Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding

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.