A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation

Abstract

We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^ε$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,ω) = K(p) - V(x/ε,ω)$ in dimension $d \geq 2$. It is known that homogenization occurs as $ε\to 0$, but little is known about the statistical fluctuations of $w^ε$. Our main result shows that the variance of the solution $w^ε$ is bounded by $O(ε/|\log ε|)$. The proof relies on a modified Poincaré inequality of Talagrand.