Section: New Results

Quantitative stochastic homogenization

A. Gloria, S. Neukamm (Univ. Dresden), and F. Otto (MPI for mathematics in the sciences, Leipzig) developed in [17] a general approach to quantify ergodicity in stochastic homogenization of 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 [4] ). They also obtained the first error analysis of the popular periodization method to approximate the homogenized coefficients.

In [32] , 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 [28] , A. Gloria and D. Marahrens (MPI for mathematics in the sciences, Leipzig) extended the annealed results [51] 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.

In their recent work [29] , A. Gloria, S. Neukamm, and F. Otto developed a regularity theory for random elliptic operators inspired by the contributions of Avellaneda and Lin [39] in the periodic setting and of our visitor S. Armstrong with C. Smart [38] . This allowed them to consider coefficients with arbritarily slow decaying correlations in the form of a family of correlated Gaussian fields.

In [30] , 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 [17] on the approximation of homogenized coefficients by periodization by characterizing the limiting law in addition to the scaling.