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arXiv:math/0407359v1[math.PR]21Jul2004ThesemigroupoftheGlauberdynamicsofacontinuoussystemoffreeparticlesYuriKondratievFakult¨atf¨urMathematik,Universit¨atBielefeld,Postfach100131,D-33501Bielefeld,Germany;InstituteofMathematics,Kiev,Ukraine;BiBoS,Univ.Bielefeld,Germany.e-mail:kondrat@mathematik.uni-bielefeld.deEugeneLytvynovDepartmentofMathematics,UniversityofWalesSwansea,SingletonPark,SwanseaSA28PP,U.K.e-mail:e.lytvynov@swansea.ac.ukMichaelR¨ocknerFakult¨atf¨urMathematik,Universit¨atBielefeld,Postfach100131,D-33501Bielefeld,Germany;BiBoS,Univ.Bielefeld,Germany.e-mail:roeckner@mathematik.uni-bielefeld.deAbstractWestudypropertiesofthesemigroup(e−tH)t≥0onthespaceL2(ΓX,π),whereΓXistheconfigurationspaceoveralocallycompactsecondcountableHausdorfftopologicalspaceX,πisaPoissonmeasureonΓX,andHisthegeneratoroftheGlauberdynamics.WeexplicitlyconstructthecorrespondingMarkovsemigroupofkernels(Pt)t≥0and,usingit,weprovethemainresultsofthepaper:theFellerpropertyofthesemigroup(Pt)t≥0withrespecttothevaguetopologyontheconfigurationspaceΓX,andtheergodicpropertyof(Pt)t≥0.FollowinganideaofD.Surgailis,wealsogiveadirectconstructionoftheGlauberdynamicsofacontinuousinfinitesystemoffreeparticles.Themainpointhereisthatthisprocesscanstartineveryγ∈ΓX,willneverleaveΓXandhascadlagsamplepathsinΓX.2000AMSMathematicsSubjectClassification:60K35,60J75,60J80,82C21Keywords:Birthanddeathprocess;Continuoussystem;Poissonmeasure;Glauberdynamics1IntroductionTheGlauberdynamics(GD)ofacontinuousinfinitesystemofparticles,eitherfreeorinter-acting,isaspecialcaseofaspatialbirthanddeathprocessontheEuclideanspaceRd,oronamoregeneraltopologicalspaceX.ForasystemofparticlesinaboundedvolumeinRd,suchprocesseswereintroducedandstudiedbyC.Prestonin[16],seealso[7].Inthelattercase,thetotalnumberofparticlesisfiniteatanymomentoftime.IntherecentpaperbyL.Bertini,N.Cancrini,andF.Cesi,[3],thegeneratoroftheGDinafinitevolumewasstudied.ThisgeneratorcorrespondstoaspecialcaseofbirthanddeathcoefficientsinPreston’sdynamics.Undersomeconditionsontheinteractionbetween1particles,theauthorsof[3]provedtheexistenceofthespectralgapofthegeneratoroftheGD,whichisuniformlypositivewithrespecttoallfinitevolumesΛandboundaryconditionsoutsideΛ.AnexplicitestimateofthespectralgapinafinitevolumewasderivedbyL.Wuin[20].TheproblemofconstructionofaspatialbirthanddeathprocessintheinfinitevolumewasinitiatedbyR.A.HolleyandD.W.Stroockin[7],whereitwassolvedinaveryspecialcaseofnearestneighborbirthanddeathprocessesontherealline.In[11],theGDininfinitevolumewasdiscussed.TheprocessnowtakesvaluesintheconfigurationspaceΓRdoverRd,i.e.,inthespaceofalllocallyfinitesubsetsinRd,whichisequippedwiththevaguetopology.UsingthetheoryofDirichletforms[13,14],theauthorsof[11]provedtheexistenceofaHuntprocessMonΓRdthatisproperlyassociatedwiththegeneratoroftheGDwithaquitegeneralpairpotentialofinteractionbetweenparticles.Inparticular,MisaconservativeMarkovprocessonΓRdwithcadlagpaths.AnestimateofthespectralgapofthegeneratoroftheGDininfinitevolumewasalsoproved.Inthecasewheretheinteractionbetweenparticlesisabsent(i.e.,theparticlesarefree),thePoissonmeasureπonΓRdisastationarymeasureoftheGD.LetusrecallthatthePoissonmeasurepossessesthechaosdecompositionproperty,andhencethespaceL2(ΓRd,π)isunitarilyisomorphictothesymmetricFockspaceoverL2(Rd),seee.g.[18].Itcanbeshownthat,underthisisomorphism,thegeneratoroftheGDoffreeparticlesgoesoverintothenumberoperatorNontheFockspace.Thelatteroperatorisevidentlythesecondquantizationoftheidentityoperator,i.e.,N=dExp(111).Ontheotherhand,aconstructionofaMarkovprocesswhichcorrespondstothePoissonspacerealizationofthesecondquantizationofadoublysub-MarkovgeneratoronRd(oronamoregeneralspace)wasproposedbyD.Surgailisin[19].However,D.Surgailisdidnotdiscussthefollowingquestion:Fromwhichconfigurationsistheprocessallowedtostartsothatitneverleavestheconfigurationspace?Itwasonlyprovedin[19]that,forπ-a.e.configurationγ,theprocessstartingatγwillbea.s.intheconfigurationspaceatsomefixedtimet0.InthecaseoftheBrownianmotionontheconfigurationspace,i.e.,inthecaseoftheindependentmotionofinfiniteBrownianparticles(cf.[1]),asolutiontotheabovestatedproblemwasproposedbytheauthorsin[12].Moreexactly,asubsetΓ∞oftheconfigurationspaceΓXoveracomplete,connected,oriented,andstochasticallycompletemanifoldXofdimension≥2wasconstructedsuchthattheprocesscanstartatanyγ∈Γ∞,willneverleaveΓ∞,andhascontinuoussamplepathsinthevaguetopology(andeveninastrongerone).Inthecaseofaone-dimensionalunderlyingmanifoldX,onecannotexcludecollisionsofparticles,sothatamodificationoftheconstructionofΓ∞isnecessary,see[12]fordetails.Inthispaper,westudypropertiesofthesemigroupoftheGDofacontinuousinfinitesystemoffreeparticles.So,wefixalocallycompactsecondcountableHausdorfftopologicalspaceX.WedenotebyπmthePoissonmeasureonΓXwithintensitymbeingaRadonnon-atomicmeasureonX.InSection2,weconstruct,onthespaceL2(ΓX,πm),theDirichletformE,thegeneratorH,andthesemigroup(e−tH)t≥0fortheGDoffreeparticlesinX.Inparticular,wederiveanexplicitformulaoftheactionofthesemigrouponexponentialfunctions(Corollary1).Theresultsofthissecti
本文标题:The semigroup of the Glauber dynamics of a continu
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