fy(t)2445 2671 y FF(:)915 b FH(\(4.137\))118 2943 y(Let)31 b FF(\013)338 2910 y Fy(\034)382 2943 y FG(\()p FF(t;)15 b(s)p FG(\))25 b(=)689 2870 y Fr(R)750 2896 y Fy(t)732 2975 y(s)795 2943 y FF(a)843 2910 y Fy(\034)886 2943 y FG(\()p FF(u)p FG(\))15 b(d)p FF(u)31 b FH(and)f(denote)i(b)m(y)1363 3213 y FF(v)1410 3175 y Fy(\034)1454 3213 y FG(\()p FF(t)p FG(\))25 b(=)1688 3151 y FF(\033)1743 3118 y FC(2)p 1688 3192 95 4 v 1714 3275 a FF(")1808 3089 y Fr(Z)1899 3115 y Fy(t)1858 3295 y(\034)1944 3213 y FG(e)1984 3175 y FC(2)p Fy(\013)2064 3152 y Fj(\034)2104 3175 y FC(\()p Fy(t;s)p FC(\))p Fy(=")2323 3213 y FG(d)p FF(s)968 b FH(\(4.138\))118 3471 y(the)31 b(v)-5 b(ariance)30 b(of)g FF(y)779 3438 y FC(0)776 3493 y Fy(t)818 3471 y FH(.)40 b(Again)30 b(w)m(e)h(in)m(tro)s(duce)g(and)g(in)m(v)m(estigate)f(a)g (function)997 3731 y FF(\020)1044 3693 y Fy(\034)1087 3731 y FG(\()p FF(t)p FG(\))c(=)1431 3669 y(1)p 1321 3710 265 4 v 1321 3793 a(2)p FE(j)q FG(~)-46 b FF(a)q FG(\()p FF(\034)10 b FG(\))p FE(j)1611 3731 y FG(e)1652 3693 y FC(2)p Fy(\013)1732 3670 y Fj(\034)1771 3693 y FC(\()p Fy(t;\034)e FC(\))p Fy(=")1997 3731 y FG(+)2078 3669 y(1)p 2078 3710 46 4 v 2080 3793 a FF(")2149 3607 y Fr(Z)2240 3633 y Fy(t)2199 3813 y(\034)2284 3731 y FG(e)2325 3693 y FC(2)p Fy(\013)2405 3670 y Fj(\034)2444 3693 y FC(\()p Fy(t;s)p FC(\))p Fy(=")2664 3731 y FG(d)p FF(s:)602 b FH(\(4.139\))118 3984 y Fq(Lemma)33 b(4.12.)41 b FB(The)33 b(function)f FF(\020)1328 3951 y Fy(\034)1370 3984 y FG(\()p FF(t)p FG(\))h FB(satis\034es)f(the)h(fol)5 b(lowing)31 b(r)-5 b(elations)33 b(for)f FF(\034)j Fn(6)25 b FF(t)g Fn(6)g FF(T)13 b FB(:)1023 4226 y FF(\020)1070 4189 y Fy(\034)1113 4226 y FG(\()p FF(t)p FG(\))26 b(=)1449 4165 y(1)p 1348 4205 248 4 v 1348 4289 a(2)p FE(j)q FG(~)-46 b FF(a)p FG(\()p FF(t)p FG(\))p FE(j)1625 4226 y FG(+)20 b FE(O)1791 4125 y Fr(\020)1870 4165 y FF(")p 1855 4205 73 4 v 1855 4289 a(t)1888 4262 y FC(3)1938 4125 y Fr(\021)2012 4226 y FG(+)g FE(O)2178 4125 y Fr(\020)2242 4165 y FG(1)p 2242 4205 46 4 v 2248 4289 a FF(t)2313 4226 y FG(e)2353 4189 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")2702 4125 y Fr(\021)3385 4226 y FH(\(4.140\))1154 4403 y FG(1)p 1033 4444 287 4 v 1033 4527 a(2)p FE(j)p FF(a)1151 4501 y Fy(?)1191 4527 y FG(\()p FF(t)p FG(\))p FE(j)1355 4465 y Fn(6)25 b FF(\020)1498 4427 y Fy(\034)1541 4465 y FG(\()p FF(t)p FG(\))h Fn(6)1885 4403 y FG(1)p 1776 4444 265 4 v 1776 4527 a(2)p FE(j)q FG(~)-46 b FF(a)p FG(\()p FF(\034)10 b FG(\))p FE(j)3385 4465 y FH(\(4.141\))1023 4703 y FG(\()p FF(\020)1105 4666 y Fy(\034)1148 4703 y FG(\))1183 4666 y FD(0)1207 4703 y FG(\()p FF(t)p FG(\))26 b Fn(6)1442 4642 y FG(1)p 1442 4682 46 4 v 1444 4765 a FF(")1497 4703 y(:)1863 b FH(\(4.142\))118 5002 y Fh(Pr)m(oof:)156 5139 y FH(1.)47 b(By)30 b(in)m(tegration)h(b)m(y)f(parts,)h(w)m(e)g (\034nd)1097 5395 y FF(\020)1144 5357 y Fy(\034)1187 5395 y FG(\()p FF(t)p FG(\))25 b(=)1522 5333 y(1)p 1421 5374 248 4 v 1421 5457 a(2)p FE(j)q FG(~)-46 b FF(a)q FG(\()p FF(t)p FG(\))p FE(j)1699 5395 y(\000)1800 5333 y FG(1)p 1800 5374 46 4 v 1800 5457 a(2)1870 5271 y Fr(Z)1961 5297 y Fy(t)1921 5477 y(\034)2016 5333 y FG(\()p FF(a)2099 5300 y Fy(\034)2143 5333 y FG(\))2178 5300 y FD(0)2201 5333 y FG(\()p FF(s)p FG(\))p 2016 5374 299 4 v 2043 5457 a FF(a)2091 5431 y Fy(\034)2135 5457 y FG(\()p FF(s)p FG(\))2248 5431 y FC(2)2340 5395 y FG(e)2380 5357 y FC(2)p Fy(\013)2460 5334 y Fj(\034)2500 5357 y FC(\()p Fy(t;s)p FC(\))p Fy(=")2719 5395 y FG(d)p FF(s:)547 b FH(\(4.143\))1845 5871 y(41)p eop %%Page: 42 42 42 41 bop 273 328 a FH(The)31 b(relation)f FE(j)p FF(a)864 295 y Fy(\034)907 328 y FG(\()p FF(s)p FG(\))p FE(j)c Fn(>)f FE(j)q FG(~)-46 b FF(a)p FG(\()p FF(s)p FG(\))p FE(j)26 b Fn(>)f FF(\021)s FE(j)p FF(a)p FG(\()p FF(s)p FG(\))p FE(j)31 b FH(together)h(with)e(\(4.135\))h(yields)416 456 y Fr(\014)416 511 y(\014)416 566 y(\014)416 620 y(\014)447 465 y(Z)537 491 y Fy(t)497 671 y(\034)592 527 y FG(\()p FF(a)675 494 y Fy(\034)719 527 y FG(\))754 494 y FD(0)777 527 y FG(\()p FF(s)p FG(\))p 592 567 299 4 v 620 651 a FF(a)668 624 y Fy(\034)711 651 y FG(\()p FF(s)p FG(\))824 624 y FC(2)916 588 y FG(e)956 551 y FC(2)p Fy(\013)1036 527 y Fj(\034)1076 551 y FC(\()p Fy(t;s)p FC(\))p Fy(=")1296 588 y FG(d)o FF(s)1389 456 y Fr(\014)1389 511 y(\014)1389 566 y(\014)1389 620 y(\014)1444 588 y Fn(6)25 b FB(c)-5 b(onst)1756 465 y Fr(Z)1847 491 y Fy(t)1806 671 y(\034)1876 487 y Fr(\020)1959 527 y FG(1)p 1941 567 83 4 v 1941 651 a FF(s)1984 624 y FC(2)2053 588 y FG(+)2154 527 y(1)p 2154 567 46 4 v 2156 651 a FF(")2224 588 y FG(e)2265 551 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(s;\034)8 b FC(\))p Fy(=")2621 487 y Fr(\021)2690 588 y FG(e)2731 551 y FD(\000)p FC(2)p Fy(\021)r(\013)p FC(\()p Fy(t;s)p FC(\))p Fy(=")3124 588 y FG(d)o FF(s:)143 b FH(\(4.144\))273 857 y(The)40 b(second)f(term)g(in)f(brac)m(k)m(ets)i(giv)m(es)e(a)h (con)m(tribution)h(of)e(order)2674 821 y FC(1)p 2674 836 36 4 v 2679 888 a