Communications in Mathematical Sciences

Volume 13 (2015)

Number 3

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Limiting models for equations with large random potential: A review

Pages: 729 – 748

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n3.a7

Authors

Guillaume Bal (Department of Applied Physics and Applied Mathematics, Columbia University, New York N.Y., U.S.A.)

Yu Gu (Department of Applied Physics and Applied Mathematics, Columbia University, New York N.Y., U.S.A.)

Abstract

This paper reviews several results obtained recently in the convergence of solutions to elliptic or parabolic equations with large highly oscillatory random potentials. Depending on the correlation properties of the potential, the resulting limit may be either deterministic and solution of a homogenized equation or random and solution of a stochastic PDE. In the former case, the residual random fluctuations of the heterogeneous solution may also be characterized, or at least the rate of convergence to the deterministic limit established. We present several results that can be obtained by the methods of asymptotic perturbations, diagrammatic expansions, probabilistic representations, and the multiscale method.

Keywords

propagation of stochasticity, homogenization, stochastic partial differential equations

2010 Mathematics Subject Classification

35B27, 35R60, 60H15

Published 3 March 2015