Stochastic Homogenization of a Class of Quasiconvex and Possibly Degenerate Viscous HJ Equations in 1D

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 $\R$. 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''.