appears to be quite a technical assumption. Let us mention that it is redundant in the first-order case ($A=0$) or in the case that the Hamiltonian grows faster than linearly ($p>1$). The full condition is necessary to allow for $p=1$ in the viscous setting, which includes the forced mean-curvature model. We show below in Section \[s.assumptions\] that  is satisfied by each of our motivating examples. It is a generalization of the condition  for the forced mean curvature equation, which, as mentioned above, has been shown to be necessary for homogenization, even in the periodic case [@CM]. The main role of  is to ensure that Lipschitz estimates hold for solutions of the equations we consider (this is proved in Appendix \[s.metricproblem\]). The reason it is so technical is because it is the condition needed to ensure that Bernstein’s method for obtaining gradient bounds is applicable.

We work with the probability space $\Omega$, defined to be the set of all such coefficient fields: $$\Omega:= \left\{ 
(\sigma,H) \,:\, 
\mbox{$\sigma$ and $H$ satisfy~\eqref{e.Asigma},~\eqref{e.sigma},~\eqref{e.H},~\eqref{e.Hposhom},~\eqref{e.HLip} and~\eqref{e.LScoercivity}}
 \right\}.$$ We endow $\Omega$ with a family $\{ {\mathcal{F}}(U) \}$ of $\sigma$–algebras, indexed by the family of Borel subsets $U$ of ${\mathbb{R}^d}$, and defined by $$\begin{gathered}
\label{e.FU}
{\mathcal{F}}(U):= \mbox{the $\sigma$--algebra 