Log-H\"older regularity of stationary measures

TL;DR

This paper proves that stationary measures of certain Lipschitz and Hölder continuous random dynamical systems are necessarily log-Hölder continuous, given specific non-degeneracy conditions and finite logarithmic moments.

Contribution

It establishes the log-Hölder regularity of stationary measures for a class of random dynamical systems, extending understanding of measure regularity under minimal assumptions.

Findings

Stationary measures are log-Hölder continuous under non-degeneracy.

Finite logarithmic moments are sufficient for regularity results.

Applicable to Lipschitz and Hölder continuous systems.

Abstract

We consider Lipschitz and H\"{o}lder continuous random dynamical systems defined by a distribution with a finite logarithmic moment. We prove that under suitable non-degeneracy conditions every stationary measure must be $\log$-H\"{o}lder continuous.