  We prove homogenization for a class of nonconvex (possibly degenerate)
viscous Hamilton-Jacobi equations in stationary ergodic random environments in
one space dimension. The results concern Hamiltonians of the form
$G(p)+V(x,\omega)$, where the nonlinearity $G$ is a minimum of two or more
convex functions with the same absolute minimum, and the potential $V$ is a
bounded stationary process satisfying an additional scaled hill and valley
condition. This condition is trivially satisfied in the inviscid case, while it
is equivalent to the original hill and valley condition of A. Yilmaz and O.
Zeitouni [31] in the uniformly elliptic case. Our approach is based on PDE
methods and does not rely on representation formulas for solutions. Using only
comparison with suitably constructed super- and sub- solutions, we obtain tight
upper and lower bounds for solutions with linear initial data $x\mapsto \theta
x$. Another important ingredient is a general result of P. Cardaliaguet and
P.E. Souganidis [13] which guarantees the existence of sublinear correctors for
all $\theta$ outside "flat parts" of effective Hamiltonians associated with the
convex functions from which $G$ is built. We derive crucial derivative
estimates for these correctors which allow us to use them as correctors for
$G$.
