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ModeratedeviationsandfunctionalLILforsuper-BrownianmotionAlexanderSchiedHumboldt-Universit atzuBerlinAbstract:AmoderatedeviationprincipleandaStrassentypelawoftheiteratedloga-rithmforthesmall-timepropagationofsuper-Brownianmotionarederived.OneofthemaintoolsisanembeddingtheorembetweencertainH olderandOrliczspaces.ItenablesonetocontroltheH oldernormofareal-valuedstochasticprocess,andmaybeofindepen-dentinterest.Toprovethelawoftheiteratedlogarithm,moderatedeviationestimatesthatareuniformwithrespecttothestartingpointarederived.1.IntroductionLetM+(IRd)denotethespaceofpositive nitemeasuresonIRdandendowitwiththeusualweaktopology.Thensuper-BrownianmotionXisanM+(IRd)-valueddi usion.ThelawIP ofthisprocessstartingfrom 2M+(IRd)maybecharacterizedasfollowsbytheLaplacefunctionalsofitstransitionprobabilities:(1:1)IE hexp( hf;Xti)i=exp( hu(t); i);wheret 0,fisanon-negativefunctioninCb(IRd),thespaceofboundedandcontinuousfunctionsonIRd,hg; idenotestheintegralofafunctiongwithrespecttosomemeasure ,anduistheuniquepositivemildsolutionofthereaction-di usionequation(1:2)@@tu(t;x)=12 u(t;x) u(t;x)2u(0;x)=f(x):AMS1991subjectclassi cation.Primary60F10,60F17,Secondary60G57;60G17Keywordsandphrases.Moderatedeviations,Strassentypelawoftheiteratedlogarithm,H olderspace,Orliczspace,uniformlargedeviations.1Thereadermay ndacomprehensiveintroductiontosuperprocessesinDawson(1993).ThegoalofthispaperistoderiveamoderatedeviationprincipleandalocallawoftheiteratedlogarithmofStrassentypeforsuper-Brownianmotion.Thereforeitisnecessarytocentertheprocess.Itfollowsfrom(1.1)and(1.2)thatIE hf;Xti =hf; Pti(t 0;f2Cb(IRd));wherePt(x;dy)=(2 t)d=2exp( jx yj2=2t)dyistheBrowniantransitionsemigroup.Thecenteredprocessisthusgivenby(1:3)bXt=Xt X0Pt(t 0):IttakesvaluesinthesetM(IRd)of nitesignedmeasures.Thisspacewillbeendowedwiththecoarsesttopologysuchthatthemappings 7!hf; iarecontinuousforeachf2BL,thespaceofboundedLipschitzfunctionsonIRd.Notethat,incontrasttotheconvexconeM+(IRd),thetopologicalspaceM(IRd)isnotmetrizable;seeDudley(1966).ThesetC [0;1]:M(IRd) ofallcontinuousM(IRd)-valuedpathswillbeendowedwiththecompactopentopology.WhenM(IRd)isendowedwiththevariationnorm,itbecomesa(non-separable)Banachspace,andonecande neBochnerintegrals.For 2M+(IRd),wede neH tobethesetofall!2C [0;1]:M(IRd) thatareoftheform!(t)=Zt0_!(s)ds(t 0);forsomeBochnerintegrablemapping_!:[0;1]!M(IRd)suchthat_!(t) ,foralmosteveryt.Thenwede netheGaussianratefunction(1:4)I (!)=8:14Z10 d_!(t)d 2L2( )dtif!2H 1otherwise.Theorem1.1:Suppose :(0;1]!(0;1]isafunctionsuchthat ( )&0as #0.ThenthedistributionsoftheprocessesbX ; ( )t:= ( ) 1bX ( )2t(0 t 1)satisfyalargedeviationprincipleonC [0;1]:M(IRd) withscale andgoodratefunctionI .I.e.,ifU C [0;1]:M(IRd) isopen,thenlim #0 logIP bX ; ( )2U inf!2UI (!);ifA C [0;1]:M(IRd) isclosed,thenlim #0 logIP bX ; ( )2A inf!2AI (!);thelevelsetsfI cgarecompact,foreachc 0.2Duetothecondition‘ ( )&0as #0’,Theorem1.1describesthemoderatedeviationsfromthe‘lawoflargenumbers’Xt! IP -almostsurely,ast#0.Thecorrespondinglargedeviations,i.e.thecase 1,aretreatedasCorollary3ofSchied(1996).Inthiscaseonegetsanon-Gaussianratefunction,namelytheenergywithrespecttotheKakutani-Hellingermetric.If 1and #0,weareintheregimeofafunctionalcentrallimittheorem;seetheremarkattheendofSection4.Otherresultsonsamplepathlargedeviationsforsuper-BrownianmotioncanbefoundinDeuschelandWang(1993),Fleischmannetal.(1996)andSchied(1996).LetAdenotethesetofallclosedsubsetsofC [0;1]:M(IRd) .Iffdjjj2Jgisa lteringfamilyofsemi-distancesonC [0;1]:M(IRd) ,theHausdor topologyonAisgeneratedbythesemi-metrics j(A;B):=supx2Adj(x;B)_supy2Bdj(y;A)(j2J;A;B2A):SeeCastaingandValadier(1977),ChapterII.For 1=e,de neanA-valuedrandomvariable asclosureoftherandomset(bXtp2loglog 1(0 t 1) 0 ):Thenourlawoftheiteratedlogarithmreadsasfollows:Theorem1.2:IfU AisopenandK :=fI 1=2g2U,then 2U,forsmall ,IP -almostsurely.Inparticular, convergesinIP -probabilitytoK as #0.Notethatwecannotconcludealmostsureconvergenceof toK ,becausethe rstcountabilityaxiomfailsforM(IRd)andhenceforA.DawsonandVinogradov(1994)giveatypeofalocalLILfortheset-valuedsupportprocessofa(2;d; )-superprocess.Theygetadi erentscalefunction,thatlooksmoreliketheglobalmodulusofcontinuityofaBrownianmotion.Toexplainthisfact,theyarguethat,becauseofthebranching,onemustexpectthealmost-sureoccurrenceofverylargeincrements(or‘fastpoints’)ofindividualBrownianmotions,evenonarbitrarysmalltimeintervals.SeeRemark(iii)followingTheorem1.3oftheabovearticle.Inadditiontothisresult,ourTheorem1.2nowstatesthat,forsmalltimeintervals,themajorityofparticlesinthepopulationstaysclosetotheplaceoftheirorigin,andthatthebranchingbehaviourfollowsalocaliteratedlogarithm.3Thereisavastliteratureonfunctionallawsoftheiteratedlogarithm.TheoriginalstatementisduetoStrassen(1964).HerewewilladoptStroock’sideaofusinglargedeviationstogetthekeyestimates(seee.g.DeuschelandStroock(1989)).ForordinaryBrownianmotion,thelocallawhasbeenstatedinGantert(1993).Inthemeasure-valuedsetting,moderatedeviationshavebeenusedinWu(19