{#s.outline}
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In this section, we state the main result of the paper and reduce its proof to four statements which are the focus of the rest of the paper.

Statement of the main result
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We begin by stating the main result of the paper. Given $L,T\geq 1$ and ${\varepsilon}>0$, we consider solutions $u,u^{\varepsilon}\in W^{1,\infty}({\mathbb{R}^d}\times [0,T])$ of $$\label{e.ptfmpdes}
\left\{ 
\begin{aligned}
& \partial_t u^{\varepsilon}-{\varepsilon}\operatorname{tr}\left( A\left(Du^{\varepsilon},\frac x{\varepsilon}\right)D^2u^{\varepsilon}\right) + H\left(Du^{\varepsilon},\frac x{\varepsilon}\right) = 0 &  \mbox{in} & \ {\mathbb{R}^d}\times (0,T], \\
& \partial_t u +  \overline H(Du) = 0 & \mbox{in} & \ {\mathbb{R}^d}\times (0,T], \\
& u^{\varepsilon}(\cdot,0) = u(\cdot,0) & \mbox{on} & \ {\mathbb{R}^d},
\end{aligned}
\right.$$ such that $u^{\varepsilon}$ and $u$ satisfy the following Lipschitz estimate: for every $x,y\in{\mathbb{R}^d}$ and $t,s\in [0,T]$, $$\label{e.LipLs}
\left| u^{\varepsilon}(x,t) - u^{\varepsilon}(y,s) \right| \vee \left| u(x,t) - u(y,s) \right| \leq L \left( |x-y| + |t-s| \right).$$ (See Remark \[r.lipschitz\] for comments on removing the Lipschitz hypothesis .)

\[t.main\] Consider a probability measure ${\mathbb{P}}$ on $(\Omega,{\mathcal{F}})$ satisfying (P1) and (P2). Then there exists $\overline H:{\mathbb{R}^d}\to [0,\infty)$, depending only on ${\mathbb{P}}$, such that:

-   For every $L\geq 1$ and $\alpha \in \left( 0,\tfrac27\right)$, there exists $C=C({\mathrm{data}},L,\alpha)\geq 1$ such that, for every $\xi,\eta \in \overline B_L$, $$\label{e.Hbarprops}
    c_0 \left| \xi 