This book is devoted to the asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces. It is divided into three parts: Markovian dynamical systems; invariant measures for stochastic evolution equations; and invariant measures for specific models. The focus is on models of dynamical processes affected by white noise, which are described by partial differential equations such as the reaction-diffusion equations or Navier-Stokes equations. Besides existence and uniqueness questions, the authors pay special attention to the asymptotic behavior of the solutions, to invariant measures and ergodicity. The authors present some of the results found here for the first time. For all whose research interests involve stochastic modeling, dynamical systems, or ergodic theory, this book will be an essential purchase.
Author(s): G. Da Prato, J. Zabczyk
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1996
Language: English
Pages: 351
Contents......Page 5
Preface......Page 9
I Markovian Dynamical Systems......Page 13
1.1 Basic concepts......Page 15
1.2 Ergodic Systems and the Koopman-von Neumann Theorem......Page 17
2.1 Markovian semigroups......Page 23
2.2 Canonical systems and their continuity.......Page 26
3.1 The Krylov-Bogoliubov existence theorem.......Page 32
3.2 Characterizations of ergodic measures......Page 34
3.3 The strong law of large numbers......Page 42
3.4 Mixing and recurrence......Page 45
3.5 Limit behaviour of Pt, t > 0.......Page 51
4.1 Regular, strong Feller and irreducible semigroups......Page 53
4.2 Doob's theorem......Page 55
II Invariant measures for stochastic evolution equations......Page 61
5.1 Introduction......Page 63
5.2 Wiener and Ornstein-Uhlenbeck processes......Page 64
5.2.1 Stochastic integrals and convolutions......Page 68
5.3 Stochastic evolution equations.......Page 77
5.4.1 Differentiable dependence on initial datum......Page 81
5.4.2 Kolmogorov equation......Page 82
5.5 Dissipative stochastic systems......Page 83
5.5.1 Generalities about dissipative mappings......Page 84
5.5.2 Existence of solutions for deterministic equations......Page 87
5.5.3 Existence of solutions for stochastic equations in Hilbert spaces.......Page 92
5.5.4 Existence of solutions for stochastic equations in Banach spaces......Page 99
6.1 Existence from boundedness.......Page 101
6.2 Linear systems......Page 108
6.2.1 A description of invariant measures.......Page 109
6.2.2 Invariant measures and recurrence.......Page 114
6.3 Dissipative systems......Page 116
6.3.1 General noise.......Page 117
6.3.2 Additive noise.......Page 120
6.4 Genuinely dissipative systems......Page 126
6.5 Dissipative systems in Banach spaces......Page 129
7.1 Strong Feller property for non-degenerate diffusions......Page 133
7.2 Strong Feller property for degenerate diffusion......Page 141
7.3 Irreducibility for non-degenerate diffusions.......Page 149
7.4 Irreducibility for equations with additive noise......Page 152
8.1 Introduction......Page 159
8.2 Sobolev spaces......Page 161
8.3 Properties of the semigroup Rt, t > 0, on L2(H, u)......Page 165
8.4 Existence and absolute continuity of the invariant measure of Pt, t > 0, with respect to p......Page 168
8.5 Locally Lipschitz nonlinearities......Page 171
8.6 Gradient systems......Page 172
8.7 Regularity of the density when C is variational......Page 177
8.8 Further regularity results in the diagonal case......Page 180
III Invariant measures for specific models......Page 187
9.1 Introduction......Page 189
9.2.1 General properties......Page 190
9.2.2 Second order dissipative systems......Page 193
9.2.3 Comments on nonlinear equations......Page 195
9.3 Ornstein-Uhlenbeck processes in finance......Page 196
9.4 Ornstein-Uhlenbeck processes in chaotic environment......Page 198
9.4.1 Cylindrical noise......Page 199
9.4.2 Chaotic noise......Page 206
10.1 Introduction.............Page 211
10.2 Linear case......Page 212
10.3 Nonlinear equations......Page 215
11.1 Introduction......Page 223
11.2 Finite interval. Lipschitz coefficients......Page 225
11.2.1 Existence and uniqueness of solutions......Page 226
11.2.2 Existence and uniqueness of invariant measures......Page 227
11.3 Equations with non-Lipschitz coefficients......Page 229
11.4 Reaction-diffusion equations on d dimensional spaces......Page 231
12.1 Introduction......Page 237
12.2 Classical spin systems......Page 239
12.3 Quantum lattice systems......Page 247
13.1 Introduction......Page 253
13.2 Equations with non-homogeneous boundary conditions......Page 255
13.3 Equations with Neumann boundary conditions......Page 260
13.4 Ergodic solutions.........Page 266
14.1 Introduction......Page 269
14.2 Existence of solutions......Page 272
14.3 Strong Feller property......Page 277
14.4.1 Existence......Page 280
14.4.2 Uniqueness......Page 288
15.1 Preliminaries......Page 293
15.2 Local existence and uniqueness results......Page 296
15.3 A priori estimates and global existence......Page 303
15.4 Existence of an invariant measure......Page 307
IV Appendices......Page 317
A.1......Page 319
B.1......Page 323
C.1......Page 329
Bibliography......Page 333
Index......Page 350
