Gaussian Measures in Finite and Infinite Dimensions

Description

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