Decay Properties of Invariant Measure and Application to Elliptic Homogenization of Non-divergence Form with an Interface

TL;DR

This paper studies the decay of invariant measures in elliptic homogenization with interfaces, providing new proofs and quantitative estimates for the homogenization process in non-divergence form equations.

Contribution

It offers an alternative PDE-based proof for invariant measure decay and derives quantitative homogenization estimates for elliptic equations with interfaces.

Findings

Invariant measure existence and decay properties established.

Quantitative homogenization estimates obtained.

Provides an alternative proof approach compared to previous work.

Abstract

Using the self-contained PDE analysis, this paper investigates the existence and the decay properties of the invariant measure in elliptic homogenization of non-divergence form with an interface assumptions on the leading coefficient $A$ and the drift $b$ for $b_1\equiv 0$, which partially provides an alternative proof of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682]. Moreover, as a direct application after using the analysis by the second author [Calc. Var. Partial Differ. Equ. 64(2025) No. 114], we obtain the quantitative estimates for the homogenization problem.