Minimal jointly uniform attractor for nonautonomous random dynamical systems

TL;DR

This paper introduces a new concept of minimal jointly uniform attractors for nonautonomous random dynamical systems, providing a unified framework that accounts for time and randomness, with applications to differential equations and stochastic flows.

Contribution

It defines the minimal jointly uniform attractor, compares it with existing notions, and develops methods for compactifying symbol spaces and analyzing stability in nonautonomous random systems.

Findings

Introduces the minimal jointly uniform attractor concept.

Provides examples and compares with existing attractors.

Establishes stability and natural occurrence in stochastic differential equations.

Abstract

We introduce a notion of minimal uniform attractor for nonautonomous random dynamical systems, which depends jointly on time and on a random parameter. Several examples are provided to illustrate the concept and to compare it with existing notions of uniform attractors in the literature. We further apply the abstract theory to nonautonomous random differential equations with a non-compact symbol space. In particular, we develop a method to compactify the symbol space, by adapting techniques from the theory of deterministic nonautonomous differential equations. We also establish the stability of the minimal jointly uniform attractor by exploiting the relationship between deterministic and random dynamics. Finally, we show that such structures arise naturally in stochastic differential equations whose noise terms carry additional time dependence, by establishing a topological conjugacy between the resulting stochastic flows and suitable random differential equations.