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 + ϵΔ u 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. © 2007 Wiley Periodicals, Inc.