On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

TL;DR

This paper investigates a variant of stochastic homogenization involving oscillatory coefficients composed of periodic matrices and stochastic diffeomorphisms, establishing convergence results and approximation strategies for the residual process and homogenized coefficients.

Contribution

It provides the first convergence rate for the residual process in one dimension and proves almost sure convergence of an approximation method for the homogenized matrix in multiple dimensions.

Findings

Explicit convergence rate for the residual process in 1D.

Almost sure convergence of the approximation of the homogenized matrix.

Extension of results to multidimensional stochastic homogenization.

Abstract

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006 and Journal de Mathematiques Pures et Appliquees 2007]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.