  This article is a sequel to [M.Z.Z.1] aimed at completing the
characterization of the pathwise local structure of solutions of semilinear
stochastic evolution equations (see's) and stochastic partial differential
equations (spde's) near stationary solutions. Stationary solution are viewed as
random points in the infinite-dimensional state space, and 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 semilinear 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).
