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1、Chapter9HarrisandtopologicalrecurrenceInthischapterweconsiderstrongerconceptsofrecurrenceandlinkthemwiththedichotomyprovedinChapter8.Wealsoconsiderseveralobviousdefinitionsofglobalandlocalrecurrenceandtransienceforchainsontopologicalspaces,andshowthattheyalsolinktothefundamentaldichotomy.Ind
2、evelopingconceptsofrecurrenceforsetsA∈B(X),wewillconsidernotjustthefirsthittingtimeτA,ortheexpectedvalueU(·,A)ofηA,butalsotheeventthatΦ∈Ainfinitelyoften(i.o.),orηA=∞,definedby∞*∞{Φ∈Ai.o.}:={Φk∈A}N=1k=NwhichiswelldefinedasanF-measurableeventonΩ.Forx∈X,A∈B(X)wewriteQ(x,A):=Px{Φ∈Ai.o.}:(9.1)obvio
3、usly,foranyx,AwehaveQ(x,A)≤L(x,A),andbythestrongMarkovpropertywehaveQ(x,A)=Ex[PΦτ{Φ∈Ai.o.}I{τA<∞}]=UA(x,dy)Q(y,A).(9.2)AAHarrisrecurrenceThesetAiscalledHarrisrecurrentifQ(x,A)=Px(ηA=∞)=1,x∈A.AchainΦiscalledHarris(recurrent)ifitisψ-irreducibleandeverysetinB+(X)isHarrisrecurrent.199200Harris
4、andtopologicalrecurrenceWewillseeinTheorem9.1.4thatwhenA∈B+(X)andΦisHarrisrecurrenttheninfactwehavetheseeminglystrongerandperhapsmorecommonlyusedpropertythatQ(x,A)=1foreveryx∈X.ItisobviousfromthedefinitionsthatifasetisHarrisrecurrent,thenitisrecurrent.Indeed,intheformulationabovethestrengthe
5、ningfromrecurrencetoHarrisrecurrenceisquiteexplicit,indicatingamovefromanexpectedinfinityofvisitstoanalmostsurelyinfinitenumberofvisitstoaset.ThisdefinitionofHarrisrecurrenceappearsonthefaceofittobestrongerthanrequiringL(x,A)≡1forx∈A,whichisastandardalternativedefinitionofHarrisrecurrence.Inone
6、ofthekeyresultsofthissection,Proposition9.1.1,weprovethattheyareinfactequivalent.ThehighlightoftheHarrisrecurrenceanalysisisTheorem9.0.1.IfΦisrecurrent,thenwecanwriteX=H∪N(9.3)whereHisabsorbingandnon-emptyandeverysubsetofHinB+(X)isHarrisrecur-rent;andNisψ-nullandtransient.ProofThisisproved,
7、inaslightlystrongerform,inTheorem9.1.5.Hencearecurrentchaindiffersonlybyaψ-nullsetfromaHarrisrecurrentchain.IngeneralwecanthenrestrictanalysistoHandderiveverymuchstrongerresultsusingpropertiesofHarrisrecurrentchains.ForchainsonacountablespacethenullsetNi
