Regularity of Measures Generated by Random Dynamics and Applications

A stationary measure in random dynamics is an analog of an invariant measure for a classical dynamical system. While there is no reason, in general, for an invariant measure of a diffeomorphism to have any regularity, it turns out that, under mild nondegeneracy assumptions, every stationary measure of a smooth random dynamical system must be Holder continuous. I will describe the main idea behind the proof of this result, as well as some of its applications.