Contents Introduction 1 Part 1. The stochastic semiflow §1.1 Basic concepts 6 §1.2 Flows and cocycles of semilinear see's 7 (a) Linear see's 8 (b) Semilinear see's 27 §1.3 Semilinear spde's: Lipschitz nonlinearity 35 §1.4 Semilinear spde's: Non-Lipschitz nonlinearity 44 (a) Stochastic reaction diffusion equations 44 (b) Burgers equation with additive noise 58 Part 2. Existence of stable and unstable manifolds §2.1 Hyperbolicity of a stationary trajectory 69 §2.2 The nonlinear ergodic theorem 73 §2.3 Proof of the local stable manifold theorem 77 §2.4 The local stable manifold theorem for see's and spde's 95 (a) See's: Additive noise 95 (b) Semilinear see's: Linear noise 98 (c) Semilinear parabolic spde's: Lipschitz nonlinearity 100 (d) Stochastic reaction diffusion equations: Dissipative nonlinearity 100 (e) Stochastic Burgers equation: Additive noise 101 Acknowledgments 102 Bibliography 103 v
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