Hyers-Ulam stability for hyperbolic random dynamics

TL;DR

This paper establishes the Hyers-Ulam stability for hyperbolic random linear dynamics with small nonlinear perturbations, demonstrating the preservation of exponential dichotomy and Lyapunov exponents under these conditions.

Contribution

It introduces a random version of the shadowing property and proves Hyers-Ulam stability for hyperbolic random dynamics with nonlinear perturbations.

Findings

Small nonlinear perturbations preserve exponential dichotomy.

Random linear dynamics exhibit Hyers-Ulam stability under perturbations.

Lyapunov exponents remain conserved despite nonlinear perturbations.

Abstract

We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations.