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Normal approximation for the net flux through a random conductor

  • James Nolen
Article

Abstract

We consider solutions of 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 \(L \rightarrow \infty \), 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 (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.

Keywords

Stochastic homogenization Random media Normal approximation Random conductance 

Mathematics Subject Classification

35B27 35J15 60F05 60H25 

Notes

Acknowledgments

I am grateful to Felix Otto for very helpful discussions. This work was partially funded by Grant DMS-1007572 from the US National Science Foundation.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

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