Quantitative Estimates in Elliptic Homogenization of Non-divergence Form with Unbounded Drift and an Interface

TL;DR

This paper provides quantitative estimates for elliptic homogenization in non-divergence form with unbounded drift and an interface, extending prior work by determining effective equations and convergence rates.

Contribution

It introduces a method to analyze non-divergence form problems by transforming them into divergence form with exponentially decaying coefficients near an interface.

Findings

Derived size estimates for Green function gradients.

Established optimal convergence rates in homogenization.

Extended homogenization techniques to non-divergence form with interfaces.

Abstract

This paper investigates quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, which continues the study of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682], where they investigated the limiting long time/large scale behavior of such a process under diffusive rescaling. We determine the effective equation and obtain the size estimates of the gradient of Green functions as well as the optimal (in general) convergence rates. The proof relies on transferring the non-divergence form into the divergence-form with the coefficient matrix decaying exponentially to some (different) periodic matrix on the different sides of the interface first and then investigating this special structure in homogenization of divergence form.