Pathwise Random Periodic Solutions of Stochastic Differential Equations

TL;DR

This paper establishes the existence of random periodic solutions for semilinear stochastic differential equations by transforming the problem into coupled infinite horizon stochastic integral equations and applying fixed point theorems.

Contribution

It introduces a novel approach to prove the existence of random periodic solutions using coupled integral equations and advanced fixed point techniques.

Findings

Existence of random periodic solutions proved under general conditions.

Method applicable to stationary solutions as a special case.

Weakened conditions on the function F for solution existence.

Abstract

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(\Omega))$ and generalized Schauder's fixed point theorem to prove the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations. The condition on $F$ is then further weakened by applying the coupling method of forward and backward Gronwall inequalities. The results are also valid for stationary solutions as a special case when the period $\tau$ can be an arbitrary number.\