On the averaged Green's function of an elliptic equation with random coefficients

TL;DR

This paper refines Bourgain's harmonic analysis techniques to improve decay rate estimates for the averaged Green's function of a random elliptic operator on a lattice, advancing understanding of its higher derivatives.

Contribution

It improves the decay rate bounds of the averaged Green's function for a random elliptic operator, extending previous results and providing new estimates on higher derivatives.

Findings

Decay rate improved from -2d+ε to -3d+ε.

Established estimates on higher derivatives of the Green's function.

Enhanced understanding of the Green's function behavior in random elliptic equations.

Abstract

We consider a divergence-form elliptic difference operator on the lattice $\mathbb{Z}^d$, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which improves the key decay rate from $-2d+\epsilon$ to $-3d+\epsilon$. (The optimal decay rate is conjectured to be $-3d$.) As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works.