On the paths Holder continuity in models of Euclid

OnthepathsHoldercontinuityinmodelsofEuclideanQuantumFieldTheorybySergioAlbeverioFakultatfurMathematik,RuhrUniversitatBochumD{44780Bochum,GermanySFB237;BiBoS;CERFIM(Locarno);Acc.Arch.USI(Mendrisio)RomanGielerakInstituteofTheoreticalPhysics,UniversityofWroclawPL{50{205Wroclaw,PolandFrancescoRussoDepartementdeMathematiques,InstitutGalileeUniversiteParisNord,Av.J.B.Clement,F{93430Villetaneuse,FranceAMSClassication:60G60,35Q99,60H15Keywords:QuantumFieldTheory,HolderContinuity,TracesofdistributionsAbstractSamplepathspropertiesofcertainstochasticprocessesconnectedtomod-elsofEuclideanQuantumFieldTheoryarestudied.Inparticular,theHoldercontinuityofpathsofthecoordinateprocessesandtraceprocessesisproven.TheresultsareobtainedbyanapplicationofclassicalprobabilisticcriteriatogetherwithbasicestimatesproveninConstructiveQuantumFieldTheory.1IntroductionThesamplepathsspacepropertiesoftheEuclideanFieldTheorymodelswhichhavebeenconstructed([Si74,GJ81,AFHKL86,BaeSeZh97]andref-erencestherein),werestudiedinthepastquiteintensivelyseei.e.[Can74,CL73,ReRo74,AHK77,Car77,BeGN80,Ha79].However,mosteortshavebeendevotedtothecaseoftheNelsonfreeeld[Nel73a,Nel73b].Inthecaseofd=1,detailedresultsonsamplepathpropertiescanbefoundin[RoSi76].Forthecaseofspace-dimensiond2,theonlyauthorwhostudiedwithsomegeneralitytheHoldercontinuityofsamplepathsoftheinteract-ingmodelsseemstobeHaba[Ha79].Providedthecharacteristicfunctionaloftheeld(inthecased=2),veriessomeinequality,Haba[Ha79]wasablethroughacleverbutrathercomplicatedconstructiontodeduceHoldercontinuityofthecorrespondingcoordinateprocesses.InrecentconstructedmodelsofEuclideanquantumeldtheory,[AGW96,AGW97,BGL96],thesamplepathscontinuitycaneasilyberecognized.Thisispossiblebecauseofspecicsupportpropertiesofthoseeldswhicharesolutionsofstochas-tic(pseudo)dierentialequations.Continuityresultsforthecorrespondingstochasticquantizationprocesseswerediscussedford2,see[AR91].Ford3,andwithoutregularization,wearenotawareofanyresultofthistype,exceptforthepathcontinuityofthestochasticquantizationpolymermeasureoverd=3[ARZ96a];seealso[ARZ96b]fordetailedpropertiesofthismodelinthecased=2.OneoftheobjectivesofthepresentpaperistodemonstratethattheHoldercontinuityofsamplepathsofthecoordinateprocessesconnectedtoseveralmodelsofEuclideanQuantumFieldtheoryfollowsinastraightfor-wardmannerfromwellknownestimatesonmomentsandtheclassicalKol-mogorovcontinuitycriterion.Inthiswaywegreatlysimplifyandgeneralizethemainresultsof[Ha79].1RecentlyanewapproachtotheoldproblemofEuclideanQuantumeldtheory,thesocalledGlobalMarkovPropertyproblem(seei.e.[AZe92]foranextensiveoverviewandreferencestopreviousworks),hasbeengivenbythepresentauthors[AGR96,AGR97].Inthesereferences,itwasshownthattoanysucientlyregulargeneralizedrandomeldonthespaceoftempereddistributionsS0(IRd)onecanassociateatraceprocess(Xt)t2IR;thisprocesscanbeinformallyobtainedbytakingthetracesinasuitablesenseofatypical2S0(IRd)onxedtimehyperplanest=f(x0;x)2IRdjx0=tgIRd.ForseveralmodelsofEuclideanQuantumFieldTheorytheMarkovcharacterofXtfollowsthenbytheexplicitconstructionofthecorrespondingtransitionkernelt(;ei(;f))=IEfei(Xt;)j(Xs),s0g.ThecontinuitypropertiesoftheconstructedtraceprocessesXtarestudiedinsection4ofthepresentpaper.Employingthebasicestimatesonthemomentsdescribedin([Si74,GJ81,AFHKL86]andreferencestherein)andtheGarsia-Rodemich-Rumsey,(seeforinstance[BaY82])criterionweprovetheHoldercontinuity(withmodulusarbitrarycloseto1/2)ofthecorrespondingtraceprocesses.ThisextendstheresultofRockner[R88b]tothecaseofinteractingelds.WeremarkthatapplyingourargumentstothefreecaseconsideredbyRockner,wecanevenimprovecertainaspectsofhisresults.Howeverourresultsarebynomeansoptimal.InparticularthequestionsabouttheoptimalityoftheobtainedvaluesoftheHoldercontinuitymodulusandtheuctuationproperties(i.e.largetimebehaviour)ofthepathsarenotdiscussedinthepresentpaper.Weremarkhoweverthatwhereasthepathsofinteractingmodelsareexpectedtohaveexactlythesamelocalcontinuitybehaviourasthoseofthefreeeldcase,thelargetimebehaviourofpathsdistinguishes,ingeneral,thefreeeldfromtheinteractingone(concerningthe1-dimensionalcase,i.e.theP()1processes,wereferto[RoSi76]).Itseemsworthwhilealsotoquotesomeknowncontinuitypropertiesofthe2processesconstructedbytheDirichletformapproach[AFHKL86,AHK75,AR89,AR91,HKPS93].HowevertheproblemwhethertheDirichletpro-cessesconstructedwithintheframeworkofEuclideanQuantumFieldTheorymodelscanbeidentiedwiththeprocessesdiscussedinthepresentpaperisstillunsolved(althoughanidenticationinthemodelswhoseinterac-tionisrestrictedtoaboundeddomaininIR2hasbeendone,see[ARZh93],[RZh9294].)2Preliminaries.2.1Freeeldtraceprocess(accordingto[R85,R88b])LetusdenotebyPtetpd1+1,t0thecontractionsemigroupinthespaceL2(IRd1),withgeneratorpd1+1whered1meansthed1dimensionalLaplaceoperatorwhered2.ItiswellknownthatthesemigroupPtissubordinatedtotheheatsemigroup[BlH86,Sect.V.3].ThecorrespondingMarkoviankernelwillbedenotedasPt(x;y)dy;wheredymeansLebesguemeasureonIRd1.Itisalso