Probab.TheoryRelat.Field114,309–399(1999) - PDF document

Probab.TheoryRelat.Field114,309–399(1999) Rescaledcontactprocessesconvergetosuper-BrownianmotionintwoormoredimensionsRichardDurrett,EdwinA.PerkinsDepartmentofMathematicsandORIE,278RhodesHall,CornellUniversity,Ithaca,NY14853,USA(e-mail:rtd1@cornell.edu)DepartmentofMathematics,1984MathematicsRd.,UniversityofBritishColumbia,Vancouver,B.C.V6T1Z2,Canada(e-mail:perkins@math.ubc.ca)Received:2February1998Revisedversion:28August1998Weshowthatindimensionstwoormoreasequenceoflongrangecontactprocessessuitablyrescaledinspaceandtimeconvergestoasuper-Brownianmotionwithdrift.AsaconsequenceofthisresultwecanimprovetheresultsofBramson,Durrett,andSwindle(1989)byreplacingtheirorderofmagnitudeestimatesofhowclosethecriticalvalueisto1withsharpasymptotics.MathematicsSubjectClassication(1991):Primary60K35,60G57;Secondary:60F05,60J801.IntroductionOurcontactprocessestakesplaceonanelatticez=M.Thestateoftheprocessattimeisgivenbyafunction,where.x/0indicatesthatisvacantattime.x/thatthesiteisoccupiedbyaparticle.Thedynamicsofthisright-continuouscontinuoustimeMarkovchaincanbedescribedasfollows:(a)Particlesdieatrate1andgivebirthtoonenewparticleatrate(b)Whenabirthoccursatthenewparticleissenttoasitechosenatrandomfromthewith01,where SupportedinpartbyNSFgrantDMS-93-01070andbyanNSERCcollaborativegrantSupportedinpartbyanNSERCresearchgrantandanNSERCcollaborativegrant 310R.Durrett,E.A.Perkins(c)Ifisvacantanewparticleestablishesitselfthere.Ifisoccupied,thebirthissuppressedandnochangeoccurs.1thenparticlesdiefasterthantheygivebirthandinadditionlosebirthsontooccupiedsites,sotheprocessdiesout.Tobeprecise,ifwestartwithallsitesoccupiedi.e.,considertheprocessstartingfrom.x/1thentheprobabilityofanoccupiedsite,.x/,whichdoesnotdependon,tendsto0as.Harris(1974)wasthersttoshowthatifislargeenoughthen(a).x/decreasestoapositivelimitas,and(b)thelimitofthedenesastationarydistribution,.Simplemonotonicityconsiderationstellusthatconclusion(a)willholdforalllargerthan.x/Self-dualityofthecontactprocess(seeTheoremVI.1.7ofLiggett(1985))showsthat:limP.isalsothecriticalbirthrateforsurvivalofthecontactprocessstartingwithasingleoccupiedsite.Harris'originalboundofwasverylargebutHolleyandLiggett(1978)showedthatinthenearestneighborcasethat4inalldimensions.Therehasbeenmuchworkonnumericalboundsinparticularcases,mostcommonlythenearestneighborone.SeeStacey(1994)andLiggett(1985,1995),butnotethatourparameteristhetotalbirthrateontoanysiteratherthanthantherateofbirthfromasitetoaparticularneighbouringsite.ChapterVIofLiggett(1985),andDurrett(1988),(1992b)aregoodplacestolearnaboutcontactprocesses.Bramson,Durrett,andSwindle(1989)consideredtheproblemoftheasymptoticbehaviorofthecriticalvalue.M/forthelongrangecontactprocessasTheoremA..M/.Furthermore.M/C=MM/=MC=MwheremeansifisasmalllargepositivenumberthentherighthandsideisalowerboundforlargeToexplaintheanswer,webeginbyconsideringaninverseproblem:given=N,where0,howlargedoesneedtobetoallowthecontactprocesstosurvive?Forthebranchingprocesswith Super-Brownianmotionintwoormoredimensions311(sothatnewparticlesareneverbornontooccupiedsites)themeannumberofparticlesattimeisexp.t=N/,sotheprocesswillneedtimeO.N/becomesignicantlysupercritical.Itisawellknownfactthatforthebranchingrandomwalkwhenatypicalparticleattimecountsthenumberofitsrelativeswithindistance1,theexpectedvalueoftheresultwillbeisthetimethenearbyrelativebrokeofffromtheancestraltreeofourtypicalparticleandistheprobabilitythatitstaysneartotheoriginalparticle.Theexcessbirthrateabove1isonly=Ninthecontactprocess,soforthistocompensateforthesuppressedbirthsweneedbelargeenoughsothatthefractionofoccupiedsiteswillbeoforder1Thatis,wechoosesuchthatBramson,Durrett,andSwindle(1989)usedbranchingprocessestimatesandablockconstructiontoshowthatwiththischoiceofthecontactprocessdiesoutforsmall0andsurvivesforlarge.Toapproachtheproblemofgettingsharpasymptoticswewillset=N,compresstimebyafactorof,andscalespacebyafactoroftocompensateforthetimescaling.Thatis,wedeclarethat:(a)Particlesdieatrateandgivebirthtoonenewparticleatrate(b)Whenabirthoccursatthenewparticleissenttoasitechosenatrandomfromthewith0(c)Ifisvacantanewparticleestablishesitselfthere.Ifisoccupied,thenthebirthissuppressedandnochangeoccurs.Nowisaxedrealnumber(althoughweareprimarilyinterestedinandweonlyconsidersuchthatThecase1hasbeenpreviouslystudiedbyMuellerandTribe(1995).Tostatetheirresult,werewritethecontactprocessasasetvaluedprocessby.x/andconsidertheapproximatedensityprocess.t;x/ TocheckthisscalingnotethatforsothelatticeisandtheaboveneighbourhoodwillcontainsitesbutonlyaboutO.Nparticlesby(1.1)andourspatialscalingby.Letdenotethespace 312R.Durrett,E.A.Perkinsofcontinuousfunctionsfromto[0withcompactsupportequippedwiththetopologyofuniformconvergenceandletlet;1/;C0/betheSkorokhodspaceofcadlag-valuedpaths.AspecialcaseoftheirresultshowsTheoremB.Iftheinitialconditions;x/approach;x/.t;x/convergesweaklyintothesolutionofthestochasticpartialdifferentialequation 6u00Cu2p udWdenotesdifferentiatioinwithrespecttoisaspace-timewhitenoise(seeWalsh(1986)),andwehaveconsideredarestrictedclassofinitialconditionstoavoidtheissueofgrowthconditionsat.Toexplainthelimit,resultsfromdisplacementofparticleswiththe63dictatedbythefactthattheuniformdistributionon[1]hasvariance13.Thedriftcomesfromthe“excess”birthrate,reectsthelostbirthsontooccupiedsites,andthe fromthefactthatwehavebirthsanddeathseachatrate1perparticle.ToprepareforourdiscussionofourTheorem1,notethatinprovingTheoremB,MuellerandTribeshowedtightnessoftheapproximationsinaspaceofcontinuousfunctions.Thusforlargenearbysiteshaveanalmostidenticalnumberofoccupiedneighbors,i.e.,theratioofthenumberofneighborsattwonearbysitesiscloseto1.Withoutthetheequationin(1.3)isthestochasticpartialdifferen-tialequation(SPDE)forthedensityofone-dimensionalsuper-Brownianmotion(seeReimers(1989)orKonnoandShiga(1988)).Whensuper-BrownianmotionissingularwithrespecttoLebesguemeasuresoequationslike(1.3)aremeaningless.Analternateapproachistocharacter-izesuper-Brownianmotionasthesolutionofameasure-valuedmartingaleproblem.Tothisendweintroducethespaceofboundedcontin-uousfunctionswhosepartialderivativesoforderlessthan1arealsoboundedandcontinuous().Letdenotethespaceofnitemeasuresonwiththetopologyofweakconvergence,gence,;1/;MbetheSkorokhodspaceofcadlagvaluedpaths,andbethespaceofcontinuous-valuedpathswiththetopologyofuniformconvergenceoncompacts.Integrationofawithrespecttoameasureisdenotedby./.Anadapted-valuedprocess(0)onacompletelteredprobabilityspace.;;P/isright-continuous)isan-super-Brownianmotionstartingatwithbranchingratediffusioncoefcient0anddriftifandonlyifitsatisesthe Super-Brownianmotionintwoormoredimensions313followingmartingaleproblem:Forall.MP/ ;./././1=/dsisan-martingalewithZ./. /ds:Thelawofisthenuniqueand.MP/ ;holdsforalargerclassoftestfunctionsincluding.SeeTheorem2.3ofEvansandPerkins(1994)forthelatter.TheuniquenessfollowsfromDawson'sGirsanovthe-orem(seeTheorem5.1ofDawson(1978))andtheuniquenessoftheabovemartingaleproblemwith0.ThislatterresultmaybefoundinDawson(1994)(Theorem6.1.3)foraslightlylargerclassoftestfunctionsbutasourclassoffunctionscontainsacoreforthegeneratorofBrownianmotionontheBanachspaceofcontinuousfunctionswithlimitsatinnity(EthierandKurtz(1986),Proposition5.1.1)uniquenessthenfollowsintheabove.Ifinsteadof(1.2)wechoose,thenthesetofoccupiedsitesattimeisabranchingrandomwalk.Deneameasure-valuedprocessby./ .x/forallboundedmeasurablefunctions.ResultsinChapter4ofDawson(1993)(seeTheorem4.6.2)thengiveTheoremC.Iftheinitialmeasuresapproachthenthesequenceofmeasure-valuedprocessesconvergestosuper-BrownianstartingatwithbranchingratediffusioncoefcientanddriftThepuristmaynoticethatourbranchingmechanismisslightlydifferentthanthatinDawson(1993)(onlyoneparticlejumpsateachbirthtime)andDawsonworksontheone-pointcompactication.Thenecessarymodica-tionsarestraightforward,moreoverTheoremCwillalsofollowfromtheeasypartsofourproofofTheorem1below.Sincethecontactprocesscanbedominatedbythebranchingprocessthatresultsbyignoringrule(c)andallowingbirthsontooccupiedsites,wemusthaveasingularlimitfortherescaledcontactprocessin2.Weagainassigneachparticlemass1andlookatthemeasurevaluedprocess 314R.Durrett,E.A.Perkinsdenedby./.x/.Hereforreasonsthatwillbecomeclearinamomentwehavesuppressedthedependenceon.Tostateourlimitresultweneedadenition.In3weletbei.i.d.uniformon[denetherandomwalkC


C,andletlet�1;1]d
=2dIndD2wesetTheorem1.Supposethat.Iftheinitialmeasuresapproachawithnopointmassesthenthesequenceofmeasure-valuedprocessesconvergesweaklyontoasuper-BrownianmotionstartingatwithbranchingratediffusioncoefcientanddriftToexplainTheorem1,webeginwiththeeasiercase3andagainlookattheunscaledbranchingrandomwalkinwhichbirthsanddeathseachoccuratratesO(1).Acloserlookatthereasoningthatledto(1.2)tellsusgivesanupperboundontheexpectednumberofneighborsofaran-domlychosenparticleattimesuchthatthelastcommonancestorofwasatatimebefore.Herex;yareneighborsif1.Thisimpliesthatifthenthenumberofneighborsoftwoparticlesintheunitspeed(andunscaled)branchingrandomwalkare“almostindependent”.Toseethisnotethat(1.4)showsthat,uptoasmallerrorapproaching0asweonlyneedconsidercontributionstothenumberofneighborsoffromcousinswhichbranchoffinthelasttimeintervaloflength.AsthisshowsthatmoduloasmallerrorthenumberofneighborsofthetwopointsdependsonadisjointsetofrandomwalkincrementsinthebranchingBrowniantreeandsoare“almostindependent”.Rescalingtimeandspaceweseethatthenumberofneighborsoftwoparticlesarealmostindependent.Thisandthelawoflargenumbersimpliesthattheamountofmasslostnearapoint Super-Brownianmotionintwoormoredimensions315tobirthsontoanoccupiedsiteisjustthemeannumberofneighborsofarandomlychosenpoint,,timesthemassthere.ThereasoningdescribedinthelastparagraphjustbarelyworksinTakingin(1.4)givesloglogSorescalingtimeandspace,weseemostoftheneighborsofaparticleareitsrelativeswithmostrecentcommonancestorlessthan1backintime.Thus,ifthenumberofneighborsoftwoparticlesarealmostindependent.Againthelawoflargenumbersimpliesthattheamountofmasslostnearapointduetobirthsontoanoccupiedsiteisjustaconstanttimesthemassthere,where logNNXnD1P�Un2[�1;1]2
