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The stable manifold theorum for semilinear stochastic evolution equations and stochastic partial differential equations. II: Existence of stable and unstable manifolds.

This article is a sequel, aimed at completing the characterization of
the pathwise local structure of solutions of semi-linear stochastic evolution equations (see’s)
and stochastic partial differential equations (spde’s) near stationary solutions. The characterization
is expressed in terms of the almost sure long-time behavior of trajectories of the
equation in relation to the stationary solution. More specifically, we establish local stable
manifold theorems for semi-linear see’s and spde’s (Theorems 4.1-4.4). These results give
smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution
of the underlying stochastic equation. The stable and unstable manifolds are stationary,
live in a stationary tubular neighborhood of the stationary solution and are asymptotically
invariant under the stochastic semiflow of the see/spde. The proof uses infinite-dimensional
multiplicative ergodic theory techniques and interpolation arguments (Theorem 2.1).