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Universitext Daniel W. Stroock Gaussian Measures in Finite and Infinite Dimensions Universitext SeriesEditors NathanaëlBerestycki,UniversitätWien,Vienna,Austria CarlesCasacuberta,UniversitatdeBarcelona,Barcelona,Spain JohnGreenlees,UniversityofWarwick,Coventry,UK AngusMacIntyre,QueenMaryUniversityofLondon,London,UK ClaudeSabbah,ÉcolePolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau, France EndreSüli,UniversityofOxford,Oxford,UK https://avxhm.se/blogs/hill0 Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approachtotheirsubjectmatter.Someofthemostsuccessfulandestablishedbooks intheserieshaveevolvedthroughseveraleditions,alwaysfollowingtheevolution ofteachingcurricula,intoverypolishedtexts. Thusasresearchtopicstrickledownintograduate-levelteaching,firsttextbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. https://avxhm.se/blogs/hill0 Daniel W. Stroock Gaussian Measures in Finite and Infinite Dimensions https://avxhm.se/blogs/hill0 DanielW.Stroock DepartmentofMathematics MassachusettsInstituteofTechnology Cambridge,MA,USA ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-031-23121-6 ISBN 978-3-031-23122-3 (eBook) https://doi.org/10.1007/978-3-031-23122-3 MathematicsSubjectClassification:28C20,60G15,60B11,46G12 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland https://avxhm.se/blogs/hill0 ThisbookisdedicatedtoLenGross, themanwhoseideasinspiredmuchofit https://avxhm.se/blogs/hill0 Preface Duringthespringsemesterof2022ItaughtaclassatM.I.T.basedonthematerial inthisbook.AlthoughmychoiceofGaussianmeasuresasthetopicwasinfluenced bytherenewedinterest,resultingfromOdedShramm’sworkonconformalfields,of M.I.T.studentsinGaussianmeasures,Iwasequallymotivatedbymyownadmiration ofthetheoryofGaussianmeasuresonBanachspacesdevelopedbyIrvingSegaland Leonard Gross in connection with constructive quantum field theory. I realize that the program to construct quantum fields via Gaussian measures has come on hard times,butIbelievethatideaslikeGross’snotionofabstractWienerspacesshould notbeforgottenandmaywellhavearoletoplayinthefuture. Thecontentsarebrokenintofourchapters.Chapter1containssomebasicfacts aboutcharacteristicfunctions.Allofthemareclassical,manyofthemarefamiliar, but others have been either neglected or forgotten. Be that as it may, they will be usefullater.Chapter2beginswithanumberofresultsaboutGaussianmeasuresin finitedimensions.IncludedaretheLévy-CramérTheoremforindependentrandom variableswhosesumisGaussian,thePoincaréandGross’slogarithmicinequalities forGaussianmeasures,therelationshipofGaussianmeasuretoHermitepolynomials and functions, the concentration theorem of Maury and Pisier, and the Gaussian isoperimetricinequality.Thesearefollowedbytheconstruction,viaKolmogorov’s Consistency Theorem and his continuity criterion, of Gaussian processes, and the chapterconcludeswithadiscussionofBrownianmotionandtheOrnstein-Uhlenbeck processes. Becausetheyarerequiredforanunderstandingofwhatfollows,Chap.3begins withasummaryofafewfunctionalanalyticresults,likeBochner’stheoryofinte- grationforBanachspacevaluedfunctions,X.Fernique’sremarkableinequalityfor GaussianmeasuresonaBanachspace,andGaussianmeasuresonaHilbertspace. Armed with those preliminaries, I next introduce the notion of an abstract Wiener spaceandshowthatthereisoneforeverynon-degeneratecenteredGaussianmeasure onaseparableBanachspace.Havingintroducedandprovidedexamplesofabstract Wienerspaces,IattemptinChap.4todemonstratetheimportanceoftheCameron- Martinsubspace,thatinvisiblebutcriticalHilbertspacewhichisthedistinguishing featureofanabstractWienerspace,byprovingseveralresultsinwhichitplaysan vii https://avxhm.se/blogs/hill0 viii Preface essentialpart.TheseincludetheCameron-Martinformula,theBanachspaceversion oftheGaussianisoperimetricinequality,theDonsker-Varadhanlargedeviationsprin- ciple for rescaled Gaussian measures, Rademacher’s differentiation theorem, and, somewhatlater,Strassen’slawoftheiteratedlogarithm.Alongtheway,Iderivea resultwhichshowsthatthedefinitionIuseofanabstractWienerspaceisequivalent toGross’sandleadstoaproofsofacoupleofhisfundamentaltheoremswhichare