=22D3 thelastbyalocalcentrallimittheorem(seeSection8).Toturntheheuristicsinthelasttwoparagraphsintoaproof,wewilldeneasequenceofapproximatingprocesseswiththesameinitialcon-.Likethecontactprocess,,thesedependonbutwewillnotexhibitthedependenceinthenotation.Therstprocessinthesequenceisthebranchingrandomwalkwhichresultsifweignorerule(c)inthedenitionofthecontactprocessaboveandallowbirthsontooccupiedsites.Withoutthecollisionrule,mayhavemorethanoneparticleatasitesoweregardasa“multi-set,”i.e.,asetinwhichrepetitionsofelementsisallowed.Forexample,a;a;b;b;b;cwouldrepresenttwoparticlesatthreeatandoneatFor1weletbethebranchingrandomwalkwiththecollisionrulethatbirthsontositesinaresuppressed.isanunderestimateofthecontactprocesssinceitremovesparticlesthatcollidewiththelargerset.Intheotherdirectionisanoverestimatesinceitremovesonlyparticlesthatcollidewiththesmallerset.Theprocessesareanalternatingsequenceofupperandlowerboundsthat,forxed,areequaltothecontactprocessfor.!;t/,i.e.,thenumberofiterationsneededdependsontherealizationandthetimeofinterest.Wewillnotusethisfactandsoleaveitsvericationtotheinterestedreader.Fromtheapproximatingprocesses0wecandenemeasure-valuedprocesses./.x/ 316R.Durrett,E.A.Perkinswheresitesarecountedaccordingtotheirmultiplicitiesin.In1,weconjecturethatastheseprocessesconvergetolimitsthatarealldistinct.However,wecanproveProposition1.UnderthehypothesesofTheoremthenforallasNNoteherethatsothisresultimpliesthetotalvariationofapproaches0asuniformlyin.Sincethecontactistrappedbetweenitfollowsthatisasymptoticallythesameas.Giventhis,wecanproveTheorem1bydemonstratingProposition2.UnderthehypothesesofTheoremconvergesweaklyintosuper-BrownianmotionstartingatbranchingratediffusioncoefcientanddriftTheprocessismucheasiertoanalyzethanthecontactprocesssinceitisjustthebranchingprocessminusparticlesthatarebornontositesinthebranchingprocess.However,itstilltakesquiteabitofefforttoproveProposition2.AnoutlineoftheproofcanbefoundinSection2.ThedetailsllupSections3to10.OnereasonforinterestinTheorem1isthatitallowsustosharpentheconclusionofaresultofBramson,Durrett,andSwindle(1989).Letting1bethenumberofneighborsasitehas,andrecallingarerelatedby(1.2),wecannowreneTheoremAasfollows:Theorem2..M/M/=MV/=VwheremeanstheratioapproachesoneasapproachesTheblockconstruction,asdescribed,forexample,inSection4ofDurrett(1995b),makesthisafairlystraightforwardconsequenceofTheorem1.ThedetailsofthelowerandupperboundsneededtoproveTheorem2aregiveninSections11and12.1,MuellerandTribe(1994)haveshownthatthelimitingSPDEinTheoremBhasacriticalvalue,,belowwhichthereisa.s.extinctionandabovewhichthereislongtermsurvivalwithpositiveprobability.Inviewofthisitisnaturalto Super-Brownianmotionintwoormoredimensions317Conjecture..M/ Toprovethisseemsdifcult.OurproofofTheorem2makescrucialuseoftwofacts:(i)thesupercritical-subcriticalphasetransitioninsuper-Brownianmotioncanbeidentiedbybylookingatthemeannumberofparticles,and(ii)bycomputingsecondmomentswecanidentifyasuitableblockeventforthelimitingprocess.NeitheroftheseisavailablefortheFor2,Theorem2givesthefollowingasymptoticresultforthecriticalvalueofthecontactprocessforanneighbourhoodwith.V/ logV Toinvestigatethequalityofthisapproximationforniterangewehavesimulatedtheprocesswith2,i.e.Forthesimulationitisconvenienttochangethetimescalesothatandthedeathrate,,istheparametersothatwecansimulatetheprocessforallvaluesofsimultaneouslyusingthemethodsofButtel,CoxandDurrett(1993).Inthegraphbelowwehaveplottedthefractionofoccupiedsitesattime5000asafunctionof,startingfromacongurationofallsitesoccupiedinthe10001000gridandwithperiodicboundaryconditions.Theestimateoffromtheformulaisabout125whichleadstoanestimateof45forthecriticalvalueof.Thislatterestimateishigherthanthecriticalvalueof23obtainedfromthesimulation.Note,however,thatthestraightlinefromdoesamuchbetterjobofestimatingtheequilibriumdensityofparticlesthanTheorem2ofBramson,DurrettandSwindle(1989)which,whenrewrittenintermsof,says.x/InprincipleonecoulduseourmethodstosharpenTheorem3ofBramson,DurrettandSwindle(1989)andgivecorrectionsto(1.6).Weleavethistasktoanenergeticreader.2.OutlineoftheProofandconsideronly0.Webeginbyconstructingtheprocesseswhichliveontherescaledlattices 318R.Durrett,E.A.Perkins Figure1Wehaveseparatedthescalingintotwopiecessincetherstcompensatesforthefactthatbirthsoccuratrate,whilethesecondincreasesthenumberofneighborsasitehas.Sincewewanttorelatethebehaviorofthelongrangecontactprocesstothatofabranchingprocess,wewilluseabranchingprocesstypeconstructionforthecontactprocessratherthantheusualgraphicalrepresentation.LetLetnD0�Nf0;1gn
bethesetoflabelsfortheparticlesinourprocess.Therstterminthe,labelstheinitialparticlesThroughout,wewillASSUMEthattheinitialstateconsistsofanitenumberofparticlesthatarelocatedatdistinctsitesandthat Toconstructthetimeevolutionweusuallysuppressdependenceonandworkonacompleteprobabilityspace.;;P/containingthefollow-ingindependentcollectionsofrandomvariables: Super-Brownianmotionintwoormoredimensions319arei.i.d.exponentialwithrate2arei.i.d.withP. P. arei.i.d.withP.eP.earei.i.d.uniformonDeathsoccuratrateandbirthsatrate,soisthetimeuntilabirthordeathaffectsparticle.Theeventisadeathif1andabirth1.Theparticle.;eisdisplacedfromitsparentbyan,whiletheotherparticle.;remainsatthelocationof.;i/isanewmemberofobtainedfrombyaddingacoordinateattheend.)thenwesayisingenerationandwrite.Ifwewritefortheancestorofingeneration.If0thenweusetodenoteitsparent,i.e.,itsancestoringeneration1.If0thisisthe1generationwhichdoesnotexist,soweset,theemptystring.Fromthedenitionsaboveitshouldbeclearthatistheendofthelifeofparticle.Sinceisaparticlethatdoesn'texist,itisreasonabletodeclarethatitdiedattime,i.e.,D�1.Lett/.Twhichweassumehasbeencompletedbyaddingallthenullsets.Tocomputethepositionsofparticles,webeginbynotingthatthedisplacementofthejumpingparticleattime.Thefamilylineofismovedwhendiesif,sowedenethepositionofthefamilylineofattimehereismeanttosuggestBrownianmotion,astoppedversionofwhichisthelimitof(thestoppingtime). 320R.Durrett,E.A.PerkinsThebarisaddedheresothatitcanberemovedinthenextparagraphinthedisplacementprocessthatwewilluserepeatedly.Uptothispointwehaveignoredthedeaths.TotakethemintoaccountNotethatwhenthesetoftimesisempty,wehaveinf'sproducedeaths,thefamilylinediesoutattime(orceasestomakesenseattime)andwelettistheusualcemeterystateofMarkovprocesstheoryusedtoindicatethattheparticleisnolongeralive.Inanumberofinstanceswewillbeinterestedinthespatiallocationofparticle.Sinceisaliveon[andnevermoves,wecanwriteindicatesthatthelastquantityisashorthandwewilluseforthersttwo.WewritetT.Inwords,labelsaparticlebealiveattime.TriviabuffswillwanttonotethatifD�1,sotheparticlesintheinitialcongurationareactuallyaliveatallnegativetimes.Wewilladopttheconventionthat.1/0,forallfunctionssothatdeadparticlesdon'tcontributetooursums.Countingonlytheparticlesthatareactuallyaliveleadstoourrstmeasure-valuedprocesses,thebranchingrandomwalk.Thisandallthesubsequentmeasure-valuedprocesseswillbedenedbyintegratingaboundedmeasurabletestfunctionwithrespecttotheprocess:./ .BNotethatdependsoneventhoughwehavenotrecordedthisde-pendenceinthenotation.Whenweneedtodisplaythe,wewillwriteisameasure,letsuppdenoteitsclosedsupport.Ifisameasure-valuedpathwhichiscadlag,recallthedenitionoffrom(2.4)andlet Super-Brownianmotionintwoormoredimensions321Inwordsthesecondtermidentiesthersttimethatajumpinthefamilylineoflandsonasitethatisalreadyoccupied,i.e.,in.Withthisnotationintroduced,wecandenethecontactprocesssimplyastheunique“strongsolution”of./ .B.X�/t/TheexistenceanduniquenessofthesolutionaretrivialforaninitialnitesetofparticlessincewecansuccessivelydecidewhattodoattheeventWecannowdenethesequenceofprocesses1introducedintheprevioussectionby:./ .BNotethatinthecase0thisreducestothedenitionofthebranchingrandomwalkgivenin(2.5).(Anyonewhoisconcernedthatisnotdenedwhen0canletthisprocessbe0.)asmeasures(i.e.,././forall0),comparing(2.6)and(2.7)shows,againasmeasures.RepeatingthisreasoninggivesTherststepinourderivationofPropositions1and2istowritedownastochasticequationfor1,whichinthelimitwillapproachthemartingaleproblemcharacterizingthelimitingsuper-Brownianmotion.Wewillonlyhavetodothisfor1and2butformostoftheproofitwillbeeasiertowriteouttheargumentsforageneral.Toderiveourequation,westartwiththeobservationthataspassesthroughtimelosetheparticle,butifwehaveabirtheventthenanewparticlewillexistatthesamelocation,,andasecondparticlewillexistatifitdoesnotlandonanoccupiedsite:././ .BB.B.BTakingadvantageofthefactthatwecanrewritethelastexpressionas(justcheckthetwocases) 322R.Durrett,E.A.Perkins .B.B.Bbysubtractingitsexpectedvalue,wecandeneandsplittherstterm(involving.B)intotwopieces: N1�n�B
gC /=.anddosomearithmetictowritethesecondtermin(2.9)asthesumofthefollowingthreeterms: .B.B .B.B .BTocheckthiseasily,beginbycombiningthesecondandthirdterms.Summingin(2.9)over,telescopingthesum,recallingtheabovedenitionof,andintroducing.t/t;t;/;nasshorthandfor“wasaliveinbutdiedbeforetime,”wehave././ .t/.B .t/.B .t/h/h.B.B .t/.B.B .t/.BByintroducingnotationforthevariousterms,wecanrewritethelastequa-tionbrieyas Super-Brownianmotionintwoormoredimensions323./././././././Notethisisvalidfor0,inwhichcase0byourconventionthat./isthe“collisionterm”whichcountsthenumberofbirthsontooccupiedsites.Theanalysisofthistermwillbethehardpartoftheargument,sowebeginwiththeotherfourterms.Here,andinwhatfollows,./are“error”termsthatwillgoto0,theare“drift”termsthatwillhavenon-zerolimitswhicharelocallyofboundedvariation.ThroughoutthispaperwewillASSUMEthatandlet.x/Foranumberoftheresultsthisconditioncanbeweakenedto:isboundedandmeasurable(or:isLipschitzcontinuous).However,wenditcon-venienttouseonecollectionoftestfunctionsforalltheresults.Let=..InSection3weestablishthefollowingresultsforLemma2.1../isan-martingalewith./ /dr././ //drLemma2.2.Forall././drLemma2.3.Forall./.1=/drLemma2.4.Forall0lim./Therstthreeconclusionsarestraightforwardtoproveandarewhatoneshouldguessbycomparisonwiththecorrespondingpartsof(2.10),i.e.,therst,second,andfourthlines.Tohandletheerrorterm./,notethatifweremovethemean-zerorandomvariablesfromthedenition,itisequalto././.Intuitively,if././staysbounded,thentheshouldcausecancellationsthatdrive./to0.Toturnthisideaintoaproofwebound./abovebythe“collisionterm”ofthebranchingrandomwalk. 324R.Durrett,E.A.PerkinsLemma2.5.Thereisaconstantsothat.t/C.XNoticeaboutconstants.Thisresulthastherstofalargenumberof'sthatwillappear.Alloftheseconstantsmaydependonthedrift,thetimeand(thoughitisvacuoushere)onthetestfunction.However,forxedourconstantC.;t/willbeboundedoncompactsubsetsofof;1/.Thisconditionisobviouslysatisedwhenever.;t/isacontinuousfunction.willneverdependon,orontheinitialconditionLemma2.5indicatesthatwehaveenoughneighborssothattheamountofmasslostduetointerferenceis.ThisresultisprovedinSection4,byprovingtworesults,Lemmas4.1and4.4thatinvestigatetheamountofinterferencebetween(i)unrelatedindividualsand(ii)individualswithacommonancestoringeneration0.InLemmas5.1and5.3,wesharpenthetwoboundsinSection4toshowthatiftheinitialconditionsameasurewithnopointmasses,then“collisionsbetweendistantrelativescanbeignored.”Heredistantmeansthattheirmostrecentcommonancestorwasmorethanunitsoftimeinthepastwhere(i)in(ii)inTheseobservationsarethekeytoourresultforthecollisionterm.Lemma2.6.Thenforandany././drToargueintuitively,notethatifislargethentwoindividualsthatarerelatedwithinunitsoftimeliewithin distanceinspacewithhighprobability.Fromthisweseethatthecollisiontermhasacorrelationlengththatvanishesinthelimit,andthenumberofcollisionsinaregionbecomesaconstanttimesthemassoftheprocessthere.AnoutlineoftheproofofLemma2.6canbefoundinSection6.TheretheresultisbrokendownintosevenlemmasthatareprovedinSections6–10.Ratherthandescribethosetechnicalitiesnow,wewillinsteadexplainwhyLemmas2.1–2.6areenoughtoprovePropositions1and2,andhenceTheorem1.ThenextstepinthatdirectionistheProofofProposition1.Subtracting(2.11)with1and2fromthesameformulawith1andusingLemmas2.1–2.6showsthat Super-Brownianmotionintwoormoredimensions325/dr.t/.t/0.Sinceisamartingale,takingexpectedvaluesandletting.t//X1t.1/�X2t.1/]0showsthat.t/.r/dr.t/.t/.t/.AformofGronwall'slemmaimpliesthat.s/.s/.t/andhenceE.X.t/0(2.14)Toputthesupremuminsidetheexpectedvaluewehavetolookatthemartingaledifference.Themaximalinequality(appliedto)combinedwiththefactthatisamartingalenullat0impliesthatUsingLemma2.1now,wehave //drCombining(2.14)–(2.16)wehavethatas.t/0(2.17)Returningto(2.13)now,wecanlet.t/notethatthedifferenceinsidetheabsolutevaluesisalwayspositive,and 326R.Durrett,E.A.Perkins.t/.t/.r/dr.t/Againifwelet.t/.t/.t//.t/.t/ TurningnowtoProposition2,wehavefromtheabovelemmas:Foreach./././1=/ds././isan-martingaleasinLemma2.1and./denotethelawof.!/!.t/denotethecoordinatevariableson.ToprovetightnessofweusethefollowingspecializedversionofJakubowski'sgeneralcriteriononon;1/;E/Polish(seeTheorem3.6.4ofDawson(1993)).Recallthatisaseparatingclassifftheintegrals./uniquelydetermineLemma2.7.beaseparatingclasswhichisclosedunderaddition.AsequenceofprobabilitiesistightiffthefollowingconditionsForeachT;�thereisacompactsetT;suchthatT;:(ii)lim&#x-277;/M/./thenforeachistightinin;1/;R/.Thederivationofthisresultfromthemoregeneralresultscitedaboveisstraightforward(seeTheorem3.7.1ofDawson(1993)fortheslightlysimplersettingoftheone-pointcompacticationofRecallthatisthespaceofcontinuous-valuedpathswiththecompact-opentopology.SpecializingtheaboveresultfurtherwehaveLemma2.8.satisfyhypothesisofLemmaandforeachistightinandalllimitpointsaresupportedsupported;1/;Rd/.ThenistightinandalllimitpointsaresupportedonProof.Taking1,weseethatourassumptiononreadilyimplies(ii)inLemma2.7(seeTheorem3.10.2ofEthierandKurtz(1986)).Lemma Super-Brownianmotionintwoormoredimensions3272.7showsthatistighton.Letbealimitpoint.IfP.X./isalimitpointofandsoissupported.Letbeacountablesubsetofthefunctionsinwithcompactsupportwhichisdenseinthespaceofcontinuousfunctionswithcompactsupportin.Then./iscontinuousforalla:s:aseparatingclass,thisimpliesforalla:s: Lemma2.9.denotethelawofacontinuoustimerandomwalkstartingatwhichatratetakesastepuniformlydistributedover.IfisboundedandmeasurablethenE.X.//..B.dx/andthereisaconstantC.;t/suchthatE.XXX00.1/CX00.1/4]:Proof.Thisfollowsfromtheknownmomentmeasuresofabranchingran-domwalkstartingfromasingleparticle.SeeforexampleLemma2.2ofBramson,DurrettandSwindle(1989)(andset.=N/inthatresult).Afewsimplemomentinequalitiesforsumsofi.i.d.randomvariablesareneededtoderivethesecondresultfromthelemmainBramsonetal.whichassumesasingleinitialparticle. Lemma2.10.istightonandalllimitpointsaresupportedbyProof.WeapplyLemma2.8.Let/;T�0,andfor1choosechoose;1]suchthat;R/.x/.x/andallthederivativesofofordertwoorlessareuniformlybounded.x;r/.By(2.2),Lemma2.9andtheweakconvergenceoftherandomwalksinthatresulttoBrownianmotion,wemaychoosesufcientlylargesothatE.X.B.;R///Theanalogueof(2.18)for(omitthekillingterm)gives:.h/ds 328R.Durrett,E.A.Perkinsisanmartingalesuchthat /ds0forallTherefore,applyingChebychevoneachtermandthen(2.19),weseethat�// E.X//ds E.X.B.;R///ds cT .cT/;wherethelastinequalityisvalidforsufcientlylarge.As,(i)ofLemma2.7isnowobvious..Wewilluse(2.18)toverifytheotherhypothesisofLemma2.8.If0tu,then1=/dsc./E/dsc./E.Xc./.uwherewehaveusedLemma2.9inthelastline.Thisshowsthat.t/1=/dsdenesatightsequenceofprocesseson(e.g.byTheorem8.3ofBillingsley(1968)).Turningnowtothemartingaletermsin(2.18),arguingasaboveweseefromLemma2.1that./isatightsequenceofprocesses.Notethatbydenition,sup./.TheoremVI.4.13andPropositionVI.3.26ofJacodandShiryaev(1987)nowshow./isatightsequenceinandalllimitpointsaresupportedby.Theseresultswith(2.18)andCorollaryVI.3.33ofJacodandShiryaev(1987)showthatistightinandthatalllimitpointsaresupportedon.Lemma2.8nowcompletestheproof. Super-Brownianmotionintwoormoredimensions329ItisnowstraightforwardtoproveProposition2.Letbealimitpoint.Thenisalawonandwemustshowitsatisesthemartingaleproblem.MP/fromSection1whichcharacterizessuper-Brownianmotionwiththeappropriateparameters.BySkorokhod'stheoremwemayassumethat(nowmakingdependenceonexplicit)a.s.Letandset./././1=/dsWemustshowthat././/ds,whereisthecanonicalright-continuousltrationgenerated.Weonlyshowthelatterasitisslightlymoreinvolved.Fix0st,andletbeboundedandcontinuous.Write./forthemartingaletermin(2.18)withBytakinganothersubsequenceweseefrom(2.18)thatsup././0forall0a.s.UseLemma2.9forthenecessaryuniformintegrabilitytoconclude././/dr././ 0(by(2.18))ThiscompletestheproofofProposition2andhenceprovesTheorem1,moduloLemmas2.1–2.6.3.ThefoureasyconvergencesInthissection,wehavetwoaims.Firstwewillintroducesomeusefulmartingales.ThenwewillproveLemmas2.1–2.4.Lemma3.1.isaright-continuousltration.ForeachForallare-stoppingtimes.ForallareTheseclaimsareintuitivelyobviousandformalproofsarenothardtoconstructsoweproceedto: 330R.Durrett,E.A.PerkinsLemma3.2.:[0beboundedand-predictableand.Thenthefollowingprocessisan .T;!// .r;!/drProof.bethenumberofarrivalsbytimeinaratePoissonprocess,andletbethetimeofthetharrival.Asiswellknownisamartingalewithrespectto.Tandsoisthestochasticintegral .r;!//dM .T;!// .r;!/drIfwetake1weobtainthedesiredresultforthe.Tandhencealsoforthelargerltrationobtainedbyadjoiningtheindependentinformationin.tnotanancestorof.Asthisislargerthan,theresultfollows. OursecondclassofmartingalesisLemma3.3.beboundedandmeasurableandrecallwhere=..ThenthefollowingprocessisanJ.t/t;T.BProof.Recallthatisgeneratedbyaclassofsetsclosedunderniteintersections.ItisthenstraightforwardtocheckthatE.J.Tt;T.B/E.gJ.t/isconstantexceptforajumpat,equaltoJ.TitfollowsJ.t/isan BeforeextendingtheclassofmartingalesinLemma3.3,weneedatechnicalresultwhichwillenableustocomputeorboundvariousrandomvariables.Letr.;t/ Super-Brownianmotionintwoormoredimensions331Lemma3.4.foranyE.Nt;T/r.;t/XProof.Toprove(a)consider,thenumberofparticlesthatwouldcontributetoourbranchingrandomwalkifwestartwithindividualsandhavenodeaths.Thatis,ateacheventweignorethe'ssoanewparticleisbornandtheoldonedoesnotdie.Clearly,.Sinceisabranchingprocessinwhicheachparticlegivesbirthatrate2,itisaspeededupversionoftheYuleprocess.Ithaslongbeenknown,seee.g.,Kendall(1949),thathasageometricdistribution,soforallToprove(b)webeginbyobservingthat=..t/t;Tmultiplyingtheaboveequationby1,andsummingover,whichislegitimatebecauseof(a),wehave .t/g .t/Thersttermontheright-handsideisamartingalebyLemma3.3(part(a)provedabovejustiesintegrability).TakingtheexpectedvalueandusingLemma2.9givesfor.t/ /r.;t/XTheresultisnowalsoimmediatefor0bymonotonicityinoftheleft-handsideof(b)andcontinuityinoftheright-handside. Lemma3.5.AssumethatismeasurablewithrespecttoE.Gt/Gisan-martingaleandthereisasothat 332R.Durrett,E.A.PerkinsProof..t/t/G.WithourassumptionswecanfollowtheproofofLemma3.3toconcludethat.t/isamartingale.Summingupthesemartingaleswith(a)ofLemma3.4tocheckintegrabilityshowsthatisamartingale.Toprovetheboundwerecallthatsinceourmartingalehaspathsofboundedvariation,[.UsingtheinequalityformartingaleswiththefactthatthatM]tisamartingalethatisnullat0givesesM]t(3.1)Combiningthiswiththeformulafor[wehave t/G t;T r.;t/Xand(b)inLemma3.4.Theresultisnowimmediatebyourconventiononconstants. RecallfromSection2,that./ t;T/.Bandwehaveassumed,althoughthenextresultonlyrequirestobeboundedandmeasurable.Lemmas2.1–2.4fromSection2willnowbeproved.Werestatethemforthereader'sconvenience.Lemma2.1../isan-martingalewith./ /dr;././ /dr:Proof.Thefactthat./isan-martingalefollowsfromLemmas3.3and3.4(a)(thelatterforthenecessaryintegrability).Clearly,,Zn./t;T Super-Brownianmotionintwoormoredimensions333Toconvertthisinto./,wewillreplacebyitsmeanandthenthesumbyitscompensator.Recallingvariousdenitionsweseethatvar.EAsinLemma3.3onemayreadilycheckthatE.gsoanapplicationofLemma3.5impliesthefollowingisamartingale:t;TApplyingLemma3.2(andLemma3.4(a))with .r;!