  The aim of this paper is to prove the existence and smoothness of stable and
unstable invariant manifolds for a stochastic delayed partial differential
equation of parabolic type. The stochastic delayed partial differential
equation is firstly transformed into a random delayed partial differential
equation by a conjugation, which is then recast into a Hilbert space. For the
auxiliary equation, the variation of constants formula holds and we show the
existence of Lipschitz continuous stable and unstable manifolds by the
Lyapunov-Perron method. Subsequently, we prove the smoothness of these
invariant manifolds under appropriate spectral gap condition by carefully
investigating the smoothness of auxiliary equation, after which, we obtain the
invariant manifolds of the original equation by projection and inverse
transformation. Eventually, we illustrate the obtained theoretical results by
their application to a stochastic single-species population model.
