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December  2010, 28(4): 1311-1343. doi: 10.3934/dcds.2010.28.1311

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, United States

Received  October 2009 Revised  February 2010 Published  June 2010

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.
Keywords: central limit theorem, transport equation, spatial Poisson point process., Homogenization theory, random media, corrector theory.
Mathematics Subject Classification: Primary: 35B40, 35B27, 35R60; Secondary: 60G55, 60F9.
Citation: Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
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