Stochastic homogenization of quasiconvex degenerate viscous HJ equations in 1d

We prove homogenization for degenerate viscous Hamilton-Jacobi equations in dimension one in stationary ergodic environments with a quasiconvex and superlinear Hamiltonian of fairly general type. We furthermore show that the effective Hamiltonian is quasiconvex. This latter result is new even in the periodic setting, despite homogenization has been known for quite some time.