On shadowing and Stochastic Stability

TL;DR

This paper investigates how stochastic perturbations affect the long-term behavior of dynamical systems with shadowing, showing that stationary measures converge to physical measures as noise diminishes.

Contribution

It establishes the convergence of stationary measures to physical measures in shadowing dynamical systems under small random perturbations.

Findings

Stationary measures converge to physical measures as noise approaches zero.

Time averages along pseudo-trajectories align with stationary measures.

Shadowing property ensures stability of physical measures under stochastic perturbations.

Abstract

In this paper, we study stochastic stability of a dynamical system with shadowing property, which evolves under small random perturbation. We prove that time averages along the pseudo-trajectory converge with respect to stationary measure for the randomly perturbed dynamics. In particular, we prove that stationary measures converge to physical measures when the noise level goes to zero if dynamical system has a shadowing property.