Affiliations: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA. E-mail: [email protected] | Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 2 Place Jussieu, 75251 Paris Cedex 5, France. E-mail: [email protected] | Laboratoire MIP, Université Paul Sabatier, 31062 Toulouse Cedex 9, France. E-mail: [email protected] | Institut de Mathématiques de Bordeaux and Centre des Lasers Intenses et Applications, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France. E-mail: [email protected]
Abstract: This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.
Keywords: homogenization, partial differential equations with random coefficients, long-range memory effects, central limit, Gaussian processes
