Memoirs of the American Mathematical Society 2008; 105 pp; softcover Volume: 196 ISBN10: 0821842501 ISBN13: 9780821842508 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/196/917
 The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the longterm behavior of the solution field near stationary points. The analysis is in two parts. In Part 1, the authors prove general existence and compactness theorems for \(C^k\)cocycles of semilinear see's and spde's. The results cover a large class of semilinear see's as well as certain semilinear spde's with Lipschitz and nonLipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinitedimensional noise. In Part 2, stationary solutions are viewed as cocycleinvariant random points in the infinitedimensional state space. The pathwise local structure of solutions of semilinear see's and spde's near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution. Table of Contents Part 1. The stochastic semiflow  Basic concepts
 Flows and cocycles of semilinear see's
 Semilinear spde's: Lipschitz nonlinearity
 Semilinear spde's: NonLipschitz nonlinearity
Part 2. Existence of stable and unstable manifolds  Hyperbolicity of a stationary trajectory
 The nonlinear ergodic theorem
 Proof of the local stable manifold theorem
 The local stable manifold theorem for see's and spde's
 Acknowledgments
 Bibliography
