Random periodic solutions for stochastic differential equations with non-uniform dissipativity

TL;DR

This paper investigates the existence and uniqueness of random periodic solutions for stochastic differential equations with non-uniform dissipativity, using coupling methods to handle different dissipativity conditions.

Contribution

It introduces new methods to establish the existence of random periodic solutions under non-uniform dissipativity conditions in both SDEs and functional SDEs.

Findings

Existence of distributional random periodic solutions for SDEs with dissipative drifts.

Pathwise random periodic solutions for functional SDEs with finite and infinite time lag.

Applicable coupling strategies for non-uniform dissipativity scenarios.

Abstract

This paper is concerned with the existence and uniqueness of random periodic solutions for stochastic differential equations (SDEs), where the drift terms involved need not to be uniformly dissipative. On the one hand, via the reflection coupling approach, we investigate the existence of random periodic solutions in the sense of distribution for SDEs without memory, where the drifts are merely dissipative at long distance. On the other hand, via the synchronous coupling strategy, we establish respectively the existence of pathwise random periodic solutions for functional SDEs with a finite time lag and an infinite time lag, in which the drifts are only dissipative on average rather than uniformly dissipative with respect to the time parameters.