Influence of Lower-Order Terms on the Convergence Rates in Stochastic Homogenization of Elliptic Equations

TL;DR

This paper examines how lower-order terms influence the convergence rates in stochastic homogenization of elliptic equations, revealing their significant impact in the full space and developing new localization techniques for analysis.

Contribution

It introduces a novel localization method to analyze the effect of lower-order terms on convergence rates in stochastic homogenization.

Findings

Lower-order terms change the convergence rate to O(ε^{d/(d+2)}) in full space.

In bounded domains, the convergence rate remains at O(ε^{1/2}) regardless of lower-order terms.

A new technique localizes analysis within small grids, facilitating effective estimates.

Abstract

In this study, we investigate the convergence rates for the homogenization of elliptic equations with lower-order terms under the spectral gap assumption, in both bounded domains and the entire space. Our analysis demonstrates that lower-order terms significantly affect the convergence rate, particularly in the full space, where the rate changes from \(O(\epsilon)\) (observed without lower-order terms) to \(O(\epsilon^{d/({d+2})})\) due to their influence. In contrast, in bounded domains, the convergence rate remains \(O(\epsilon^{1/2})\), as boundary conditions exert a stronger influence than the lower-order terms. To manage the complexities introduced by lower-order terms, we developed a novel technique that localizes the analysis within small grids, enabling the application of the Poincar\'e inequality for effective estimates. This work builds upon existing frameworks, offering a refined approach to quantitative homogenization with lower-order terms.