Table of contents for An introduction to infinite-dimensional analysis / Giuseppe Da Prato.


Bibliographic record and links to related information available from the Library of Congress catalog
Note: Electronic data is machine generated. May be incomplete or contain other coding.


Counter
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  W iener integral ....................................  42
3.4 Law of the Brownian motion in L2(0, T) ............... 45
3.4.1 Brownian bridge .............................  47
3.5 Multidimensional Brownian motions ................. 48
4   Stochastic perturbations of a dynamical system ...... 51
4.1 Introduction ................. .................. . 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
8.1 Introduction.... ............................. 109
8.2 The heat semigrofip ... .......................... 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   L2 spaces with respect to a Gaussian measure........ 125
9.1  Notations .........................................  125
9.2 Orthonormal basis in L2(H, ) ....................... 126
9.2.1  The one-dimensional case ....................  126
9.2.2  The infinite dimensional case ................... 129
9.3  W iener-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 W1,2(H, y) .................. 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 W12(H, p) .. 144
10.2 Expansions in  W iener chaos  ......................... 145
10.2.1 Compactness of the embedding
of W 1'2(H ,j ) in  L2(H,  ) ..................... 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 p ................ .. 151
10.5 Poincare and log-Sobolev inequalities ................. 155
10.5.1 Hypercontractivity............................ 159
10.6 The Sobolev  space W 2,2(H, )  ....................... 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 L2(0, 1) ................... 170
11.3 The Sobolev  space W 1,2(H,)  ....................... 172
11.4 Symmetry of the operator No ........................ 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 N2 ............................. 182
11.7 Asymptotic behaviour of Pt  ......................... 187
11.7.1 Poincare and log-Sobolev inequalities ............ 188
11.7.2 Compactness of the embedding
of W '12(H, v) in  L2(H, v) ...................... 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



Library of Congress subject headings for this publication: Dimensional analysis, Functional analysis