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

TL;DR

This paper establishes a sublinear bound on the variance of solutions to a random Hamilton-Jacobi equation, revealing that fluctuations diminish faster than previously understood as the scale parameter approaches zero.

Contribution

It provides the first sublinear variance bound for solutions of a random Hamilton-Jacobi equation using a modified Poincaré inequality, advancing understanding of statistical fluctuations in homogenization.

Findings

Variance of solution bounded by O(ε/|log ε|)

Homogenization occurs as ε approaches zero

Fluctuations diminish faster than linear bounds

Abstract

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