Fy(t)2734 857 y FG(e)2775 824 y FD(\000)p Fy(\021)r(\013)p FC(\()p Fy(t;\034)8 b FC(\))p Fy(=")3124 857 y FH(.)66 b(In)38 b(order)i(to)273 970 y(estimate)d(the)h(con)m(tribution)h(of)f(the)g(\034rst)g(term,)i(w)m (e)e(p)s(erform)g(the)g(c)m(hange)i(of)d(v)-5 b(ariables)37 b FF(u)h FG(=)273 1083 y FF(\021)s FG(\()p FF(t)389 1050 y FC(2)450 1083 y FE(\000)19 b FF(s)583 1050 y FC(2)622 1083 y FG(\))p FF(=)p FG(2)p FF(")p FH(,)32 b(thereb)m(y)g(obtaining) 430 1226 y Fr(Z)521 1252 y Fy(t)481 1432 y(\034)594 1288 y FG(1)p 576 1329 83 4 v 576 1412 a FF(s)619 1386 y FC(2)683 1350 y FG(e)724 1312 y FD(\000)p Fy(\021)r FC(\()p Fy(t)868 1289 y Fw(2)904 1312 y FD(\000)p Fy(s)992 1289 y Fw(2)1026 1312 y FC(\))p Fy(=)p FC(2)p Fy(")1176 1350 y FG(d)o FF(s)25 b FG(=)1439 1288 y FF(")p 1400 1329 121 4 v 1400 1412 a(\021)s(t)1481 1386 y FC(3)1546 1226 y Fr(Z)1637 1252 y Fy(\030)s FD(\000)p Fy(\030)1757 1261 y Fw(0)1596 1432 y FC(0)1989 1288 y FG(e)2030 1255 y FD(\000)p Fy(u)p 1820 1329 479 4 v 1820 1416 a FG(\(1)c FE(\000)f FF(u=\030)t FG(\))2188 1389 y FC(3)p Fy(=)p FC(2)2324 1350 y FG(d)p FF(u)25 b Fn(6)2597 1288 y FF(")p 2558 1329 121 4 v 2558 1412 a(\021)s(t)2639 1386 y FC(3)2688 1249 y Fr(h)2731 1350 y FG(2)2776 1312 y FC(3)p Fy(=)p FC(2)2907 1350 y FG(+)20 b(2)3053 1288 y FF(\030)3097 1255 y FC(3)p Fy(=)p FC(2)3222 1288 y FG(e)3263 1255 y FD(\000)p Fy(\030)s(=)p FC(2)p 3053 1329 373 4 v 3162 1347 a FE(p)p 3238 1347 80 4 v 68 x FF(\030)3278 1429 y FC(0)3436 1249 y Fr(i)3479 1350 y FF(;)3385 1515 y FH(\(4.145\))273 1628 y(where)37 b FF(\030)j FG(=)35 b FF(\021)s(t)807 1595 y FC(2)846 1628 y FF(=)p FG(2)p FF(")j FH(and)e FF(\030)1237 1642 y FC(0)1312 1628 y FG(=)f FF(\021)s(\034)1516 1595 y FC(2)1556 1628 y FF(=)p FG(2)p FF(")p FH(.)59 b(The)37 b(last)e(inequalit)m(y)g(is)g(obtained)h(b)m(y)h(splitting)d(the)273 1741 y(in)m(tegral)29 b(at)g FF(\030)t(=)p FG(2)p FH(.)40 b(Using)28 b(the)h(fact)g(that)g FF(t)1720 1708 y FC(3)1774 1741 y FG(e)1815 1708 y FD(\000)p Fy(\021)r(t)1932 1684 y Fw(2)1967 1708 y Fy(=)p FC(4)p Fy(")2100 1741 y Fn(6)c FG(\(6)p FF("=\021)s FG(\))2446 1708 y FC(3)p Fy(=)p FC(2)2574 1741 y FG(e)2614 1708 y FD(\000)p FC(3)p Fy(=)p FC(2)2807 1741 y FH(for)k(all)e FF(t)e Fn(>)g FG(0)p FH(,)k(w)m(e)h(reac)m(h)273 1854 y(the)e(conclusion)f(that)h(this)f(in) m(tegral)g(is)f(b)s(ounded)j(b)m(y)f(a)f(constan)m(t)i(times)d FF("=t)2925 1821 y FC(3)2965 1854 y FH(,)i(whic)m(h)g(completes)273 1967 y(the)j(pro)s(of)f(of)37 b(\(4.140\))q(.)156 2080 y(2.)47 b(W)-8 b(e)31 b(no)m(w)g(use)f(the)h(fact)f(that)h FF(\020)1350 2047 y Fy(\034)1393 2080 y FG(\()p FF(t)p FG(\))g FH(solv)m(es)e(the)h(ODE)1088 2265 y FG(d)o FF(\020)1185 2232 y Fy(\034)p 1088 2306 141 4 v 1116 2389 a FG(d)p FF(t)1263 2326 y FG(=)1369 2265 y(1)p 1369 2306 46 4 v 1371 2389 a FF(")1424 2253 y Fr(\000)1466 2326 y FG(2)p FF(a)1559 2289 y Fy(\034)1603 2326 y FG(\()p FF(t)p FG(\))p FF(\020)1753 2289 y Fy(\034)1816 2326 y FG(+)20 b(1)1952 2253 y Fr(\001)1994 2326 y FF(;)196 b(\020)2262 2289 y Fy(\034)2305 2326 y FG(\()p FF(\034)10 b FG(\))26 b(=)2667 2265 y(1)p 2557 2306 265 4 v 2557 2389 a(2)p FE(j)q FG(~)-46 b FF(a)q FG(\()p FF(\034)10 b FG(\))p FE(j)2832 2326 y FF(:)528 b FH(\(4.146\))273 2579 y(Then,)42 b(\(4.142\))d(is)e(an)i (immediate)d(consequence)j(of)f(this)g(relation,)h(and)g(\(4.141\))h (is)d(obtained)273 2692 y(from)30 b(the)g(fact)h(that)1356 2769 y FG(d)o FF(\020)1453 2736 y Fy(\034)1496 2769 y FG(\()p FF(t)p FG(\))p 1356 2809 244 4 v 1436 2893 a(d)o FF(t)1635 2830 y FG(=)1741 2769 y(1)p 1741 2809 46 4 v 1743 2893 a FF(")1796 2729 y Fr(\020)1850 2830 y FE(\000)1931 2769 y(j)p FF(a)2004 2736 y Fy(\034)2047 2769 y FG(\()p FF(t)p FG(\))p FE(j)p 1931 2809 246 4 v 1944 2893 a(j)q FG(~)-46 b FF(a)p FG(\()p FF(\034)10 b FG(\))p FE(j)2206 2830 y FG(+)20 b(1)2342 2729 y Fr(\021)2422 2830 y Fn(6)25 b FG(0)p FF(;)797 b FH(\(4.147\))273 3046 y(whenev)m(er)32 b FF(\020)717 3013 y Fy(\034)760 3046 y FG(\()p FF(t)p FG(\))25 b(=)g(1)p FF(=)p FG(2)p FE(j)q FG(~)-46 b FF(a)r FG(\()p FF(\034)10 b FG(\))p FE(j)p FH(,)31 b(and)897 3243 y FG(d)p 873 3283 84 4 v 873 3367 a(d)o FF(t)966 3203 y Fr(\020)1020 3304 y FF(\020)1067 3267 y Fy(\034)1110 3304 y FG(\()p FF(t)p FG(\))21 b FE(\000)1455 3243 y FG(1)p 1334 3283 287 4 v 1334 3367 a(2)p FE(j)p FF(a)1452 3340 y Fy(?)1493 3367 y FG(\()p FF(t)p FG(\))p FE(j)1631 3203 y Fr(\021)1711 3304 y FG(=)1817 3243 y(1)p 1817 3283 46 4 v 1819 3367 a FF(")1872 3203 y Fr(\020)1927 3304 y FE(\000)2008 3243 y(j)p FF(a)2081 3210 y Fy(\034)2124 3243 y FG(\()p FF(t)p FG(\))p FE(j)p 2008 3283 246 4 v 2010 3367 a(j)p FF(a)2083 3340 y Fy(?)2122 3367 y FG(\()p FF(t)p FG(\))p FE(j)2283 3304 y FG(+)e(1)2418 3203 y Fr(\021)2493 3304 y FE(\000)2625 3243 y FF(a)2673 3210 y Fy(?)2713 3205 y FD(0)2736 3243 y FG(\()p FF(t)p FG(\))p 2594 3283 276 4 v 2594 3367 a(2)p FF(a)2687 3340 y Fy(?)2727 3367 y FG(\()p FF(t)p FG(\))2830 3340 y FC(2)2905 3304 y Fn(>)25 b FG(0)p FF(;)314 b FH(\(4.148\))273 3557 y(whenev)m(er)37 