On the Averaged Green’s Function of an Elliptic Equation with Random Coefficients

We consider a divergence-form elliptic difference operator on the lattice 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 + ϵ to -3d + ϵ. (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.