Self-similar stable processes arising from high-de

arXiv:0709.0773v1[math.PR]6Sep2007Self-SimilarStableProcessesArisingFromHigh-DensityLimitsofOccupationTimesofParticleSystems(Self-SimilarStableProcessesandParticleSystems)T.BOJDECKI1,∗,L.G.GOROSTIZA2,∗∗andA.TALARCZYK3,∗1InstituteofMathematics,UniversityofWarsaw,ul.Banacha2,02-097Warszawa,Poland(e.mail:[e-mail])2CentrodeInvestigaci´onydeEstudiosAvanzados,A.P.14-740,Mexico07000D.F.,Mexico(e.mail:[e-mail])3InstituteofMathematics,UniversityofWarsaw,ul.Banacha2,02-097Warszawa,Poland(e.mail:[e-mail])Abstract.Weextendresultsontime-rescaledoccupationtimefluctuationlimitsofthe(d,α,β)-branchingparticlesystem(0α≤2,0β≤1)withPoissoninitialcondition.Theearlierresultsinthehomogeneouscase(i.e.,withLebesgueinitialintensitymeasure)wereobtainedfordimensionsdα/βonly,sincetheparticlesystembecomeslocallyextinctifd≤α/β.InthispaperweshowthatbyintroducinghighdensityoftheinitialPoissonconfiguration,limitsareobtainedforalldimensions,andtheycoincidewiththepreviousonesifdα/β.Wealsogivehigh-densitylimitsforthesystemswithfiniteintensitymeasures(withouthighdensitynolimitsexistinthiscaseduetoextinction);theresultsaredifferentandhardertoobtainduetothenon-invarianceofthemeasurefortheparticlemotion.Inbothcases,i.e.,Lebesgueandfiniteintensitymeasures,forlowdimensions(dα(1+β)/βanddα(2+β)/(1+β),respectively)thelimitsaredeterminedbynon-L´evyself-similarstableprocesses.Forthecorrespondinghighdimensionsthelimitsarequalitativelydifferent:S′(Rd)-valuedL´evyprocessesintheLebesguecase,stableprocessesconstantintimeon(0,∞)inthefinitemeasurecase.Forhighdimensions,thelawsofalllimitprocessesareexpressedintermsofRieszpotentials.Ifβ=1,thelimitsareGaussian.Limitsarealsogivenforparticlesystemswithoutbranching,whichyieldsinparticularweightedfractionalBrownianmotionsinlowdimensions.TheresultsareobtainedinthesetupofweakconvergenceofS′(Rd)-valuedprocesses.MathematicsSubjectClassifications(2000):60G52,60G18,60F17,60J80Keywords:self-similarstableprocess,long-rangedependence,branchingparticlesystem,occupationtime,functionallimittheorem,S′(Rd)-valuedprocess.∗ResearchsupportedbyMNiSWgrant1P03A1129(Poland).∗∗ResearchsupportedbyCONACyTgrant45684-F(Mexico).11.IntroductionInordertoexplainthemotivationsforthispaper,wereferbrieflytopreviousresultsonoccupationtimesofthe(d,α,β)-branchingparticlesystem,whichhasbeenwidelystudied,andisdescribedasfollows.Attimet=0particlesaredistributedinRdaccordingtoaPoissonrandommeasure,andthentheyevolvemovingandbranchingindependentlyofeachother.Themotionisgivenbythesymmetricα-stableL´evyprocess,0α≤2(calledstandardα-stableprocess),thelifetimeisexponentiallydistributedwithparameterV,andthebranchinglawhasgeneratingfunctions+1(1+β)(1−s)1+β,0s1,(1.1)where0β≤1.Thislawiscriticalandbelongstothedomainofattractionofastablelawwithexponent1+β.Thecaseβ=1correspondstobinarybranching(0or2particles).Thisisthesimplestinaclassofbranchingparticlesystemsthatyieldessentiallythesameresults.Wealsoconsiderthesystemwithoutbranching(V=0).IftheinitialparticleconfigurationisgivenbyahomogeneousPoissonrandommeasure,i.e.,whoseintensityistheLebesguemeasureλ,thenthesystemwithoutbranchingisinequilibrium,thebranchingsystemconvergestowardsanon-trivialequilibriumstateastimetendstoinfinityfordα/β,anditbecomeslocallyextinctford≤α/β[13].Let(Nt)t≥0denotetheempiricalmeasureprocessofthesystem(withorwithoutbranching),i.e.,Nt(A)isthenumberofparticlesinthesetA⊂Rdattimet.TherescaledoccupationtimefluctuationprocesswithacceleratedtimeisdefinedbyXT(t)=1FTZTt0(Ns−ENs)ds,t≥0,(1.2)whereFTisasuitablenormingforconvergenceasT→∞.NotethatifλistheintensityoftheinitialPoissonconfiguration,thenENt=λforalltduetotheinvarianceofλforthestandardα-stableprocessandthecriticalityofthebranching(ornobranching).WithhomogeneousPoissoninitialcondition,functionallimittheoremsfortheprocessXTinthebranchingcasewereobtainedin[4,5]forβ=1,wherethelimitprocessesareGaussian,andin[6,7]forβ1,with(1+β)-stablelimitprocesses.Thelimitsaredimension-dependent,theirmainqualitativepropertiesbeingthatfortheintermediatedimensions,α/βdα(1+β)/β,theprocesshaslong-rangedependence,whileforthecriticalandhighdimensions,d=α(1+β)/βanddα(1+β)/β,respectively,theprocesseshaveindependentincrements.ForhighdimensionsthelimitsareS′(Rd)-valued(S′(Rd)isthespaceoftempereddistributions,thedualofS(Rd),thespaceofsmoothrapidlydecreasingfunctions),andtheirlawsareexpressedintermsofRieszpotentials.Thereisafunctionalergodictheoremford=α/β[16].ForintermediatedimensionsthelimithastheformX=Kλξ,whereKisaconstant,and(ξt)t≥0isarealnon-L´evyself-similar(1+β)-stableprocess,whichforβ=1isasub-fractionalBrownianmotion,whosepropertiesaredescribedin[3].Thefirstmotivationforthispapercomesfromthefactthatinthehomogeneouscasewithβ=1anddα,thecovarianceoftheprocessXThasanon-triviallimitasT→∞,whichcorrespondstoaprocessXofthesameformasabove,withadifferentGaussianprocessinsteadofsub-fractionalBrownianmotion,butXisnotthelimitofXTbecause,asrecalledabove,theparticlesystembecomeslocallyextinctifdα.Thereforethequestionarisesifitispossibletogiveaproba