| Gaussian measures in Hilbert spaces | p. 1 |
| Notations and preliminaries | p. 1 |
| One-dimensional Hilbert spaces | p. 2 |
| Finite dimensional Hilbert spaces | p. 3 |
| Product probabilities | p. 3 |
| Definition of Gaussian measures | p. 4 |
| Measures in Hilbert spaces | p. 5 |
| Gaussian measures | p. 8 |
| Some results on countable product of measures | p. 9 |
| Definition of Gaussian measures | p. 12 |
| Gaussian random variables | p. 15 |
| Changes of variables involving Gaussian measures | p. 17 |
| Independence | p. 18 |
| The Cameron-Martin space and the white noise mapping | p. 21 |
| The Cameron-Martin formula | p. 25 |
| Introduction and setting of the problem | p. 25 |
| Equivalence and singularity of product measures | p. 26 |
| The Cameron-Martin formula | p. 30 |
| The Feldman-Hajek theorem | p. 32 |
| Brownian motion | p. 35 |
| Construction of a Brownian motion | p. 35 |
| Total variation of a Brownian motion | p. 39 |
| Wiener integral | p. 42 |
| Law of the Brownian motion in L[superscript 2](O, T) | p. 45 |
| Brownian bridge | p. 47 |
| Multidimensional Brownian motions | p. 48 |
| Stochastic perturbations of a dynamical system | p. 51 |
| Introduction | p. 51 |
| The Ornstein-Uhlenbeck process | p. 56 |
| The transition semigroup in the deterministic case | p. 57 |
| The transition semigroup in the stochastic case | p. 59 |
| A generalization | p. 66 |
| Invariant measures for Markov semigroups | p. 69 |
| Markov semigroups | p. 69 |
| Invariant measures | p. 72 |
| Ergodic averages | p. 75 |
| The Von Neumann theorem | p. 76 |
| Ergodicity | p. 78 |
| Structure of the set of all invariant measures | p. 80 |
| Weak convergence of measures | p. 83 |
| Some additional properties of measures | p. 83 |
| Positive functionals | p. 85 |
| The Prokhorov theorem | p. 89 |
| Existence and uniqueness of invariant measures | p. 93 |
| The Krylov-Bogoliubov theorem | p. 93 |
| Uniqueness of invariant measures | p. 95 |
| Application to stochastic differential equations | p. 98 |
| Existence of invariant measures | p. 98 |
| Existence and uniqueness of invariant measures by monotonicity | p. 101 |
| Uniqueness of invariant measures | p. 105 |
| Examples of Markov semigroups | p. 109 |
| Introduction | p. 109 |
| The heat semigroup | p. 110 |
| Initial value problem | p. 113 |
| The Ornstein-Uhlenbeck semigroup | p. 115 |
| Smoothing property of the Ornstein-Uhlenbeck semigroup | p. 118 |
| Invariant measures | p. 121 |
| L[superscript 2] spaces with respect to a Gaussian measure | p. 125 |
| Notations | p. 125 |
| Orthonormal basis in L[superscript 2](H, [mu]) | p. 126 |
| The one-dimensional case | p. 126 |
| The infinite dimensional case | p. 129 |
| Wiener-Ito decomposition | p. 131 |
| The classical Ornstein-Uhlenbeck semigroup | p. 134 |
| Sobolev spaces for a Gaussian measure | p. 137 |
| Derivatives in the sense of Friedrichs | p. 138 |
| Some properties of W[superscript 1,2] (H, [mu]) | p. 140 |
| Chain rule | p. 141 |
| Gradient of a product | p. 142 |
| Lipschitz continuous functions | p. 142 |
| Regularity properties of functions of W[superscript 1,2] (H, [mu]) | p. 144 |
| Expansions in Wiener chaos | p. 145 |
| Compactness of the embedding of W[superscript 1,2] (H, [mu]) in L[superscript 2] (H, [mu]) | p. 148 |
| The adjoint of D | p. 149 |
| Adjoint operator | p. 149 |
| The adjoint operator of D | p. 149 |
| The Dirichlet form associated to [mu] | p. 151 |
| Poincare and log-Sobolev inequalities | p. 155 |
| Hypercontractivity | p. 159 |
| The Sobolev space W[superscript 2,2] (H, [mu]) | p. 161 |
| Gradient systems | p. 165 |
| Introduction and setting of the problem | p. 165 |
| Assumptions and notations | p. 166 |
| Moreau-Yosida approximations | p. 168 |
| A motivating example | p. 168 |
| Random variables in L[superscript 2] (0, 1) | p. 170 |
| The Sobolev space W[superscript 1,2] (H, [nu]) | p. 172 |
| Symmetry of the operator N[subscript 0] | p. 174 |
| Some complements on stochastic differential equations | p. 176 |
| Cylindrical Wiener process and stochastic convolution | p. 176 |
| Stochastic differential equations | p. 179 |
| Self-adjointness of N[subscript 2] | p. 182 |
| Asymptotic behaviour of P[subscript t] | p. 187 |
| Poincare and log-Sobolev inequalities | p. 188 |
| Compactness of the embedding of W[superscript 1,2] (H, [nu]) in L[superscript 2] (H, [nu]) | p. 190 |
| Linear semigroups theory | p. 193 |
| Some preliminaries on spectral theory | p. 193 |
| Closed and closable operators | p. 193 |
| Strongly continuous semigroups | p. 195 |
| The Hille-Yosida theorem | p. 199 |
| Cores | p. 203 |
| Dissipative operators | p. 204 |
| Bibliography | p. 207 |
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