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 ε 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 ε. Our main result shows that the variance of the solution w ε is bounded by O(ε/{pipe}log ε {pipe}). The proof relies on a modified Poincaré inequality of Talagrand. © 2012 Springer Science+Business Media, LLC.

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Author James H. Nolen Mathematics