Gaussian Measures in Finite and Infinite Dimensions

Description

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. 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