j. reine angew. Math. 456 (1994), 19—5l Journal f r die reine und angewandte Mathematik © Walter de Gruyter Berlin · New York 1994 Stochastic two-scale convergence in the mean and applications By Alain Bourgeat at St.-Etienne, Andro Mikelic at Lyon and Steve Wright at Rochester 1. Introduction It is well known that the modelling of physical processes in strongly inhomogeneous media leads to the study of differential equations with rapidly varying coefficients. More precisely, if the scale of inhomogeneity of the medium is of order å, then the coefficients of the differential operators are of the form a(s~ 1 x), where the function a depends on the microstructure. For small å, a direct numerical solution of such problems is for all practical purposes impossible, and this necessitates the construction of averaged models. The main goal of homogenization theory is the analysis of these averaged models s å -» 0, with particular emphasis on the establishment of convergence to a limit in an appropriate sense and the derivation of averaged equations for the limit. One expects such results if there is a certain regularity in the microstructure, present for example when the coefficients are periodic, almost periodic, or are homogeneous random fields. There is a vast literature on homogenization, but most of the papers are devoted to the case of periodic inhomogeneities, with comparatively little treatment of the almost- periodic and general stochastic cases. The Stochastic results for linear, second-order, elliptic equations were obtained by Kozlov [15], [16] and Papanicolaou and Varadhan [19], and the general linear setting was treated in Zhikov, Kozlov, Olenik, and Ngoan [23]. Papanicolaou and Varadhan use Tartar's energy method, s explained for instance in Bensoussan, Lions, and Papanicolaou [3]. Kozlov establishes convergence by direct con- struction of the corrector, and Zhikov, Kozlov, Oleinik and Ngoan use strong G-con- vergence of operators and the TV^-condition. More recently, Dal Maso and Modica [6], [7] analyze stochastic homogenization of convex integral functions by means of Ã-conver- gence. The various techniques just mentioned are somewhat difficult to apply when the homogenized problem has a nonlocal structural coupling between the micro- and macro- structures or when the coefficients are of the form a(x, å" 1 *). In order to capture the two- scale structure of the most general homogenization problems, the idea of "two-scale convergence" was recently introduced by Nguetseng [18], and Allaire [1], [2]. It has been Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/31/15 6:35 AM