In this contribution we prove optimal rates of convergence in stochastic homogenization of monotone operators with p-growth for all 2 ≤ p < ∞ in dimensions d ≤ 3 (and in periodic homogenization for all p ≥ 2 and d ≥ 1). Previous contributions were so far restricted to p = 2. The main issues to treat superquadratic growth are the potential degeneracy and the unboundedness of the coefficients when linearizing the equation. To deal with this we develop a perturbative regularity theory in the large for random linear operators with unbounded coefficients. From the probabilistic point of view we rely on functional inequalities, and make crucial use of the central limit theorem scaling to absorb nonlinear contributions. Combining these two ingredients we obtain sharp bounds on the growth of correctors and of corrector differences, from which the quantitative two-scale expansion follows. Our approach bypasses the need of non-perturbative large-scale regularity theory -which might not hold true in the general setting we consider here.

We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble • and the corresponding linear elliptic operator −∇ • a∇. In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called "representative" volume element, i. e. a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations.By this we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble • . We make this point by investigating the bias (or systematic error), i. e. the difference between a hom and the expected value of the RVE method, in terms of its scaling w. r. t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L −1 ). In case of a suitable periodization of • , we rigorously show that it is O(L −d ). In fact, we give a characterization of the leading-order error term for both strategies, and argue that even in the isotropic case it is generically non-degenerate.We carry out the rigorous analysis in the convenient setting of ensembles • of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. Contents

We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R ≥ 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.

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We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale $$R\ge 1$$ R ≥ 1 . For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.