  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.
