Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium
Elena Kosygina, Srinivasa R. S. Varadhan · Communications on Pure and Applied Mathematics · 2007
Abstract We consider a family {uϵ(t, x, ω)}, ϵ < 0, of solutions to the equation ∂uϵ/∂t+ ϵΔuϵ/2 +H(t/ϵ,x/ϵ, ∇uϵ, ω) = 0 with the terminal datauϵ(T, x, ω) =U(x). Assuming that the dependence of the HamiltonianH(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and thatHis convex inpand satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, ofuϵ(t, x, ω) as ϵ → 0 to the solutionu(t, x) of a deterministic averaged equation ∂u/∂t+H̄(∇u) = 0,u(T, x) =U(x). The “effective” HamiltonianH̄is given by a variational formula. © 2007 Wiley Periodicals, Inc.