used in the derivation of the Donsker-Varadhan large deviations principle as well astheconstructionofBanachspacevaluedBrownianmotion.Thefinaltopicisthe construction of Euclidean free fields, first for one dimension and then for higher dimensions.Inordertodoso,IhavetogiveabriefaccountintermsofHermitefunc- tionsofL.Schwartz’stheoryoftempereddistributions,afterwhichtheconstruction closelyresemblesonesgivenearlier. Idonotknowhow,orevenif,thisbookwillbeused.Itcoversonlyasmallpart of the subject and does so from a particular perspective. A much more complete treatmentisgiveninV.Bogachev’sGaussianMeasures,whichcontainsnotonlyan enormousamountofmaterialbutalsoanadmirablesummaryofandreferencesto othersources.Nonetheless,Ithinkthatthecontentsofthisbookconstitutethebasis foraonesemester,specialtopicscourseinprobabilitytheory.Italsomightbeused asreferencematerialinalessspecializedcourseorinresearch.Inanycase,Ihave enjoyedwritingitandhopethattherewillbeatleastoneotherpersonwhowillread andenjoyit. Nederland,CO,USA DanielW.Stroock https://avxhm.se/blogs/hill0 Contents 1 CharacteristicFunctions ........................................ 1 1.1 SomeBasicFacts ........................................... 1 1.2 InfinitelyDivisibleLaws .................................... 11 2 GaussianMeasuresandFamilies ................................. 19 2.1 GaussianMeasuresonR .................................... 19 2.2 Cramér–LévyTheorem ...................................... 21 2.2.1 GaussainMeasuresandCauchy’sEquation .............. 23 2.3 GaussianSpectralProperties ................................. 26 2.3.1 ALogarithmicSobolevInequality ...................... 29 2.3.2 HermitePolynomials ................................. 33 2.3.3 HermiteFunctions ................................... 35 2.4 GaussianFamilies .......................................... 39 2.4.1 AFewBasicFacts ................................... 40 2.4.2 AConcentrationPropertyofGaussianMeasures .......... 42 2.4.3 TheGaussianIsoperimetricInequality .................. 45 2.5 ConstructingGaussianFamilies .............................. 51 2.5.1 ContinuityConsiderations ............................. 54 2.5.2 SomeExamples ...................................... 60 2.5.3 StationaryGaussianProcesses ......................... 63 3 GaussianMeasuresonaBanachSpace ........................... 69 3.1 Motivation ................................................ 69 3.2 SomeBackground .......................................... 70 3.2.1 ALittleFunctionalAnalysis ........................... 71 3.2.2 Fernique’sTheorem .................................. 76 3.2.3 GaussianMeasuresonaHilbertSpace .................. 78 3.3 AbstractWienerSpaces ..................................... 80 3.3.1 TheCameron–MartinSubspaceandFormula ............ 86 3.3.2 SomeExamplesofAbstractWienerSpaces .............. 94 ix https://avxhm.se/blogs/hill0 x Contents 4 FurtherPropertiesandExamplesofAbstractWienerSpaces ....... 101 4.1 WienerSeriesandSomeApplications ......................... 101 4.1.1 AnIsoperimetricInequalityforAbstractWienerSpace .... 104 4.1.2 Rademacher’sTheoremforAbstractWienerSpace ....... 106 4.1.3 Gross’sOperatorExtentionProcedure ................... 109 4.1.4 OrthogonalInvariance ................................ 114 4.1.5 LargeDeviationsinAbstractWienerSpaces ............. 117 4.2 BrownianMotiononaBanachSpace .......................... 121 4.2.1 AbstractWienerFormulation .......................... 121 4.2.2 Strassen’sTheorem ................................... 124 4.3 OneDimensionalEuclideanFields ............................ 129 4.3.1 SomeBackground ................................... 129 4.3.2 AnAbstractWienerSpacefor L2(λR;R) ................ 130 4.4 EuclideanFieldsinHigherDimensions ........................ 133 4.4.1 AnAbstractWienerSpacefor L2(λRN;R) ............... 134 4.4.2 TheOrnstein–UhlenbeckFieldinHigherDimensions ..... 136 4.4.3 IsThereanyPhysicsHere? ............................ 137 References ........................................................ 141 Index ............................................................. 143 https://avxhm.se/blogs/hill0