  We prove homogenization for viscous Hamilton-Jacobi equations with a
Hamiltonian of the form $G(p)+V(x,\omega)$ for a wide class of stationary
ergodic random media in one space dimension. The momentum part $G(p)$ of the
Hamiltonian is a general (nonconvex) continuous function with superlinear
growth at infinity, and the potential $V(x,\omega)$ is bounded and Lipschitz
continuous. The class of random media we consider is defined by an explicit
hill and valley condition on the diffusivity-potential pair which is fulfilled
as long as the environment is not ``rigid''.