b FF(\020)722 3524 y Fy(\034)765 3557 y FG(\()p FF(t)p FG(\))d(=)g(1)p FF(=)p FG(2)p FE(j)p FF(a)1215 3524 y Fy(?)1256 3557 y FG(\()p FF(\034)10 b FG(\))p FE(j)p FH(.)57 b(Here)36 b(w)m(e)g(used)g(\(4.133\))h(and)f(the)g(monotonicit) m(y)e(of)j FG(~)-47 b FF(a)q FG(\()p FF(t)p FG(\))35 b FH(for)273 3670 y(small)28 b FF(t)p FH(.)p 3596 3670 4 62 v 3600 3612 55 4 v 3600 3670 V 3653 3670 4 62 v 259 3858 a(W)-8 b(e)22 b(note)g(that)h(Lemma)e(4.12)h(and)g(the)g(b)s (ounds)g(\(4.133\))g(on)g FF(a)2364 3825 y Fy(\034)2429 3858 y FH(imply)d(the)j(existence)f(of)g(constan)m(ts)118 3971 y FF(c)157 3985 y FC(+)242 3971 y Fn(>)k FF(c)377 3985 y FD(\000)461 3971 y FF(>)g FG(0)p FH(,)31 b(dep)s(ending)f(only)g (on)g FF(f)40 b FH(and)31 b FF(T)13 b FH(,)30 b(suc)m(h)g(that)1263 4130 y FF(c)1302 4144 y FD(\000)p 1263 4170 99 4 v 1295 4254 a FF(t)1396 4191 y Fn(6)25 b FF(\020)1539 4154 y Fy(\034)1582 4191 y FG(\()p FF(t)p FG(\))h Fn(6)1816 4130 y FF(c)1855 4144 y FC(+)p 1816 4170 V 1849 4254 a FF(t)2106 4191 y FE(8)p FF(t)e FE(2)h FG([)p FF(\034)5 b(;)15 b(T)e FG(])p FF(:)859 b FH(\(4.149\))118 4429 y(W)-8 b(e)33 b(can)g(no)m(w)g(easily)e(pro)m(v)m(e)i(that)h FF(y)1378 4396 y FC(0)1375 4452 y Fy(t)1449 4429 y FH(remains)d(in)h(a) g(strip)g(of)g(width)h FF(h)2603 4361 y FE(p)p 2679 4361 90 4 v 68 x FF(\020)2726 4403 y Fy(\034)2801 4429 y FH(with)f(high)g (probabilit)m(y)-8 b(,)118 4542 y(in)30 b(m)m(uc)m(h)h(the)f(same)g(w)m (a)m(y)h(as)f(in)g(Prop)s(osition)f(3.3.)118 4730 y Fq(Prop)s(osition) 35 b(4.13.)41 b FB(F)-7 b(or)33 b(su\036ciently)f(smal)5 b(l)32 b FF(T)45 b FB(and)32 b FF(")p FB(,)g(and)g(al)5 b(l)33 b FF(t)25 b FE(2)f FG([)p FF(\034)5 b(;)15 b(T)e FG(])p FB(,)716 4989 y Fo(P)777 4952 y Fy(\034)t(;)p FC(0)871 4888 y Fr(n)967 4989 y FG(sup)932 5066 y Fy(\034)8 b Fx(6)p Fy(s)p Fx(6)p Fy(t)1242 4928 y FE(j)p FF(y)1315 4895 y FC(0)1312 4950 y Fy(s)1354 4928 y FE(j)p 1164 4969 294 4 v 1164 4987 a Fr(p)p 1255 4987 203 4 v 77 x FF(\020)1302 5038 y Fy(\034)1344 5064 y FG(\()p FF(s)p FG(\))1493 4989 y Fn(>)25 b FF(h)1641 4888 y Fr(o)1727 4989 y Fn(6)g FF(C)1895 4952 y Fy(\034)1938 4989 y FG(\()p FF(t;)15 b(")p FG(\))g(exp)2278 4888 y Fr(n)2338 4989 y FE(\000)2419 4928 y FG(1)p 2419 4969 46 4 v 2419 5052 a(2)2486 4928 y FF(h)2538 4895 y FC(2)p 2485 4969 95 4 v 2485 5052 a FF(\033)2540 5026 y FC(2)2589 4916 y Fr(\002)2627 4989 y FG(1)21 b FE(\000)f FF(r)s FG(\()p FF(")p FG(\))2940 4916 y Fr(\003)2978 4888 y(o)3039 4989 y FF(;)321 b FH(\(4.150\))118 5260 y FB(wher)-5 b(e)34 b FF(r)s FG(\()p FF(")p FG(\))26 b(=)f FE(O)s FG(\()p FF(")p FG(\))33 b FB(and)1405 5421 y FF(C)1477 5383 y Fy(\034)1519 5421 y FG(\()p FF(t;)15 b(")p FG(\))27 b(=)1837 5359 y FE(j)p FF(\013)1920 5326 y Fy(\034)1964 5359 y FG(\()p FF(t;)15 b(\034)10 b FG(\))p FE(j)p 1837 5400 347 4 v 1969 5483 a FF(")2011 5457 y FC(2)2213 5421 y FG(+)20 b(2)p FF(:)1011 b FH(\(4.151\))1845 5871 y(42)p eop %%Page: 43 43 43 42 bop 118 328 a Fh(Pr)m(oof:)47 b FH(Let)29 b FF(K)j FG(=)25 b FE(dj)p FF(\013)966 295 y Fy(\034)1010 328 y FG(\()p FF(t;)15 b(\034)10 b FG(\))p FE(j)p FF(=)p FG(2)p FF(")1360 295 y FC(2)1402 328 y FE(e)29 b FH(and)g(de\034ne)h(a) e(partition)g FF(\034)36 b FG(=)25 b FF(u)2577 342 y FC(0)2641 328 y FF(<)g FE(\001)15 b(\001)g(\001)27 b FF(<)e(u)3017 342 y Fy(K)3110 328 y FG(=)g FF(t)j FH(of)h FG([)p FF(\034)5 b(;)15 b(t)p FG(])29 b FH(b)m(y)1084 532 y FE(j)p FF(\013)1167 494 y Fy(\034)1211 532 y FG(\()p FF(u)1298 547 y Fy(k)1341 532 y FF(;)15 b(\034)10 b FG(\))p FE(j)26 b FG(=)f(2)p FF(")1700 494 y FC(2)1740 532 y FF(k)s(;)196 b(k)29 b FG(=)c(1)p FF(;)15 b(:)g(:)g(:)i(;)e(K)27 b FE(\000)20 b FG(1)p FF(:)690 b FH(\(4.152\))118 736 y(Since)35 b FF(a)407 703 y Fy(\034)450 736 y FG(\()p FF(s)p FG(\))e Fn(6)g FG(~)-46 b FF(a)p FG(\()p FF(s)p FG(\))33 b Fn(6)f FE(\000)p FF(\021)s(s=)p FG(2)p FH(,)37 b(w)m(e)e(obtain)g FF(u)1785 751 y Fy(k)1851 736 y FE(\000)23 b FF(u)1997 751 y Fy(k)r FD(\000)p FC(1)2162 736 y Fn(6)33 b FG(4)p FF(")2353 703 y FC(2)2393 736 y FF(=)p FG(\()p FF(\021)s(u)2573 751 y Fy(k)r FD(\000)p FC(1)2707 736 y FG(\))i FH(for)f(all)f FF(k)s FH(.)54 b(No)m(w)35 b(w)m(e)h(can)118 849 y(pro)s(ceed)31 b(as)f(in)g(the)g(pro)s(of)g(of)g(Prop)s(osition)g (3.3.)p 3596 849 4 62 v 3600 791 55 4 v 3600 849 V 3653 849 4 62 v 259 1037 a(W)-8 b(e)30 b(can)g(no)m(w)h(compare)f(the)g (solutions)e(of)h(the)h(linear)f(and)h(the)g(nonlinear)f(equation.)40 b(T)-8 b(o)31 b(do)e(so,)118 1150 y(w)m(e)i(de\034ne)g(the)g(ev)m(en)m (ts)1095 1354 y FG(\012)1161 1368 y Fy(t)1190 1354 y FG(\()p FF(h)p FG(\))26 b(=)1434 1280 y Fr(\010)1487 1354 y FF(!)13 b FG(:)31 b FE(j)p FF(y)1686 1316 y Fy(\034)1683 1376 y(s)1729 1354 y FE(j)26 b FF(<)f(h)1928 1272 y Fr(p)p 2019 1272 203 4 v 82 x FF(\020)2066 1328 y Fy(\034)2108 1354 y FG(\()p FF(s)p FG(\))h FE(8)p FF(s)e FE(2)h FG([)p FF(\034)5 b(;)15 b(t)p FG(])2619 1280 y Fr(\011)3385 1354 y FH(\(4.153\))1085 1510 y FG(\012)1151 1473 y FC(0)1151 1533 y Fy(t)1190 1510 y FG(\()p FF(h)p FG(\))26 