Homogenization of non-divergence form operators in i.i.d. random environments

TL;DR

This paper improves the convergence rates for homogenization of non-divergence form operators in i.i.d. random environments, providing sharper bounds than previously known, especially in higher dimensions.

Contribution

It establishes new, faster convergence rates for the homogenization process in i.i.d. environments, surpassing the standard $O(R^{-1})$ rate, with explicit bounds depending on the dimension.

Findings

Convergence rate of $O(R^{-3/2})$ for $d=3$

Convergence rate of $O(R^{-2}\log R)$ for $d extgreater 3$

Improved homogenization bounds in i.i.d. environments

Abstract

We study random walks in a balanced, i.i.d. random environment in $\mathbb Z^d$ for $d\geq 3$. We establish improved convergence rates for the homogenization of the Dirichlet problem associated with the corresponding non-divergence form difference operators, surpassing the $O(R^{-1})$ rate, which is expected to be optimal for environments with a finite range of dependence. In particular, the improved rates are $O(R^{-3/2})$ when $d=3$, and $O(R^{-2}\log R)$ when $d\geq 4$.