Introduction to Stochastic Homogenization.

Besides being of mathematical interest, PDEs with random coefficient field appear naturally in a number of applications, for instance as models for materials with random heterogeneities. On large scales, such PDEs with random coefficient field often behave like a PDE with constant coefficients. The field of stochastic homogenization - the derivation of such effective macroscopic descriptions for PDEs with microscopically random coefficient fields - has seen a series of breakthroughs in the last decade.

In this lecture series, we give an introduction to the Gloria-Otto approach to quantitative stochastic homogenization. We start with a short introduction to homogenization at the example of linear elliptic PDEs with periodic coefficient field, introducing basic notions of homogenization theory. Subsequently, we give a brief introduction to concentration inequalities. In the main part, we illustrate how concentration inequalities and elliptic regularity estimates play together to provide quantitative error estimates in stochastic homogenization of linear elliptic equations.

Non-equilibrium stationary states (NESS) are stationary states of Markov chains that are characterized by the presence of non-zero currents. From a probabilistic point of view, these are stationary, non-reversible states. Unless some combinatorial miracle happens, these states can not be described explicitly. The aim of this course is to explain how entropy methods can be used in order to derive law of large numbers and central limit theorems for a family of NESS arising from driven diffusive systems. This provides a first step into a general strategy to study more general NESS.

A number of important models from mathematical physics are given by stochastic PDEs, i.e. by partial differential equations which include a random term to describe some noise which is inherent to the system. A prominent example is the KPZ equation which describes the fluctuations around 1+1 dimensional moving interfaces, but there are many more.

Developing a mathematical theory for equations of this type was open for many years, the key difficulty being the ``roughness’’ of the natural noise terms. This irregularity forces the appearance of “infinite counter-terms” in the equation and makes standard solution methods inapplicable. Over the last 10 years this field has been revolutionized, most notably in works by Hairer, Gubinelli and their collaborators. A rigorous solution theory for many equations has been developed, and these solutions have in turn be applied e.g. to study scaling limits for microscopic particle models. On the mathematical level, these new methods constitute a beautiful mixture of ideas from several very different branches of mathematics, including the theory of rough paths, Gaussian analysis, PDE methods and perturbative field theory.

The aim of this course is to give an introduction to some of these ideas and to show how they work together.