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

2025/08/11 by Lv, Xiang
#34C10 #37A30 #37H05 #60G10 #60H10 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2508.08497

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 ω-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 \citeCKS. Furthermore, in the applications of stability analysis for SDEs, our conditions are not only sufficient but indeed sharp.

Citations

• A small-gain theorem for nonlinear stochastic systems with inputs and outputs I: Additive white noise
• Synchronization by noise

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