Effective conductivity in random media

We are interested in computing the electric field generated by a neutral, localized charge distribution in a random medium, modeled by a linear elliptic equation in divergence form. First, I will present results in the homogenization regime, where the medium exhibits scale separation. I will then explain how ideas from homogenization theory can be used to address the case where no scale separation is present. Finally, I will discuss some interesting possible extensions to nonlinear settings. This talk is based on joint work with Felix Otto, Marc Josien, Qiang Xu, and Lihan Wang.