Date: 17. - 19. February 2016.

Lectures will start on the morning of the 17th and the school will end on the 19th after the first afternoon session. Here is a first sketch of the program:

Wednesday (17th)Thursday (18th)Friday (19th)
9:00-10:00 Felix Otto 1 Felix Otto 3 Felix Otto 5
10:00-10:30 Coffee Coffee Coffee
10:30-12:00 Scott Armstrong 1 Scott Armstrong 3 Mitia Duerinckx
Arianna Giunti
Matthias Ruf
12:00-13:30 Lunch Lunch Lunch
13:30-15:00 Felix Otto 2 Felix Otto 4 Felix Otto 6
15:00-15:30 Coffee Coffee Coffee
15:30-17:00 Scott Armstrong 2 Antoine Gloria End of Winterschool
17:00-17:30 Alberto Chiarini

For people arriving early, we will set up an informal meeting on the 16th in the evening at Kaffeehaus im Thalia (Obstmarkt 5), from 7pm (but you are welcome to join us any time later).

On the 18th we have organized a joint conference dinner at the local brewery Riegele Brauhaus (Frölichstrasse 26).

In the morning of the 20th, we have organized a guided city tour through (registration during the school).

Titles and Abstracts

Main lecture course: Felix Otto (MPI Leipzig)
A quantitative theory of stochastic homogenization

The minicourse is about homogenization of linear elliptic equations in divergence form. These equations are determined by a uniformly elliptic coefficient field a=a(x). Homogenization means that on large scales, the solution operator behaves like that corresponding to a constant coefficient ahom. Homogenization is known to occur almost surely if the coefficient field a is randomly distributed according to a probability measure,
the distribution of which is translation invariant ("stationary'') and
decorrelates over large distances ("ergodic''). The fact that the reduction of complexity expressed by homogenization occurs in the presence of incomplete information expressed by randomness is of importance in the engineering practise.

Stochastic homogenization is a classical area when it comes to the qualitative theory, but of current research when it comes to a quantitative theory. A key object in homogenization is the corrector, which provides harmonic coordinates, and a key property is its sublinear growth on large scales. The goal of the course is the optimal quantification
of this sublinear growth, both in terms of rates and stochastic integrability.

We will start the course by reviewing the general framework for homogenization of elliptic operators in divergence form, namely the notion of H-convergence of uniformly elliptic coefficient fields introduced by Murat & Tartar (F. Murat, L. Tartar, H-convergence, Topics in the mathematical
modelling of composite materials, Birkhäuser), which amounts to a convergence of the solution operators on the level of the weak topology in the gradients and the fluxes of the solution. We then will quantify this qualitative concept by metricizing weak convergence of gradients and fluxes in a way that is suitable for the stochastic application. This quantification requires the introduction of an augmented notion of corrector (scalar and vector potentials of the harmonic coordinates seen as differential forms). Next, we shall apply this quantification to the corrector itself, which yields a self-averaging property of the flux of the corrector. We then pass to the stochastic framework and use this self-averaging property to set up an iteration by which we may treat larger and larger scales, reminiscent of a renormalization group argument.

This course is based on recent joint work (A. Gloria, F. Otto, The corrector in stochastic homogenization, near-optimal rates with optimal stochastic integrability) with A. Gloria.

Short lecture course: Scott Armstrong (CNRS / Université Paris-Dauphine)
Regularity theory in stochastic homogenization

Recently, elliptic regularity techniques have taken a more central role in the theory of stochastic homogenization, in particular because of their importance for quantitative estimates. In this short course, I will be motivated by this new regularity theory itself, although (fortunately) this requires a detour through some quantitative estimates obtained by variational techniques.

Invited lecture: Antoine Gloria (Brüssel, INRIA Lille)
A path-wise theory of fluctuations in stochastic homogenization

Abstract: In this lecture I will consider the model problem of a discrete elliptic equation with independent and identically distributed conductances. I shall identify a single quantity, which we call the corrected energy density of the corrector, that drives the fluctuations in stochastic homogenization in the following sense. On the one hand, when properly rescaled, this quantity satisfies a functional central limit theorem, and converges to a Gaussian field. On the other hand, the fluctuations of the corrector and the fluctuations of the solution of the stochastic PDE (that is, the solution of the discrete elliptic equation with random coefficients) are characterized at leading order by the fluctuations of this corrected energy density. As a consequence, when properly rescaled, the corrector and the solution satisfy a functional central limit theorem, and converge to (a variant of) a Gaussian free field. Compared to previous contributions, the approach I shall present yields the first path-wise results, quantifies the CLT in Wasserstein distance, and only relies on arguments that extend to the continuum setting.
This is based on a joint work with M. Duerinckx and F. Otto.

Contributed Talks

Alberto Chiarini (Université d’Aix-Marseille)
Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

Abstract: We study a continuous-time random walk X on ℤd in an environment of dynamic random conductances. We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle for the Markov process X under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser’s iteration scheme.

Mitia Duerinckx (Université libre de Bruxelles)
Quantitative stochastic homogenization via weighted functional inequalities

Abstract: The current quantitative theory of stochastic homogenization essentially either assumes strong mixing or strong functional inequalities for the coefficient random field. In practice, most interesting examples of random fields (random inclusions with unbounded random radii, random parking process, Gaussian random fields, etc.) are constructed as transformations of product (i.i.d.) structures. Provided that the transformation has some localization properties, we show how the functional inequalities satisfied by the underlying product structure (spectral gap in probability, etc.) are deformed into families of weighted inequalities. Even though mixing properties for the random fields of interest may be quite poor, this calculus directly implies nice concentration properties, which are the clue for optimal quantitative stochastic homogenization results. For the study of fluctuations in stochastic homogenization, for instance, optimal results have been proven for product structures but no general theory is available yet; the calculus we introduce allows to extend these optimal results to all our practical examples. This talk is based on a joint work with A. Gloria.

Arianna Giunti (MPI Leipzig)
Green’s function for elliptic systems: existence and Delmotte-Deuschel bounds

Abstract: We study the Green function G associated to the operator −∇ · a∇ in ℝd , when a = a(x) is a (measurable) uniformly elliptic tensor field. An example of De Giorgi implies that in the case of systems, the existence of a Green’s function is not ensured by such a broad class of coefficients a. We give a more general definition of G and with this definition show that for every uniformly elliptic tensor field a, a unique G exists. In addition, given a stationary ensemble <·> on a, we extend to the case of systems the Delmotte-Deuschel bounds for |∇G| and |∇∇G| . This talk is based on a joint work with Felix Otto and Joseph Conlon.

Matthias Ruf (TU München)
Stochastic homogenization of discrete surface energies

Abstract: We consider the energy of ferromagnetic Ising-type spin systems, when the position of the interacting particles is given by a suitable random stationary point process. Scaling the point process by a small parameter, we perform a discrete-to-continuum analysis via Gamma-convergence. By an appropriate energy renormalization we obtain functionals of perimeter type, where the surface tension is given by an asymptotic homogenization formula. If time permits, we will also consider interacting particles close to a lower dimensional subspace combining stochastic homogenization and dimension reduction. This talk is based on joint works with R. Alicandro, A. Braides and M. Cicalese.