  The current paper is devoted to the asymptotic behavior of a class of
stochastic PDE. More precisely, with the help of the theory of integrated
semigroups and a crucial estimate of the random Stieltjes convolution, we study
the existence and smoothness of center-unstable invariant manifolds and
center-stable foliations for a class of stochastic PDE with non-dense domain
through the Lyapunov-Perron method. Finally, we give two examples about a
stochastic age-structured model and a stochastic parabolic equation to
illustrate our results.
