White Noise Calculus and Fock Space(Obata)

LectureNotesinMathematicsEditors:A.Dold,HeidelbergB.Eckmann,ZtirichF.Takens,Groningen1577NobuakiObataWhiteNoiseCalculusandFockSpaceSpringer-VerlagBerlinHeidelbergNewYorkLondonParisTokyoHongKongBarcelonaBudapestAuthorNobuakiObataDepartmentofMathematicsSchoolofScienceNagoyaUniversityNagoya,464-01,JapanMathematicsSubjectClassification(1991):46F25,46E50,47A70,47B38,47D30,47D40,60H99,60J65ISBN3-540-57985-0Springer-VerlagBerlinHeidelbergNewYorkISBN0-387-57985-0Springer-VerlagNewYorkBerlinHeidelbergCIP-DataappliedforThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.ViolationsareliableforprosecutionundertheGermanCopyrightLaw.9Springer-VerlagBerlinHeidelberg1994PrintedinGermanySPIN:1013001946/3140-543210-Printedonacid-freepaperContentsIntroductionviiIPrerequisites11.1Locallyconvexspaces!ngeneral......................11.2CountablyHilbert~paces..........................31.3Nuclear,'pacesandkerneltheorem....................71.4StandardCH-spacescffunctions......................111.5Bochner-Minlostheorem..........................161.6Furtlcernotationalremarks.........................17Bibliographical~otes............................182WhiteNoiseSpace192.1Gaussianmeasure..............................192.2Wick-orderedpolynomials.........................232.3Wiener-ItS-SegalisomorphismandFockspace..............28Bibliographicalnotes............................323WhiteNoiseFunctionals333.1StandardConstruction...........................333.2Continuousversiontheorem........................383.3S-transform.................................483.4Contractionoftensorproducts.......................533.5Wienerproduct...............................583.6Characterizationtheorems.........................~5Bibliographicalnotes............................694OperatorTheory4.14.24.34.44.54.671Hidsdifferentialoperator.........................71Translationoperators............................76Integralkerneloperators..........................79Symbolsofoperators............................88Fockexpansion...............................98Someexamples...............................100Bibliographicalnotes............................107viCONTENTS5TowardHarmonicAnalysis1095.1Firstorderdifferentialoperators......................1095.2R(gularone-parametertransformationgrcup...............1185.3InfinitedimensionalLaplacians......................1215.4Infinitedimensionalrotationgroup....................1265.5Rotation-invariantoperaters........................1325.6Fouriertransform..............................1405.7IntertwiningpropertycfFouriertransform................145Bibliographicalnotes............................1496Addendum1516.1Integral-sumkezneloperators.......................1516.2Reductiontofinitedegreeoffreedom...................1546.3Vector-valuedwhitenoisefunctionals...................159Appendices167APolarizationformula............................167BHermitepolynomials............................168CNormestimatesofcontracticns......................169References171Index181IntroductionThewhitenoisecalculus(oranalysis)waslaunchedoutbyHida[1]in1975withhislecturenotesongeneralizedBrownianfunctionals.ThisnewapproachtowardaninfinitedimensionalanalysiswasdeeplymotivatedbyLdvy[1]whoconsiderablydevelopedfunctionalanalysisonL2(0,1)andactuallyanalysisofBrownianfunction-als.TherootofwhitenoisecalculusistoswitchafunctionalofBrownianmotionf(B(t);tER)withoneofwhitenoiserteR),whereB(t)isatimederiva-tiveofaBrownianmotionB(t).AlthougheachBrownianpathB(t)isnotsmoothenough,/~(t)isthoughtofasageneralizedstochasticprocessandrisrealizedasageneralizedwhitenoisefunctionalinourlanguage.Wemaytherebyregard{/~(t)}asacollectionofinfinitelymanyindependentrandomvariablesandhenceacoordinatesystemofaninfinitedimensionalspace.Themathematicalframeworkofthewhitenoisecalculusisbaseduponaninfi-nitedimensionalanalogueoftheSchwartzdistributiontheory,wheretheroleoftheLebesguemeasureonR~isplayedbytheGanssianmeasure#onthedualofacertainnuclearspaceE.IntheclassicalcasewhereB(t)isformulated,wetakeE=S(R)andtheGanssianmeasure#onE*definedbythecharacteristicfunctional:where[~]istheusualL2-normof~.ThentheHilbertspace(L2)=L2(E*,~)iscanonicallyisomorphictothe(Boson)FockspaceoverL2(R)throughtheWiener-It6-Segalisomorphismandlinksthetestandgeneralizedfunctionals.Namely,inaspecificway(calledstandardconstruction)weconstructanuclearFr~chetspace(E)denselyandcontinuouslyimbeddedin(L2),andbydualityweobtainaGelfandtriple:(E)C(L2)=L2(E,#)C(E)*.Anelementin(E)isatestwhitenoisefunctionalandhenceanelementin(E)*isageneralizedwhitenoise.functional.TheabovepictureiseasilyunderstoodasadirectanalogyofS(R~.)CL2(R~)CS'(]~~)whi