nonnegative definite for each $(e,x) \in \partial B_1\times{\mathbb{R}^d}$; in particular, the diffusive term may vanish or be the Laplacian. The Hamiltonian $H(\xi,x)$ is assumed to be positively homogeneous of order $p\in[1,\infty)$ in $\xi$, but is not necessarily convex in $\xi$. Both $A(e,\cdot)$ and $H(\xi,\cdot)$ are assumed to be stationary random fields sampled by a probability measure ${\mathbb{P}}$ which satisfies a finite range of dependence.

The interest is in describing the behavior of solutions of  for $0<{\varepsilon}\ll 1$. The main result is a characterization of the limit of $u^{\varepsilon}(x,t)$, subject to suitable initial conditions, as ${\varepsilon}\to 0$. We show that $u^{\varepsilon}$ converges locally uniformly, with ${\mathbb{P}}$–probability one, to the solution $u$ of a deterministic equation of the form $$\label{e.homogenized}
\partial_t u+ \overline H(Du) = 0 \quad \mbox{in} \ {\mathbb{R}^d}\times (0,\infty),$$ with an effective Hamiltonian $\overline H:{\mathbb{R}^d}\to{\mathbb{R}}$ which has sublevel sets that are star-shaped with respect to the origin. This is the first stochastic homogenization result for a viscous equation with a Hamiltonian which may be nonconvex in $Du^{\varepsilon}$, or for a diffusion matrix which may have dependence on $Du^{\varepsilon}$.

A particular case of  satisfying our assumptions is the *equation of forced mean curvature motion* $$\label{e.MCM}
\partial_t u^{\varepsilon}- {\varepsilon}\operatorname{tr}\left( \left( I_d - 