The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

TL;DR

This paper investigates how small bounded noise affects the hyperbolic structure of autonomous semilinear differential equations, demonstrating the persistence and convergence of hyperbolic solutions under stochastic perturbations.

Contribution

It establishes the existence and robustness of random hyperbolic solutions for perturbed systems, extending hyperbolicity results to stochastic nonautonomous settings.

Findings

Existence of bounded random hyperbolic solutions under small noise

Convergence of these solutions to autonomous equilibria

Application to damped wave equations with noise

Abstract

In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded \textit{random hyperbolic solution} for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.