Table Of ContentUniversitext
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
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Universitext is a series of textbooks that presents material from a wide variety
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Thusasresearchtopicstrickledownintograduate-levelteaching,firsttextbooks
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Daniel W. Stroock
Gaussian Measures
in Finite and Infinite
Dimensions
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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
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ThisbookisdedicatedtoLenGross,
themanwhoseideasinspiredmuchofit
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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
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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
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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
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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
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