Abstract
In this paper, we discuss a stochastic analogue ofAubry-Mather theory in which a deterministic control problemis replaced by a controlled diffusion. We prove theexistence of a minimizing measure (Mather measure) and discussits main properties using viscosity solutionsof Hamilton-Jacobi equations. Then we prove regularity estimateson viscosity solutions of the Hamilton-Jacobi equationusing the Mather measure. Finally, we apply these results toprove asymptotic estimates on the trajectories of controlled diffusions andstudy the convergence of Mather measures as the rate of diffusion vanishes.
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