b(=)1434 1437 y Fr(\010)1487 1510 y FF(!)13 b FG(:)31 b FE(j)p FF(y)1686 1473 y FC(0)1683 1533 y Fy(s)1725 1510 y FE(j)26 b FF(<)f(h)1924 1428 y Fr(p)p 2015 1428 V 82 x FF(\020)2062 1484 y Fy(\034)2105 1510 y FG(\()p FF(s)p FG(\))g FE(8)p FF(s)g FE(2)g FG([)p FF(\034)5 b(;)15 b(t)p FG(])2616 1437 y Fr(\011)2669 1510 y FF(:)691 b FH(\(4.154\))118 1715 y(The)31 b(follo)m(wing)e(prop)s(osition)g(sho)m(ws)h(that)h FF(y)1664 1682 y Fy(\034)1661 1737 y(t)1737 1715 y FH(and)g FF(y)1961 1682 y FC(0)1958 1737 y Fy(t)2031 1715 y FH(di\033er)e(only)g (sligh)m(tly)-8 b(.)118 1902 y Fq(Prop)s(osition)39 b(4.14.)k FB(L)-5 b(et)35 b FF(\015)h FG(=)30 b(1)37 b FE(_)g FG(48)p FF(M)10 b FG(\(2)25 b(+)1848 1847 y FE(p)p 1923 1847 99 4 v 1923 1902 a FF(c)1962 1916 y FC(+)2022 1902 y FG(\))p FF(c)2096 1869 y FC(2)2096 1925 y(+)2155 1902 y FF(=)2200 1847 y FE(p)p 2277 1847 V 2277 1902 a FF(c)2316 1916 y FD(\000)2410 1902 y FB(and)35 b(assume)h FF(h)30 b(<)g(\034)10 b(=\015)41 b FB(as)35 b(wel)5 b(l)35 b(as)118 2015 y FF(h)26 b Fn(6)f FG([)p FF(d)20 b FE(\000)g FF(x)527 1982 y Fy(?)567 2015 y FG(\()p FF(t)p FG(\)])695 1950 y FE(p)p 771 1950 51 4 v 65 x FF(\034)10 b(=)p FG(\(2)946 1960 y FE(p)p 1023 1960 99 4 v 1023 2015 a FF(c)1062 2029 y FC(+)1122 2015 y FG(\))p FB(.)41 b(Then)1385 2271 y FG(\012)1451 2285 y Fy(t)1480 2271 y FG(\()p FF(h)p FG(\))1629 2204 y FC(a)p Fy(:)p FC(s)p Fy(:)1644 2271 y FE(\032)g FG(\012)1822 2234 y FC(0)1822 2294 y Fy(t)1861 2171 y Fr(\020h)1958 2271 y FG(1)21 b(+)f FF(\015)2177 2210 y(h)p 2177 2251 53 4 v 2178 2334 a(\034)2239 2171 y Fr(i)2282 2271 y FF(h)2334 2171 y Fr(\021)3385 2271 y FH(\(4.155\))1375 2492 y FG(\012)1441 2454 y FC(0)1441 2514 y Fy(t)1480 2492 y FG(\()p FF(h)p FG(\))1629 2424 y FC(a)p Fy(:)p FC(s)p Fy(:)1644 2492 y FE(\032)41 b FG(\012)1822 2506 y Fy(t)1851 2391 y Fr(\020)q(h)1949 2492 y FG(1)20 b(+)g FF(\015)2167 2430 y(h)p 2167 2471 V 2168 2554 a(\034)2230 2391 y Fr(i)2273 2492 y FF(h)2325 2391 y Fr(\021)2379 2492 y FF(:)981 b FH(\(4.156\))118 2804 y Fh(Pr)m(oof:)47 b FH(Assume)40 b(\034rst)j(that)f FF(!)48 b FE(2)d FG(\012)1507 2771 y FC(0)1507 2827 y Fy(t)1546 2804 y FG(\()p FF(h)p FG(\))p FH(.)77 b(W)-8 b(e)43 b(in)m(tro)s(duce)f(the)h(di\033erence)f FF(z)2977 2818 y Fy(s)3059 2804 y FG(=)j FF(y)3223 2771 y Fy(\034)3220 2827 y(s)3294 2804 y FE(\000)28 b FF(y)3441 2771 y FC(0)3438 2827 y Fy(s)3480 2804 y FH(,)45 b(set)118 2917 y FF(\016)29 b FG(=)c FF(\015)5 b(h=\034)36 b(<)25 b FG(1)p FH(,)31 b(and)g(de\034ne)g(the)g(\034rst)f(exit)f(time)885 3121 y FG(^)-50 b FF(\034)35 b FG(=)25 b(inf)1162 3048 y Fr(\010)1215 3121 y FF(s)g FE(2)f FG([)p FF(\034)5 b(;)15 b(t)p FG(])10 b(:)32 b FE(j)p FF(z)1670 3135 y Fy(s)1707 3121 y FE(j)26 b Fn(>)f FF(\016)s(h)1949 3039 y Fr(p)p 2041 3039 203 4 v 2041 3121 a FF(\020)2088 3095 y Fy(\034)2130 3121 y FG(\()p FF(s)p FG(\))2259 3048 y Fr(\011)2337 3121 y FE(2)g FG([)p FF(\034)5 b(;)15 b(t)p FG(])21 b FE([)f(f1g)p FF(:)486 b FH(\(4.157\))118 3325 y(On)31 b FF(A)25 b FG(=)g(\012)524 3293 y FC(0)524 3348 y Fy(t)563 3325 y FG(\()p FF(h)p FG(\))d FE(\\)d(f)5 b FG(^)-50 b FF(\034)36 b(<)25 b FE(1g)p FH(,)31 b(w)m(e)g(get)f(b)m(y)h(the)g(estimate)e (\(4.136\))i(on)g FF(b)2594 3293 y Fy(\034)2637 3325 y FH(,)f(Lemma)g(4.12)h(and)f(\(4.149\))369 3586 y FE(j)p FF(z)436 3600 y Fy(s)474 3586 y FE(j)25 b Fn(6)630 3525 y FG(1)p 630 3565 46 4 v 632 3649 a FF(")701 3462 y Fr(Z)791 3489 y Fy(t)751 3669 y(\034)836 3586 y FG(e)877 3549 y Fy(\013)922 3525 y Fj(\034)961 3549 y FC(\()p Fy(s;u)p FC(\))p Fy(=")1181 3586 y FE(j)p FF(b)1245 3549 y Fy(\034)1288 3586 y FG(\()p FF(y)1368 3600 y Fy(u)1413 3586 y FF(;)15 b(u)p FG(\))p FE(j)g FG(d)q FF(u)524 3839 y Fn(6)25 b FG(6)p FF(M)10 b FG(\(1)22 b(+)d FF(\016)s FG(\))1033 3802 y FC(2)1074 3738 y Fr(\020)1128 3839 y FG(2)p FF(c)1212 3853 y FC(+)1282 3778 y FF(h)p 1282 3818 53 4 v 1283 3901 a(\034)1365 3839 y FG(+)h(\(1)h(+)f FF(\016)s FG(\))p FF(c)1765 3791 y FC(3)p Fy(=)p FC(2)1765 3862 y(+)1886 3778 y FF(h)1938 3745 y FC(2)p 1886 3818 92 4 v 1887 3901 a FF(\034)1937 3875 y FC(2)1988 3738 y Fr(\021)2090 3778 y FF(c)2129 3792 y FC(+)p 2052 3818 175 4 v 2052 3846 a FE(p)p 2128 3846 99 4 v 55 x FF(c)2167 3915 y FD(\000)2251 3839 y FF(h)2303 3757 y Fr(p)p 2395 3757 203 4 v 2395 3839 a FF(\020)2442 3813 y Fy(\034)2484 3839 y FG(\()p FF(s)p FG(\))26 b FF(<)f(\016)s(h)2814 3757 y Fr(p)p 2906 3757 V 2906 3839 a FF(\020)2953 3813 y Fy(\034)2996 3839 y FG(\()p FF(s)p FG(\))p FF(;)251 b FH(\(4.158\))118 4101 y(for)29 b(all)f FF(s)d FE(2)f FG([)p FF(\034)5 b(;)21 b FG(^)-51 b FF(\034)11 b FG(])p FH(,)30 b(whic)m(h)f(leads)g(to)g(a)h(con)m(tradiction)g(for)f FF(s)c FG(=)k(^)-49 b FF(\034)10 b FH(.)40 b(W)-8 b(e)29 b(conclude)h(that)g Fo(P)p FG(\()p FF(A)p FG(\))c(=)f(0)30 b FH(and)118 4214 y(th)m(us)f FE(j)p FF(z)382 4228 y Fy(s)420 4214 y FE(j)c Fn(6)g FF(\015)5 b(h)670 4181 y FC(2)710 4137 y Fr(p)p 801 4137 V 77 x FF(\020)848 4188 y Fy(\034)891 4214 y FG(\()p FF(s)p FG(\))p FF(=\034)39 b