  In this paper, we study the existence of random periodic solutions for
semilinear SPDEs on a bounded domain with a smooth boundary. We identify them
as the solutions of coupled forward-backward infinite horizon stochastic
integral equations on $L^2(D)$ in general cases. For this we use Mercer's
Theorem and eigenvalues and eigenfunctions of the second order differential
operators in the infinite horizon integral equations. We then use the argument
of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T],
L^2(\Omega\times D))$ and generalized Schauder's fixed point theorem to prove
the existence of a solution of the integral equations. This is the first paper
in literature to study random periodic solutions of SPDEs. Our result is also
new in finding semi-stable stationary solution for non-dissipative SPDEs, while
in literature the classical method is to use the pull-back technique so
researchers were only able to find stable stationary solutions for dissipative
systems.
