SPD

Abstract

I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behaviour. For example, imagine a conductor with an electric potential imposed at the boundary. Some current will flow through the material...what is the net current per unit volume? For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant (homogenization). I will describe a recent result about normal approximation: the probability law of the net current is very close to that of a normal random variable having the same mean and variance. Closeness is quantified by an error estimate in total variation.