Linearization of random dynamical systems in infinite dimensions

Mathematical models of real-life problems often lead to nonlinear differential or difference equations. In general, they cannot be solved explicitly and even a qualitative analysis might be difficult. At the same time, we have a well-studied and quite complete theory of linear systems. A very powerful and useful tool for investigating the qualitative properties of systems of non-linear differential equations is the Hartman-Grobman theorem, also known as linearization theorem. It states that the behaviour of a given dynamical system near a hyperbolic fixed point is qualitatively the same as a behaviour of its linearization close to origin.
Our goal is to investigate the assumptions when the homeomorphism between a nonlinear system and its linearization satisfies a Hölder condition and how is the Hölder exponent related to the spectrum of the linear operator. We are also interested in obtaining the smoothness (or differentiable) properties of topological equivalence between the system and its linearization in the infinite-dimensional Banach spaces and in extending this to the random dynamical systems.