On the linearization of infinite‐dimensional random dynamical systems

Abstract

AbstractWe 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 nonlinear perturbations, we do not 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 do not prescribe any behavior for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are Hölder continuous when restricted to bounded subsets of the space.