An introduction to infinite-dimensional analysis / Giuseppe Da Prato.

Format:

Book

Author/Creator:

Da Prato, Giuseppe.

Series:

Universitext

Language:

English

Subjects (All):

Dimensional analysis.

Functional analysis.

Physical Description:

x, 208 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, 2006.

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 b" for an audience knowing basic functional analysis and measure theory but not necessarily probability theory b" to analysis in a separable Hilbert space of infinite dimension. Starting from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate some basic stochastic dynamical systems (including dissipative nonlinearities) and Markov semi-groups, paying special attention to their long-time behavior: ergodicity, invariant measure. Here fundamental results like the theorems of Prokhorov, Von Neumann, Krylov-Bogoliubov and Khas'minski are proved. The last chapter is devoted to gradient systems and their asymptotic behavior.

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.

Notes:

Includes bibliographical references (pages [207]-208) and index.

ISBN:

3540290206

OCLC:

70267950

Publisher Number:

9783540290209