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

Homogenization and corrector theory for linear transport in random media


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


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.
Citation: Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190


Enkhbat Rentsen, N. Tungalag, J. Enkhbayar, O. Battogtokh, L. Enkhtuvshin. Application of survival theory in Mining industry. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 443-448. doi: 10.3934/naco.2020036


Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451


Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021025


Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003


W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349


Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020


Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002


Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016


Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015


John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004


Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031


Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25


Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002


Yves Capdeboscq, Shaun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228


Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037


Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006


Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032


Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233


Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

2019 Impact Factor: 1.338


  • PDF downloads (46)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]