  We study the qualitative homogenization of second order viscous
Hamilton-Jacobi equations in space-time stationary ergodic random environments.
Assuming that the Hamiltonian is convex and superquadratic in the momentum
variable (gradient) we establish a homogenization result and characterize the
effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion
matrices. The result extends previous work that required uniform ellipticity
and space-time homogeneity for the diffusion.
