Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium

We consider a family {u_ε (t, x, ω)}, ε > 0, of solutions to the equation ∂u_ε/∂t + εΔu_ε/2 + H(t/ε, x/ε, ∇u _ε, ω) = 0 with the terminal data u_ε(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_ε(t, x, ω) as ε → 0 to the solution u(t, x) of a deterministic averaged equation ∂u/∂t + H̄(∇u) = 0, u(T, x) = U(x). The "effective" Hamiltonian H̄ is given by a variational formula.