Occupation time fluctuations of Poisson and equili

arXiv:math/0512414v3[math.PR]12May2006OupationtimeutuationsofPoissonandequilibriumnitevarianebranhingsystemsPiotrMi“o–InstituteofMathematisPolishAademyofSienesWarsaw2ndFebruary2008AbstratFuntionallimittheoremsarepresentedfortheresaledoupationtimeutuationproessofaritialnitevarianebranhingpartilesysteminRdwithsymmetriα-stablemotionstartingofromeitherastandardPoissonrandomeldorfromtheequilibriumdistributionforintermediatedimensionsαd2α.Thelimitproessesaredeterminedbysub-frationalandfrationalBrownianmotions,respetively.AMSsubjetlassiation:primary60F17,60G20,seondary60G15Keywords:Funtionalentrallimittheorem;Oupationtimeutuations;Branhingpartilessystems;FrationalBrownianmotion;Sub-frationalBrow-nianmotion;equilibriumdistribution.1IntrodutionConsiderasystemofpartilesinRdstartingoattimet=0fromaertaindistribution(astandardPoissonandequilibriumeldsareinvestigatedinthispaper).Theyevolveindependently,movingaordingtoasymmetriα-stableLØvyproessandundergoingnitevarianebranhingatrateV(V0).Weobtainfuntionallimittheoremsfortheresaledoupationtimeutuationsofthissystemwhenαd2α.Thisisanextensionof[4,Theorem2℄wherethestartingdistributionisaPoissoneldandthebranhinglawisritialandbinary.1.1BranhinglawInthe[3,4,5℄thelawofbranhingisritialandbinary.Inthispaperan1extendedmodelisinvestigated.ThepartilesbranhaordingtothelawgivenbyamomentgeneratingfuntionF.Ffulllstworequirements:1.F′(1)=1,whihmeansthatthelawisritial(theexpetednumberofpartilesspawningfromonepartileis1),2.F′′(1)+∞,whihstatesthattheseondmomentexists.(Noteherethatthebranhinglawin[4℄isgivenbyF(s)=121+s2
andobviouslyfulllsthetworequirements.)AlthoughonstraintsimposedonFarenotveryrestritiveandquitenatural(sothatthelassofthebranhinglawssatisfyingthemisbroad)stillthereremainotherinterestingasestobeinvestigated.Oneofthemisthelassofbranhinglawsinthedomainofattrationofthe(1+β)−stablelaw(i.e.,themomentgeneratingfuntionisF(s)=s+12(1+s)1+β),theasestudiedin[6,7℄.AremarkablefeatureofthelatteraseisthatthelimitproessesarestableonesandnotGaussianasitoursinthenitevarianease.1.2EquilibriumdistributionAnotheroneptnaturallyrelatedtopartilesystemsisanequilibriumdistri-bution.Ithasbeenshownthatinertainirumstanesthesystemonvergestotheequilibriumdistribution[12℄.ItisbothaninterestingandimportantquestionwhetherthetheoremsshownbyBojdekietalstillholdintheasewhentheequilibriumstateistakenastheinitialondition.Aonjeturein[3℄statesthatthetemporalstrutureofthelimitisgivenbyfrationalBrownianmotion.ItisofinteresttonotiethatthelimitisdierentfromtheoneintheaseofthesystemstartingofromthePoissoneld(wheretemporalstru-tureissub-frationalBrownianmotion).WestudybehaviorofthesystemforabranhinglawgivenbyF.Butthereisstillbroadareaforfurtherstudies.Noattempthasbeenmadetodevelopmoregeneraltheoryonerningsystemswithageneralstartingdistribution(oralargelassofdistributions).1.3GeneraloneptsandnotationLetusdenoteNPoisstandNeqt,theempirialproessesforthesystemstartingofromthePoissoneldwithLebesgueintensitymeasureandtheequilibriumrespetively.ForameasurablesetA⊂Rd,NPoisst(A),Neqt(A),respetivelyarethenumbersofpartilesofthesysteminsetAattimet.Notethattheyaremeasure-valuedproessesbutwewillonsiderthemasproesseswithvaluesinS′(thespaeoftempereddistributions)beausethisspaehasgoodanalytialproperties.Theequilibriumdistributionisdenedbylimt→+∞NPoisst=Neq,2wherethelimitisunderstoodinweaksense.TheLaplaefuntionaloftheequilibriumdistributionisgivenbyEexp{−hNeq,ϕi}=expλ,e−ϕ−1+VZ∞0hλ,H(j(·,s))ids,(1.1)wherej(x,l):=Eexp(−hNxl,ϕi)(1.2)H(s)=F(s)−s,ϕ:Rd→R+,ϕ∈L1(Rd)∩C(Rd)andjsatisestheintegralequationj(x,l)=Tle−ϕ(x)+VZl0Tl−sH(j(·,s))(x)ds,Thisequationsanbeobtainedinthesamewayas[12,(2.4)℄.Notethatin[12℄funtionϕisontinuouswithompatsupport.Weapproximateϕ∈L1usingfuntionsϕnwithompatsupportϕnրϕ.UsingLebesgue’smonotoneon-vergenetheoremitiseasytoobtaintheaboveequationsforϕ(Hisdereasingbeauseoftheritialityofthebranhinglaw).ForanempirialproessNttheresaledoupationtimeutuationproessisdenedbyXT(t)=1FTZTt0(Ns−ENs)ds,t≥0,(1.3)whereT0andFTisasuitablenorming.WeareinterestedintheweakfuntionallimitofXTwhentimeisaelerated(i.e.,Ttendsto∞).Theα-stableproessstartingfromxwillbedenotedbyηxtitssemigroupbyTtanditsinnitesimaloperatorbyΔα.TheFouriertransformofTtisbTtϕ(z)=e−t|z|αbϕ(z).(1.4)ForbrevityletusdenoteK=VΓ(2−h)2d−1πd/2αΓ(d/2)h(h−1),(1.5)whereh=3−d/α(1.6)(inthispaperwealwaysassumethatαd2αsoh1)andM=F′′(1).(1.7)WewillnowintroduetwoenteredGaussianproesses.Oneofthemissub-frationalBrownianmotionwithparameterhwiththeovarianefuntionChCh(s,t)=sh+th−12h(s+t)h+|s−t|hi(1.8)andtheseondoneisfrationalBrownianmotionwithparameterhandtheovarianefuntionchch(s,t)=12sh+th−|s−t|h.(1.9)31.4Spae-timemethodThespae-timemethodisaveryonvenienttehniqueforinvestigatingtheweakonvergeneintheC[0,τ],S′Rd

spae.ItwasdevelopedbyBojdekietalandanbefoundin[2℄.IfX=(X(t))t∈[0,τ]isaontinuousS′Rd
-valuedproesswedenearandomelement˜XofS′Rd+1
byD˜X,ΦE=Zτ0hX(t),Φ(·,t)idt,(1.10)whereΦ∈SRd+1
.InordertoprovethatXTonvergesweaklytoXinC[0,τ],S′Rd

itsuestoshowthatD˜XT,ΦE⇒D˜X