目录
 前辅文
 Preface
 Chapter 1. Finite Dimensional Gaussian Distributions
  1.1. Gaussian measures on the real line
  1.2. Multivariate Gaussian distributions
  1.3. Hermite polynomials
  1.4. The Ornstein-Uhlenbeck semigroup
  1.5. Sobolev classes
  1.6. Hypercontractivity
  1.7. Several useful estimates
  1.8. Convexity inequalities
  1.9. Characterizations of Gaussian measures
  1.10. Complements and problems
 Chapter 2.Infinite Dimensional Gaussian Distributions
  2.1. Cylindrical sets
  2.2. Basic definitions
  2.3. Examples
  2.4. The Cameron-Martin space
  2.5. Zero-one laws
  2.6. Separability and oscillations
  2.7. Equivalence and singularity
  2.8. Measurable seminorms
  2.9. The Ornstein-Uhlenbeck semigroup
  2.10.Measurable linear functionals
  2.11. Stochastic integrals
  2.12. Complements and problems
 Chapter 3. Radon Gaussian Measures
  3.1. Radon measures
  3.2. Basic properties of Radon Gaussian measures
  3.3. Gaussian covariances
  3.4. The structure of Radon Gaussian measures
  3.5. Gaussian series
  3.6. Supports of Gaussian measures
  3.7. Measurable linear operators
  3.8. Weak convergence of Gaussian measures
  3.9. Abstract Wiener spaces
  3.10. Conditional measures and conditional expectations
  3.11. Complements and problems
 Chapter 4. Convexity of Gaussian Measures
  4.1. Gaussian symmetrization
  4.2. Ehrhard's inequality
  4.3. Isoperimetric inequalities
  4.4. Convex functions
  4.5. H-Lipschitzian functions
  4.6. Correlation inequalities
  4.7. The Onsager-Machlup functions
  4.8. Small ball probabilities
  4.9. Large deviations
  4.10. Complements and problems
 Chapter 5. Sobolev Classes over Gaussian Measures
  5.1. Integration by parts
  5.2. The Sobolev classes Wp,r and Dp,n
  5.3. The Sobolev classes HP,r
  5.4. Properties of Sobolev classes and examples
  5.5. The logarithmic Sobolev inequality
  5.6. Multipliers and Meyer's inequalities
  5.7. Equivalence of different definitions
  5.8. Divergence of vector fields
  5.9. Gaussian capacities
  5.10 Measurable polynomials
  5.11. Differentiability of H-Lipschitzian functions
  5.12. Complements and problems
 Chapter 6. Nonlinear Transformations of Gaussian Measures
  6.1. Auxiliary results
  6.2. Measurable linear automorphisms
  6.3. Linear transformations
  6.4. Radon-Nikodym densities
  6.5. Examples of equivalent measures and linear transformations
  6.6. Nonlinear transformations
  6.7. Examples of nonlinear transformations
  6.8. Finite dimensional mappings
  6.9. Malliavin's method
  6.10. Surface measures
  6.11. Complements and problems
 Chapter 7. Applications
  7.1. Trajectories of Gaussian processes
  7.2. Infinite dimensional Wiener processes
  7.3. Logarithmic gradients
  7.4. Spherically symmetric measures
  7.5. Infinite dimensional diffusions
  7.6. Complements and problems
 Appendix A. Locally Convex Spaces, Operators, and Measures
  A.1. Locally convex spaces
  A.2. Linear operators
  A.3. Measures and measurability
 Bibliographical Comments
 References
 Index
 前辅文
 Preface
 Chapter 1. Finite Dimensional Gaussian Distributions
  1.1. Gaussian measures on the real line
  1.2. Multivariate Gaussian distributions
  1.3. Hermite polynomials
  1.4. The Ornstein-Uhlenbeck semigroup
  1.5. Sobolev classes
  1.6. Hypercontractivity
  1.7. Several useful estimates
  1.8. Convexity inequalities
  1.9. Characterizations of Gaussian measures
  1.10. Complements and problems
 Chapter 2.Infinite Dimensional Gaussian Distributions
  2.1. Cylindrical sets
  2.2. Basic definitions
  2.3. Examples
  2.4. The Cameron-Martin space
  2.5. Zero-one laws
  2.6. Separability and oscillations
  2.7. Equivalence and singularity
  2.8. Measurable seminorms
  2.9. The Ornstein-Uhlenbeck semigroup
  2.10.Measurable linear functionals
  2.11. Stochastic integrals
  2.12. Complements and problems
 Chapter 3. Radon Gaussian Measures
  3.1. Radon measures
  3.2. Basic properties of Radon Gaussian measures
  3.3. Gaussian covariances
  3.4. The structure of Radon Gaussian measures
  3.5. Gaussian series
  3.6. Supports of Gaussian measures
  3.7. Measurable linear operators
  3.8. Weak convergence of Gaussian measures
  3.9. Abstract Wiener spaces
  3.10. Conditional measures and conditional expectations
  3.11. Complements and problems
 Chapter 4. Convexity of Gaussian Measures
  4.1. Gaussian symmetrization
  4.2. Ehrhard's inequality
  4.3. Isoperimetric inequalities
  4.4. Convex functions
  4.5. H-Lipschitzian functions
  4.6. Correlation inequalities
  4.7. The Onsager-Machlup functions
  4.8. Small ball probabilities
  4.9. Large deviations
  4.10. Complements and problems
 Chapter 5. Sobolev Classes over Gaussian Measures
  5.1. Integration by parts
  5.2. The Sobolev classes Wp,r and Dp,n
  5.3. The Sobolev classes HP,r
  5.4. Properties of Sobolev classes and examples
  5.5. The logarithmic Sobolev inequality
  5.6. Multipliers and Meyer's inequalities
  5.7. Equivalence of different definitions
  5.8. Divergence of vector fields
  5.9. Gaussian capacities
  5.10 Measurable polynomials
  5.11. Differentiability of H-Lipschitzian functions
  5.12. Complements and problems
 Chapter 6. Nonlinear Transformations of Gaussian Measures
  6.1. Auxiliary results
  6.2. Measurable linear automorphisms
  6.3. Linear transformations
  6.4. Radon-Nikodym densities
  6.5. Examples of equivalent measures and linear transformations
  6.6. Nonlinear transformations
  6.7. Examples of nonlinear transformations
  6.8. Finite dimensional mappings
  6.9. Malliavin's method
  6.10. Surface measures
  6.11. Complements and problems
 Chapter 7. Applications
  7.1. Trajectories of Gaussian processes
  7.2. Infinite dimensional Wiener processes
  7.3. Logarithmic gradients
  7.4. Spherically symmetric measures
  7.5. Infinite dimensional diffusions
  7.6. Complements and problems
 Appendix A. Locally Convex Spaces, Operators, and Measures
  A.1. Locally convex spaces
  A.2. Linear operators
  A.3. Measures and measurability
 Bibliographical Comments
 References
 Index