$$\label{}
\operatorname{tr}^*(A(0,x)X) := \limsup_{\xi\to 0, \ \xi\neq 0,\  x'\to x}\operatorname{tr}(A(\xi,x')X), 
$$
$$
 \operatorname{tr}_*(A(0,x)X) := \liminf_{\xi\to 0, \ \xi\neq 0, \ x'\to x} \operatorname{tr}(A(\xi,x') X).$$ The definitions of solution for other equations encountered in this paper (such as the metric problem and approximate corrector problem) are completely analogous.

Outline of the paper
--------------------

In the next section, we state the main result, reduce it to auxiliary results which are the focus of the rest of the paper and show that Theorem \[t.homogenization\] is a corollary of it. Sections \[s.fluctuations\] and \[s.means\] are the heart of the paper: there we give the proof of homogenization for the planar metric problem. We give an estimate on the stochastic fluctuations in Section \[s.fluctuations\] and then Section \[s.means\] contains the convergence of the statistical bias. The proof of Theorem \[t.homogenization\] is completed in the final two sections, where we provide a deterministic link between the planar metric problem and the approximate correctors (Section \[s.correctors\]) and between the approximate correctors and the full time-dependent, initial-value problem (Section \[s.timedependent\]). Appendix \[s.metricproblem\] contains some auxiliary results on well-posedness and global Lipschitz bounds for the metric, approximate corrector and full time-dependent problems.

Overview of the proof of Theorem \[t.homogenization\] 