  We prove homogenization for possibly degenerate viscous Hamilton-Jacobi
equations with a Hamiltonian of the form $G(p)+V(x,\omega)$, where $G$ is a
quasiconvex, locally Lipschitz function with superlinear growth, the potential
$V(x,\omega)$ is bounded and Lipschitz continuous, and the diffusion
coefficient $a(x,\omega)$ is allowed to vanish on some regions or even on the
whole $\mathbb{R}$. The class of random media we consider is defined by an
explicit scaled hill condition on the pair $(a,V)$ which is fulfilled as long
as the environment is not ``rigid''.
