  We prove the homogenization of a class of one-dimensional viscous
Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the
gradient variable. Due to the special form of the Hamiltonians, the solutions
of these PDEs with linear initial conditions have representations involving
exponential expectations of controlled Brownian motion in a random potential.
The effective Hamiltonian is the asymptotic rate of growth of these exponential
expectations as time goes to infinity and is explicit in terms of the tilted
free energy of (uncontrolled) Brownian motion in a random potential. The proof
involves large deviations, construction of correctors which lead to exponential
martingales, and identification of asymptotically optimal policies.