FH(for)29 b(all)e FF(s)h FH(in)g FG([)p FF(\034)5 b(;)15 b(t)p FG(])p FH(,)29 b(whic)m(h)h(pro)m(v)m(es)f(\(4.156\))q(.)40 b(The)29 b(inclusion)e(\(4.155\))j(is)d(a)118 4327 y(straigh)m(tforw)m (ard)32 b(consequence)f(of)f(the)h(same)e(estimates.)p 3596 4327 4 62 v 3600 4269 55 4 v 3600 4327 V 3653 4327 4 62 v 259 4515 a(No)m(w,)h(the)g(follo)m(wing)d(corollary)h(is)g(a)h (direct)g(consequence)h(of)e(the)i(t)m(w)m(o)g(preceding)g(prop)s (ositions.)118 4702 y Fq(Corollary)37 b(4.15.)k FB(Ther)-5 b(e)33 b(exists)g FF(h)1394 4716 y FC(0)1466 4702 y FB(such)g(that)g (if)e FF(h)26 b(<)f(h)2169 4716 y FC(0)2209 4702 y FF(\034)10 b FB(,)32 b(then)232 4972 y Fo(P)293 4934 y Fy(\034)t(;)t FC(~)-39 b Fy(x)p FC(\()p Fy(\034)8 b FC(\))486 4843 y Fr(\032)589 4972 y FG(sup)554 5048 y Fy(\034)g Fx(6)p Fy(s)p Fx(6)p Fy(t)786 4910 y FE(j)p FF(x)863 4924 y Fy(s)920 4910 y FE(\000)20 b FF(x)1063 4866 y FC(det)q Fy(;\034)1063 4922 y(s)1224 4910 y FE(j)p 786 4951 464 4 v 871 4969 a Fr(p)p 961 4969 203 4 v 961 5047 a FF(\020)1008 5020 y Fy(\034)1051 5047 y FG(\()p FF(s)p FG(\))1284 4972 y FF(>)25 b(h)1432 4843 y Fr(\033)1526 4972 y Fn(6)g FF(C)1694 4934 y Fy(\034)1737 4972 y FG(\()p FF(t;)15 b(")p FG(\))g(exp)2077 4871 y Fr(n)2138 4972 y FE(\000)2219 4910 y FG(1)p 2219 4951 46 4 v 2219 5034 a(2)2285 4910 y FF(h)2337 4877 y FC(2)p 2284 4951 95 4 v 2284 5034 a FF(\033)2339 5008 y FC(2)2388 4871 y Fr(h)2431 4972 y FG(1)21 b FE(\000)f(O)s FG(\()p FF(")p FG(\))h FE(\000)f(O)2962 4871 y Fr(\020)3026 4910 y FF(h)p 3026 4951 53 4 v 3027 5034 a(\034)3088 4871 y Fr(\021)q(io)3246 4972 y FF(;)114 b FH(\(4.159\))118 5242 y FB(wher)-5 b(e)34 b FF(C)445 5209 y Fy(\034)487 5242 y FG(\()p FF(t;)15 b(")p FG(\))34 b FB(is)e(given)h(by)41 b FH(\(4.151\))q FB(.)1845 5871 y FH(43)p eop %%Page: 44 44 44 43 bop 118 328 a FI(App)t(endix)118 531 y FH(The)34 b(app)s(endix)e(pro)m(vides)h(t)m(w)m(o)h(lemmas)d(needed)j(in)e (Sections)g(3)h(and)h(4.)48 b(The)34 b(\034rst)f(one)g(uses)f(exp)s(o-) 118 643 y(nen)m(tial)i(martingales)g(to)h(deduce)g(an)g(exp)s(onen)m (tial)g(b)s(ound)g(on)g(the)g(probabilit)m(y)f(that)h(a)g(sto)s(c)m (hastic)118 756 y(in)m(tegral)30 b(exceeds)h(a)f(giv)m(en)g(v)-5 b(alue.)118 933 y Fq(Lemma)33 b(A.1.)42 b FB(L)-5 b(et)32 b FF(')p FG(\()p FF(u)p FG(\))i FB(b)-5 b(e)34 b(a)e(Bor)-5 b(el-me)g(asur)g(able)37 b(deterministic)32 b(function)g(such)h(that) 1507 1180 y FG(\010\()p FF(t)p FG(\))25 b(=)1797 1057 y Fr(Z)1888 1083 y Fy(t)1848 1263 y FC(0)1933 1180 y FF(')p FG(\()p FF(u)p FG(\))2114 1143 y FC(2)2170 1180 y FG(d)o FF(u)1181 b FH(\(A.1\))118 1412 y FB(exists.)42 b(Then)998 1557 y Fo(P)1053 1456 y Fr(n)1147 1557 y FG(sup)1114 1634 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1333 1434 y Fr(Z)1423 1460 y Fy(s)1383 1640 y FC(0)1475 1557 y FF(')p FG(\()p FF(u)p FG(\))15 b(d)q FF(W)1809 1571 y Fy(u)1879 1557 y Fn(>)25 b FF(\016)2018 1456 y Fr(o)2105 1557 y Fn(6)g FG(exp)2340 1429 y Fr(\032)2408 1557 y FE(\000)2554 1496 y FF(\016)2597 1463 y FC(2)p 2489 1536 215 4 v 2489 1620 a FG(2\010\()p FF(t)p FG(\))2713 1429 y Fr(\033)3453 1557 y FH(\(A.2\))118 1794 y Fh(Pr)m(oof:)47 b FH(Let)30 b FF(P)44 b FH(denote)31 b(the)g(left-hand)f(side)f(of)37 b(\(A.2\).)j(F)-8 b(or)31 b(an)m(y)g FF(\015)f(>)25 b FG(0)p FH(,)31 b(w)m(e)g(ha)m(v)m(e)356 2029 y FF(P)38 b FG(=)25 b Fo(P)603 1929 y Fr(n)698 2029 y FG(sup)665 2106 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)883 2029 y FG(exp)1022 1929 y Fr(n)1082 2029 y FF(\015)1150 1906 y Fr(Z)1241 1932 y Fy(s)1200 2112 y FC(0)1292 2029 y FF(')p FG(\()p FF(u)p FG(\))15 b(d)q FF(W)1626 2043 y Fy(u)1671 1929 y Fr(o)1757 2029 y Fn(>)25 b FG(e)1893 1992 y Fy(\015)t(\016)1972 1929 y Fr(o)2057 2029 y Fn(6)g Fo(P)2208 1929 y Fr(n)2303 2029 y FG(sup)2270 2106 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)2488 2029 y FF(M)2576 2043 y Fy(s)2638 2029 y Fn(>)g FG(e)2774 1992 y Fy(\015)t(\016)r FD(\000)2912 1961 y Fj(\015)2947 1940 y Fw(2)p 2913 1977 71 3 v 2933 2018 a(2)2994 1992 y FC(\010\()p Fy(t)p FC(\))3129 1929 y Fr(o)3190 2029 y FF(;)238 b FH(\(A.3\))118 2275 y(where)972 2410 y FF(M)1060 2424 y Fy(s)1122 2410 y FG(=)25 b(exp)1357 2309 y Fr(n)1418 2286 y(Z)1509 2312 y Fy(s)1468 2492 y FC(0)1561 2410 y FF(\015)5 b(')p FG(\()p FF(u)p FG(\))15 b(d)q FF(W)1947 2424 y Fy(u)2012 2410 y FE(\000)2113 2374 y FC(1)p 2113 2389 36 4 v 2113 2441 a(2)2173 2286 y Fr(Z)2264 2312 y Fy(s)2224 2492 y FC(0)2316 2410 y FF(\015)2368 2372 y FC(2)2407 2410 y FF(')p FG(\()p FF(u)p FG(\))2588 2372 y FC(2)2644 2410 y FG(d)p FF(u)2747 2309 y Fr(o)3453 2410 y FH(\(A.4\))118 2616 y(is)31 b(an)i(\(exp)s(onen)m (tial\))f(martingale,)g(satisfying)e Fo(E)11 b FE(f)p FF(M)1970 2630 y Fy(t)2005 2616 y FE(g)30 b FG(=)e Fo(E)11 b FE(f)p FF(M)2368 2630 y FC(0)2413 2616 y FE(g)29 b FG(=)g(1)p FH(,)34 b(whic)m(h)f(implies)c(b)m(y)j(Do)s(ob's)118 2729 y(submartingale)d(inequalit)m(y)-8 b(,)29 b(that)1194 2953 y Fo(P)1249 2852 y Fr(n)1343 2953 y FG(sup)1310 3030 y FC(0)p Fx(6)p Fy(s)p Fx(6)p