| Description |
1 online resource (x, 208 pages) |
| Series |
Universitext |
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Universitext.
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| Contents |
1 Gaussian measures in Hilbert spaces 1 -- 1.1 Notations and preliminaries 1 -- 1.2 One-dimensional Hilbert spaces 2 -- 1.3 Finite dimensional Hilbert spaces 3 -- 1.3.1 Product probabilities 3 -- 1.3.2 Definition of Gaussian measures 4 -- 1.4 Measures in Hilbert spaces 5 -- 1.5 Gaussian measures 8 -- 1.5.1 Some results on countable product of measures 9 -- 1.5.2 Definition of Gaussian measures 12 -- 1.6 Gaussian random variables 15 -- 1.6.1 Changes of variables involving Gaussian measures 17 -- 1.6.2 Independence 18 -- 1.7 The Cameron-Martin space and the white noise mapping 21 -- 2 The Cameron-Martin formula 25 -- 2.1 Introduction and setting of the problem 25 -- 2.2 Equivalence and singularity of product measures 26 -- 2.3 The Cameron-Martin formula 30 -- 2.4 The Feldman-Hajek theorem 32 -- 3 Brownian motion 35 -- 3.1 Construction of a Brownian motion 35 -- 3.2 Total variation of a Brownian motion 39 -- 3.3 Wiener integral 42 -- 3.4 Law of the Brownian motion in L[superscript 2](O, T) 45 -- 3.4.1 Brownian bridge 47 -- 3.5 Multidimensional Brownian motions 48 -- 4 Stochastic perturbations of a dynamical system 51 -- 4.2 The Ornstein-Uhlenbeck process 56 -- 4.3 The transition semigroup in the deterministic case 57 -- 4.4 The transition semigroup in the stochastic case 59 -- 4.5 A generalization 66 -- 5 Invariant measures for Markov semigroups 69 -- 5.1 Markov semigroups 69 -- 5.2 Invariant measures 72 -- 5.3 Ergodic averages 75 -- 5.4 The Von Neumann theorem 76 -- 5.5 Ergodicity 78 -- 5.6 Structure of the set of all invariant measures 80 -- 6 Weak convergence of measures 83 -- 6.1 Some additional properties of measures 83 -- 6.2 Positive functionals 85 -- 6.3 The Prokhorov theorem 89 -- 7 Existence and uniqueness of invariant measures 93 -- 7.1 The Krylov-Bogoliubov theorem 93 -- 7.2 Uniqueness of invariant measures 95 -- 7.3 Application to stochastic differential equations 98 -- 7.3.1 Existence of invariant measures 98 -- 7.3.2 Existence and uniqueness of invariant measures by monotonicity 101 -- 7.3.3 Uniqueness of invariant measures 105 -- 8 Examples of Markov semigroups 109 -- 8.2 The heat semigroup 110 -- 8.2.1 Initial value problem 113 -- 8.3 The Ornstein-Uhlenbeck semigroup 115 -- 8.3.1 Smoothing property of the Ornstein-Uhlenbeck semigroup 118 -- 8.3.2 Invariant measures 121 -- 9 L[superscript 2] spaces with respect to a Gaussian measure 125 -- 9.1 Notations 125 -- 9.2 Orthonormal basis in L[superscript 2](H, [mu]) 126 -- 9.2.1 The one-dimensional case 126 -- 9.2.2 The infinite dimensional case 129 -- 9.3 Wiener-Ito decomposition 131 -- 9.4 The classical Ornstein-Uhlenbeck semigroup 134 -- 10 Sobolev spaces for a Gaussian measure 137 -- 10.1 Derivatives in the sense of Friedrichs 138 -- 10.1.1 Some properties of W[superscript 1,2] (H, [mu]) 140 -- 10.1.2 Chain rule 141 -- 10.1.3 Gradient of a product 142 -- 10.1.4 Lipschitz continuous functions 142 -- 10.1.5 Regularity properties of functions of W[superscript 1,2] (H, [mu]) 144 -- 10.2 Expansions in Wiener chaos 145 -- 10.2.1 Compactness of the embedding of W[superscript 1,2] (H, [mu]) in L[superscript 2] (H, [mu]) 148 -- 10.3 The adjoint of D 149 -- 10.3.1 Adjoint operator 149 -- 10.3.2 The adjoint operator of D 149 -- 10.4 The Dirichlet form associated to [mu] 151 -- 10.5 Poincare and log-Sobolev inequalities 155 -- 10.5.1 Hypercontractivity 159 -- 10.6 The Sobolev space W[superscript 2,2] (H, [mu]) 161 -- 11 Gradient systems 165 -- 11.1 Introduction and setting of the problem 165 -- 11.1.1 Assumptions and notations 166 -- 11.1.2 Moreau-Yosida approximations 168 -- 11.2 A motivating example 168 -- 11.2.1 Random variables in L[superscript 2] (0, 1) 170 -- 11.3 The Sobolev space W[superscript 1,2] (H, [nu]) 172 -- 11.4 Symmetry of the operator N[subscript 0] 174 -- 11.5 Some complements on stochastic differential equations 176 -- 11.5.1 Cylindrical Wiener process and stochastic convolution 176 -- 11.5.2 Stochastic differential equations 179 -- 11.6 Self-adjointness of N[subscript 2] 182 -- 11.7 Asymptotic behaviour of P[subscript t] 187 -- 11.7.1 Poincare and log-Sobolev inequalities 188 -- 11.7.2 Compactness of the embedding of W[superscript 1,2] (H, [nu]) in L[superscript 2] (H, [nu]) 190 -- A Linear semigroups theory 193 -- A.1 Some preliminaries on spectral theory 193 -- A.1.1 Closed and closable operators 193 -- A.2 Strongly continuous semigroups 195 -- A.3 The Hille-Yosida theorem 199 -- A.3.1 Cores 203 -- A.4 Dissipative operators 204 |
| Summary |
"In this revised and extended version of his course notes from a 1-year course at Scuola Normale Superiore, Pisa, the author provides an introduction - for an audience knowing basic functional analysis and measure theory but not necessarily probability theory - to analysis in a separable Hilbert space of infinite dimension."--Jacket |
| Notes |
Introduction |
| Bibliography |
Includes bibliographical references (pages 207-208) and index |
| Notes |
Print version record |
| In |
Springer eBooks |
| Subject |
Dimensional analysis.
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Functional analysis.
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Distribution (Probability theory)
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distribution (statistics-related concept)
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Análisis dimensional
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Análisis funcional
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Distribución (Teoría de probabilidades)
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Dimensional analysis
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Functional analysis
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Stochastische analyse.
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Oneindige dimensie.
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Hilbertruimten.
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| Form |
Electronic book
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| LC no. |
2006924566 |
| ISBN |
9783540290216 |
|
3540290214 |
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9783540290209 |
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3540290206 |
|
