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?CJacobi equation with random Hamiltonian H(p,x,??)=K(p)?V(x/?,??) in dimension d??2. It is known that homogenization occurs as ???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(?/|log?|). The proof relies on a modified Poincaré inequality of Talagrand.