This text provides a concise introduction, suitable for a one-semester special topics
course, to the remarkable properties of Gaussian measures on both finite and infinite
dimensional spaces. It begins with a brief resumé of probabilistic results in which Fourier
analysis plays an essential role, and those results are then applied to derive a few basic
facts about Gaussian measures on finite dimensional spaces. In anticipation of the analysis
of Gaussian measures on infinite dimensional spaces, particular attention is given to those
properties of Gaussian measures that are dimension independent, and Gaussian processes
are constructed. The rest of the book is devoted to the study of Gaussian measures on
Banach spaces. The perspective adopted is the one introduced by I. Segal and developed
by L. Gross in which the Hilbert structure underlying the measure is emphasized.
The contents of this book should be accessible to either undergraduate or graduate
students who are interested in probability theory and have a solid background in Lebesgue
integration theory and a familiarity with basic functional analysis. Although the focus is
on Gaussian measures, the book introduces its readers to techniques and ideas that have
applications in other contexts.
Author(s): Daniel W. Stroock
Series: Universitext
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 144
City: Cham, Switzerland
Tags: Gaussian Measures, Wiener Space, Brownian Motion, Euclidian Fields
Preface
Contents
Notation
General
Sets, Functions, and Spaces
Measure Theoretic
1 Characteristic Functions
1.1 Some Basic Facts
1.2 Infinitely Divisible Laws
2 Gaussian Measures and Families
2.1 Gaussian Measures on mathbbR
2.2 Cramér–Lévy Theorem
2.2.1 Gaussain Measures and Cauchy's Equation
2.3 Gaussian Spectral Properties
2.3.1 A Logarithmic Sobolev Inequality
2.3.2 Hermite Polynomials
2.3.3 Hermite Functions
2.4 Gaussian Families
2.4.1 A Few Basic Facts
2.4.2 A Concentration Property of Gaussian Measures
2.4.3 The Gaussian Isoperimetric Inequality
2.5 Constructing Gaussian Families
2.5.1 Continuity Considerations
2.5.2 Some Examples
2.5.3 Stationary Gaussian Processes
3 Gaussian Measures on a Banach Space
3.1 Motivation
3.2 Some Background
3.2.1 A Little Functional Analysis
3.2.2 Fernique's Theorem
3.2.3 Gaussian Measures on a Hilbert Space
3.3 Abstract Wiener Spaces
3.3.1 The Cameron–Martin Subspace and Formula
3.3.2 Some Examples of Abstract Wiener Spaces
4 Further Properties and Examples of Abstract Wiener Spaces
4.1 Wiener Series and Some Applications
4.1.1 An Isoperimetric Inequality for Abstract Wiener Space
4.1.2 Rademacher's Theorem for Abstract Wiener Space
4.1.3 Gross's Operator Extention Procedure
4.1.4 Orthogonal Invariance
4.1.5 Large Deviations in Abstract Wiener Spaces
4.2 Brownian Motion on a Banach Space
4.2.1 Abstract Wiener Formulation
4.2.2 Strassen's Theorem
4.3 One Dimensional Euclidean Fields
4.3.1 Some Background
4.3.2 An Abstract Wiener Space for L2(λmathbbR;mathbbR)
4.4 Euclidean Fields in Higher Dimensions
4.4.1 An Abstract Wiener Space for L2(λmathbbRN;mathbbR)
4.4.2 The Ornstein–Uhlenbeck Field in Higher Dimensions
4.4.3 Is There any Physics Here?
Appendix References
Index
