An abstract criterion on the existence and global stability of stationary solutions for random dynamical systems and its applications

TL;DR

This paper introduces a simple, verifiable criterion for the existence and global stability of stationary solutions in random dynamical systems, with applications to stochastic differential equations exhibiting complex noise-driven behaviors.

Contribution

It provides a novel, concise criterion for stability in RDSs that is easier to verify and sharper than previous conditions, applicable to a broad class of SDEs.

Findings

All pullback trajectories' omega-limit sets consist of nontrivial random equilibria.

The criterion is both sufficient and sharp for stability analysis of SDEs.

The proof differs from classical RDS approaches, offering new insights.

Abstract

We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs). As a consequence, we can show that the $\omega$-limit sets of all pullback trajectories of semilnear/nonlinear stochastic differential equations (SDEs) with additive/multiplicative white noise are composed of nontrivial random equilibria. The proof is different from the classical RDS scheme, which was established in \cite{CKS}. Furthermore, in the applications of stability analysis for SDEs, our conditions are not only sufficient but indeed sharp.