rely on subadditivity, we demonstrate here that the ideas in fact do not require it.

Very recently (and several months after this paper was written), Ziliotto [@Zil] produced an example of a first-order, coercive (and nonconvex) Hamilton-Jacobi equation with stationary-ergodic coefficients for which homogenization fails. Thus, in addition to negatively resolving the question of whether a general homogenization result holds for nonconvex Hamilton-Jacobi equations, Ziliotto demonstrated conclusively the impossibility of obtaining homogenization results by soft or qualitative arguments. This progress provides further motivation for developing quantitative approaches to this and similar problems.

The assumption of finite range dependence is well-motivated physically and is analogous to the standard assumption of i.i.d. in discrete probability models. It is not an assumption made for simplicity: we do not know how to relax it even to allow very quick decaying correlations of the coefficients. However, by stability arguments, we can obtain homogenization results for coefficients fields which are uniform limits of finite-range fields. This covers many typical examples, including for instance coefficients fields built by convolutions of smooth (but not compactly supported) functions against Poisson point clouds.

On a technical level, as we will see in Section \[s.fluctuations\], the finite range assumption gives us 