Normal approximation for the net flux through a random conductor

Abstract

We consider solutions to an elliptic partial differential equation in $\mathbb{R}^d$ with a stationary, random conductivity coefficient. The boundary condition on a square domain of width $L$ is chosen so that the solution has a macroscopic unit gradient. We then consider the average flux through the domain. It is known that in the limit $L \to \infty$, this quantity converges to a deterministic constant, almost surely. Our main result is about normal approximation for this flux when $L$ is large: we give an estimate of the Kantorovich-Wasserstein distance between the law of this random variable and that of a normal random variable. This extends a previous result of the author to a much larger class of random conductivity coefficients. The analysis relies on elliptic regularity, on bounds for the Green's function, and on a normal approximation method developed by S. Chatterjee based on Stein's method.