A Regularity Theory for Random Elliptic Operators

Abstract

The qualitative theory of stochastic homogenization of uniformly elliptic linear (but possibly non-symmetric) systems in divergence form is well-understood. Quantitative results on the speed of convergence and on the error in the representative volume method, like those recently obtained by the authors for scalar equations, require a type of stochastic regularity theory for the corrector (e.g., higher moment bounds). One of the main insights of the very recent work of Armstrong and Smart is that one should separate these error estimates, which require strong mixing conditions in order to yield the best rates possible, from the (large scale) regularity theorỳ a la Avellaneda & Lin for a-harmonic functions that should hold under milder mixing conditions. In this paper, we establish an intrinsinc C 1,1 -version of the improved regularity theory for non-symmetric random systems, that is qualitative in the sense it yields a new Liouville theorem under mere ergodicity, and that is quantifiable in the sense that it holds under a condition that has high stochastic integrabiliity provided the coefficients satisfy quantitative mixing assumptions. We introduce such a class of quantitative mixing condition, that allows for arbitrarily slow-decaying correlations, and under which we derive a new family of optimal estimates in stochastic homogenization.

Higher-order pathwise theory of fluctuations in stochastic homogenization

“…• PDE ingredients: It remains to estimate the infinitesimal variations of interest from their representation formulas. Based on the large-scale Lipschitz regularity theory for aharmonic functions as developed in [7,6,5,21], we introduce in Section 6 a new annealed Calderón-Zygmund theory for linear elliptic equations in divergence form with random coefficients. This constitutes an upgrade of the quenched large-scale Calderón-Zygmund theory of [4,3,21] (see also [31] for a direct approach); it happens to be particularly well suited for our purposes in this article and is of independent interest.…”

Higher-Order Linearization and Regularity in Nonlinear Homogenization

“…This large-scale regularity theory introduced in [6] was further developed in the case of (1.3) in [17,5,15,2,3] and now plays an essential role in the quantitative theory of stochastic homogenization. Whether one employs functional inequalities [17,13] or renormalization arguments [2,3,19], it is a crucial ingredient in the proof of the optimal error estimates in homogenization for (1.3): see the monograph [4] and the references therein for a complete presentation of these developments.…”

“…See [4,Theorem 3.8] for the full statement, which was first proved in the periodic setting by Avellaneda and Lin [8]. Subsequent versions of this result, which are based on the ideas of [7,8] in their more quantitative formulation given in [6], were proved in various works [17,15,2], with the full statement here given in [3,10]. In all of its various forms, higher regularity in stochastic homogenzations is based on the simple idea that solutions of the heterogeneous equation should be close to those of the homogenized equation, which should have much better regularity.…”

“…Part (iii) of the theorem is a quantitative version of this Liouville-type result, which we call a "large-scale C 1,1 estimate" since it states that, on large scales, an arbitrary solution of the nonlinear equation can be approximated by an elements of L 1 with the same precision as harmonic functions can be approximated by affine functions. It can be compared to similar statements in the linear case (see for instance [4,17]).…”

Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids

“…A related approach based on the parabolic flow was put forward in [38], see also [12,Chapter 9], and will give us the most convenient statement for us to build upon here. A different approach based on concentration inequalities was put forward in [35,36,33,47,37,34], inspired by earlier insights from statistical mechanics [54,55].…”