A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

TL;DR

This paper establishes Gevrey regularity of order 2 for homogenized coefficients in Poisson point processes, extending previous results by overcoming the challenge of non-uniform local finiteness.

Contribution

It provides the first proof of Gevrey regularity for homogenized coefficients in the Poisson setting, utilizing independence properties of Poisson processes.

Findings

Gevrey regularity of order 2 proven for Poisson homogenized coefficients

Extension of previous qualitative regularity results to quantitative Gevrey regularity

Utilization of independence properties of Poisson processes in the proof

Abstract

In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order~2. The new ingredient is the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.