Section: New Results

Quantitative stochastic homogenization

Discrete equations

Decay of the semi-group

A. Gloria, S. Neukamm (Univ. Dresden), and F. Otto (MPI for mathematics in the sciences, Leipzig) developed in [15] a general approach to quantify ergodicity in stochastic homogenization of scalar discrete elliptic equations. Using a parabolic approach, they obtained optimal estimates on the time-decay of the so-called environment seen from the particle. This allowed them to prove optimal bounds on the corrector gradient and the corrector itself in any dimension (thus improving on [5] ). They also obtained the first error analysis of the popular periodization method to approximate the homogenized coefficients.

Quantitative CLT

In [16] , A. Gloria and J. Nolen (Duke Univ.) proved a quantitative central limit theorem for the effective conductance on the discrete torus. In particular, they quantified the Wasserstein distance between a normal random variable and the CLT-like rescaling of the difference between the approximation of the effective conductance by periodization and the effective conductance. Their estimate is sharp and shows that the Wasserstein distance goes to zero (up to logarithmic factors) as if the energy density of the corrector was iid (which it is not). This completes and settles the analysis started in [15] on the approximation of homogenized coefficients by periodization by characterizing the limiting law in addition to the scaling.

Continuum equations

Scalar equations with random coefficients

In [17] , A. Gloria and F. Otto extended their results [4] , [5] on discrete elliptic equations to the continuum setting. They treated in addition the case of non-symmetric coefficients, and obtained optimal estimates in all dimensions by the elliptic approach (whereas [4] ,[5] were suboptimal for $d=2$).

In [14] , A. Gloria and D. Marahrens (MPI for mathematics in the sciences, Leipzig) extended the annealed results [14] on the discrete Green function by D. Marahrens and F. Otto to the continuum setting. As a by-product of their result, they obtained new results in uncertainty quantification by estimating optimally the variance of the solution of an elliptic PDE whose coefficients are perturbed by some noise with short range of dependence.

Systems with random coefficients

In a revised version of [58] , A. Gloria, S. Neukamm, and F. Otto developed a regularity theory for random elliptic operators inspired by the contributions of Avellaneda and Lin [43] in the periodic setting and of S. Armstrong with C. Smart [42] . This allowed them to consider coefficients with arbritarily slow decaying correlations in the form of a family of correlated Gaussian fields, and obtain (in the new version of this paper) a family of estimates with optimal rates and exponential-type integrability.

In [35] , A. Gloria and F. Otto obtained the first nearly-optimal estimates with optimal stochastic integrability on the corrector for linear elliptic systems whose coefficients satisfy a finite range of dependence assumption (thus avoiding the functional inequalities they considered so far).

Systems with almost periodic coefficients

In [23] , S. Armstrong, A. Gloria and T. Kuusi (Aalto University) obtained the first improvement over the thirty year-old result by Kozlov [60] on almost periodic homogenization. In particular they introduced a class of almost periodic coefficients which are not quasi-periodic (and thus strictly contains the Kozlov class) and for which almost periodic correctors exist. Their approach combines the regularity theory developed by S. Armstrong and C. Smart in [42] and adapted to the almost periodic setting by S. Armstrong and Z. Shen [41] , a new quantification of almost-periodicity, and a sensitivity calculus in the spirit of [4] .

Clausius-Mossotti formulas

In the mid-nineteenth century, Clausis, Mossotti and Maxwell essentially gave a first order Taylor expansion for (what is now understood as) the homogenized coefficients associated with a constant background medium perturbed by diluted spherical inclusions. Such an approach was recently used and extended by the team MATHERIALS to reduce the variance in numerical approximations of the homogenized coefficients, cf. [39] , [38] , [62] . In [12] , M. Duerinckx and A. Gloria gave the first rigorous proof of the Clausius-Mossotti formula and provided the theoretical background to analyze the methods introduced in [62] .