//.Bweseethat t;T NZt01 ;r/drisamartingale.Recallingthedenitionofandusingthefactthatisamartingalewehaveshown./ /drForthesecondassertionnotethat././ .Bandmakeminorchangesintheaboveargument. Wenextconsider:./ t;T/.BThefollowingresultagainonlyrequirestoboundedandmeasurable.WewillneedthisgeneralizationfortheproofofLemma2.3.Lemma2.2.Forall0lim././drProof.Lemma3.2with .r;!//.Band(b)ofLemma3.4,thelattertocheckintegrability,showthatthefollowingisamartingale(notethatrT 334R.Durrett,E.A.Perkins t;T/.B rT/.B/dr t;T/.B./drUsingthedenitionof[andreversingthelastsimplicationwehave /dr t;T NCt01 rT/drisamartingale,whence /drUsingthemaximalinequalityformartingaleswiththefactthatisamartingalethatisnullat0givesCombiningthiswiththeformulaforandusingLemma2.9,weget /dr././dr,thedesiredresultfollows. ThethirdtermfromSection2thatwewillconsideris:./ t;T.B.BLemma2.3.Forall./.1=/drProof.Given,wecanapplytheone-dimensionalTaylor'stheoremwithremaindertof.t/.yt.zy//toget Super-Brownianmotionintwoormoredimensions335.z/.y/.y/.z .v/.zdenotepartialderivativesandisapointonthelinesegmentfrom.Usingthisresult,takingconditionalexpectation,andrecallingthatthevectorisindependentofE.W0and0forwehave.B.B /E.W.!/.!/ .v.!//.!/foreach,andfollowsthat.!/C.NNowas convergestoauniformdistributionon[whichhassecondmoment13,soifwelet.B.B 1.B.!/0as.ApplyingLemma3.5withiscertainlyLipschitzcontinuous)weseethatt/Gisamartingalewith Usingthemartingalewith(3.5)wecanwrite./ ./././ 2NCN�11 t;T 1.B./ t;T.!/ 336R.Durrett,E.A.PerkinsTohandle./weobservethatand(b)inLemma3.4imply./ r.;t/X.For./,wenotethatLemma2.2(whichonlyrequirestheboundednessof)implies./ N1 .1=/drTheresultisnowaneasyconsequenceoftheaboveestimatesandLemma Weturnourattentionnowtothefourthandnalterm:./ .t/h.B.B.t/t;T/=.Lemma2.4.Forall0lim./Proof.Lemma3.5impliesthat./isamartingale.ToapplyLemma3.5here,rstconditionthesummandwithrespectto.W.Usingmaximalinequality,(3.1),andthetrivialcomparison,wehave./CE..En;1./.t/.B.BSinceanyisLipschitzcontinuousand.B.B Using(b)ofLemma3.4,and.t/.t/,itfollowsthatthersttermin(3.6)isboundedby Super-Brownianmotionintwoormoredimensions337.t/Cr.;t/X0asThesecondtermin(3.6)ismorecomplicatedbecausewehavetoshowitissmallbyshowingthattheindicatorfunctionissmall,i.e.,itis0mostofthetime.TheproofofLemma2.4willbecompletedonceweestablishLemma2.5.Section4isdevotedtothattask.4.UpperboundsforthecollisiontermThissectionisdevotedtotheproofofLemma2.5.Tobeginwenotethatwhenacollisionoccurs,someparticlegavebirthatthetimeofitsdeath,,ontoasiteoccupiedbyatleastoneotherparticlewhomusthavebeenbornearlierandisstillalive.Insymbols,theexpectedvalueinLemma2.5canbeboundedaboveby; t;TDenethe-eldofalleventsinthefamilylineofstrictlybefore,plusthevalueof.t.tConditioningon; ,wecanrewrite(4.1)as; t;T .N//�N�1=2;N�1=2]d\ZN�f0gisthesetofneighborsof0, .N/isthenumberofneighbors.Herenotethattheinequalityimpliesthatisnotastrictdescendantofandsoonthis; -measurablesetwehaveP.W2
j; P.WThereadershouldnotethatimpliesthatinparticularthat,i.e.,arealiveinthebranchingrandomwalk. 338R.Durrett,E.A.PerkinsOurnalbitofnotationbeforegettingdowntotheworkofdoingtheestimatesistolet; .r/betheindicatoroftheeventthatarealiveattimeinthebranchingrandomwalkandtheyareneighbors.Withthisnotation,wecanuseLemma3.2with .r/; .r/(andwith(a)ofLemma3.4tojustifyintegrability)torewrite(4.2)as; .N//; .r/ .N/; ; .r/Toestimateestimate .r/]weneedtoconsiderthetimeandlocationofthemostrecentcommonancestorof.ThesimplestsituationiswhenLemma4.1.Thereisaconstantsothatforall; ; .r//C4.NC/rBeforetacklingtheproofofthisresultweneedsomepreliminaries.betherandomwalkthatwithprobability1/2staysput,andwithprobability1/2takesajumpuniformlydistributedon.Wemultiplyheresothatasconvergesto,thatwithprobability1/2staysput,andwithprobability1/2takesajumpuniformlydistributedon[.ThelocalcentrallimittheoremforimpliesthatP.VV�1;1]d/Cm�d=2asm!1Thenextresultwhichis(4)inSection2ofBramson,Durrett,andSwin-dle(1989),givesanupperboundthatisuniformin.ItcomesfromaconcentrationfunctioninequalityofKesten(1969).Lemma4.2.ThereisaconstantindependentofsothatifP.VV�1;1]d/C.1Cm/�d=2ToconvertthediscretetimeestimateinLemma4.2tocontinuoustime,wewillusethefollowingeasilyprovedfactaboutthePoissondistribution. Super-Brownianmotionintwoormoredimensions339Lemma4.3..Thereisaconstantsothatif ProofofLemma4.3.Thetrivialinequality1andstandardlargedeviationsresultsforthePoissondistribution(seee.g.,Ex.1.4onpage82ofDurrett(1995a))implythat m!.1Cm/�pXmD0e�m forsome0.Thedesiredresultfollowssincewehave�m= m!.1Cm/�p1C �m= m!1C 2�p ProofofLemma4.1.4.1.; .r/P.T1;T�1;1]d(4.4)Breakingthingsdownaccordingtothevaluesof,wecanwritethelastsumas /r/r/ P.NN�1;1]d/(4.5)Therstfactorgivestheprobabilityofnodeathalongeachline.Thesecondandthirdthenumberofchoicesfor.ThefourthgivestheprobabilitythatbotharealiveattimeChangingvariablestoandrecalling,wemayconvert(4.5)into /r/r/ n! 340R.Durrett,E.A.PerkinsP.NN�1;1]d/(4.6)DoingsomearithmeticandthenusingtheupperboundinLemma4.2weseethisisnomorethan/r/r/ /r/byLemma4.3.Thisboundholdsforeachpairofvalues.Multiply-ingbythesquareofthenumberofinitialparticles,,givesthedesiredconclusion. Tostatetheboundforthemoredifcultcaseinwhich,weneedtodeneI.u/Thiswilloftenbecomparedwith.N/ .N/=N,sowenotenowthatthedenitionof .N/andalittlecalculusshowthatthereareconstantssothat.N/I.N/.N/1(4.7)denotesthemostrecentcommonancestorof,i.e.,theuniqueancestorofwhichmaximizes,andifLemma4.4.Thereisaconstantsothat; ; .r//r/BeforewegetinvolvedinthedetailsoftheproofofLemma4.2,wewilldotheProofofLemma2.5.UsingLemmas4.1and4.4with(4.3)wemayboundtheexpectaioninLemma2.5byC .N/ /r/C .N//r/dr Super-Brownianmotionintwoormoredimensions341Changingvariables/rds=.intherstintegralandusingatrivialboundonthesecondwhichhasanincreasingintegrandweseetheaboveequals .N//tCte .N//t/TheintegralisboundedbyI../t/,sothedesiredresultfollowsfrom(4.7)(recallourconventionaboutconstants ProofofLemma4.4.Notethatisnot,or.Letsuchthat.Let1besuchthat.Onwehave/.r/.r .`�1/!
NC C..`Therstfactorontheright-handsidereectsthefactthattheremustbeexactly1moregenerationsinthelineattime.Thesecondthattherecanbenodeathsalongtheway.Thethird,2,givesthenumberofwiththepropertiesstatedinthesum.ThefourthcomesfromLemma4.2.Summingthelastresultover1gives/.r/.r.NC/.r UsingLemma4.3now,itfollowsthat(4.8)isatmost/.r 342R.Durrett,E.A.PerkinsSummingover0wehavethatonH.;r/H.;r//C2.NC/.r.Since�rThaveH.;r//C2.NC/.TH./Summingoverweseethat; ; .r/H./Toestimate(4.10)wewillbreakthesumdownaccordingtothevalueof.(Notethatbytheremarksatthebeginningoftheproofwemusthave1.)Letbeindependentmean1exponentials,andletC


C.Toexplainourchoiceofnotation,observethathasagammadistribution.Usingournewsymbols,wecanboundtheright-handsideof(4.10)(throughanowfamiliarargument)by /r//C0m�0kC1]�d=2!(4.12)Tochecktheindexingofthe'shere,notethatforisthetimeofthethdeathalongthelineofdescentofEvaluating(4.12)andthenthesumin(4.11)isasimple(thoughsome-whattedious)exerciseabouttherateonePoissonprocess.Tomaketheresultavailableforlateruse,werecall=.andthefunctiondenedpriortoLemma4.4,andstate Super-Brownianmotionintwoormoredimensions343Lemma4.5.ForallI.u//rinLemma4.5,andusing(4.10)–(4.12)givesLemma4.4.ThuswecancompleteitsproofbydoingtheProofofLemma4.5.Webeginbybitingoffasmallpartoftheproblem.Thereasonfordoingthiswillbecomeclearwhenwetacklethemainpiece.andtheterminthesumis11.LetP.0Summingthiscontributionofthe1termfor1wehaveP.0 expTobegintotacklethemainpiecewenotethat .m�2�yk k!D.xC�2 Summingover2nowwehave .m�2�yk y/.Usingthefactthathaveindependentgammadistributions,onecanwrite .m�k�2/!
e�yyk dxdySummingthelastestimateandusing(4.15)givesdyey/. 344R.Durrett,E.A.PerkinsTheinsideintegralisdyey/.,sotherighthandsideof(4.17)isboundedbyAdding(4.18)to(4.13)givesthedesiredresult. Latertoestimatesecondmomentsofthecollisionterm,wewillneedanestimateforforC0m�0kC1]�d=2!29=;(4.19)Themethodsaresimilartotheproofjustcompleted,sowewillgivetheproofhere.Lemma4.6.ThereisasuchthatforallI.u/Proof.Wemayassume0(sothat0)becausetheresultforclearlyfollowsfromtheresultfor0.ReversingtheorderoftherstincrementswecanbewriteWritingthesquareasadoublesumandcountingthediagonaltwice,theaboveisatmostkmMultiplyingby,summingoveranddoingsomerearrangementgives Super-Brownianmotionintwoormoredimensions345Toboundthiswebeginwiththeinnersum.Onwecanchangevariablestogetisindependentof,theaboveequals wherewehaveusedthehypothesis0inthelastinequality.Usingthisin(4.20)andisolatingthe0termswehaveu/.u/.u/.Since11,therstsumontherightissmallerthanthesecond.Toboundthesecondsumin(4.21)wenotethatu/. dx;againusing0inthelastinequality.Usingthefactthathaveindependentgammadistributions,weseethatthedoublesumin(4.21)is 346R.Durrett,E.A.Perkins .j�1/!yk�j�1 Usinganaloguesof(4.14)and(4.15)nowwiththefactthat,weseethatthedoublesumin(4.23)issmallerthan.Replacingandenlargingthedomainofintegration,(4.23)isboundedbydy.I.u/Combining(4.21)–(4.24)givesLemma4.6. 5.CollisionsbetweendistantrelativescanbeignoredInthissectionwewillrenetheboundsinLemmas4.1and4.4provingtheclaiminthesection'sname.Imitating(4.2),wecanboundthecollisiontermforunrelatedindividualsby.t/ N .N/; t;TIfwestartwithalltheparticlesinoneneighborhoodthenthistermwillnotbesmall.Thustohave.t/0,wemustassumethattheparticlesaresufcientlyspreadoutintheinitialdistribution.Thenextresultgivesasimplesufcientcondition.Lemma5.1.convergestowhereisanatomlessmeasurethenforanyE.J.t//Proof.Asweconverted(4.2)into(4.3),wecanboundE.J.t// .N/; ; .r/ Super-Brownianmotionintwoormoredimensions347TheboundonthisthatresultsfromLemma4.1is .N//C4.NC/rThisisalmostgoodenoughbyitself,butclearlyweneedtogetabetterestimateforsmall(i.e.,=N/)valuesofforwhichwewillusetheatomlessassumptionbelow.Repeatingthecomputationsin(4.4)–(4.6)weseethatifif; .r/ /r/r/ P.NN�1;1]d/(5.3)UsingLemma4.2toboundthelastprobability,thenintegratingover[0andsummingover,weseethatthecontributionto.t/isatmost .N//r/r/ n!