Fy(t)1528 2953 y FF(M)1616 2967 y Fy(s)1678 2953 y Fn(>)c FF(\025)1827 2852 y Fr(o)1913 2953 y Fn(6)2023 2891 y FG(1)p 2019 2932 54 4 v 2019 3015 a FF(\025)2082 2953 y Fo(E)2137 2879 y Fr(\010)2196 2953 y FF(M)2284 2967 y Fy(t)2314 2879 y Fr(\011)2392 2953 y FG(=)2502 2891 y(1)p 2498 2932 V 2498 3015 a FF(\025)2561 2953 y(:)867 b FH(\(A.5\))118 3199 y(This)30 b(giv)m(es)f(us)1556 3346 y FF(P)38 b Fn(6)25 b FG(e)1789 3308 y FD(\000)p Fy(\015)t(\016)r FC(+)1982 3278 y Fj(\015)2017 3257 y Fw(2)p 1982 3293 71 3 v 2002 3334 a(2)2063 3308 y FC(\010\()p Fy(t)p FC(\))2198 3346 y FF(;)1230 b FH(\(A.6\))118 3506 y(and)31 b(w)m(e)g(obtain)f(the)h(result)f(b)m(y)g(optimizing)f (\(A.6\))i(o)m(v)m(er)g FF(\015)5 b FH(.)p 3596 3506 4 62 v 3600 3448 55 4 v 3600 3506 V 3653 3506 4 62 v 259 3692 a(The)31 b(follo)m(wing)e(lemma)f(allo)m(ws)h(to)i(estimate)e (exp)s(ectation)h(v)-5 b(alues)29 b(b)m(y)i(in)m(tegration)f(b)m(y)h (parts.)118 3868 y Fq(Lemma)38 b(A.2.)44 b FB(L)-5 b(et)37 b FF(\034)43 b Fn(>)33 b FF(s)1113 3882 y FC(0)1188 3868 y FB(b)-5 b(e)38 b(a)e(r)-5 b(andom)37 b(variable)h(satisfying)d FF(F)2523 3882 y Fy(\034)2567 3868 y FG(\()p FF(s)p FG(\))e(=)g Fo(P)p FE(f)p FF(\034)43 b(<)33 b(s)p FE(g)g Fn(>)f FF(G)p FG(\()p FF(s)p FG(\))37 b FB(for)118 3981 y(some)c(c)-5 b(ontinuously)33 b(di\033er)-5 b(entiable)34 b(function)e FF(G)p FB(.)40 b(Then)776 4223 y Fo(E)831 4150 y Fr(\010)890 4223 y FG(1)935 4242 y FC([)p Fy(s)988 4251 y Fw(0)1022 4242 y Fy(;t)p FC(\))1099 4223 y FG(\()p FF(\034)10 b FG(\))p FF(g)s FG(\()p FF(\034)g FG(\))1385 4150 y Fr(\011)1465 4223 y Fn(6)25 b FF(g)s FG(\()p FF(t)p FG(\))1710 4150 y Fr(\002)1749 4223 y FF(F)1807 4237 y Fy(\034)1851 4223 y FG(\()p FF(t)p FG(\))c FE(\000)f FF(G)p FG(\()p FF(t)p FG(\))2241 4150 y Fr(\003)2299 4223 y FG(+)2390 4100 y Fr(Z)2481 4126 y Fy(t)2440 4306 y(s)2473 4315 y Fw(0)2527 4223 y FF(g)s FG(\()p FF(s)p FG(\))p FF(G)2758 4186 y FD(0)2781 4223 y FG(\()p FF(s)p FG(\))15 b(d)p FF(s)450 b FH(\(A.7\))118 4464 y FB(holds)33 b(for)f(al)5 b(l)32 b FF(t)25 b(>)g(s)819 4478 y FC(0)890 4464 y FB(and)32 b(al)5 b(l)32 b(functions)g FG(0)26 b Fn(6)f FF(g)k Fn(6)c FG(1)33 b FB(satisfying)e(the)i(two)f(c)-5 b(onditions)181 4601 y FE(\017)48 b FB(ther)-5 b(e)31 b(exists)f(an)f FF(s)909 4615 y FC(1)973 4601 y FE(2)c FG(\()p FF(s)1137 4615 y FC(0)1176 4601 y FF(;)15 b FE(1)p FG(])30 b FB(such)h(that)e FF(g)k FB(is)c(c)-5 b(ontinuously)30 b(di\033er)-5 b(entiable)32 b(and)d(incr)-5 b(e)g(asing)31 b(on)274 4713 y FG(\()p FF(s)352 4727 y FC(0)391 4713 y FF(;)15 b(s)474 4727 y FC(1)514 4713 y FG(\))p FB(;)181 4826 y FE(\017)48 b FF(g)s FG(\()p FF(s)p FG(\))26 b(=)f(1)33 b FB(for)f(al)5 b(l)32 b FF(s)25 b Fn(>)g FF(s)1113 4840 y FC(1)1152 4826 y FB(.)118 5003 y Fh(Pr)m(oof:)47 b FH(First)29 b(note)i(that)g(for)f(all)e FF(t)d Fn(6)g FF(s)1555 5017 y FC(1)1594 5003 y FH(,)649 5122 y Fr(Z)740 5148 y Fy(t)700 5328 y(s)733 5337 y Fw(0)786 5245 y FF(g)832 5208 y FD(0)856 5245 y FG(\()p FF(s)p FG(\))p Fo(P)p FE(f)p FF(\034)37 b Fn(>)25 b FF(s)p FE(g)15 b FG(d)p FF(s)24 b FG(=)h Fo(E)1614 5144 y Fr(n)1681 5122 y(Z)1771 5148 y Fy(t)p FD(^)p Fy(\034)1731 5328 y(s)1764 5337 y Fw(0)1902 5245 y FF(g)1948 5208 y FD(0)1972 5245 y FG(\()p FF(s)p FG(\))15 b(d)p FF(s)2194 5144 y Fr(o)1463 5449 y FG(=)25 b Fo(E)11 b FE(f)p FF(g)t FG(\()p FF(t)26 b FE(^)20 b FF(\034)10 b FG(\))p FE(g)21 b(\000)f FF(g)s FG(\()p FF(s)2247 5463 y FC(0)2287 5449 y FG(\))1463 5587 y(=)25 b Fo(E)11 b FE(f)p FG(1)1705 5606 y FC([)p Fy(s)1758 5615 y Fw(0)1798 5606 y Fy(;t)p FC(\))1874 5587 y FG(\()p FF(\034)f FG(\))p FF(g)s FG(\()p FF(\034)g FG(\))p FE(g)23 b FG(+)d FF(g)s FG(\()p FF(t)p FG(\))p Fo(P)p FE(f)p FF(\034)38 b Fn(>)24 b FF(t)p FE(g)d(\000)f FF(g)s FG(\()p FF(s)3055 5601 y FC(0)3095 5587 y FG(\))323 b FH(\(A.8\))1845 5871 y(44)p eop %%Page: 45 45 45 44 bop 118 328 a FH(whic)m(h)31 b(implies,)c(b)m(y)j(in)m(tegration) h(b)m(y)f(parts,)562 568 y Fo(E)11 b FE(f)p FG(1)708 586 y FC([)p Fy(s)761 595 y Fw(0)801 586 y Fy(;t)p FC(\))877 568 y FG(\()p FF(\034)f FG(\))p FF(g)s FG(\()p FF(\034)g FG(\))p FE(g)28 b FG(=)1332 444 y Fr(Z)1423 470 y Fy(t)1383 650 y(s)1416 659 y Fw(0)1469 568 y FF(g)1515 530 y FD(0)1539 568 y FG(\()p FF(s)p FG(\))1652 494 y Fr(\002)1690 568 y FG(1)21 b FE(\000)f FF(F)1905 582 y Fy(\034)1948 568 y FG(\()p FF(s)p FG(\))2061 494 y Fr(\003)2115 568 y FG(d)o FF(s)g FE(\000)g FF(g)s FG(\()p FF(t)p FG(\))2468 494 y Fr(\002)2507 568 y FG(1)h FE(\000)f FF(F)2722 582 y Fy(\034)2765 568 y FG(\()p FF(t)p FG(\))2868 494 y Fr(\003)2927 568 y FG(+)g FF(g)s FG(\()p FF(s)3142 582 y FC(0)3182 568 y FG(\))1236 831 y Fn(6)1332 707 y Fr(Z)1423 733 y Fy(t)1383 913 y(s)1416 922 y Fw(0)1469 831 y FF(g)s FG(\()p FF(s)p FG(\))p FF(G)1700 793 y FD(0)1724 831 y FG(\()p FF(s)p FG(\))15 b(d)p FF(s)20 b FG(+)g FF(g)s FG(\()p FF(t)p FG(\))2206 757 y Fr(\002)2245 831 y FF(F)2303 845 y Fy(\034)2346 831 y FG(\()p FF(t)p FG(\))h FE(\000)f FF(G)p FG(\()p FF(t)p FG(\))2736 757 y Fr(\003)2774 831 