On the linearization of infinite-dimensional random dynamical systems

TL;DR

This paper extends the Grobman-Hartman linearization theorem to infinite-dimensional random dynamical systems, allowing for non-invertible linear parts and central directions, with results on conjugacy regularity.

Contribution

It introduces a new linearization theorem for infinite-dimensional random systems, relaxing hyperbolicity and invertibility assumptions, and establishes Hölder continuity of conjugacies under growth conditions.

Findings

Linearization holds for non-invertible linear parts.

Existence of conjugacies without requiring nonuniform hyperbolicity.

Hölder continuity of conjugacies on bounded sets.

Abstract

We present a new version of the Grobman-Hartman's linearization theorem for random dynamics. Our result holds for infinite dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the non-linear perturbations, we don't require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we don't prescribe any behaviour for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are H\"older continuous when restricted to bounded subsets of the space.