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

Abstract

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.

arXiv admin note: text overlap with arXiv:1502.02945

References in corpus (3)

• Random Periodic Solutions of Random Dynamical Systems
• Pathwise Random Periodic Solutions of Stochastic Differential Equations
• Stationary Solutions of SPDEs and Infinite Horizon BDSDEs

Cited by in corpus (5)

• Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations
• Random periodic solutions of non-autonomous stochastic differential equations
• Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations
• The Galerkin analysis for the random periodic solution of semilinear stochastic evolution equations
• The random periodic solution of a stochastic differential equation with a monotone drift and its numerical approximation