Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems

Abstract

In this paper, we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces. We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations, which preserve the CN,τ(ω) HOlder smoothness of the random system in the space variable and the measurability of the random system in the sample point. Moreover, we also prove that the stable foliation is CN−1,τ(ω) in the base point.