Weak Random Periodic Solutions of Random Dynamical Systems

TL;DR

This paper introduces the concept of weak random periodic solutions in random dynamical systems, explores their existence and relationship with invariant measures, and illustrates these phenomena through concrete examples involving stochastic differential equations.

Contribution

It is the first to define and analyze weak random periodic solutions and measures, establishing their role in invariant measure existence within random dynamical systems.

Findings

Existence of weak random periodic solutions demonstrated.

Relationship between weak solutions and invariant measures established.

Concrete examples illustrate weak random periodic phenomena.

Abstract

We first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their relationship with weak random periodic solutions. In particular, we establish the existence of invariant measures of random dynamical systems by virtue of their weak random periodic solutions. Finally, we use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.