A Probabilistic view on the homogenization of the heat equation

Consider the heat equation with random coefficients on Z^d. The randomness of the coefficients models the inhomogeneous nature of the medium where heat propagates. We assume that the distribution of these coefficients is invariant under spatial translations, and has a finite range of dependence. It is known that if a solution to this equation is rescaled diffusively, then it converges to the solution of a heat equation with constant coefficients. I will first explain how this convergence can be rephrased in terms of the long-time behaviour of an associated random walk. Based on this representation, I will then present recent progress on the estimation of the speed of this convergence.