Central limit theorem for stochastic Hamilton-Jacobi equations

We study the asymptotic behavior of u(epsilon)(x, t) = epsilon u (x/epsilon, t/epsilon), where u solves the Hamilton-Jacobi equation u(t) + H(x, u(x)) = 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou-Tarver [RT] that u(epsilon) converges to a deterministic function (u) over bar provided H(x, p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by showing that for a class of examples, u(epsilon)(x, t) can be (stochastically) represented as (u) over bar(x, t) + root epsilon Z(x, t) + o(root epsilon), where Z(x, t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and H(x, p) = 1/2p(2) - omega(x), where omega is a random function that enjoys some mild regularity.