Discrete approximation of the viscous HJ equation

We consider a stochastic discretization of the stationary viscous Hamilton-Jacobi equation on the flat d-dimensional torus T-d associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on T-d, up to additive constants. By additionally assuming a technical condition on the associated Lagrangian, we show that each solution of the viscous Hamilton-Jacobi equation is the limit of solutions of the discrete problems, as the discretization step goes to zero.