;N0.1/g2 .N/Thelastintegralisboundedby.Soconsideringthetwopossibilities:inwhichcase .N/,and2inwhichcase .N/weseethat0asThecontributionto.t/canbeboundedby .N//r/r/ �1;1]d
Changingvariables/rds=intheintegral,andthennoticingthegammadensity!hastotalmass1,weseetheaboveisnomorethan .N/ 2NCNXnD0Xi6DjP�N1=2.xj�xi/CVNn2[�1;1]d
(5.5) 348R.Durrett,E.A.PerkinsThisquantityiseasytoestimatein2.Usingthefactthat .N/andthenusingLemma4.2,wehavethat(5.5)islessthanorequalto.x;y/whereinthesecondtermwehaveusedthefactthatiftakesatleast1stepsoftogettoto�1;1]d.Sincef.x;y/isaclosedset,thelimsupofthersttermasisboundedbyC.X.x;y/Sincewehavesupposedthathasnoatoms,theaboveexpressionissmallis.Nowin2thesecondtermin(5.6)tendsto0forany0,sincethesumconverges.(5.5)offersmoreresistanceintheborderlinecase2.Using .N/.N/,andthenLemma4.2,butdecomposingthingsintothreepiecesnow,weseethat(5.5)isatmost .x;y/ N=. N=.AgainthelimsupofthersttermasisboundedbyC.X.x;y/whichissmallifis.Boundingthesumofbytheintegralofweseethatthethirdtermisnomorethan 3loglogTohandlethesecondtermin(5.7)wewilluseastandardestimatefor“small”largedeviationsofrandomwalks. Super-Brownianmotionintwoormoredimensions349Lemma5.2.Thereareconstants;c;Csothatif=nexp=n/Proof.C


Cwherethearei.i.d.realrandomvariableswithmean0and1a.s.,and0thenP.S�z/exp.ATaylorexpansionshowsthatexp.2.Usingthisfactandtheinequalitywithz=nleadstoP.S�z/exp �e0=2]Applythistoeachcoordinateofandtheirnegativestoobtainthedesired Turnnowtothesecondtermin(5.7).LetN/nandusethefactthat1inthesecondsumin(5.7)toseethatN/=nN/=.Nforlarge,sotheconditionsofLemma5.2hold.PlugginginthechosenvalueofN/nexpToconvertthisintotheresultweneedfor(5.7)notethatN=.N/n soitfollowsthatifif�1;1]d
CN�clogNThisismorethanenoughtosendthesecondtermin(5.7)to0.Thiscom-pletestheestimationof(5.5)inthecase2andwehaveestablishedLemma5.1. 350R.Durrett,E.A.PerkinsWeturnnowtothemoredifculttaskofestimatingtheprobabilityofcollisionforlineswith.IfwexanamountoftimethenwecandenethecollisionsofrelatedparticlesmoredistantlyrelatedthanJ.t;/ N .N/; t;Titisimpossibletosatisfyalltheconditionsinsidethesum,soJ.t;/Lemma5.3.ThereisaconstantdependingoncordingtoourusualconventionsothatforallE.J.t;// .N//t/dy.Proof.Asweconverted(4.2)into(4.3),wecanboundJ.t;/ .N/; ; .r/Now,andimplythatisnot,or.Let0besuchthatByconditioningonandusingLemma4.2wecanbound(5.9)by .N/;TTheconditionalexpectationisjusttheexpectednumberofchildrenattimeoftheparticle,andsobyLemma2.9itis.UsingthisandthenevaluatingP.B,webound(5.10)by .N/ P.T;T/dr Super-Brownianmotionintwoormoredimensions351,usingourstandardGammarandomvariables,and=.,(5.11)canbewrittenas .N/P.0/.r/;0/r0/dr1thelastprobabilityisjustP.0/.r//r0/.r /r1wehavetointegrateoutthevalueofandtheresultis/.rdxe /rdye /rInterchangingtheorderofsummation,setting2,whichruns1(for1)to,andusingtheaboveexpressions,wecanwritethedoublesumin(5.12)as/r/.r /r Doingthesumover,estimatingthesumoverusingLemma4.3,andthenabsorbingtheextraintotheweboundtheaboveby/r/.rdxe/rdyeAlittlecalculus(lefttothereader)showsthatLemma5.4.Thereisaconstantsothatif 352R.Durrett,E.A.PerkinsUsingthiswith/r,and,weseethat(5.14)isnomorethan/r/.r/r/rChangingvariables/r,wemaybound(5.13)by/r/Insertingthelastresultinto(5.12),wehaveanupperboundEJ.t;/ .N//t/whicheasilyconvertsintotheboundgiveninLemma5.3. 6.ConvergenceofthecollisiontermThegoalofthissectionistoanalyzethelimitingbehaviorofthecollision./ .t/.B.t/t;is1iftheparticlewasoncealiveinbutdiedbeforetime.MorespecicallywewillcommencetheproofofLemma2.6whichwenowrestateasTheorem6.1.Recallfrom(2.12)thatourtestfunctionsbelongtoTheorem6.1.Forandany././dr0(6.1)Wereallydomeanintheaboveandnot.OfcourseProposition1andtheorderingofthe's(see(2.8))showthedifferenceisunimportant.ToproveTheorem6.1,wewillslowlychange./intotheintegral.WerstoutlinethemainstepsinasequenceofLemmasandthenwillprovidetheproofsinthisandthenextfoursections.Intherststep,wetidyuptheexpressionalittlereplacing.B.B.Wealsomakea Super-Brownianmotionintwoormoredimensions353moresignicantchangebyreplacingthecollisioneventsthemselvesbytheirconditionalprobabilitiesgiventheinformationavailablejustbeforethedisplacementoccurred.Recallthatthat�N�1=2;N�1=2]d\ZN�f0gdenotestheneighborsof0inourlattice .N/isthenumberofneighbors.Let./Djfbethenumberofneighborsofoccupiedinattime.Writeisanancestorofanduse ifitisastrictancestor.ForfuturereferencenotethattheconditionsWedeneourrstmodicationofthecollisiontermby./ .t/.B./ .N/Lemma6.1.Foranyandany././Thecollisionterm./countsthenumberofoccupiedsites,./foreachparticlewhodiedbeforetime.Lemmas5.1and5.3implythatmostofthethecollisiontermcomesfromcloserelatives.Tosayhowclosetheserelativesare,welookattheconclusionsofLemma5.3,andintroduceasequenceofcutoffsdenedbytherequirementsthat:(i)in(ii)in.(6.3)Ournextgoalistoshowthatcollisionsinvolvingtwoindividualsmoredistantlyrelatedthanintimecanbeignored.Tosaythisinsymbols,we; Wethencandenetheinterferencetermforcloserelativesby./ .t/.B .N/; wherewerecallthatisthemostrecentcommonancestorofD�1 354R.Durrett,E.A.PerkinsLemma6.2.Forany././MostoftheworkforthishasalreadybeendoneinSection5.However,noticethat./; sincetheleft-handsidecountsmultiplyoccupiedsitesonlyonce.Ournextstepistoreplacetherequirementsbythecondition1;anddene./ .N/ .Bt;Lemma6.3.Forany././Toprepareforthenextstep,wenotethat.Thus,when1or2Atthispoint,wearenallyreadytoconvert./intoanintegral.Let.r/ .N/ Super-Brownianmotionintwoormoredimensions355.N/ .N/=NisintroducedtomakethisO(1).Notefromtheabovethatfor1or2./ t;.B./Thismotivatesthedenitionof./ .B.r/Notethathasbeenreplacedbythepositionofitsfamilylineattime.Moreimportantlyhasturnedinto.r/,andpassingfromthePoissonprocesstoitscompensatorviaLemma3.2hasremovedthefactorof1Lemma6.4.Forandany././dr.s/betheparticlesaliveattime.Thecontributionstothesumin./fromthevariousparticlesin,areindependent,soitisnaturaltolet;Tbethesetofdescendantsthatarealiveattimeinthebranchingrandomwalk,anduseournewnotationtowrite./ .B.r/.r/.r/.Comparingwith./ .B .Bsuggeststhatwedene(here1labelstherstindividualingeneration0)andconsider 356R.Durrett,E.A.Perkins././ .B.r/Bycomputingthevarianceofthelastdifference,wewillconcludethatLemma6.5.Forany././drRecallingnowthat./ rT.Bweseethatitremainstoremovethesuperscript'sfromandcom-pletetheproofofthelimittheoremforthecollisionterm.Thisisatwo-stepLemma6.6.Lemma6.7.Forany././drTheorem6.1,andhenceTheorem1,isanimmediateconsequenceofLemmas6.1–6.7(technicallyonealsoneedsthetrivialboundE.X//fromLemma2.9).TheproofsofLemmas6.1–6.7willkeepusoccupieduntiltheendofSection10.Therestofthissectionisdevotedtoproofsofthersttwooftheselemmas.ProofofLemma6.1.bedenedasbutwith.Binplace.B.Let.x.x//�N�1=2;N�1=2]d AnyisLipschitzcontinuous,so0as.Ifweletbetheindicatoroftheeventthatthesupportofishitbythebirthattimewecanwrite Super-Brownianmotionintwoormoredimensions357././ .t/byLemma2.5.Toestimatethedifferencebetweennow,wenotethat,asintheproofofLemma2.3,./ .N/soLemma3.5impliesthatthedifference./ 1C ./ .t/.B./ .N/isamartingale.Toestimatetherighthandsidenotethat(i)thesquaresofthejumpsofaresmallerthan./= .N/,(ii).t/.t/,and(iii)since,var.Sowehavebythemaximalinequality,(3.1),(3.1),M]tCN�2E0@Xa0.t/ byLemma2.5.Thedesiredconclusionnowfollowsfrom(6.8)–(6.10)andtheinequality =N E.K0as(thelastbyLemma2.5again) ProofofLemma6.2../Djf 358R.Durrett,E.A.Perkinsbethenumberofneighborsofoccupiedinbycloserelativesofattime.Tobridgethegapbetween./ .t/.B./ .N/Lemmas5.1and5.3implythatforany././Toestimatethecontributiontothecollisiontermfrombirthsontomultiplyoccupiedsites,werecallthat.t/t;Bistheeventthatwasoncealiveinthebranchingrandomwalkbutdiedbeforetimeandlet.t/ .t/ .N/; ;Themotivationforthisdenitionisthat././.t/Tocheckthisobservethatifthereare2closerelativesofatonesitethentheleft-handsidecontributesatmost/.B / .N/buttherightcontributesatleastk.k N .N/.Thelatterinequalitycomesfromthefactthatthedenitionof.t/inadditionweakenstherequirementofcloserelativesfromthatofhavingarecentcommonancestorandofbeingaliveintojustbeingrelatedandaliveinToestimate.t/wewillhavetosumandintegrateoverallthepossi-bilities.Therststepistousethesymmetryof.; /andsupposewithoutlossofgeneralitythat,i.e.,thatthelinedidnotsplitofffromthelineafterthelinedid.Justtokeepontopofthingsthereadershouldnotethattherearetwosomewhatdifferentsub-casesofthissituation:(a) ,or(b).Inwords,(b)saysthatthemostrecentcommonancestorofoccursaftertheircommonlineofdescentjoinsTotackle(6.11)webeginwiththeinsidesumandbreakthingsdownaccordingtothevalueofsothat,notingthattheindicatorfunctionsinvolvingruleoutandso; bethe-eldgeneratedbyallthebranchingevents,andtherandomwalkeventsforthelinesonly,butomittingthevalueofthe(whichmighthavemovedtheortheline).Thenwehave Super-Brownianmotionintwoormoredimensions359; .N/ToprovethisweuseLemma4.2toconcludethat NfB�B.Tjk/ �1;1]djH; andobservethattheprobabilitythemissingwillhavetheexactvalueneededtomakeisatmost1= .N/Inordertouse(6.12),wewanttotaketheconditionalexpectationof(6.11)withrespectto; .Unfortunately,nbr; isnotmeasurablewithrespectto; .Thisproblemiseasytox.Let)betheposition)withthevalueofsubtractedifitappearsinthesum.Clearly,; �3;3]dg(6.13)andtherighthandsideis; Modifying(6.11)using(6.13)thentakingtheexpectationofthecon-ditionalexpectationofthesummandsin(6.11)withrespectto; ,wehave.t/ .t/ .N/ �3;3]dgX:6D ; .N/Toevaluatetheinsidesumwebreakthingsdownaccordingtothevalueofsothat,andthevalueof1.Sincetherebirthsinthelineafteritsplitsfrom,thereare2choicesforandeachofthemisalivewithprobability/=..Recallingtheremustalsobeexactlyarrivalsintherelevantrateprocess,wearriveat ; .N/; 360R.Durrett,E.A.Perkins 2NC`
C .N/exp[ `!C .N/exp[.TbyLemma4.3.Pluggingthisinto(6.14)wemaybound.t/ N .N/ �3;3]dg
j j�1XkD0.1C.T�T jk/.2NC2//UsingLemma3.2tochangefromthePoissonjumpstotheircompensator(thisintroducesafactorof2),andputtingbackinthevariableleftoutof,weseetheaboveisatmost .N/ N.BB�5;5]dg
