thus easier to read. It asserts that the initial value problem for  homogenizes almost-surely at an algebraic rate, at least for Lipschitz continuous solutions (the Lipschitz assumption can be removed, see Remark \[r.lipschitz\] below).

\[t.homogenization\] Let ${\mathbb{P}}$ be a probability measure on $(\Omega,{\mathcal{F}})$ which satisfies (P1) and (P2). Then there exists a universal exponent $\alpha>0$ and a function $\overline H\in C^{0,\frac 27-}_{\mathrm{loc}}({\mathbb{R}^d})$ satisfying, for every $\xi\in{\mathbb{R}^d}$, $$c_0 \left| \xi \right|^p \leq \overline H(\xi) \leq C_0 \left| \xi \right|^p$$ such that, for every $T\ge1$ and $u,u^{\varepsilon}\in W^{1,\infty} ({\mathbb{R}^d}\times [0,T])$ satisfying $$\left\{ 
\begin{aligned}
& \partial_t u^{\varepsilon}-{\varepsilon}\operatorname{tr}\left( A\left(Du^{\varepsilon},\frac x{\varepsilon}\right)D^2u^{\varepsilon}\right) + H\left(Du^{\varepsilon},\frac x{\varepsilon}\right) = 0 &  \mbox{in} & \ {\mathbb{R}^d}\times (0,T], \\
& \partial_t u +  \overline H(Du) = 0 & \mbox{in} & \ {\mathbb{R}^d}\times (0,T], \\
& u^{\varepsilon}(\cdot,0) = u(\cdot,0) & \mbox{on} & \ {\mathbb{R}^d},
\end{aligned}
\right.$$ we have $${\mathbb{P}}\left[ \sup_{R\geq1} \limsup_{{\varepsilon}\to 0} \, \sup_{(x,t) \in B_R\times [0,T]}  {\varepsilon}^{-\alpha} \left| u^{\varepsilon}(x,t)  - u(x,t) \right| =  0 \right]  = 1.$$

The proof of Theorem \[t.homogenization\] is given at the end of Section \[s.outline\].

As we will see from the argument, the exponent $\alpha$ in Theorem \[t.homogenization\] can be taken to be any positive number smaller than $\frac{1}{90}$. Needless to say, this is not optimal, 