2014 DiPerna Lectures

The 2014 DiPerna Lecture was given by Takis Souganidis (University of Chicago) on January 23, 2014, 4PM in 60 Evans.

Stochastic homogenization

Abstract : Homogenization is about the study of the effective (averaged) macroscopic behavior of phenomena (equations) which depend on many scales and take place at the microscopic level in self-averaging environments. In this context, periodicity is often very restrictive and, from the modeling point of view, it is necessary to consider more general, stationary ergodic media. Typical examples of such problems are first time percolation and random walks, diffusions in random environments, fronts moving in oscillatory media, materials with several phases and defects, and others. The random setting lacks compactness, and this leads to rather challenging mathematical problems and requires the development of new methods. In this lecture I will discuss developments of the theory of stochastic homogenization of first- and second-order degenerate elliptic equations. I will provide examples, discuss the main difficulties, present some of the important results and state a few open problems.