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Normal approximation for a random elliptic equation


We consider solutions of an elliptic partial differential equation in \(\mathbb{R }^d\) with a stationary, random conductivity coefficient that is also periodic with period \(L\). Boundary conditions on a square domain of width \(L\) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit \(L \rightarrow \infty \), this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size \(L\) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.

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I am grateful to Felix Otto whose insight and helpful comments led to improvement of the main argument. I also thank Jan Wehr, Sourav Chatterjee, and Jonathan Mattingly for stimulating discussion in the early stages of this work. The anonymous referees also provided very helpful comments. The author’s research is partially funded by grant DMS-1007572 from the US National Science Foundation.

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Correspondence to James Nolen.

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Nolen, J. Normal approximation for a random elliptic equation. Probab. Theory Relat. Fields 159, 661–700 (2014). https://doi.org/10.1007/s00440-013-0517-9

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Mathematics Subject Classification

  • 35B27
  • 35J15
  • 60F05
  • 60H25