Central Limit Theorem for a Class of Linear System

arXiv:0805.0342v1[math.PR]3May2008CentralLimitTheoremforaClassofLinearSystems1YukioNAGAHATAandNobuoYOSHIDA2AbstractWeconsideraclassofinteractingparticlesystemswithvaluesin[0,∞)Zd,ofwhichthebinarycontactpathprocessisanexample.Ford≥3andunderacertainsquareintegrabilityconditiononthetotalnumberoftheparticles,weproveacentrallimittheoremforthedensityoftheparticles,togetherwithupperboundsforthedensityofthemostpopulatedsiteandthereplicaoverlap.AbbreviatedTitle:CLTforLinearSystems.AMS2000subjectclassification:Primary60K35;secondary60F05,60J25.Keywordsandphrases:centrallimittheorem,linearsystems,binarycontactpathprocess,diffusivebehavior,delocalization.Contents1Introduction11.1Thebinarycontactpathprocess(BCPP).........................11.2Theresults..........................................32Lemmas62.1Markovchainrepresentationsforthepointfunctions...................62.2CentrallimittheoremsforMarkovchains.........................102.3ANashtypeupperboundfortheSchr¨odingersemi-group................123ProofofTheorem1.2.1andTheorem1.2.3133.1ProofofTheorem1.2.1....................................133.2ProofofTheorem1.2.3....................................141IntroductionWewriteN={0,1,2,...},N∗={1,2,...}andZ={±x;x∈N}.Forx=(x1,..,xd)∈Rd,|x|standsfortheℓ1-norm:|x|=Pdi=1|xi|.Forη=(ηx)x∈Zd∈RZd,|η|=Px∈Zd|ηx|.Let(Ω,F,P)beaprobabilityspace.WewriteP[X]=RXdPandP[X:A]=RAXdPforar.v.(randomvariable)XandaneventA.1.1Thebinarycontactpathprocess(BCPP)Westartwithamotivatingsimpleexample.Letηt=(ηt,x)x∈Zd∈NZd,t≥0bebinarycontactpathprocess(BCPPforshort)withparameterλ0.Roughlyspeaking,theBCPPisanextendedversionofthebasiccontactprocess,inwhichnotonlythepresence/absenceoftheparticlesateachsite,butalsotheirnumberisconsidered.TheBCPPwasoriginallyintroducedbyD.Griffeath[3].Here,weexplaintheprocessfollowingtheformulationinthebookofT.Liggett[4,ChapterIX].LetNz=(Nzt)t≥0,(z∈Zd)beindependentrateonePoissonprocesses.Wesupposethattheprocess(ηt)startsfromadeterministicconfigurationη0=(η0,x)x∈Zd∈NZdwith|η0|∞.Atthei-thjumptimet=Tz,iofNz,ηt−isreplacedbyηtrandomlyasfollows:foreache∈Zdwith|e|=1,ηt,x=ηt−,x+ηt−,zifx=z+e,ηt−,xifotherwisewithprobabilityλ2dλ+1,1May6,20082SupportedinpartbyJSPSGrant-in-AidforScientificResearch,Kiban(C)175401121(alltheparticlesatsitezareduplicatedandaddedtothoseonthesitez=x+e),andηt,x=0ifx=z,ηt−,xifx6=zwithprobability12dλ+1(alltheparticlesatsitezdisappear).Thereplacementoccursindependentlyfordifferent(z,i)andindependentlyfromthePoissonprocesses.AmotivationtostudytheBCPPcomesfromthefactthattheprojectedprocess(ηt,x∧1)x∈Zd,t≥0isthebasiccontactprocess[3].Letκp=2dλ−12dλ+1andηt=(exp(−κ1t)ηt,x)x∈Zd.Then,(|ηt|)t≥0isamartingaleandtherefore,thefollowinglimitexistsalmostsurely:|η∞|def=limt|ηt|.Moreover,P(|η∞|0)0ifd≥3andλ12d(1−2πd),(1.1)whereπdisthereturnprobabilityforthesimplerandomwalkonZd[3].Itisknownthatπd≤π3=0.3405...ford≥3[5,page103].Wedenotethedensityoftheparticlesby:ρt,x=ηt,x|ηt|=ηt,x|ηt|,t0,x∈Zd.(1.2)Interestingobjectsrelatedtothedensitywouldbeρ∗t=maxx∈Zdρt,x,andRt=Xx∈Zdρ2t,x.(1.3)ρ∗tisthedensityatthemostpopulatedsite,whileRtistheprobabilitythatagivenpairofparticlesattimetareatthesamesite.WecallRtthereplicaoverlap,inanalogywiththespinglasstheory.Clearly,(ρ∗t)2≤Rt≤ρ∗t.Thesequantitiesconveyinformationonlocalization/delocalizationoftheparticles.Roughlyspeaking,largevaluesofρ∗torRtindicatesthatthemostoftheparticlesareconcentratedonsmallnumbersof“favoritesites”(localization),whereassmallvaluesofthemimpliesthattheparticlesarespreadoutoverlargenumberofsites(delocalization).AsaspecialcaseofCorollary1.2.2below,wehavethefollowingresult,whichshowsthediffusivebehaviorandthedelocalizationoftheBCPPunderthecondition(1.1):Theorem1.1.1Suppose(1.1).Then,foranyf∈Cb(Rd),limt→∞Xx∈Zdfx/√tρt,x=ZRdfdνinP(·||η∞|0)-probability,whereCb(Rd)standsforthesetofboundedcontinuousfunctionsonRd,andνistheGaussianmeasurewithZRdxidν(x)=0,ZRdxixjdν(x)=λ2dλ+1δij,i,j=1,..,d.Furthermore,Rt=O(t−d/2)astր∞inP(·||η∞|0)-probability.21.2TheresultsWegeneralizeTheorem1.1.1toacertainclassoflinearinteractingparticlesystemswithvaluesin[0,∞)Zd[4,ChapterIX].RecallthattheparticlesinBCPPeitherdie,ormakebinarybranching.Todescribemoregeneral“branchingmechanism”,weintroducearandomvectorK=(Kx)x∈ZdwhichisboundedandoffiniterangeinthesensethatKx∈[0,bK]a.s.forallx∈Zd,=0,a.s.if|x|rKforsomenon-randombK,rK∈[0,∞).(1.4)LetNz=(Nzt)t≥0,(z∈Zd)beindependentrateonePoissonprocesses,andletKz,i=(Kz,ix)x∈Zd(z∈Zd,i∈N∗)bei.i.d.randomvectorswiththesamedistributionsasK,independentof{Nz}z∈Zd.Wesupposethattheprocess(ηt)t≥0startsfromadeterministicconfigurationη0=(η0,x)x∈Zd∈[0,∞)Zdwith|η0|∞.Atthei-thjumptimet=Tz,iofNz,ηt−isreplacedbyηt,whereηt,x=(Kz,i0ηt−,zifx=z,ηt−,x+Kz,ix−zηt−,zifx6=z.(1.5)TheBCPPisaspecialcaseofthisset-up,inwhichK=0withprobability12dλ+1(δx,0+δx,e)x∈Zdwithprobabilityλ2dλ+1,foreach2dneighboureof0.(1.6)Itisconvenienttointroducethefollowingmatrixrepresentationofthelineartransformationηt−7→ηt:ηt,x=Xy∈ZdAz,ix,yηt−,y,whereAz,ix,y=(1+δx,z(Kz,i0−1)ifx=y,δy,zKz,ix−yifx6=y.(1.7)Theprecisedefinitionoftheprocess(ηt)