Gaussian Measures (Mathematical Surveys and Monographs)

This book gives a systematic exposition of the modern theory of Gaussian measures. It presents with complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes. Suitable for use as a graduate text and/or a reference work, this volume contains many examples, exercises, and an extensive bibliography. It brings together many results that have not appeared previously in book form. Front Cover......Page 1 Title Page......Page 6 Copyright......Page 7 Contents......Page 8 Preface......Page 12 1.1. Gaussian measures on the real line......Page 14 1.2. Multivariate Gaussian distributions......Page 16 1.3. Hermite polynomials......Page 20 1.4. The Ornstein-Uhlenbeck semigroup......Page 22 1.5. Sobolev classes......Page 25 1.6. Hypercontractivity......Page 29 1.7. Several useful estimates......Page 32 1.8. Convexity inequalities......Page 38 1.9. Characterizations of Gaussian measures......Page 43 1.10. Complements and problems......Page 46 2.1. Cylindrical sets......Page 52 2.2. Basic definitions......Page 55 2.3. Examples......Page 61 2.4. The Cameron-Martin space......Page 72 2.5. Zero-one laws......Page 77 2.6. Separability and oscillations......Page 80 2.7. Equivalence and singularity......Page 84 2.8. Measurable seminorms......Page 87 2.9. The Ornstein-Uhlenbeck semigroup......Page 91 2.10. Measurable linear functionals......Page 92 2.11. Stochastic integrals......Page 96 2.12. Complements and problems......Page 103 3.1. Radon measures......Page 110 3.2. Basic properties of Radon Gaussian measures......Page 113 3.3. Gaussian covariances......Page 117 3.4. The structure of Radon Gaussian measures......Page 122 3.5. Gaussian series......Page 125 3.6. Supports of Gaussian measures......Page 132 3.7. Measurable linear operators......Page 135 3.8. Weak convergence of Gaussian measures......Page 142 3.9. Abstract Wiener spaces......Page 149 3.10. Conditional measures and conditional expectations......Page 153 3.11. Complements and problems......Page 155 4.1. Gaussian symmetrization......Page 170 4.2. Ehrhard's inequality......Page 175 4.3. Isoperimetric inequalities......Page 180 4.4. Convex functions......Page 184 4.5. H-Lipschitzian functions......Page 187 4.6. Correlation inequalities......Page 190 4.7. The Onsager-Machlup functions......Page 194 4.8. Small ball probabilities......Page 200 4.9. Large deviations......Page 208 4.10. Complements and problems......Page 210 5.1. Integration by parts......Page 218 5.2. The Sobolev classes Wp'' and Dr"......Page 224 5.3. The Sobolev classes HP2'......Page 228 5.4. Properties of Sobolev classes and examples......Page 231 5.5. The logarithmic Sobolev inequality......Page 239 5.6. Multipliers and Meyer's inequalities......Page 242 5.7. Equivalence of different definitions......Page 247 5.8. Divergence of vector fields......Page 251 5.9. Gaussian capacities......Page 256 5.10. Measurable polynomials......Page 262 5.11. Differentiability of H-Lipschitzian functions......Page 274 5.12. Complements and problems......Page 279 6.1. Auxiliary results......Page 292 6.2. Measurable linear automorphisms......Page 295 6.3. Linear transformations......Page 298 6.4. Radon-Nikodym densities......Page 301 6.5. Examples of equivalent measures and linear transformations......Page 308 6.6. Nonlinear transformations......Page 311 6.7. Examples of nonlinear transformations......Page 321 6.8. Finite dimensional mappings......Page 327 6.9. Malliavin's method......Page 329 6.10. Surface measures......Page 334 6.11. Complements and problems......Page 337 7.1. Trajectories of Gaussian processes......Page 346 7.2. Infinite dimensional Wiener processes......Page 349 7.3. Logarithmic gradients......Page 352 7.4. Spherically symmetric measures......Page 357 7.5. Infinite dimensional diffusions......Page 359 7.6. Complements and problems......Page 366 A.1. Locally convex spaces......Page 374 A.2. Linear operators......Page 378 A.3. Measures and