Admissibility and generalized nonuniform dichotomies for nonautonomous Random Dynamical Systems

TL;DR

This paper introduces generalized dichotomies for nonautonomous random linear systems on Banach spaces, characterizes them via admissibility, and proves their robustness under small perturbations, extending classical exponential dichotomy concepts.

Contribution

It develops a broad framework for generalized dichotomies in random systems, including growth rates beyond exponential, and establishes their stability under perturbations.

Findings

Generalized dichotomies characterized by admissibility.

Includes classical exponential behavior as a special case.

Proves robustness of these dichotomies under small linear perturbations.

Abstract

In this paper, we introduce generalized dichotomies for nonautonomous random linear dynamical systems acting on arbitrary Banach spaces, and obtain their complete characterization in terms of an appropriate admissibility property. These generalized dichotomies are associated to growth rates satisfying mild conditions and they include the standard exponential behavior as a very particular case. As a nontrivial application, we establish the robustness property of such dichotomies under small (linear) perturbations.