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

Abstract

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 $C^0([0,T],L^2(\Omega \times \mathcal{O}))$ 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.

References in corpus (5)

• Random Periodic Solutions of Random Dynamical Systems
• Pathwise Random Periodic Solutions of Stochastic Differential Equations
• Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding
• Random Periodic Processes, Periodic Measures and Ergodicity
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Cited by in corpus (3)

• Random Periodic Processes, Periodic Measures and Ergodicity
• Nonlinear Feynman-Kac formulae for SPDEs with space-time noise
• The random periodic solution of a stochastic differential equation with a monotone drift and its numerical approximation