Gaussian Measures (Mathematical Surveys and Monographs)

✍ Scribed by Vladimir I. Bogachev

• Publisher: American Mathematical Society
• Year: 1998
• Tongue: English
• Leaves: 449
• Series: Mathematical Surveys and Monographs
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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