j j�1XkD0.1C.r�T jk/.2NC2//Ifwesumoverrstandconditionon,andthenuse(4.9),weseethattheaboveisatmost .N///Recallingtherearechoicesfor,thenbreakingthingsdownaccordingtothevalueof,andusingthereasoningweappliedto(6.14),weboundtheaboveby Super-Brownianmotionintwoormoredimensions361 .N/ /r0/rUsingLemma4.6and(4.7)itfollowsthattheaboveisnomorethan .N//r/ .N//t/Recallingthat/t/= .N/C=Nitfollowsthat.t/ andtheproofofLemma6.2iscomplete. 7.ProofsofLemmas6.4and6.5ForthemomentwewillskipLemma6.3,closingtheloopwiththeproofofthatresultandthecloselyrelatedLemma6.7inSection10.ProofofLemma6.4.Recallthat.N/ .N/=N.r/ .N/./ .B.r/andthecollisiontermsofinterest,.Ourrstobservationisthat./ /Nt;.Biscloselyrelatedto 362R.Durrett,E.A.Perkins./ .B.r/Inparticular,Lemma3.2(seealsoLemma3.4(a))impliesthat././drisamartingale(7.1)ToboundthesizeofthismartingalewenotethattheaboveLemmasalso .B.r//drFromtheformulafor,andverycrudebounds,wegetInthesecondequation,wehaveweakenedthesurvivalconditionstobeingaliveinthebranchingrandomwalk,sotheaboveisatmostT=NThelastinequalityisimmediatefromLemma2.9andHolder'sinequality.maximalinequality,(3.2),nowshowsthat/T=NItremainstoestimatethedifferencebetweenthetwointegrals.Tothisendwenotethat././ .B.B.r/Plugginginthedenitionof.r/thentakingtheconditionalexpectationwithrespectto(recallitsdenitionfromthebeginningofsection4),theaboveisnomorethan Super-Brownianmotionintwoormoredimensions363 .N/.B.BNotethatwehaveeliminatedandreplaced1by1.By(4.9),theaboveisboundedby .N/.B.B/.rOurnextstepistoconditionon.t.ttheinformationaboutthebranchingeventsinthefamilylineofplusthedeathtimeof.Breakingthingsdownaccordingtothevalueofandthenaccordingtothevaluesofthetimestheaboveisatmost .N/ .B.B/r0andthearethegammarandomvariablesintroducedintheproofofLemma4.5.UsingtheCauchy-SchwarzinequalityweconcludeE.6Wehavesupposedthat,andhenceisLipschitzcontinuous,soUsing(7.5)and(7.6)weseethat(7.4)issmallerthan 364R.Durrett,E.A.Perkins .N/ /r0UsingLemma4.6now,and(4.7)weseethattheaboveisboundedby .N/I../r/ThiscompletestheproofofLemma6.4. ProofofLemma6.5.Webeginbyrecallingformula(6.7):././ .B.r/Plugginginthedenitionof.r/fromSection6weget././ .B.r/Ifweconditionon,thenasimpleargumentusingtheMarkovprop-ertyandourbasicindependenceassumptionsshowsthattheindividualsum-mandsin(7.8)areindependent.Thedenitionofnowimpliesthattheir(conditional)meansare0.HereweuseanobvioustranslationinvariancetoseethatonP..Z.r/;2
jP..ZToshowthatthedifferencein(7.8)issmallwewillcomputethevarianceofthisrandomsum.Forthisitisclearfromtheaboveindependenceandequivalenceinlawthatthefollowingtwolemmaswillbeneeded.Lemma7.1.ThereisasothatC.N/Lemma7.2.ThereisasothatforanysN/ Super-Brownianmotionintwoormoredimensions365ThesecondresultisastandardfactaboutcriticalbranchingprocessesandagainisacorollaryofLemma2.2inBramson,Durrett,andSwindle(1989).BeforeenteringintothesomewhatlengthydetailsoftheproofofLemma7.1,letuscheckthatit,andLemma6.6,willbeenoughtonishtheproofofLemma6.5.Conditioningthesumin(7.8)on,wehavefromtheaboveobservationsthatif././ CombiningLemmas7.1,7.2and6.6(thelattertoshowthatboundedas)andusingthefactthatwehavechosenwehaveCNsotheexpressionis(7.10)isboundedbybyLemma2.9.Fromthisitfollowsthat././././0asTohandletheintegralfrom0towenotethatif./.B.r/./.BUsingsometrivialinequalitiesandthenthedenitionof.r/,wehave 366R.Durrett,E.A.Perkins././//dr/dr .N/UsingLemmas2.9and6.6onthersttermandLemma4.4and(4.7)onthesecond,theaboveisboundedby0(7.12)Combining(7.11)and(7.12)weseethattheproofofLemma6.5willbecompletewhenwedothe(independent!)proofsofLemma6.6inSection8andLemma7.1inSection9.Thelatterresultconcernsthesecondmoment.s/,sowewillrstcomputethemean.s/,whichisneededtoproveLemma6.6.8.MeanoftheinterferencetermWeclaimthatfor.r/ .N/Toseethisnotethatforthecondition(appearinginthedenitionof.r/)holdsiff(whichappearsinthedenitionof.r/).Forthisequivalence,observeandsincearebothancestorsofthisforcesandso.Theconverseimpli-cationisobviousand(8.1)nowfollowsfromthedenitionsof.r/.r/.If,thenandso(8.1)with)impliesthat.N/EZ Super-Brownianmotionintwoormoredimensions367Notethatforaxedintheabovesumifthenthefollowingaremutuallyindependent;.t/;.t /;/;.Breakingthingsdownaccordingtothevalueof,usingtheaboveindependence,andconditioningon.tandthenshowsthat.N/EZk;䀀; 䀀; P.TP.TT�1;1]d�f0g
)(8.2)Startingatthebottomof(8.2),ifisuniformonanindependentrandomwalkthatwithprobability1/2staysputandwithprobability1/2takesastepuniformon,thenthen�1;1]d�f0g
DP�WNCVN`Cm2[�1;1]d�f0g
CombinethiswiththeusualPoissonprocessformulasfortheprobabilityarealiveattime,toequate(8.2)tok; /. /. 368R.Durrett,E.A.Perkins=..Changingvariablesfrom.`;m/.n;m/,givesk; 2NCnn! /. /.P.WW�1;1]d�f0g/)(8.3)Summing�nm
overfrom0togives2.Alittlearithmeticturnsthesumover./. /.P.WW�1;1]d�f0g/(8.4)Let.u/beaPoissonrandomvariablewithmeanthatisindependentof,andlet.u/.u/�1;1]d�f0g
Usingournewnotationwecanwrite(8.4)as././/:Pluggingthisinto(8.3)weseethat(8.3)equals./.UsingLemma3.2andLemma3.4(a)forintegrability,wemayconverttheaboveto./.r// Super-Brownianmotionintwoormoredimensions369Summingovergivesalltheindividualsaliveinthebranchingprocessat,sousingLemma2.9,wehaveshown.N/EZ./.r///drWecansimplifyourcalculationsbynotingthat(indicatestheratioap-proaches1as.N/EZ/./dr /.s/dswhereinthesecondstepwehavechangedvariables/.beuniformover[.s/beanindependentcontin-uoustimerandomwalkthatatrate1/2takesastepuniformon[ElementaryweakconvergenceargumentsshowthatLemma8.1.thenash.s/.s//�1;1]d
Wenowwishtointerchangethelimitasandtheintegraloverin(8.6).In2thisiseasytojustify.Lemma2.4inBramson,Durrett,andSwindle(1989)gives.t//�1;1]d
C.1Ct/�d=2(8.7)Sincetherighthandsideisindependentof,thesameboundholdswhenisputinplaceofontheleft.Thisgivesusthedominationweneedtoletin(8.6)andconcludethatif0then.N/EZ .s//�1;1]d
dsIn�d2, .N/.N/.Fortheright-handside,wenotethatthecontinuoustimerandomwalk.s/staysineachstateforanexponentialamountoftimewithmean1/2beforemoving,so,recallingthedenitionofinSection1priortoTheorem1,wecanrewritethelastformulaasP.U 370R.Durrett,E.A.PerkinsThingsarealittlemoredelicatein2sincethelimitingintegralisdivergent.Fortunately,muchoftheworkhasbeendoneinLemma4.6ofBramson,Durrett,andSwindle(1989).HeredenotestheLebesguemeasureofLemma8.2.=.s=thenforanyBorel.s=P.V.s/.x/wheren.x/expisthenormaldensitywithvari-2comesfromthefactthatinBramson,Durrett,andSwindle(1989),whattheycalltakesjumpsatrate1whileour.s/takesjumpsatrate2,soweneedtosetintheirLemma4.6.Lemma8.3./=h.sProof.Usetheclassicallocalcentrallimittheoremtoseethat(Problem1inSection10.4ofBreiman(1968)andasimplecalculationwillsufce.)Notethatbyconditioningonon;1]dandusingLemma8.2and(8.7)tointegrateouttheconditioning,wegetTheresultfollows. CombiningthiswithLemma8.1,onecanconcludeeasilythatLemma8.4..Ifislargethen.s/=h.s//�;1C]forallProof.0andsupposethatthereisasequenceofexceptionstheinequality.Thereiseitherasubsequenceconvergingtoanitelimitorthesequenceconvergesto.IntherstcasewecontradictLemma8.1,inthesecondwecontradictLemma8.3. FromLemma8.4itisimmediatethat/.s/ds/h.s/ds;thatis,theratioapproaches1as.Tocomputetheright-handside,weuse(8.9)(with2)toget Super-Brownianmotionintwoormoredimensions371h.s/.s= .N/4log,so(8.6),(8.10),theaboveasymptoticestimate,andthetrivialboundh.s/1implythatas 4log /=loglog 3 Formulas(8.11)and(8.8)givetheasymptoticbehaviorof2and2,respectively.TocompletetheproofofLemma6.6now,wenotethatbyLemma2.9andthefactthat1(8.12).Thereforefrom(8.8),(8.11)and(8.12)weseethatas andLemma6.6isproved.9.SecondmomentoftheinterferencetermInthissectionwewillproveLemma7.1.Wewillusetoindicatethat.Usingthisin(8.1)wecanwrite.N/whereeach1.Tosuppressnuisancetermslateritisusefultonote:(i)Sinceallthearealiveattimewecannothave(ii)Since0wemusthave(iii)From(i),(ii),and1itfollowsthat1forallThereareseveralcasesintheestimationof(9.1)dependingontherela-tiverelationshipofthethe.Tosorttheseoutweneedsomenotation.For 372R.Durrett,E.A.Perkinsanitesetofpossibleindividualslet.33; withmax;D�1.Inwords,.3isthenumberofthelastgenerationinwhichhadanancestorincommonwithsomeindividualin.For3,letbethecontributiontothesumin(9.1)from.Thecontributionswehavedenedoverlapbutthetermsinthesumarenonnegativeso.N/whereinthesecondstepwehaveusedsymmetrytoconcludeToestimatetheright-handsideof(9.2)wehavetodoeachofthefoursumsin(9.1).Tostructuretheproofwewilldividethissectionintothecorrespondingsubsections.a.Sumover.For3,weletandnotethatconditioningonE.RBreakingthingsdownaccordingtothevalueofandusingLemma4.2,wehavethatE.R1isatmost Herethe1correspondstotheterm(whichcontributesifandwehaveused(i)tojustify.Ifu. //.thenontheaboveisnomorethan 2NCj4j�k�1 Super-Brownianmotionintwoormoredimensions373 1andtakingintoaccountthenumberofpossiblewemayboundtheaboveon 2NC`.1C`/�
e�u.jjjj .k//H.CH.byLemma4.3andadenitiongivenafter(4.9).Wecancombine(9.4)and(9.5)togetCH.ItistemptingtouseH.H.H.1tosimplifytherighthandsidetoCH.butthatupperbounddoesnotworkwellinb.Sumon.Using(9.2),(9.3),and(9.6),thenconditioningon.N//CCH.CCH.Usingsymmetrywecanreplace1intherstsumby;.andputanotherfactorof2intothe.Todealwiththeright-handsideof(9.7)welet 374R.Durrett,E.A.Perkins;.H.;.2.Separatingoutthepossibilityofrst,andplugginginthedenitionofH.,weseethatE.RisboundedbyH... Cu.3jj/]�d=2H12CE1fT3NT3gH12
k�1XjD0[1Cu.2jj/]�d=2)(9.8)wherewehaveusedthefactthatforjk1,changingvariables,andusingourstandardgammarandomvariablesdenedintheproofofLemma4.5,therstterminthesetbracesin(9.8)isatmost/.UsingtheusualPoissonreasoningwiththetrivialbound,andrecallingthedenitionofH.,weseethatthesecondterminthesetbracesin(9.8)isboundedby H.Using(9.9)and(9.10)in(9.8),andcountingthenumberof'sforeachwehaveE.RH. /. H. Super-Brownianmotionintwoormoredimensions375Recallingthenotationfrom(4.12)weseethattherstterminsetbracesin(9.11)isatmost2/..Herethefactor2isusedtohandlethe0termwhichdoesn'tappearinthesumdening.DoingthesumovernowandusingtheaboveandLemma4.5,withthetrivialbound0,weseethattherstterminsetbracesin(9.11)whensummedovercontributesatmostI../Thesecondterminsetbracesin(9.11)whensummedcontributesatmost H.expk//H.H.=./.Using(9.12)and(9.13)in(9.11),thenrecalling1by(iii),wehaveE.RI../H.Ournextstepistoconsider.WithH.beingreplacedby1in,theanalysisismucheasier.Imitating(9.8)wewriteforE.Q P.T1,weseetheaboveisatmost 2NC`e�u.jjjj u.k/=.,and(iii)tellsusthat(9.14)and(9.15)handletherstterminsquarebracketsin(9.7).Totakecareofthesecondtermthere,wenotethat 376R.Durrett,E.A.Perkinssousing(9.15),H.H.H.1wehaveeCCH.H.Using(9.14)and(9.16),weseethat(9.7)isboundedbyI../H.H.Havingsummedoverandthen,ourthirdstepistoc.Sumover.Ifweconditiononin(9.17)thenwewillbeleftwithtwotypesofterms:R.Usingournewnotation,isnomorethanH.E.R.I../H.E.Q.(4.9)impliesthatE.Q.CH.TocopewiththeextrafactorofR.,wenotethat,adaptingtheproofof(4.8),onecaneasilyshowE.R. 2NC`e�u.2j2j `! Super-Brownianmotionintwoormoredimensions377Dividingthesuminto(a),where,and(b),andthenusingLemma4.3oneachpiece,weseethattheaboveislessthanorequaltok//k//ThesecondsumisatmostH..Fortherstweusek//whichholdsin2,andH.1togetE.R.H.Atlast,wearereadyforthefourthandnalstep.d.Sumon.Using(9.19)and(9.20)weseethatthemeanvalueof(9.18)isatmostH./I../H.Breakingthingsdownaccordingtothevalueofwemayboundtheaboveby(recallthenotationfrom(4.12)and(4.19)) /I../I..////andwritefortheabovesumwhenisrestricted.If,then,andsoLemmas4.5and4.6implyI.v/CNI.N/ 378R.Durrett,E.A.PerkinsUsingthetrivialbound0,weseethat I.v/`CI.v/.v/.v/!.Ifislarge,.v/=a.v/forall,so.v/.v/exponentiallyfastasbystandardlargedeviationsestimatesforthePoissondistribution.