Quantitative stochastic homogenization of the G equation

Abstract

We prove a quantitative rate of homogenization for the G equation in a random environment with finite range of dependence. Using ideas from percolation theory, the proof bootstraps a result of Cardaliaguet–Souganidis, who proved qualitative homogenization in a more general ergodic environment.

Appendix A. Proof of Theorem 3.3

Step 1. We show that if $x \in {\mathbb {Q}}^d$ with $|x| \ge C$, then there is $\alpha \in [c, 1]$ such that $\alpha x$ lies in the convex hull of $G_x$. Let $n \in {\mathbb {N}}$ be large enough so that

$$\begin{aligned} |n^{-1}f(nx) - {\overline{f}}(x)| \le 1. \end{aligned}$$

Let $x_0, x_1, \dots , x_m$ be an $G_x$-skeleton for nx. Then

$$\begin{aligned} x = \frac{1}{n}\sum _{k=1}^m (x_k-x_{k-1}), \end{aligned}$$

and $n \le m \le Cn$, where the first part of the inequality follows from applying ${\overline{f}}_x$ to both sides of the equation.

Step 2. We show that if $x \in {\mathbb {Q}}^d$, $|x| \ge K$, $t \ge 1$, and $tx \in {\mathbb {Z}}^d$, then there is a $z \in {\mathbb {Z}}^d$ with $f(tx) - {\overline{f}}(tx) \le f(z) - {\overline{f}}(z) + tC|x|^\nu \varphi (|x|)$. Using the previous step, write $tx = z + \sum _{k=1}^m v_k$, where $|z| \le C|x|$, ${\overline{f}}(z) \le {\overline{f}}_x(z) + C$, $v_k \in G_x$, and $m \le Ct$. Indeed, for some $\alpha \in [c, 1]$ we first write

$$\begin{aligned} \alpha x = \sum _{i=1}^{d+1} p_i v_i, \end{aligned}$$

where $v_i \in G_x$ and $p_i \ge 0$, $\sum _i p_i = 1$. Note that the sum only requires $d+1$ terms by Caratheodory’s theorem on convex hulls, since we are working in ${\mathbb {R}}^d$. To decompose tx, we write

$$\begin{aligned} tx = \sum _{i=1}^{d+1}(t\alpha ^{-1}p_i - \lfloor t\alpha ^{-1}p_i \rfloor )v_i + \sum _{i=1}^{d+1}\lfloor t\alpha ^{-1}p_i \rfloor v_i =: z + (tx-z), \end{aligned}$$

so z satisfies the required properties. By subadditivity of f and linearity of ${\overline{f}}_x$,

$$\begin{aligned} f(tx)&\le f(z) + \sum _{k=1}^m f(v_k)\\&\le f(z) + \sum _{k=1}^m {\overline{f}}_x(v_k) + C|x|^\nu \varphi (|x|)\\&= f(z) + {\overline{f}}_x(tx - z) + tC|x|^\nu \varphi (|x|). \end{aligned}$$

Finally, we write ${\overline{f}}(tx) = {\overline{f}}_x(z) + {\overline{f}}_x(tx-z)$ and subtract from both sides of the inequality above to get

$$\begin{aligned} f(tx) - {\overline{f}}(tx) \le f(z) - {\overline{f}}(z) + Ct, \end{aligned}$$

where we used the fact that ${\overline{f}}(z) \le \overline{f_x}(z) + C$.

Step 3. For some large $M > 1$ (and possibly enlarged K), the previous step yields

$$\begin{aligned} \sup _{|x| \le M^{k+1}K} f(x)-{\overline{f}}(x)&\le \sup _{|x| \le M^k K} f(x)-{\overline{f}}(x) + CM|x|^\nu \varphi (|x|)\\&\le \sup _{|x| \le M^k K} f(x)-{\overline{f}}(x) + C(M^\nu -1)|x|^\nu \varphi (|x|), \end{aligned}$$

where we made C larger in the second line to account for the change in constant. By induction on k, we conclude that

$$\begin{aligned} f(x)-{\overline{f}}(x) \le C|x|^\nu \varphi (|x|). \end{aligned}$$