Asymptotic expansion of the annealed Green's function and its derivatives

TL;DR

This paper derives high-precision asymptotic expansions for the annealed Green's function and its derivatives in random elliptic equations in dimensions three and higher, advancing stochastic homogenization theory.

Contribution

It provides the first detailed asymptotic expansions of the annealed Green's function up to high order, surpassing previous results in stochastic homogenization.

Findings

Asymptotic expansion of Green's function up to order 4 in 3D

Expansion up to order d+2 for dimensions d≥4

Enhanced precision beyond prior stochastic homogenization results

Abstract

We consider random elliptic equations in dimension $d\geq 3$ at small ellipticity contrast. We derive the large-distance asymptotic expansion of the annealed Green's function up to order $4$ in $d=3$ and up to order $d+2$ for $d\geq 4$. We also derive asymptotic expansions of its derivatives. The obtained precision lies far beyond what is established in prior results in stochastic homogenization theory. Our proof builds on a recent breakthrough in perturbative stochastic homogenization by Bourgain in a refined version shown by Kim and the second author, and on Fourier-analytic techniques of Uchiyama.