(Seee.g.,page82ofDurrett(1995a).)FromthelastresultitfollowsthatI.v/.v/Combining(9.24)with(9.22)andrecalling/,itfollowsthat(9.21),andhence.N/E.Z(recall(9.7)),isatmostI.N/CI.N/Nowuse(4.7)toobtaintheconclusionofLemma7.1. 10.ProofsofLemmas6.3and6.7FirstconsiderLemma6.3.Inthedenitionof.t/weimplicitlyusedthefactthatforsometoseethatThesamereasoningforshowsthatHenceinthedenitionofwecanreplacetheconditions.Takingdifferencesandreplacingwethereforehave././ / .N/;.B Super-Brownianmotionintwoormoredimensions379t;TUseLemma3.2(andLemma3.4(a)forintegrability)toboundthemeanvalueoftheaboveby .N/;r;whichisanicerformsinceplayexactlysymmetricroles.Subtractingandadding1r;andusingsymmetry,weseethatinordertodemonstrateLemma6.3,itisenoughtoestablishforall .N/;r�N;r] dr!0(10.1).Sincea.s.thisistrivialforTostarttoworkon1weintroduce.r/ .N/Herewedivideby.N/tomake.r/.Notethatandsotoestablish(10.1)itisenoughtoshowLemma10.1.Forany r�N;r];B6D1g�I.r/Here,the1isnotneededforLemma6.3,butisincludedfortheProofofLemma6.7.RecallingthedenitionsgiveninSection6,andusingthefactthatourtestfunctionsareLipschitzcontinuous,andiffa.s.,wehave 380R.Durrett,E.A.Perkins././ Thersttermtendsto0byLemma10.1.Forthesecondterm,conditiononandusetheMarkovpropertyandLemma2.9with.x/toseethatif.t/isthecontinuoustimerandomwalkinLemma2.9thenthesecondtermisatmostdrE Combinetheselasttwoobservationswiththefactthat0tocompletetheproof. ItremainsthentodotheProofofLemma10.1.Nowifisaliveinthebranchingprocessattimebuthashasr�N;r]/,thenthereisasothat//;e.iInwordsthelastconditionsaysthatattimelineexperiencedthedispersaleventandcollidedwithaparticlealreadypresentin.Letdenotetheindexofoneoftheparticleswithwhichcollidesattime;iasshorthandfortheawkward.i,andreadingthesymbolas“therewasadisplacementinthefamilylineofatthedeathof,”wecanboundthequantityofinterestinLemma10.1by ;i.r/Therststepinboundingthisistolet; ,recallH.//.Tandgeneralizetheproofof(4.9)toshow Super-Brownianmotionintwoormoredimensions381Lemma10.2..r/; .N/H./H. /Proof.Recallthat.On;T,set;.r/ .N/r;B; Sinceeithermustsplitofffromlast(itcanbeatieifthebranchesoffbeforethelinesseparate)wehave.r/; ;.r/; ; .r/; wherethesecondtermcanonlycontributeif.Thusitsufcestoshowthatfor;.r/; .N/H./Breakthingsdownaccordingtothevalueof; ;/,isolatethecaserst,andthenobservethatwhen.Thisgives.N/;.r/; ; ; ; .t.tisthe-eldgeneratedbythebranchingeventsinthefamilylineof.Letk;r//.randnotethatforthetermtocontributeintheabovesumwemusthave.UsingLemma4.2nowandsetting1,wemayboundtheabovebyk;r/k;r/ `! 382R.Durrett,E.A.PerkinsUsingLemma4.3nowwiththetrivialinequalitiesk;r//.T(fromouroriginalhypothesison),andk;r//r,theaboveisboundedby/r/.TandthedesiredresultfollowsfromthedenitionofH./ Conditioning(10.2)on; ,usingLemma10.2,andthrowingawaytheevent,weseethat(10.2)isatmost .N/.H./H. //Doingtheintegralovernowandrecallingwemayboundtheaboveby t;B.N/.H./H. //Togetridofthe,weconditionontherandomvariablesgenerating; butwithout,andnotethatifisnotadescendantof(whichwemayassumeinlightofthecondition)thenthisinformationisindependentof.Wethusmayconcludethattheaboveisnomorethan t;BBC 0.N/.H./H. // Super-Brownianmotionintwoormoredimensions383; ; butwithouttheinformationaboutthevalueof.Sinceatmostoneofthe .N/valuesofcanmakeaperfecthit,andnonewillhitunlessthesourceofthebirth,,isaneighbor,wehave; .N/Tousethis,wecondition(10.3)on; toseethatitisboundedby t;B .N//C 0.N/.H./H. //Ournextstepistobreakthingsdownaccordingtothevalueofj;jD�1)whichmustbelessthan,andconditionon,where.tistheinformationaboutthebranchingtimesinthelineof.UsingLemma4.2,thelastdisplayisboundedby .N/ 2NCjjjj�1XiD01.TjiN/i�1XkD�1X ^jD j1.Tji/
.j NC H./H. / .N/Toboundtheright-handsideof(10.4),wewillhandletheH./= .N/and1H. /= .N/termsinthelastsquarebracketsseparately,andcalltheresultingsums(10.4a)and(10.4b).Fortherstwewillconditiononandbreakthingsdownaccordingtothevalueof1.We0if1.Then /.TBytheusualPoissonreasoning,thisequals 384R.Durrett,E.A.Perkins/.T/.T.2NC/.T /.TbyLemma4.3.Pluggingthisboundinweseethat(10.4a)isnomorethan .N/ H./ .N//.TBreakingthingsdownaccordingtothevalueof1,writingoutthedenitionofH./,andintroducingourstandardgammarandomvariables,weboundtheaboveby .N//t/ .N/Bounding(10.5)isaPoissonprocessexercise,whichwewillattendtolater,soweturnnowtotheotherpieceof(10.4).Tohandle(10.4b),webeginbyinterchangingtheorderofsummationstoget .N/H. / .N/ 2NCj jX^ jDj1.TNC Conditioningon,andintroducing/.T,and/.T1then0asabove),wehave Super-Brownianmotionintwoormoredimensions385 P.0/t;u/,andputtingthecaseintothesecondterm,wemayboundthepreviousdisplayby/t;u/ .j�1/!y`�j�1 dxdy/t;u Usingtheidentity .j�1/!y`�j�1 .`�j�1/!D.xC�2 wecanrewritethedoublesuminsquarebracketsin(10.7)as /t.Evaluatingthesinglesuminthesameway,andthrowingawaytherstrestrictiononweseethat(10.7)isatmost//dxdy/dxRecallthat/t,whereisthelifetimeof,andthatourchoicesin(6.3)implythat/,toboundtheaboveby//t 386R.Durrett,E.A.PerkinsUsingthelastinequality,weseethat(10.6)isboundedby .N/H. / .N/ //tAsbeforeonecanconditiononeverythingbutanduse/ttogetridofthatterm.Breakingthingsdownaccordingtothevalueof1,separatingoutthecontributionfrom0andnotingthatH. /0if0,using,andllinginthedenitionofH. /,weseetheaboveisboundedby N .N//t/ .N/Itremainstoshowthatthequantitiesin(10.5)and(10.9)approach0as.Webeginbyeliminatingthecontributionfromlarge.Standardlargedeviationsestimatesforthesumofexponentialmeanonerandomvariables(seeSection1.9ofDurrett(1995a))implythatLemma10.3.ischosenlargeenoughthenforall/tUsingthisresultwiththetrivialfactthatshowsthatthecontributionsto(10.5)and(10.9)from�`AN0as.Toestimatethecontributionsfrom,webeginwithasimpleestimateLemma10.4.thereisaconstantsothatifProof.ByasimpleapplicationofJensen'sinequalitywemayassumeisapositiveinteger.Clearly,E.0.IntegrationshowsthatE.0 .m�1/!e�x.m�1� .m�1/!Cp.1Cm/�p Super-Brownianmotionintwoormoredimensions387Let(10.9b)denotethepartof(10.9)thatcomesfrom.Recall=.andhence.UsingLemma10.4now,andthrowingawaytheindicatorof/t,weseethat(10.9b)isboundedby .N/.AN/ .N/Thequantityinsquarebracketsisboundedand.AN/.N/,sotheaboveisatmostwhichapproaches0as,andso(10.9)alsoapproaches0asLet(10.5b)denotethepartof(10.5)thatcomesfrom.Recallthat.N/ .N/=N.Discardingtheindicatorfunctionof/tasabove,wemaybound(10.5b)by .N//Toattackthiswewillusethefactthatifisanindicatorfunction(so)andarenonnegative,thentwoapplicationsoftheCauchy-SchwarzinequalityimplyE.XYZ/This,togetherwithLemma10.4,showsthat(10.10)isatmost .N/P.0/Thesumsoverareeachsmallerthan.AN/.N/,sotheaboveisboundedby P.0/Todealwiththisprobability,notethatastandardlargedeviationsresultforsumsofexponentiallydistributedrandomvariables(againseeSection1.9ofDurrett(1995a))implies 388R.Durrett,E.A.PerkinsLemma10.5.Thereareconstants ;CsothatifP.0FromthisitfollowsthatifP.0///Usingthisin(10.11),thendoingthesumover,whichgivesafactorof,weendupwithanupperboundofofNCe� .2NC/0asThisshowsthat(10.5)approaches0asandsocompletestheproofofLemma10.1andhencetheproofsofLemmas6.3and6.7.Thisnishesourtreatmentoftheinterferencetermandhencetheproofofourconvergencetheorem,Theorem1.11.Lowerboundonthecriticalvaluebandletdenotetherescaledcontactprocessstartingfromasin-gleparticleattheorigin.Fix0andletbethediscretetimebranchingrandomwalkinwhichindividualsingivebirthtoindependentcopiesandhencemultipleoccupancyofsitesisallowed.Weviewasaninteger-valuedmeasureon.Itiseasytocouplesothat.Herenotethatparticlesinthecontactprocessesunderlyingonlyhaveanoffspringsuppressediftheyjumpontoasiteoccupiedbyanoffspringofthesameparentinandthereisnosuchancestralre-strictioninthesuppressionofoffspring.If1,forsufcientlylarge,thenthesubcriticalGalton-Watsonbranchingprocessdiesoutforlargena:s:andsothesameholdstruefor.Recallisthecriticalvalueofforwhichthereispositiveprobabilityofsurvivalasthecontactprocessstartingwithasingleoccupiedsite.Thus,toprovethelowerboundhalfofourasymptoticsforthecriticalvalueinTheorem2,itsufcestoshowLemma11.1.Proof..i/;ii:i:d:copiesofstartingfromandlet.i/.Thendiffersfromstartingatinthat Super-Brownianmotionintwoormoredimensions389jumpsontoanoccupiedsitearesuppressedonlyifthetwocollidingparticlesdescendedfromthesameancestorat0andhencemultipleoccupanciesareallowed.Thisinfactsimpliestheproofofthemainconvergenceresult(Theorem1)asLemma5.1isnolongerneededsinceisnowincorporatedintothekillingterm.Asthisistheonlyplacethenon-atomicnatureofisusedwecandropthisrestriction,allowconvergesweaklyto,super-BrownianstartingatandwithdriftThis,combinedwiththefactthatE.YYX0;Nt.1/2jX0;N0D0]staysboundedas(seeLemma2.9),showsthat 12.UpperboundonthecriticalvalueThroughoutthissectionweassume�b.Toprovetheexistenceofanon-trivialstationarydistributionandhencederiveupperboundsonthecriticalvalue,wewillusearescalingargumenttocomparethelongrangecontactprocesswithorientedpercolation.Toestablishtheconnectionwebeginbyintroducingthelatticeonwhichpercolationtakesplace:.m;n/iseven,0and .LetLet�L;Lbethecubeofradius,andbetherstunitvector.Itwillbeconvenienttoassumeandsowewillonlyconsiderandfor2replacebyitsintegerpartinthedenitionofandthroughouttheconvergencetheorem.ThiswillensurethatandinparticularGivenarealizationofthecontactprocess,andasite.m;n/willsaythatis“occupied”attimeifthecontactprocesswhenrestricted,andtranslatedinspacetobeafunctionon,liesinasetof“happy”congurations.Inwords,thesetwillbechosensothat(i)ifisoccupiedattimethenwithhighprobability1and1willbeoccupiedattime1,and(ii)theeventsthatcause(i)tooccuraredeterminedbythebehaviorofthecontactprocessmodiedsothat 390R.Durrett,E.A.Perkinsparticleswhichlandoutsideof2KL;KL/arekilled,forsomexednaturalnumber.Thesetwillbedenedbelowbutfornowwenotethatthecongurationwhichis0onisnotinwhilethecongurationofall1'sonMoreformally,wewillcheckthecomparisonassumptionsonp.140ofSection4ofDurrett(1995b).Let/.x/.xdenotethetranslation(orshift)of.Foreach0andweintroduce.CA/ ;K:Foreachthereisanevent,measurablewithrespecttothecontactprocesswithkillingoutsideKL;KL//;T],andP.G,sothatonliesinandinHereweconsiderrescaled'swhicharethereforesubsetsof,orequivalently-valuedfunctionson,andidentifywiththemeasure./wichassignsmass1toeachsiteinLegalscholarsmayhavenotedthatpage140ofDurrettinsteadsays“measurablewithrespecttothegraphicalrepresentation,”whileinthispaperwehaveusedabranchingprocessconstruction.However,itiseasytoseethattheconstructionusedherehasthepropertythatifthespacetimeboxesaredisjointthenthesubprocessesthatresultfromthecontactprocessrestrictedtotheseboxesareconditionallyindependentgiventheirinitialconditions.ThisisenoughsothatwecanrepeattheproofofTheorem4.3giveninDurrett(1995b)inournewsetting,andconcludethatif.m;n/dominatesan2-dependentorientedpercolationprocess(see(4.1)ofDurrett(1995b)),,withinitialcongurationanddensityatleast1,i.e.,forallIf,foraxedvalueof,wecancheck.CA/ ;Kforall0then.x/1(whichisinandhenceassuresthatistheentireintegerlattice)andusingTheorem4.2inDurrett(1995b)givesliminfFromthisandthefactthatthecongurationwhichis0onisnotinfollowsthattheupperinvariantmeasuremustbenontrivial.Ifnot,then0andP..x/�0forsomeThustocompletetheproofoftheupperboundinTheorem2itsufcestocheckthecomparisonassumption.Intuitively,toverify.CA/ ;Kforthelongrangecontactprocess,wewillrstverify.CA/forthelimitingsuper-Brownianmotionwith Super-Brownianmotionintwoormoredimensions3910andthenuseourconvergencetheoremtoconcludethatif�b.CA/ ;Kholdsforthecontactprocessforlarge.Thesetofthatwewillchooseforthesuper-Brownianmotion,andfortherescaledcontactprocesses,arethosethathaveenoughmassandarenottooconcentrated.Specically,ifthereisasubcongurationwithcorrespondingmeasure.I.I/,andQ./,whereisaquadraticformdenedin(12.7)below,isanaturalnumberselectedinChoice3belowandisaconstantselectedinChoice4below.Clearlydoesnotcontainthecongurationofall0's.Moreoveraswewillbeabletochooseaslargeaswelikeandafterthechoiceof(seeChoice4below),itisclearthatwillcontaincongurationofall1's.Tocheckthecomparisonassumptionwehavetochooseourconstantstomaketheconstructionsuccessfulwithhighprobability.Tobegin,wenotethatthelimitingsuper-BrownianmotionwithdriftisaBrownianmotionwithvariance13perunittime.Easycal-culationswiththetransitionprobabilityofBrownianmotionshowthatliminf0(12.2)Thisbringsustotherstofseveralchoicesofparameterswewillmake.Choice1.