\right|^p \leq \overline H(\xi) \leq C_0 \left| \xi \right|^p
    \quad \mbox{and} \quad 
    \left| \overline H(\xi) - \overline H(\eta) \right| \leq C\left| \xi - \eta \right|^{\alpha}.$$

-   For every $L,R,T\geq1$, there exists $C({\mathrm{data}},L,R,T)\geq 1$ and $q({\mathrm{data}}) < \infty$ such that, for every $k\in{\mathbb{N}}$ and $\lambda\in (0,1]$, we have $$\begin{gathered}
    \label{e.bigestimate}
    \begin{multlined}
    {\mathbb{P}}\bigg[ \mbox{there exist} \ {\varepsilon}\in \left[ 2^{-(k+1)}, 2^{-k} \right) \ \mbox{and} \ u,u^{\varepsilon}\in W^{1,\infty}({\mathbb{R}^d}\times [0,T])  \\
    \mbox{satisfying~\eqref{e.ptfmpdes},~\eqref{e.LipLs} and} \ \sup_{(x,t) \in B_R\times [0,T]} \left| u^{\varepsilon}(x,t) - u(x,t) \right| \geq \lambda \bigg]
    \end{multlined} \\
     \leq C 2^{kq} \exp\left( -\frac{2^{k/5} \lambda^{18}}{C} \right).\end{gathered}$$

In this section we give an overview of the proof of Theorem \[t.main\]. Similar to the strategy in [@AT1; @AC], the homogenization of time-dependent problems is reduced to the convergence of a *metric problem*. Unlike in the convex case, however, we need to study the *planar* metric problem, which roughly measures the distance from a point to a plane (rather than between points, as in the usual metric problem considered in the convex case [@AS2]).

Ingredients in the proof of Theorem \[t.main\]
----------------------------------------------

In this subsection, we state the key auxiliary propositions which are used to prove Theorem \[t.main\]. Recall from the introduction that the planar metric problem is $$\label{e.planarmetric}
