Laplace operators and diffusions in tangent bundle

arXiv:math/0608337v1[math.PR]14Aug2006LAPLACEOPERATORSANDDIFFUSIONSINTANGENTBUNDLESOVERPOISSONSPACESSERGIOALBEVERIO,ALEXEIDALETSKII,ANDEUGENELYTVYNOVAbstractSpacesofdifferentialformsoverconfigurationspaceswithPoissonmea-suresareconstructed.ThecorrespondingLaplacians(ofBochneranddeRhamtype)on1-formsandassociatedsemigroupsareconsidered.Theirprobabilisticinterpretationisgiven.1IntroductionStochasticdifferentialgeometryofinfinite-dimensionalmanifoldshasbeenaveryactivetopicofresearchinrecenttimes.Oneoftheimportantandintriguingprob-lemsdiscussedconcernstheconstructionofspacesofdifferentialformsoversuchmanifoldsandthestudyofthecorrespondingLaplaceoperatorsandassociated(stochastic)cohomologies.AcentralroleinthisframeworkisplayedbytheconceptoftheDirichletoperatorofadifferentiablemeasure,whichisactuallyaninfinite-dimensionalgeneralizationoftheLaplace–Beltramioperatoronfunctions,respec-tivelytheLaplace–Witten–deRhamoperatorondifferentialforms.Thestudyofthelatteroperatorandtheassociatedsemigrouponfinite-dimensionalmanifoldswasthesubjectofmanyworks,anditleadstodeepresultsontheborderofstochasticanalysis,differentialgeometryandtopology,andmathematicalphysics,see,e.g.,[22],[19],[23].DirichletformsonCliffordalgebraswereconsideredin[26].Inaninfinite-dimensionalsituation,suchquestionswerediscussedintheflatcasein[11],[12],[13],[4].Aregularizedheatsemigroupondifferentialformsovertheinfinite-dimensionaltoruswasstudiedin[15].Astudyofsuchquestionsongeneralinfiniteproductmanifoldswasgivenin[2],[3].Thecaseofloopspaceswasconsideredin[28],[30].Atthesametime,thereisagrowinginterestingeometryandanalysisonPoissonspaces,i.e.,onspacesoflocallyfiniteconfigurationsinnon-compactmanifolds,equippedwithPoissonmeasures.In[5],[6],[7],anapproachtothesespacesastoinfinite-dimensionalmanifoldswasinitiated.Thisapproachwasmotivatedbytheconnectionofsuchspaceswiththetheoryofrepresentationsofdiffeomorphismgroups,see[25],[36],[27](thesereferencesand[7],[9]alsocontaindiscussionofrelationswithquantumphysics).Wereferthereaderto[8],[9],[35],andreferencesthereinforfurtherdiscussionofanalysisonPoissonspacesandapplications.Inthepresentwork,wedevelopthispointofview.Wedefinespacesofdif-ferentialformsoverPoissonspaces.Next,wedefineandstudyLaplaceoperatorsactinginthespacesof1-forms.Weshow,inparticular,thatthecorrespondingdeRhamLaplaciancanbeexpressedintermsoftheDirichletoperatoronfunctionsonthePoissonspaceandtheWittenLaplacianontheinitialmanifoldassociatedwiththeintensityofthecorrespondingPoissonmeasure.Wegiveaprobabilisticinterpretationandinvestigatesomepropertiesoftheassociatedsemigroups.LetusremarkthatthestudyofLaplaciansonn-formsbyourmethodsisalsopossible,butitleadstomorecomplicatedconstructions.Itwillbegiveninaforthcomingpaper.ThemaingeneralaimofourapproachistodevelopaframeworkwhichextendstoPoissonspaces(asinfinite-dimensionalmanifolds)thefinite-dimensionalHodge–deRhamtheory.Adifferentapproachtotheconstructionofdifferentialformsandrelatedob-jectsoverPoissonspaces,basedonthe“transferprinciple”fromWienerspaces,isproposedin[34],seealso[32]and[33].2DifferentialformsoverconfigurationspacesTheaimofthissectionistodefinedifferentialformsoverconfigurationspaces(asinfinite-dimensionalmanifold).First,werecallsomeknownfactsanddefinitionsconcerning“manifold-like”structuresandfunctionalcalculusonthesespaces.2.1FunctionalcalculusonconfigurationspacesOurpresentationinthissubsectionisbasedon[7],howeverforlateruseinthepresentpaperwegiveadifferentdescriptionofsomeobjectsandresultsoccurringin[7].LetXbeacomplete,connected,oriented,C∞(non-compact)Riemannianman-ifoldofdimensiond.Wedenotebyh•,•ixthecorrespondinginnerproductinthetangentspaceTxXtoXatapointx∈X.Theassociatednormwillbedenotedby|•|x.Letalso∇XstandforthegradientonX.TheconfigurationspaceΓXoverXisdefinedasthesetofalllocallyfinitesubsets(configurations)inX:ΓX:={γ⊂X||γ∩Λ|∞foreachcompactΛ⊂X}.Here,|A|denotesthecardinalityofthesetA.Wecanidentifyanyγ∈ΓXwiththepositiveinteger-valuedRadonmeasureXx∈γεx⊂M(X),2whereεxistheDiracmeasurewithmassatx,Px∈∅εx:=zeromeasure,andM(X)denotesthesetofallpositiveRadonmeasuresontheBorelσ-algebraB(X).ThespaceΓXisendowedwiththerelativetopologyasasubsetofthespaceM(X)withthevaguetopology,i.e.,theweakesttopologyonΓXsuchthatallmapsΓX∋γ7→hf,γi:=ZXf(x)γ(dx)≡Xx∈γf(x)arecontinuous.Here,f∈C0(X)(:=thesetofallcontinuousfunctionsonXwithcompactsupport).LetB(ΓX)denotethecorrespondingBorelσ-algebra.Following[7],wedefinethetangentspacetoΓXatapointγastheHilbertspaceTγΓX:=L2(X→TX;dγ),orequivalentlyTγΓX=Mx∈γTxX.(1)ThescalarproductandthenorminTγΓXwillbedenotedbyh•,•iγandk•kγ,respectively.Thus,eachV(γ)∈TγΓXhastheformV(γ)=(V(γ)x)x∈γ,whereV(γ)x∈TxX,andkV(γ)k2γ=Xx∈γ|V(γ)x|2x.Letγ∈ΓXandx∈γ.ByOγ,xwewilldenoteanarbitraryopenneighborhoodofxinXsuchthattheintersectionoftheclosureofOγ,xinXwithγ\{x}istheemptyset.Foranyfixedfinitesubconfiguration{x1,...,xk}⊂γ,wewillalwaysconsideropenneighborhoodsOγ,x1,...,Oγ,xkwithdisjointclosures.Now,forameasurablefunctionF:ΓX→R,γ∈ΓX,and{x1,...,xk}⊂γ,wedefineafunctionFx1,...,xk(γ,•):Oγ,x1×···×Oγ,xk→RbyOγ,x1×···×Oγ,xk∋(y1,...,yk)7→Fx1,...,xk(γ,y1,.