Ergodicity for Infinite Dimensional Systems

Transcription

1 London Mathematical Society Lecture Note Series. 229 Ergodicity for Infinite Dimensional Systems G. DaPrato Scuola Normale Superiore, Pisa J. Zabczyk Polish Academy of Sciences, Warsaw If CAMBRIDGE UNIVERSITY PRESS

2 Preface ix Markovian Dynamical Systems 1 General Dynamical Systems Basic concepts Ergodic Systems and the Koopman-von Neumann Theorem 5 Canonical Markovian Systems Markovian semigroups Canonical systems and their continuity.14 Ergodic and mixing measures The Krylov-Bogoliubov existence theorem Characterizations of ergodic measures The strong law of large numbers Mixing and recurrence Limit behaviour of P t, t > 0 39 Regular Markovian systems Regular, strong Feller and irreducible semigroups Doob's theorem 43 II Invariant measures for stochastic evolution equations 49

3 vi 5 Stochastic Differential Equations Introduction Wiener and Ornstein-Uhlenbeck processes Stochastic integrals and convolutions Stochastic evolution equations Regular dependence on initial conditions and Kolmogorov equations Differentiable dependence on initial datum Kolmogorov equation Dissipative stochastic systems Generalities about dissipative mappings Existence of solutions for deterministic equations Existence of solutions for stochastic equations in Hilbert spaces Existence of solutions for stochastic equations in Banach spaces 87 6 Existence of invariant measures Existence from boundedness Linear systems A description of invariant measures Invariant measures and recurrence Dissipative systems General noise Additive noise Genuinely dissipative systems Dissipative systems in Banach spaces Uniqueness of invariant measures Strong Feller property for non-degenerate diffusions Strong Feller property for degenerate diffusion Irreducibility for non-degenerate diffusions Irreducibility for equations with additive noise Densities of invariant measures Introduction Sobolev spaces Properties of the semigroup R t, t > 0, on L 2 (H,/J,).. 153

4 vii 8.4 Existence and absolute continuity of the invariant measure of P t, t > 0, with respect to fi 'Locally Lipschitz nonlinearities Gradient systems Regularity of the density when C is variational Further regularity results in the diagonal case 168 III Invariant measures for specific models Ornstein Uhlenbeck processes Introduction Ornstein-Uhlenbeck processes of wave type General properties Second order dissipative systems Comments on nonlinear equations Ornstein-Uhlenbeck processes in finance Ornstein-Uhlenbeck processes in chaotic environment Cylindrical noise Chaotic noise Stochastic delay systems Introduction Linear case Nonlinear equations Reaction Diffusion equations Introduction Finite interval. Lipschitz coefficients Existence and uniqueness of solutions Existence and uniqueness of invariant measures Equations with non-lipschitz coefficients Reaction-diffusion equations on d dimensional spaces Spin systems Introduction Classical spin systems.., Quantum lattice systems 235

5 viii 13 Systems perturbed through the boundary Introduction Equations with non-homogeneous boundary conditions Equations with Neumann boundary conditions Ergodic solutions Burgers equation Introduction Existence of solutions Strong Feller property Invariant measure Existence Uniqueness Navier-Stokes equations Preliminaries Local existence and uniqueness results A priori estimates and global existence Existence of an invariant measure 295 IV Appendices 305 A Smoothing properties of convolutions 307 A.I 307 B An estimate on modulus of continuity 311 B.I 311 C A result on implicit functions 317 C.I 317 Bibliography 321