Absolute Continuity Theorem for Random Dynamical Systems on $R^d$

TL;DR

This paper proves an absolute continuity theorem for random dynamical systems on R^d with invariant measures, addressing challenges from non-compactness and randomness by constructing local stable manifolds and analyzing Lebesgue measure transformations.

Contribution

It provides a novel proof of the absolute continuity theorem for random dynamical systems on R^d, including the construction of local stable manifolds in non-compact settings.

Findings

Establishment of local stable manifolds for systems on R^d

Proof of absolute continuity of Lebesgue measures under holonomy maps

Handling non-compactness and randomness in the proof

Abstract

In this article we provide a proof of the so called absolute continuity theorem for random dynamical systems on $R^d$ which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincar\'e map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.