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 $w^{\epsilon }$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\omega )=K(p)-V(x/\epsilon ,\omega )$ in dimension $d\ge 2$. 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 Poincaré inequality of Talagrand.

References in corpus (3)

• Subgaussian concentration and rates of convergence in directed polymers
• Stochastic homogenization of level-set convex Hamilton-Jacobi equations
• Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations

Cited by in corpus (5)

• Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample
• Stochastic homogenization of a nonconvex Hamilton-Jacobi equation
• Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension
• Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
• Quantitative stochastic homogenization of viscous Hamilton-Jacobi equations