Nonlocal nature of stochastic homogenization
A talk in the Nonlocal Equations: Analysis and Numerics 2023 series by
Tuomo Kuusi from Helsinki
| Abstract: | One of the principal difficulties in stochastic homogenization is
transferring quantitative ergodic information from the coefficients to
the solutions since the latter are nonlocal functions of the former.
The qualitative theory follows from quite general ergodic theorems, and
it was already established in the 1980s. The quantitative theory,
however, is much more subtle and it has been a very active research
topic for over a decade. I will describe some of our recent
contributions in trying to find the right objects to quantify ergodicity
to unify and streamline the theory. One of our key observations is that
one can use regularity theory for stochastic homogenization to
accelerate the weak convergence of the energy density, flux, and
gradient of the solutions as we pass to larger and larger length scales
until it saturates at the CLT scaling. I will discuss our approach and
give, at the same time, a brief introduction to the mathematical theory
of stochastic homogenization. Within the CRC this talk is associated to the project(s): A7 |