Homogenization and corrector theory for lineartransport in random media

Guillaume Bal; Wenjia Jing

doi:10.3934/dcds.2010.28.1311

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2010, Volume 28, Issue 4: 1311-1343. Doi: 10.3934/dcds.2010.28.1311

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Homogenization and corrector theory for linear transport in random media

Guillaume Bal^1, and
Wenjia Jing^2,

1.
Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027
2.
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027

Received: October 2009

Revised: February 2010

Published: December 2010

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Abstract

We consider the theory of correctors to homogenization in stationary transport equations with rapidly oscillating, random coefficients. Let ε << 1 be the ratio of the correlation length in the random medium to the overall distance of propagation. As ε $ \downarrow 0$, we show that the heterogeneous transport solution is well-approximated by a homogeneous transport solution. We then show that the rescaled corrector converges in (probability) distribution and weakly in the space and velocity variables, to a Gaussian process as an application of a central limit result. The latter result requires strong assumptions on the statistical structure of randomness and is proved for random processes constructed by means of a Poisson point process.

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Mathematics Subject Classification: Primary: 35B40, 35B27, 35R60; Secondary: 60G55, 60F99.

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