Measure rigidity for random dynamics on surfaces and related skew products

Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$ we study $\nu$-stationary probability measures on $M$. We prove for hyperbolic stationary measures the following trichotomy: the stable distributions are non-random, the measure is SRB, or the measure is supported on a finite set and is hence almost-surely invariant. In the proof of the above results, we study skew products with surface fibers over a measure-preserving transformation equipped with a decreasing sub-$\sigma$-algebra $\hat {\mathcal F}$ and derive a related result. A number of applications of our main theorem are presented.