Feynman-Kac functionals and positive solutions of

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1、ZeitschriftftirZ.Wahrscheinlichkeitstheorieverw.GebieteWahrscheinlichkeitstheorie65,19-33(1983)undverwandteGebiete9Springer-Verlag1983Feynman-KacFunctionalsandPositiveSolutionsofAu+qu=ONeilFalkner*DepartmentofMathematics,OhioStateUniversity,Columbus,Ohio43210,USASummary.Le

2、tDbeaboundedC2domaininIRdandletqbeaboundedBorelfunctioninD.ForxsDandze~?Dsuppose(X,)underthelawpx;zisBrownianmotioninDstartingatxandconditionedtoconvergetoz.Letbethelifetimeof(Xt).WeshowthatifthequantityE=zexpq(Xs)dsisfiniteforonexeDandonez~c~D,thenthisquantityremainsbounde

3、dasxvariesoverDandzvariesover0D.Thismaybeconsideredonequanti-tativeexpressionofthequalitativestatementthatnomatterwhereBrow-nianmotioninDeventuallyhits~D,itgoesalloverDbeforeitgetsthere.Weapplythisresulttoshowthatiftheequation
89admitsanon-negativesolutioninD,whichisstrictl

4、ypositiveonasubsetof0Dofpositiveharmonicmeasure,thenforanynon-negativeboundedBorelfunc-tionfonODitadmitsauniqueboundedsolutionusatisfyingu=fonOD,andthissolutionuisnon-negative.1.IntroductionLetDbeaboundeddomaininNdandletq:D--+RbeaboundedBorelfunction.Let((X~);px,x~lR~)beBro

5、wnianmotioninIRaandlet~=inf{t>0:Xt6D}.Recallthat"c

6、hensoisuf.Thefunctionu1obtainedbysettingf=lin(1.1)shallbecalledthe*ResearchsupportedinpartbyNSFgrantMCS-810347320N.FalknergaugeofD.IfthegaugeofDisfiniteandfisbounded,thenu=uyistheuniqueboundedsolutionoftheboundaryvalueproblem:
89inD,(1.2a)u=fon~?D.(1.2b)(SeeSect.3andtherefe

7、rencescitedthereforfurtherdiscussionofthis.)WepointoutthatthegaugeofDcanbeidentically+ooinspiteofthebounded-nessofDandq.Thisisnotpathologicalandisrelatedtothepossibilityofnon-existenceofpositivesolutionsof(1.2a).(See[-8,4],andSect.3.)NowWilliams[13]hasshownthatif~Disofclass

8、C2andifforsomenon-emptyrelativelyopensubsetAof~Dwehaveu~Afinite,thenthegaugeofDisf