Stochastic homogenization of nonconvex viscous Hamilton-Jacobi equations in one space dimension

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’'.