Continuity and topological structural stability for nonautonomous random attractors

TL;DR

This paper investigates how nonautonomous random attractors in differential equations behave under small random perturbations, focusing on their continuity and structural stability, with applications to stochastic and damped wave equations.

Contribution

It establishes the lower semicontinuity and persistence of gradient structures of attractors under nonautonomous random perturbations, extending stability theory.

Findings

Lower semicontinuity of nonautonomous random attractors

Persistence of gradient structure under perturbations

Application to stochastic and damped wave equations

Abstract

In this work, we study continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish lower semicontinuity of nonautonomous random attractors and to show that the gradient structure persists under nonautonomous random perturbations. Finally, we apply the abstract results in a stochastic differential equation and in a damped wave equation with a perturbation on the damping.