Anticipating Random Periodic Solutions--II. SPDEs with Multiplicative Linear Noise

In this paper, we study the existence of random periodic solutions for semilinear stochastic partial differential equations with multiplicative linear noise on a bounded open domain $\mathcal{O}\subset {\mathbb{R}}^d$ with smooth boundary. We identify them with the solutions of coupled forward-backward infinite horizon stochastic integral equations in $L^2(\mathcal{O})$. We then use generalized Schauder’s fixed point theorem, the relative compactness of Wiener-Sobolev spaces in $C^0([0,T],L^2(\mathrm{\Omega }\times \mathcal{O}))$ and a localization argument to prove the existence of solutions of the infinite horizon integral equations, which immediately implies the existence of the random periodic solution to the corresponding SPDEs. As an example, we apply our result to the stochastic Allen-Cahn equation with a periodic potential and prove the existence of a random periodic solution using a localisation argument.