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,ω), where G is a quasiconvex, locally Lipschitz function with superlinear growth, the potential V(x,ω) is bounded and Lipschitz continuous, and the diffusion coefficient a(x,ω) 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''.