Section: New Results
Quantitative results in stochastic homogenization
Quantitative results for almost periodic coefficients
In [6], S. Armstrong, A. Gloria and T. Kuusi (Aalto University) obtained the first improvement over the thirty year-old result by Kozlov [70] 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 [49] and adapted to the almost periodic setting by S. Armstrong and Z. Shen [48], a new quantification of almost-periodicity, and a sensitivity calculus in the spirit of [3].
Optimal stochastic integrability in stochastic homogenization
In [40] A. Gloria and F. Otto consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. They show that the gradient and flux
A theory of fluctuations in stochastic homogenization
In [39], M. Duerinckx, A. Gloria, and F. Otto establish a path-wise theory of fluctuations in stochastic homogenization of linear elliptic equations in divergence form. More precisely they consider the model problem of a discrete equation with independent and identically distributed conductances (as considered in [27]). They identify a single quantity, which they call the homogenization commutator, that drives the fluctuations in stochastic homogenization in the following sense. On the one hand, this tensor-valued stationary random field satisfies a functional central limit theorem, and (when suitably rescaled) converges to a Gaussian white noise. On the other hand, the fluctuations of the gradient of the corrector, the fluctuations of the flux of the corrector, and the fluctuations of any solution of the PDE with random coefficients and localized right-hand side are characterized at leading order by the fluctuations of this homogenization commutator in a path-wise sense. As a consequence, when properly rescaled, the solution satisfies a functional central limit theorem, the gradient of the corrector converges to the Helmholtz projection of a Gaussian white noise, and the flux of the corrector converges to the Leray projection of the same white noise. Compared to previous contributions, our approach, based on the homogenization commutator, unravels the complete structure of fluctuations. It holds in any dimension