The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations II: Existence of stable and unstable manifolds

摘要

This article is a sequel to [M.Z.Z.1] aimed at completing thecharacterization of the pathwise local structure of solutions of semilinearstochastic evolution equations (see's) and stochastic partial differentialequations (spde's) near stationary solutions. Stationary solution are viewed asrandom points in the infinite-dimensional state space, and the characterizationis expressed in terms of the almost sure long-time behavior of trajectories ofthe equation in relation to the stationary solution. More specifically, weestablish local stable manifold theorems for semilinear see's and spde's(Theorems 4.1-4.4). These results give smooth stable and unstable manifolds inthe neighborhood of a hyperbolic stationary solution of the underlyingstochastic equation. The stable and unstable manifolds are stationary, live ina stationary tubular neighborhood of the stationary solution and areasymptotically invariant under the stochastic semiflow of the see/spde. Theproof uses infinite-dimensional multiplicative ergodic theory techniques andinterpolation arguments (Theorem 2.1).