  We consider a stochastic discretization of the stationary viscous Hamilton
Jacobi equation on the flat d dimensional torus, associated with a Hamiltonian,
convex and superlinear in the momentum variable. We show that each discrete
problem admits a unique continuous solution on the torus, 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.