�bwecanpicklargeenoughsothat 5(12.3)Toachieveaniterangeofdependenceinourdominatedpercolationpro-cess,weneedtoimposeacutoffinspace.InBramson,DurrettandSwindle(1989)thiswasdonebyconsideringamodiedcontactprocessinwhichparticlesarekillediftheymoveoutofanitestrip.However,havingworkedforninesectionstoprovetheconvergenceoftherescaledcontactprocessinthefullspacetosuper-Brownianmotion,wedonotwanttorepeattheproofforprocesseswithkillingoutsideofastrip,oraskthereadertobelievewecandoso.Thuswewilltakeanapproachthatonlyrequiresuseoftheconvergencetheoremonthewholespace.Letbethethrescaledcon-tactprocessmodiedsothatparticlesbornoutsideofKL;KL/areimmediatelykilled.Nowthenumberofparticlesthatarelostfromthecontactprocessbythistruncationisatmostthenumberthatarelostinthedominatingbranchingprocess(with“drift”Thelatterlossiseasytoestimate.Letbethethbranchingran-domwalk,modiedsothatnoparticlesareallowedtobebornoutsideof 392R.Durrett,E.A.PerkinsKL;KL/.Forthisitiswellknown,see,e.g.,Sections2and6ofBramson,Durrett,andSwindle(1989),thatthecounterpartofLemma2.9withkilling.Namelyistherandomwalkthattakesjumpsuniformonatrateandiskilled(i.e.,senttothestate)whenitleavesKL;KL/UsingthemaximalinequalityontherstcomponentofitiseasytoseethatChoice2.islargeenoughthenforall1(12.5)Havingxedourtimehorizonandourspatialtruncationwidth,ournextstepistomakethesuccessprobabilityhighbyusinginitialmeasureswithlargetotalmass.Wedothisbothforourbranchingrandomwalksandsuper-BrownianmotionLemma12.1.Thereisasothatforallnaturalnumbers.I/C=J:.I/P.XC=JProof.(a)Aneasycalculationusing(12.4)withinplaceofand(12.5)showsthat.I/Turningtosecondmoments,wehaveE...X...XN�2XTX T1.0�T;�T/KL;KL/forsomes;sThecontributionfromindicessatisfyingisatmostmostE..X.Thecontributionfromindicessatisfyingistriviallyboundedby Super-Brownianmotionintwoormoredimensions393�T;�T/ wherewehaveusedawell-knownexpressionforthesecondmomentofabranchingrandomwalkinthelastline(seeLemma2.2ofBramson,Dur-rettandSwindle(1989)).TheabovecalculationsboundthevarianceofC.T/X.TheresultnowfollowsbyCheby-chev'sinequality.(b)ThisfollowsbyasimilarChebychevargumentusing(12.3)togetalowerboundonthemean,andthefactthatthevarianceofisboundedbyaconstanttimestheinitialmass(seeProposition(2.7)ofFitzsimmons WithLemma12.1inmindandleavinglotsofroomforerrorstoaccu-mulate,wecannowmakeChoice3.beasinLemmaandpickanaturallargeenoughsothatC=JInorderforthecontactprocesstobesuccessfulatavoidingextinctionwithhighprobability,itisnotsufcientthattheinitialnumberofparticlesislarge.Considerforconcretenessthesituationin3.InthiscasetheO.N/particles.Ifwelettheinitialstateconsistofallthesitesinoneormoreneighborhoodsthenthemasslostduetobirthsontooccupiedsiteswillresultinadevastatingdecrease.Sincewewillnotneedtoknowthedetails,weleaveittothereadertogureouthowmuchmassislostandhowquickly.Toavoidthisproblem,welet`.z/for00if0ordenethequadraticform.dx/.dy/`.yandthenconsiderinitialconditionsforthecontactprocessthataresupported,have.I/,and,ofcourse,atmostoneparticlepersite.Notethatwehaveset0toavoidtheinnitiesonthediagonalwhenwearedealingwithpointmassmeasures.UsingourconvergencetheoremnowwithLemma12.1andourchoicewehave 394R.Durrett,E.A.PerkinsLemma12.2..If.M/.I/P.XProof.Ifnot,thenthereisasubsequenceofintegersandasso-ciatedinitialconditionswheretheprobabilityexceeds2.Sincethehavesupportinandtotalmassthereisaweaklyconver-gentsubsequence.ThelimitmustbeatomlessbyFatou'slemma,thebound,andthelowersemicontinuityof.Ourconvergencetheoremshowsthat(recallisdenedtobeopen)limsupwhichcontradicts(b)ofLemma12.1andthechoiceof Lemma12.3..If.M/thenforall.I/wehaveProof.Asnotedabovewecanboundtheamountofmasslostinthecontactprocessbythemasslostinthebranchingprocess,so(a)ofLemma12.1C=JThedesiredresultnowfollowsfromLemma12.2andthechoiceof Havingimposedtheconditionontheinitialcondition,wearenowobligedtoshowthatwithhighprobabilityitholdsattime.TodothisitisenoughtoshowthefollowingresultforthedominatingbranchingrandomwalksLemma12.4.ForanynaturalnumberthereisaT;JsothatifthenforallwehaveEQ.XT;JThisshouldmotivatethenalChoice4.PicklargeenoughsothatT;J Super-Brownianmotionintwoormoredimensions395Toverify.CA/ ;K,forasinthedenitionofasameasure)andletbetheeventthat,startingwith,ourmodiedcontactprocesswithkilling,,satises,and.HerewechoosesothatLemmas12.3and12.4areavailablewith,respectively.Thismodiedcontactprocessusesthesameexponentialvariablestojumpordieasthefullcontactprocessstartingfromandsoitisreadilyseenthatthemodiedprocessisdominatedby.Byusingthecollectionofsitesinasourchoiceof,wethereforeseethatthat�2Le1H.FinallyLemmas12.3and12.4showthatP.GE.Q./=qT;J andso.CA/ ;Kholds.ThusthelastdetailistocompletetheProofofLemma12.4.asshorthandfor,wehaveEQ.X; T; Firstconsiderthe.Imitating(4.4)–(4.6)wecanwrite(recallT; /T/T/ istherandomwalkthatstaysputwithprobability1/2andwithprobability1/2takesajumpuniformlydistributedoverForallwehave`.z/solargedeviationsresultsforthePoissondistributionimply/T/T/T/ n!!0asN 396R.Durrett,E.A.PerkinsForthesumover/Tnotethat,sothatpartisboundedby/T/T/ E`.x../T/.u/0isaPoissonprocesswithrateone.bethetimeoftherstjumpof.u/.Using(8.7)wecanestimate/;V../T//�1;1]d
CN�d=2SinceP.T�T.///T,consideringvalueoftherstjumpshowsP.N../T/ .N/isconstantonanddecreasingfor01,themaximumvalueofE`.x../T/subjecttotheconstraintontheprobabilitiesin(12.11)canbeboundedbyx/d.x/d.y/istheuniformdistributiononthepointsofin[.Thiseasilygives../T//�1;1]2d`.y�x/dxdyWenotethatasusualthevalueofchangesfromlinetolineintheabove.Summingovernowgives; T; Turningtothetermswith,weletandnotethatas0,wetakeandsointheabovesum.1besuchthat1besuchthatArguingasinLemma4.4,wehave Super-Brownianmotionintwoormoredimensions397; T; T/`.B/.T/.T T;TSummingoverthepossiblevaluesof,changingvariables2,andnoting .n�mC1/!.m�1/!D2n wehave; T; T/`.B/.T/.T T;TAsin(12.10),largedeviationsresultsforthePoissondistributionimplythatthesumover/TisboundedbyT;Tforsome0.Continuingtoreasonasinthecaseweseethatthesumover/Tisboundedby../.TRepeatingtheproofof(12.11)nowshowsthat .N//.TAgainmaximizingwithrespecttothisconstraintontheprobabilities,gives/`.x/ .N/ 398R.Durrett,E.A.Perkins/.TSummingovernowgives; T; T/`.BT;TItfollowsfromLemma3.4(b)thattheaboveequalss///dsCombiningthiswith(12.13),wehavethedesiredboundon(12.8).ThiscompletestheproofofLemma12.4. ReferencesBillingsley,P.:ConvergenceofProbabilityMeasures.JohnWiley,NewYork,(1968)Bramson,M.,Durrett,R.,Swindle,G.:Statisticalmechanicsofcrabgrass.Ann.Prob.,444–481(1989)Breiman,L.:Probability.Addison-Wesley,Reading,Mass(1968)ButtelL.,Cox,T.,Durrett,R.:Estimatingthecriticalvaluesofstochasticgrowthmodels.J.Appl.Prob.,,445–461(1993)Dawson,D.A.:Geostochasticcalculus.Can.J.Statistics,,143–168(1978)Dawson,D.A.:Measure-valuedMarkovprocesses.Pages1–260inEcoled'edeProba-esdeSt.Flour,XXILectureNotesinMath1541,Springer,NewYork(1993)Durrett,R.:LectureNotesonParticleSystemsandPercolation.[Chapters4and11]WadsworthPub.Co.(1988)Durrett,R.:Anewmethodforprovingtheexistenceofphasetransitions.Pages141–170inSpatialStochasticProcessesEditedbyK.S.AlexanderandJ.C.Watkins.Birkhauser,Boston(1992a)Durrett,R.:Thecontactprocess:1974–89.Pages1–18inMathematicsofRandomMediaEditedbyW.KohlerandB.White.AmericanMath.Society,Providence,RI(1992b)Durrett,R.:Probability:TheoryandExamples.DuxburyPress(1995a)Durrett,R.:Tenlecturesonparticlesystems.Pages97–201inEcoled'edeProbabilitdeSt.Flour,XXIIILectureNotesinMath1608,Springer,NewYork(1995b)Ethier,S.,Kurtz,T.:MarkovProcesses:CharacterizationandConvergence.JohnWiley,NewYork(1986) Super-Brownianmotionintwoormoredimensions399Evans,S.N.,Perkins,E.:Measure-valuedbranchingdiffusionswithsingularinteractions.Can.J.Math.,,120–168(1994)Fitzsimmons,P.J.:Constructionandregularityofmeasure-valuedMarkovbranchingpro-cesses.IsraelJ.Math.,,337–361(1988)Harris,T.:Contactinteractionsonalattice.Ann.Prob.,,969–988(1974)Holley,R.,Liggett,T.:Thesurvivalofcontactprocesses.Ann.Prob.,,198–206(1978)Kendall,D.:Stochasticprocessesandpopulationgrowth.J.oftheRoy.Stat.Soc.,230–264(1949)Kesten,H.:AsharperformoftheDoeblin-Levy-Kolmogorov-Rogozininequalityforcon-centrationfunctions.Math.Scand.,,133–144(1969)Jacod,J.,Shiryaev,A.N.:LimitTheoremsforStochasticProcesses.Springer-Verlag,BerlinKonno,N.,Shiga,T.:Stochasticdifferentialequationsforsomemeasure-valueddiffusions.Prob.TheoryRel.Fields,,201–225(1988)Liggett,T.:InteractingParticleSystems.Springer-Verlag,NewYork(1985)Liggett,T.:Improvedupperboundsforthecontactprocesscriticalvalue.Ann.Prob.,697–723(1995)Mueller,C.,Tribe,R.:AphasetransitionforastochasticPDErelatedtothecontactprocess.Prob.TheoryRel.Fields,,131–156(1994)Mueller,C.,Tribe,R.:StochasticPDE'sarisingfromthelongrangecontactprocessandlongrangevotermodel.Prob.TheoryRel.Fields,,519–546(1995)Reimers,M.:Onedimensionalstochasticpartialdifferentialequationsandthebranchingmeasurediffusion.Prob.TheoryRel.Fields,,319–340(1989)Stacey,A.:Boundsonthecriticalprobabilitiesinorientedpercolationmodels,Ph.D.Thesis,CambridgeU(1994)Walsh,J.:Anintroductiontostochasticpartialdifferentialequations.Pages265–439inEcoled'edeProbabilitesdeSt.Flour,XIVLectureNotesinMath1180,Springer,NewYork(1986)