much simpler context: in other words, without Lipschitz regularity, homogenization may fail.

Another special case of  includes the general class of viscous Hamilton-Jacobi equations of the form $$\label{e.viscous}
\partial_t u^{\varepsilon}- {\varepsilon}\Delta u^{\varepsilon}+ H\left( Du^{\varepsilon},\frac x{\varepsilon}\right) = 0 \quad \mbox{in} \ {\mathbb{R}^d}\times (0,\infty),$$ where $H$ satisfies, for some $p>1$ and $0<c_0\leq C_0$, $$H(t e ,x) = t^p H(e,x) \quad \mbox{and} \quad c_0 \leq H(e,x) \leq C_0, \quad \forall t\geq 0, \ e\in\partial B_1, \ x\in{\mathbb{R}^d}.$$ Our results therefore give the first large class of non-convex Hamilton-Jacobi equations for which homogenization holds in $d>1$. Even for  with the second-order term removed the conclusions are new: the only previous results even for first-order nonconvex equations to our knowledge are found in the recent papers [@ATY1; @ATY2]. The latter articles demonstrated homogenization in one space dimension and treated special cases in higher dimensions for Hamiltonians taking the form $H(p,x) = {\widetilde}H(p)+ W(x)$, an entirely different structure from the one here.

Every proof of qualitative homogenization for a Hamilton-Jacobi equation in the random setting has been based in some way on an application of the subadditive ergodic theorem. This requires the identification of a subadditive quantity whose limiting behavior controls that of the solutions to 