Gaussian Measures (Mathematical Surveys & 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. 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.