y FF(;)654 b FH(\(A.9\))118 1075 y(where)36 b(w)m(e)g(ha)m(v)m(e)g (used)g FF(F)1005 1089 y Fy(\034)1048 1075 y FG(\()p FF(s)p FG(\))e Fn(>)f FF(G)p FG(\()p FF(s)p FG(\))i FH(and)g FF(G)p FG(\()p FF(s)1849 1089 y FC(0)1888 1075 y FG(\))f Fn(6)f FF(F)13 b FG(\()p FF(s)2210 1089 y FC(0)2250 1075 y FG(\))33 b(=)g(0)p FH(.)56 b(This)34 b(pro)m(v)m(es)i(the)g (assertion)e(in)118 1188 y(the)d(case)f FF(t)25 b Fn(6)g FF(s)662 1202 y FC(1)701 1188 y FH(.)41 b(In)30 b(the)g(case)h FF(t)25 b(>)g(s)1424 1202 y FC(1)1463 1188 y FH(,)30 b(w)m(e)h(ha)m(v)m(e)311 1376 y Fo(E)11 b FE(f)p FG(1)457 1395 y FC([)p Fy(s)510 1404 y Fw(0)550 1395 y Fy(;t)p FC(\))626 1376 y FG(\()p FF(\034)f FG(\))p FF(g)s FG(\()p FF(\034)g FG(\))p FE(g)28 b FG(=)d Fo(E)11 b FE(f)p FG(1)1227 1395 y FC([)p Fy(s)1280 1404 y Fw(0)1319 1395 y Fy(;s)1372 1404 y Fw(1)1406 1395 y FC(\))1438 1376 y FG(\()p FF(\034)f FG(\))p FF(g)s FG(\()p FF(\034)g FG(\))p FE(g)23 b FG(+)d Fo(P)p FE(f)p FF(\034)36 b FE(2)25 b FG([)p FF(s)2213 1390 y FC(1)2252 1376 y FF(;)15 b(t)p FG(\))p FE(g)985 1571 y Fn(6)1081 1447 y Fr(Z)1172 1473 y Fy(s)1205 1482 y Fw(1)1132 1653 y Fy(s)1165 1662 y Fw(0)1259 1571 y FF(g)s FG(\()p FF(s)p FG(\))p FF(G)1490 1533 y FD(0)1513 1571 y FG(\()p FF(s)p FG(\))g(d)p FF(s)20 b FG(+)g FF(g)s FG(\()p FF(s)1970 1585 y FC(1)2010 1571 y FG(\))2045 1497 y Fr(\002)2083 1571 y FF(F)2141 1585 y Fy(\034)2185 1571 y FG(\()p FF(s)2263 1585 y FC(1)2302 1571 y FG(\))h FE(\000)f FF(G)p FG(\()p FF(s)2599 1585 y FC(1)2638 1571 y FG(\))2673 1497 y Fr(\003)2731 1571 y FG(+)2822 1497 y Fr(\002)2860 1571 y FF(F)2918 1585 y Fy(\034)2961 1571 y FG(\()p FF(t)p FG(\))h FE(\000)f FF(F)3234 1585 y Fy(\034)3278 1571 y FG(\()p FF(s)3356 1585 y FC(1)3395 1571 y FG(\))3430 1497 y Fr(\003)985 1834 y FG(=)1081 1710 y Fr(Z)1172 1736 y Fy(t)1132 1916 y(s)1165 1925 y Fw(0)1218 1834 y FF(g)s FG(\()p FF(s)p FG(\))p FF(G)1449 1796 y FD(0)1473 1834 y FG(\()p FF(s)p FG(\))15 b(d)p FF(s)20 b FE(\000)1806 1760 y Fr(\002)1843 1834 y FF(G)p FG(\()p FF(t)p FG(\))h FE(\000)f FF(G)p FG(\()p FF(s)2280 1848 y FC(1)2318 1834 y FG(\))2353 1760 y Fr(\003)2412 1834 y FG(+)2503 1760 y Fr(\002)2541 1834 y FF(F)2599 1848 y Fy(\034)2642 1834 y FG(\()p FF(t)p FG(\))h FE(\000)f FF(G)p FG(\()p FF(s)3007 1848 y FC(1)3046 1834 y FG(\))3081 1760 y Fr(\003)3119 1834 y FF(;)264 b FH(\(A.10\))118 2078 y(where)37 b(w)m(e)g(ha)m(v)m(e) g(used)f(that)h FF(g)s FG(\()p FF(s)p FG(\))e(=)g(1)h FH(holds)f(for)h(all)f FF(s)f FE(2)g FG([)p FF(s)2291 2092 y FC(1)2331 2078 y FF(;)15 b(t)p FG(])p FH(.)58 b(This)35 b(pro)m(v)m(es)i(the)f(assertion)f(for)118 2191 y FF(t)25 b(>)g(s)315 2205 y FC(1)354 2191 y FH(.)p 3596 2191 4 62 v 3600 2133 55 4 v 3600 2191 V 3653 2191 4 62 v 118 2475 a FI(References)118 2678 y FH([Ar])274 b(L.)31 b(Arnold,)f FB(R)-5 b(andom)32 b(Dynamic)-5 b(al)32 b(Systems)e FH(\(Springer-V)-8 b(erlag,)31 b(Berlin,)e(1998\).)118 2825 y([Ben])223 b(E.)40 b(Beno\356t)f(\(Ed.\),)j FB(Dynamic)e(Bifur)-5 b(c)g(ations,)43 b(Pr)-5 b(o)g(c)g(e)g(e)g(dings,)46 b(Luminy)39 b(1990)h FH(\(Springer-)545 2938 y(V)-8 b(erlag,)30 b(Lecture)i(Notes)e(in)g(Mathematics)f(1493,)i(Berlin,)e(1991\).)118 3086 y([Ber])238 b(N.)43 b(Berglund,)j FB(A)-5 b(diab)g(atic)45 b(Dynamic)-5 b(al)44 b(Systems)f(and)h(Hyster)-5 b(esis)p FH(,)46 b(Thesis)c(EPFL)i(no)545 3199 y(1800)31 b(\(1998\).)g(A)-10 b(v)-5 b(ailable)28 b(at)545 3312 y Fb (http://dpwww.epfl.ch/instituts/ipt/berglund/these.html)118 3459 y FH([BK])243 b(N.)31 b(Berglund,)i(H.)e(Kunz,)i FB(Chaotic)g(hyster)-5 b(esis)35 b(in)d(an)i(adiab)-5 b(atic)g(al)5 b(ly)34 b(oscil)5 b(lating)34 b(double)545 3572 y(wel)5 b(l)p FH(,)33 b(Ph)m(ys.)f(Rev.)g(Letters)h Fq(78)p FH(:1692\0251694)g(\(1997\).)h(N.)d(Berglund,)i(H.)f(Kunz,)h FB(Memory)545 3685 y(e\033e)-5 b(cts)34 b(and)f(sc)-5 b(aling)33 b(laws)f(in)g(slow)5 b(ly)31 b(driven)i(systems)p FH(,)d(J.)h(Ph)m(ys.)f(A)g Fq(32)p FH(:15\02539)h(\(1999\).)118 3833 y([CF94])163 b(H.)27 b(Crauel,)h(F.)f(Flandoli,)f FB(A)n(ttr)-5 b(actors)30 b(for)f(r)-5 b(andom)29 b(dynamic)-5 b(al)30 b(systems)p FH(,)d(Probab.)i(The-)545 3946 y(ory)h(Related)g (Fields)e Fq(100)p FH(:365\025393)k(\(1994\).)118 4093 y([CF98])163 b(H.)20 b(Crauel,)j(F.)e(Flandoli,)g FB(A)-5 b(dditive)24 b(noise)f(destr)-5 b(oys)24 b(a)g(pitchfork)g(bifur)-5 b(c)g(ation)p FH(,)24 b(J.)d(Dynam.)545 4206 y(Di\033eren)m(tial)29 b(Equations)h Fq(10)p FH(:259\025274)i(\(1998\).)118 4354 y([FJ])272 b(W.)15 b(H.)24 b(Fleming,)g(M.)15 b(R.)23 b(James,)j FB(Asymptotic)g(series)h(and)g(exit)g(time)g(pr)-5 b(ob)g(abilities)p FH(,)27 b(Ann.)545 4467 y(Probab.)32 b Fq(20)p FH(:1369\0251384)g(\(1992\).)118 4615 y([FW])225 b(M.)15 b(I.)28 b(F)-8 b(reidlin)27 b(and)h(A.)15 b(D.)29 b(W)-8 b(en)m(tzell,)28 b FB(R)-5 b(andom)30 b(Perturb)-5 b(ations)32 b(of)e(Dynamic)-5 b(al)31 b(Systems)545 4727 y FH(\(Springer-V)-8 b(erlag,)31 b(New)g(Y)-8 b(ork,)30 b(1984\).)118 4875 y([Ga])261 b(G.)32 b(Gaeta,)g FB(Dynamic)-5 b(al)34 b(bifur)-5 b(c)g(ation)35 b(with)e(noise)p