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

Abstract

We consider random elliptic equations in dimension $d\ge 3$ 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 $d\ge 4$. 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.

15 pages. This article has been split off from arXiv:2103.17019 [math.AP] by the same authors