ABSTRACT: One of the main tools of the theory of dynamical systems are the invariant measures.

For random dynamical systems, it is replaced with stationary measures, that is, 

the measures that are equal to the average of their images.

In a recent work with A. Gorodetski and G. Monakov, we show that these measures 

almost always (under extremely mild assumptions) satisfy the Hölder regularity property: 

the measure of any ball is bounded by (a constant times) some positive power of its radius.