Jessica Lin - University of Wisconsin (Madison)

Title: Quantitative Stochastic Homogenization for Elliptic Equations

Abstract: Stochastic homogenization is concerned with identifying the asymptotic behavior of solutions to PDEs with random coefficients. Specifically, we are interested in the following: if the coefficients are randomly varying on a microscopic lengthscale, then on average, do the random solutions exhibit the same deterministic behavior? When this is indeed the case, we say that the random equation "homogenizes." Furthermore, from both the theoretical and applied perspective, an important issue is to understand the quantitative aspects of this homogenization process. In this talk, I will present an overview of the subject of stochastic homogenization for nondivergence form elliptic equations. I will discuss the interplay between PDE and probabilistic techniques used to study these types of problems.  In addition, I will present a recent quantitative result which yields the optimal error estimates on the size of the fluctuations of the random solutions. This talk is based on joint work with Scott Armstrong.