Normal approximation for the net flux through a random conductor
© Springer Science+Business Media New York 2015. We consider solutions of an elliptic partial differential equation in 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 →∞, 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 (Probab Theory Relat Fields, 2013. doi:10.1007/s00440-013-0517-9) 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 Chatterjee (Ann Probab 36:1584–1610, 2008) based on Stein’s method.
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