Discrete approximation of stochastic Mather measures

In this paper, we construct measures which minimize a discrete version of the stochastic Mather problem associated to a Tonelli Lagrangian $L:\T^d\times\R^d\to\R$, where $\T^d=\R^d/\Z^d$ is the flat $d$--dimensional torus. We show that the discrete variational problems approximate the stochastic Mather problem as the step of the discretisation goes to zero, in the sense that the minima of the discrete problems converge to the minimum of the stochastic Mather problem and the discrete minimizing measures converge to the unique stochastic Mather measure.