brownian motion in a wedge with variable skew reflection

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1、transactionsoftheamericanmathematicalsocietyVolume326,Number1,July1991BROWNIANMOTIONINAWEDGEWITHVARIABLESKEWREFLECTIONL.C.GROGERSAbstract.DoesplanarBrownianmotionconfinedtoawedgebyskewreflec-tiononthesidesapproachthevertexofthewedge?ThisquestionhasbeenansweredbyVaradhan

2、andWilliamsinthecasewherethedirectionofreflec-tionisconstantoneachofthesides,buthereweaddressthequestionwhenthedirectionreflectedisallowedtovary.Anecessarycondition,andasufficientcondition,areobtainedforthevertextobereached.Theconditionsareofageometricnature,andthegapbe

3、tweenthemisquitesmall.1.IntroductionTherehasbeenalotofrecentinterestinthebehaviourofBrownianmotioninsomedomainDcCwithskew-reflectionontheboundary—see[DeB,HLS,LeG,TW,VW,W85a,W85b]and,formulti-dimensionalgeneralizations,[LS,SV,W87,ReW].In[VW],thedomainDistakentobeawedgeD=

4、{re'e;0<60}andthedirectionofreflectionisassumedtobeconstantoneachoftheedgesofthewedge.Anumberofverynaturalquestionsariseaboutthisprocess,oneofwhichis"Willtheprocessapproach0?"Thisquestion,andthemuchmoredifficultone,'Isitpossibletoextendtheprocessbeyonditsfirsthiton

5、0?"arecompletelyansweredin[VW](seealso[R]foraproofbyexcursion-theoreticmethods).Theaimofthispaperistoexaminethequestion,"Willtheprocessapproach0?"whenthedirectionofreflectionontheedgesofDarenotassumedconstant.Thisturnsouttomakethequestionconsiderablymoredifficult,sowesh

6、allconfineourselveshereonlytothisproblemanddefertreatmentofotherrelatedinterestingquestionsuntilasubsequentpaper.Tobeginwith,itishelpfultotransformtheproblemviaprobabilisticandcomplex-analyticalmethods.Clearly,whetherornotBrownianmotioninDcanapproach0isdecidedbythedirec

7、tionsofreflectioninaneighbourhoodof0,sowelosenogeneralityinassumingthedirectionofreflectionisnor-maloutside{z:z<1}.Asiswellknown,thesamplepathsofBrownianReceivedbytheeditorsMay19,1989.1980MathematicsSubjectClassification(1985Revision).Primary30E25,60J65,60J50.Keywords

8、andphrases.Brownianmotionwithskewreflection,conformaitransformation,re-currence,transience.©1991AmericanMathem