  In this paper we discuss a stochastic analog of Aubry-Mather theory in which
a deterministic control problem is replaced by a controlled diffusion. We prove
the existence of a minimizing measure (Mather measure) and discuss its main
properties using viscosity solutions of Hamilton-Jacobi equations. Then we
prove regularity estimates on viscosity solutions of Hamilton-Jacobi equation
using the Mather measure. Finally we apply these results to prove asymptotic
estimates on the trajectories of controlled diffusions and study the
convergence of Mather measures as the rate of diffusion vanishes.