FH(,)f(In)m(t.)g(J.) h(Theoret.)g(Ph)m(ys.)f Fq(34)p FH(:595\025603)545 4988 y(\(1995\).)118 5136 y([Gr])271 b(I.)15 b(S.)23 b(Grad\262te)-10 b(\010)-35 b(\031n,)25 b FB(Applic)-5 b(ations)26 b(of)g(A.)14 b(M.)25 b(Lyapunov's)g(the)-5 b(ory)27 b(of)f(stability)f(to)h(the)h (the)-5 b(ory)545 5249 y(of)32 b(di\033er)-5 b(ential)34 b(e)-5 b(quations)34 b(with)e(smal)5 b(l)32 b(c)-5 b(o)g(e\036cients)35 b(in)d(the)h(derivatives)p FH(,)f(Mat.)f(Sb)s(ornik)545 5362 y(N.S.)f Fq(32)p FH(:263\025286)i(\(1953\).)118 5509 y([GH])238 b(J.)35 b(Guc)m(k)m(enheimer,)i(P)-8 b(.)35 b(Holmes,)f FB(Nonline)-5 b(ar)37 b(Oscil)5 b(lations,)36 b(Dynamic)-5 b(al)37 b(Systems,)g(and)545 5622 y(Bifur)-5 b(c)g(ations)33 b(of)f(V)-7 b(e)i(ctor)34 b(Fields)c FH(\(Springer-V)-8 b(erlag,)31 b(New)g(Y)-8 b(ork,)30 b(1983\).)1845 5871 y(45)p eop %%Page: 46 46 46 45 bop 118 328 a FH([IJ])298 b(G.)30 b(Io)s(oss,)f(D.)15 b(D.)30 b(Joseph,)i FB(Elementary)g(Stability)g(and)g(Bifur)-5 b(c)g(ation)33 b(The)-5 b(ory)31 b FH(\(Springer-)545 441 y(V)-8 b(erlag,)30 b(New)h(Y)-8 b(ork,)30 b(1980\).)118 591 y([JL])275 b(K.)15 b(M.)33 b(Jansons,)g(G.)15 b(D.)32 b(Lythe,)i FB(Sto)-5 b(chastic)35 b(c)-5 b(alculus:)47 b(Applic)-5 b(ation)35 b(to)f(dynamic)g(bifur-)545 704 y(c)-5 b(ations)33 b(and)f(thr)-5 b(eshold)33 b(cr)-5 b(ossings)p FH(,)31 b(J.)g(Stat.)g(Ph)m(ys.)g Fq(90)p FH(:227\025251)g(\(1998\).)118 854 y([Ku])257 b(R.)51 b(Kusk)m(e,)57 b FB(Pr)-5 b(ob)g(ability)53 b(densities)f(for)g(noisy)f (delay)h(bifur)-5 b(c)g(ations)p FH(,)58 b(J.)52 b(Stat.)g(Ph)m(ys.)545 967 y Fq(96)p FH(:797\025816)32 b(\(1999\).)118 1117 y([ME])233 b(P)-8 b(.)25 b(Mandel,)g(T.)f(Erneux,)j FB(L)-5 b(aser)27 b(L)-5 b(or)g(enz)27 b(e)-5 b(quations)27 b(with)g(a)f (time-dep)-5 b(endent)28 b(p)-5 b(ar)g(ameter)p FH(,)545 1230 y(Ph)m(ys.)31 b(Rev.)f(Letters)h Fq(53)p FH(:1818\0251820)h (\(1984\).)118 1380 y([Ne])269 b(A.)15 b(I.)24 b(Neish)m(tadt,)i FB(Persistenc)-5 b(e)28 b(of)f(stability)g(loss)f(for)h(dynamic)-5 b(al)27 b(bifur)-5 b(c)g(ations)28 b(I,)e(II)p FH(,)e(Di\033.)545 1493 y(Equ.)31 b Fq(23)p FH(:1385\0251391)h(\(1987\).)f(Di\033.)e(Equ.) i Fq(24)p FH(:171\025176)g(\(1988\).)118 1644 y([Sc)m(hm])164 b(B.)27 b(Sc)m(hmalfu\377,)g FB(Invariant)i(attr)-5 b(acting)30 b(sets)f(of)g(nonline)-5 b(ar)29 b(sto)-5 b(chastic)30 b(di\033er)-5 b(ential)30 b(e)-5 b(qua-)545 1756 y(tions)p FH(,)30 b(Math.)g(Res.)g Fq(54)p FH(:217\025228)h(\(1989\).)118 1907 y([Sh])277 b(M.)15 b(A.)43 b(Shishk)m(o)m(v)-5 b(a,)46 b FB(Examination)e(of)g(one)g(system)g(of)f(di\033er)-5 b(ential)45 b(e)-5 b(quations)46 b(with)d(a)545 2020 y(smal)5 b(l)40 b(p)-5 b(ar)g(ameter)42 b(in)d(highest)i(derivatives)p FH(,)i(Dokl.)38 b(Ak)-5 b(ad.)39 b(Nauk)f(SSSR)h Fq(209)p FH(:576\025579)545 2133 y(\(1973\).)31 b([English)f(transl.:)40 b(So)m(viet)30 b(Math.)h(Dokl.)e Fq(14)p FH(:384\025387)i(\(1973\)].) 118 2283 y([SMC])179 b(N.)15 b(G.)27 b(Sto)s(c)m(ks,)g(R.)f(Manella,)g (P)-8 b(.)15 b(V.)g(E.)29 b(McClin)m(to)s(c)m(k,)d FB(In\035uenc)-5 b(e)31 b(of)e(r)-5 b(andom)29 b(\035uctuations)545 2396 y(on)c(delaye)-5 b(d)27 b(bifur)-5 b(c)g(ations:)39 b(The)26 b(c)-5 b(ase)26 b(of)g(additive)g(white)f(noise)p FH(,)f(Ph)m(ys.)g (Rev.)e(A)g Fq(40)p FH(:5361\025)545 2509 y(5369)31 b(\(1989\).)118 2659 y([SHA])191 b(J.)15 b(B.)32 b(Swift,)g(P)-8 b(.)15 b(C.)33 b(Hohen)m(b)s(erg,)h(G.)d(Ahlers,)g FB(Sto)-5 b(chastic)35 b(L)-5 b(andau)34 b(e)-5 b(quation)35 b(with)e(time-)545 2772 y(dep)-5 b(endent)34 b(drift)p FH(,)c(Ph)m(ys.)g(Rev.)g(A)g Fq(43)p FH(:6572\0256580)i(\(1991\).)118 2922 y([Ti])286 b(A.)15 b(N.)28 b(Tihono)m(v,)g FB(Systems)i(of)g(di\033er)-5 b(ential)31 b(e)-5 b(quations)31 b(c)-5 b(ontaining)31 b(smal)5 b(l)29 b(p)-5 b(ar)g(ameters)32 b(in)545 3035 y(the)h(derivatives)p FH(,)f(Mat.)e(Sb)s(ornik)f(N.S.)i Fq(31)p FH(:575\025586)g(\(1952\).)118 3185 y([TM])229 b(M.)15 b(C.)32 b(T)-8 b(orren)m(t,)34 b(M.)d(San)h(Miguel,)e FB(Sto)-5 b(chastic-dynamics)34 b(char)-5 b(acterization)35 b(of)e(delaye)-5 b(d)545 3298 y(laser)37 b(thr)-5 b(eshold)38 b(instability)e(with)g(swept)h(c)-5 b(ontr)g(ol)38 b(p)-5 b(ar)g(ameter)p FH(,)38 b(Ph)m(ys.)e(Rev.)e(A)h Fq(38)p FH(:245\025)545 3411 y(251)c(\(1988\).)118 3810 y FQ(Nils)d(Berglund) 118 3909 y FA(Geor)n(gia)j(Institute)g(of)h(Technology)118 4009 y FQ(A)n(tlan)n(ta,)27 b(GA)h(30332-0430,)23 b(USA)118 4108 y FJ(and)118 4208 y FA(Weierstra\377)32 b(Institute)f(f)n(or)g (Applied)h(Anal)-7 b(ysis)30 b(and)h(Stochastics)118 4308 y FQ(Mohrenstra\377e)25 b(39,)i(10117)e(Berlin,)i(German)n(y)118 4407 y FJ(E-mail)k(addr)l(ess:)39 b Fa(berglund@wias-ber)o(li)o(n.d)o (e)118 4620 y FQ(Barbara)25 b(Gen)n(tz)118 4733 y FA(Weierstra\377)32 b(Institute)f(f)n(or)g(Applied)h(Anal)-7 b(ysis)30 b(and)h(Stochastics) 118 4846 y FQ(Mohrenstra\377e)25 b(39,)i(10117)e(Berlin,)i(German)n(y) 118 4959 y FJ(E-mail)k(addr)l(ess:)39 b Fa(gentz@wias-berlin)o(.d)o(e) 1845 5871 y FH(46)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0008281031761--