搜索
arXiv:math/0601350v1[math.AP]14Jan2006SmalltimeasymptoticsofdiffusionprocessesA.F.M.terElst1,DerekW.Robinson2andAdamSikora3AbstractWeestablishtheshort-timeasymptoticbehaviouroftheMarkoviansemigroupsassociatedwithstronglylocalDirichletformsunderverygeneralhypotheses.Ourresultsapplytoawideclassofstronglyel-liptic,subellipticanddegenerateellipticoperators.Inthedegeneratecasetheasymptoticsincorporatepossiblenon-ergodicity.November2005AMSSubjectClassification:35B40,58J65,35J70,35Hxx,60J60.Homeinstitutions:1.DepartmentofMathematics2.CentreforMathematicsandComputingScienceanditsApplicationsEindhovenUniversityofTechnologyMathematicalSciencesInstituteP.O.Box513AustralianNationalUniversity5600MBEindhovenCanberra,ACT0200TheNetherlandsAustralia3.DepartmentofMathematicalSciencesNewMexicoStateUniversityP.O.Box30001LasCrucesNM88003-8001,USA1IntroductionOneoftheiconicresultsinthetheoryofsecond-orderellipticoperatorsisVaradhan’s[Var67b][Var67a]identificationofthesmalltimeasymptoticlimitlimt↓0tlogKt(x;y)=−4−1d(x;y)2(1)oftheheatkernelKofastronglyellipticoperatoronRdintermsoftheintrinsicRieman-niandistanced(·;·).ThesmalltimebehaviourwassubsequentlyanalyzedatlengthbyMolchanov[Mol75]whoextended(1)toamuchwiderclassofoperatorsandmanifolds.Theseresultswerethenanalyzed,largelybyprobabilisticmethods,byvariousauthors(see,forexample,theParislectures[Aze81]).Mostoftheearlyresultswererestrictedtonon-degenerateoperatorswithsmoothcoefficientsactingonsmoothmanifolds.OptimalresultsfortheheatflowonLipschitzRiemannianmanifoldswereobtainedmuchlaterbyNorris[Nor97].Theasymptoticrelation(1)has,however,beenestablishedforcertainclassesofdegeneratesubellipticoperatorsbyseveralauthors,inparticularforsublapla-ciansconstructedfromvectorfieldssatisfyingH¨ormander’sconditionforhypoellipticity[BKRR71][L´ea87a][L´ea87b][KS88].Itisneverthelessevidentfromexplicitexamplesthat(1)failsforlargeclassesofdegenerateellipticoperators.Difficultiesarise,forexample,fromnon-ergodicbehaviour.TheproblemsintroducedbydegeneraciesareillustratedbytheoperatorH=−dcδd,whered=d/dxandcδ(x)=x21+x2δ,(2)actingontherealline.Ifδ∈[0,1/2ithentheassociateddiffusionisergodic.If,however,δ≥1/2thenthediffusionprocessseparatesintotwoprocessesontheleftandrighthalflines,respectively(see[ERSZ04],Proposition6.5).Thedegeneracyofcδattheorigincreatesanimpenetrableobstacleforthediffusion.ThereforethecorrespondingkernelKsatisfiesKt(x;y)=0forallx0,y0andt0andtheeffectivedistancebetweentheleftandrighthalflinesisinfinite.ButthisbehaviourisnotreflectedbytheRiemanniandistanced(x;y)=|Ryxc−1/2δ|whichisfiniteforallδ∈[0,1i.Hence(1)mustfailforthisdiffusionprocessifδ∈[1/2,1i.Morecomplicatedphenomenacanoccurfordegenerateoperatorsinhigherdimensions.TobespecificletI={(α,0):α∈[−1,1]}beaboundedone-dimensionalintervalinR2andconsidertheformh(ϕ)=P2i=1(∂iϕ,cδ∂iϕ)withD(h)=W1,2(R2)wherecδ(x)=(|x|2I/(1+|x|2I))δand|x|IdenotestheEuclideandistancefromx∈R2totheintervalI.Itfollowsthatifδ≥1/2thentheintervalIpresentsanimpenetrableobstacleforthecorrespondingdiffusionandtheeffectiveconfigurationspacefortheprocessisR2\I.InparticulartheappropriatedistanceforthedescriptionofthediffusionistheintrinsicRiemanniandistanceonR2\IratherthanthatonR2.AlthoughtheRiemanniandistanceonR2iswell-definedifδ∈[0,1iitisnotsuitedtothedescriptionofthediffusionifδ∈[1/2,1i.Thustheproblemofadeeperunderstandingofthesmalltimebehaviouroftheheatkernelassociatedwithdegenerateoperatorsconsistsinpartinidentifyingtheappropriatemeasureofdistance.HinoandRam´ırez[HR03](seealso[Hin02][Ram01])madeconsiderableprogressinunderstandingthesmalltimeasymptoticsofgeneraldiffusionprocessesbyexaminingtheprobleminthebroadercontextofDirichletformsonaσ-finitemeasurespace(X,B,μ)[FOT94][BH91][Mos94].Firsttheyconsideranintegratedversionof(1).SetKt(A;B)=1RAdμ(x)RBdμ(y)Kt(x;y)formeasurablesubsetsA,B∈B.Thenthepointwiseasymp-toticestimate(1)leads,underquitegeneralconditions,toaset-theoreticversionlimt↓0tlogKt(A;B)=−4−1d(A;B)2(3)foropensetswherethedistancebetweenthesetsAandBisdefinedintheusualman-nerwithinfima.Secondly,ifthemeasureμisfinitethenHino–Ram´ırezestablish(3)forthekernelofthesemigroupcorrespondingtoastronglylocal,conservative,DirichletformonL2(X;μ)andforboundedmeasurablesetsAandBbutwithaset-theoreticdistanced(A;B)defineddirectlyintermsoftheDirichletform.Thisdistancetakesvaluesin[0,∞]andisnotnecessarilythedistancearisingfromanyunderlyingRiemannianstructure.Nev-erthelesstheestimate(3)establishesthatitisthecorrectmeasureofsmalltimebehaviour.AkeyfeatureofthisformalismisthatitallowsforthepossibilitythatKt(A;B)=0forallsmalltandthend(A;B)=∞.TheprincipaldisadvantagesoftheHino–Ram´ırezresultisthatitrequires(X,B,μ)tobeaprobabilityspaceandtheformtobeconservative.Oneofouraimsistoremovetheserestrictionsandtoderivetheestimate(3)fordiffusionprocessesrelatedtoalargeclassofregular,stronglylocal,Markovianformsonageneralmeasurespace(X,B,μ).Ourformalismissuitedtoapplicationstodegenerateellipticoperators.Asecondaimistoestablishconditionswhichallowonetopassfromtheestimate(3)tothepointwiseestimate(1).Weproveageneralresultwhichcoversav