measurability......Page 384 Bibliographical Comments......Page 393 References......Page 403 Index......Page 440 Chapter 1. Finite Dimensional Gaussian Distributions 1 -- 1.1. Gaussian Measures On The Real Line 1 -- 1.2. Multivariate Gaussian Distributions 3 -- 1.3. Hermite Polynomials 7 -- 1.4. The Ornstein-uhlenbeck Semigroup 9 -- 1.5. Sobolev Classes 12 -- 1.6. Hypercontractivity 16 -- 1.7. Several Useful Estimates 19 -- 1.8. Convexity Inequalities 25 -- 1.9. Characterizations Of Gaussian Measures 30 -- 1.10. Complements And Problems 33 -- Chapter 2. Infinite Dimensional Gaussian Distributions 39 -- 2.1. Cylindrical Sets 39 -- 2.2. Basic Definitions 42 -- 2.4. The Cameron-martin Space 59 -- 2.5. Zero-one Laws 64 -- 2.6. Separability And Oscillations 67 -- 2.7. Equivalence And Singularity 71 -- 2.8. Measurable Seminorms 74 -- 2.9. The Ornstein-uhlenbeck Semigroup 78 -- 2.10. Measurable Linear Functionals 79 -- 2.11. Stochastic Integrals 83 -- 2.12. Complements And Problems 90 -- Chapter 3. Radon Gaussian Measures 97 -- 3.1. Radon Measures 97 -- 3.2. Basic Properties Of Radon Gaussian Measures 100 -- 3.3. Gaussian Covariances 104 -- 3.4. The Structure Of Radon Gaussian Measures 109 -- 3.5. Gaussian Series 112 -- 3.6. Supports Of Gaussian Measures 119 -- 3.7. Measurable Linear Operators 122 -- 3.8. Weak Convergence Of Gaussian Measures 129 -- 3.9. Abstract Wiener Spaces 136 -- 3.10. Conditional Measures And Conditional Expectations 140 -- 3.11. Complements And Problems 142 -- Chapter 4. Convexity Of Gaussian Measures 157 -- 4.1. Gaussian Symmetrization 157 -- 4.2. Ehrhard's Inequality 162 -- 4.3. Isoperimetric Inequalities 167 -- 4.4. Convex Functions 171 -- 4.5. H-lipschitzian Functions 174 -- 4.6. Correlation Inequalities 177 -- 4.7. The Onsager-machlup Functions 181 -- 4.8. Small Ball Probabilities 187 -- 4.9. Large Deviations 195 -- 4.10. Complements And Problems 197 -- Chapter 5. Sobolev Classes Over Gaussian Measures 205 -- 5.1. Integration By Parts 205 -- 5.2. The Sobolev Classes W[superscript P, R] And D[superscript P, N] 211 -- 5.3. The Sobolev Classes H[superscript P, R] 215 -- 5.4. Properties Of Sobolev Classes And Examples 218 -- 5.5. The Logarithmic Sobolev Inequality 226 -- 5.6. Multipliers And Meyer's Inequalities 229 -- 5.7. Equivalence Of Different Definitions 234 -- 5.8. Divergence Of Vector Fields 238 -- 5.9. Gaussian Capacities 243 -- 5.10. Measurable Polynomials 249 -- 5.11. Differentiability Of H-lipschitzian Functions 261 -- 5.12. Complements And Problems 266 -- Chapter 6. Nonlinear Transofrmations Of Gaussian Measures 279 -- 6.1. Auxiliary Results 279 -- 6.2. Measurable Linear Automorphisms 282 -- 6.3. Linear Transformations 285 -- 6.4. Radon-nikodym Densities 288 -- 6.5. Examples Of Equivalent Measures And Linear Transformations 295 -- 6.6. Nonlinear Transformations 298 -- 6.7. Examples Of Nonlinear Transformations 308 -- 6.8. Finite Dimensional Mappings 314 -- 6.9. Malliavin's Method 316 -- 6.10. Surface Measures 321 -- 6.11. Complements And Problems 324 -- Chapter 7. Applications 333 -- 7.1. Trajectories Of Gaussian Processes 333 -- 7.2. Infinite Dimensional Wiener Processes 336 -- 7.3. Logarithmic Gradients 339 -- 7.4. Spherically Symmetric Measures 344 -- 7.5. Infinite Dimensional Diffusions 346 -- 7.6. Complements And Problems 353 -- Appendix A. Locally Convex Spaces, Operators, And Measures 361 -- A.1. Locally Convex Spaces 361 -- A.2. Linear Operators 365 -- A.3. Measures And Measurability 371. Vladimir I. Bogachev. Includes Bibliographical References (p. 391-426) And Index. Provides a systematic exposition of the modern theory of Gaussian measures. It presents complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes. Gives a systematic exposition of the modern theory of Gaussian measures. This work presents fundamental facts about finite and infinite dimensional Gaussian distributions. It covers topics including linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes.