# Normal approximation for a random elliptic equation

Journal Article

We consider solutions of an elliptic partial differential equation in {Mathematical expression} with a stationary, random conductivity coefficient that is also periodic with period {Mathematical expression}. Boundary conditions on a square domain of width {Mathematical expression} 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 {Mathematical expression}, 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 {Mathematical expression} 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. © 2013 Springer-Verlag Berlin Heidelberg.

### Full Text

### Duke Authors

### Cited Authors

- Nolen, J

### Published Date

- 2013

### Published In

### Start / End Page

- 1 - 40

### International Standard Serial Number (ISSN)

- 0178-8051

### Digital Object Identifier (DOI)

- 10.1007/s00440-013-0517-9