Abstract

Consider a discrete elliptic equation on the integer lattice with random coefficients, arising for example as the steady state of a diffusion through the lattice with random diffusivities. Classical homogenization results show that with ergodic coefficients and on large scales, the solutions behave as the solutions to a diffusion equation with constant homogenized coefficients. The homogenized coefficients can be characterised through the solution to a so-called “corrector problem”. In contrast to periodic homogenization, the stochastic homogenization lacks compactness which makes the problem harder. Recently Gloria, Otto and Neukamm have developed tools to obtain optimal estimates for the homogenization and the corrector problem via a spectral gap inequality. In this talk, I will present how to obtain strong quantitative (optimal) estimates on the discrete Green’s function via a logarithmic Sobolev inequality and consequences from these estimates for the solutions to the discrete equation. This is joint work with Felix Otto.