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

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