Central Limit Theorem¶for Stochastic Hamilton–Jacobi Equations

Abstract:

We study the asymptotic behavior of \(\), 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 ^ɛ converges to a deterministic function \(\) 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 ^ɛ(x,t) can be (stochastically) represented as\(\) , where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and \(\), where ω is a random function that enjoys some mild regularity.