H\"older regularity of stationary measures

TL;DR

This paper proves that stationary measures of certain smooth random dynamical systems are H"older continuous, extending classical results on linear groups to more general systems with finite moment conditions.

Contribution

It generalizes the H"older regularity of stationary measures from linear groups to broader smooth dynamical systems under non-degeneracy conditions.

Findings

Stationary measures are H"older continuous under specified conditions.

The result applies to systems with finite moment of the differential norm.

Generalizes classical linear group results to broader dynamical systems.

Abstract

We consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be H\"older continuous. The result is a vast generalization of the classical statement on H\"older continuity of stationary measures of random walks on linear groups.