Grigorii Monakov, of the Institute for Advanced Study, presents this Joint IAS/PU Groups and Dynamics Seminar talk on stationary measures in random dynamical systems, the probabilistic analog of invariant measures for classical dynamics. He opens with the contrast that motivates the talk: an invariant measure for a diffeomorphism generally has no regularity at all, yet stationary measures of smooth random dynamical systems turn out to be Holder continuous under mild nondegeneracy conditions. Monakov walks through the main ideas behind the proof of this regularity result, laying out the nondegeneracy assumptions required and the mechanism that forces smoothness onto an otherwise arbitrary measure. He closes by sketching applications of the result within random dynamics and related areas. The talk assumes familiarity with dynamical systems and measure theory and is pitched at a research seminar audience at Princeton and IAS, recorded at Simonyi Hall. It runs just over an hour and stays close to the technical proof rather than offering a broad survey.