Invariant manifolds for Random Dynamical Systems on Banach Spaces exhibiting generalized dichotomies

TL;DR

This paper establishes the existence of measurable invariant manifolds for small perturbations of linear random dynamical systems on Banach spaces with generalized dichotomies, applicable in both continuous and discrete time, and describes their asymptotic behavior.

Contribution

It introduces a general framework for invariant manifolds in random dynamical systems with dichotomies, extending previous results to broader settings and perturbations.

Findings

Invariant manifolds exist under small perturbations.

Asymptotic behavior matches that of the linear system.

Applicable to both continuous and discrete time systems.

Abstract

We prove the existence of measurable invariant manifolds for small perturbations of linear Random Dynamical Systems evolving on a Banach space and admitting a general type of dichotomy, both for continuous and discrete time. Moreover, the asymptotic behavior in the invariant manifold is similar to the one of the linear Random Dynamical System.