American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[1]

Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063
[2]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[3]

Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143
[4]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[5]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
[6]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
[7]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
[8]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003
[9]

Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407
[10]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[11]

Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303
[12]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[13]

Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72
[14]

Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689
[15]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[16]

Zhongming Chen, Liqun Qi. Circulant tensors with applications to spectral hypergraph theory and stochastic process. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1227-1247. doi: 10.3934/jimo.2016.12.1227
[17]

Maroje Marohnić, Igor Velčić. Homogenization of bending theory for plates; the case of oscillations in the direction of thickness. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2151-2168. doi: 10.3934/cpaa.2015.14.2151
[18]

Yoon-Sik Cho, Aram Galstyan, P. Jeffrey Brantingham, George Tita. Latent self-exciting point process model for spatial-temporal networks. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1335-1354. doi: 10.3934/dcdsb.2014.19.1335
[19]

L. Bedin, Mark Thompson. Existence theory for a Poisson-Nernst-Planck model of electrophoresis. Communications on Pure & Applied Analysis, 2013, 12 (1) : 157-206. doi: 10.3934/cpaa.2013.12.157
[20]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences