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.
Author(s): Vladimir I. Bogachev
Series: Mathematical Surveys and Monographs
Publisher: AMS
Year: 1998
Language: English
Commentary: no p.347
Pages: 441
Title Page......Page 1
Copyright......Page 2
Contents......Page 3
Preface......Page 7
1.1. Gaussian measures on the real line......Page 9
1.2. Multivariate Gaussian distributions......Page 11
1.3. Hermite polynomials......Page 15
1.4. The Ornstein-Uhlenbeck semigroup......Page 17
1.5. Sobolev classes......Page 20
1.6. Hypercontractivity......Page 24
1.7. Several useful estimates......Page 27
1.8. Convexity inequalities......Page 33
1.9. Characterizations of Gaussian measures......Page 38
1.10. Complements and problems......Page 41
2.1. Cylindrical sets......Page 47
2.2. Basic definitions......Page 50
2.3. Examples......Page 56
2.4. The Cameron-Martin space......Page 67
2.5. Zero-one laws......Page 72
2.6. Separability and oscillations......Page 75
2.7. Equivalence and singularity......Page 79
2.8. Measurable seminorms......Page 82
2.9. The Ornstein-Uhlenbeck semigroup......Page 86
2.10. Measurable linear functionals......Page 87
2.11. Stochastic integrals......Page 91
2.12. Complements and problems......Page 98
3.1. Radon measures......Page 105
3.2. Basic properties of Radon Gaussian measures......Page 108
3.3. Gaussian covariances......Page 112
3.4. The structure of Radon Gaussian measures......Page 117
3.5. Gaussian series......Page 120
3.6. Supports of Gaussian measures......Page 127
3.7. Measurable linear operators......Page 130
3.8. Weak convergence of Gaussian measures......Page 137
3.9. Abstract Wiener spaces......Page 144
3.10. Conditional measures and conditional expectations......Page 148
3.11. Complements and problems......Page 150
4.1. Gaussian symmetrization......Page 165
4.2. Ehrhard's inequality......Page 170
4.3. Isoperimetric inequalities......Page 175
4.4. Convex functions......Page 179
4.5. H-Lipschitzian functions......Page 182
4.6. Correlation inequalities......Page 185
4.7. The Onsager-Machlup functions......Page 189
4.8. Small ball probabilities......Page 195
4.9. Large deviations......Page 203
4.10. Complements and problems......Page 205
5.1. Integration by parts......Page 213
5.2. The Sobolev classes Wp'' and Dr"......Page 219
5.3. The Sobolev classes HP2'......Page 223
5.4. Properties of Sobolev classes and examples......Page 226
5.5. The logarithmic Sobolev inequality......Page 234
5.6. Multipliers and Meyer's inequalities......Page 237
5.7. Equivalence of different definitions......Page 242
5.8. Divergence of vector fields......Page 246
5.9. Gaussian capacities......Page 251
5.10. Measurable polynomials......Page 257
5.11. Differentiability of H-Lipschitzian functions......Page 269
5.12. Complements and problems......Page 274
6.1. Auxiliary results......Page 287
6.2. Measurable linear automorphisms......Page 290
6.3. Linear transformations......Page 293
6.4. Radon-Nikodym densities......Page 296
6.5. Examples of equivalent measures and linear transformations......Page 303
6.6. Nonlinear transformations......Page 306
6.7. Examples of nonlinear transformations......Page 316
6.8. Finite dimensional mappings......Page 322
6.9. Malliavin's method......Page 324
6.10. Surface measures......Page 329
6.11. Complements and problems......Page 332
7.1. Trajectories of Gaussian processes......Page 341
7.2. Infinite dimensional Wiener processes......Page 344
7.3. Logarithmic gradients......Page 347
7.4. Spherically symmetric measures......Page 352
7.5. Infinite dimensional diffusions......Page 354
7.6. Complements and problems......Page 361
A.1. Locally convex spaces......Page 369
A.2. Linear operators......Page 373
A.3. Measures and measurability......Page 379
Bibliographical Comments......Page 388
References......Page 398
Index......Page 435
