Homogenization is a theory that describes the effective behavior of PDEs with highly oscillatory coefficients satisfying certain ergodic assumptions. It was first developed in the periodic setting in the 1970s and later extended to the stochastic setting in the 1980s. However, the whole theory largely relies on the uniform ellipticity of the operator, and little is known about cases where the uniform ellipticity is violated, especially at the quantitative level.

In this talk, we will present some quantitative stochastic homogenization results under a degenerate assumption (log-normality) on the coefficient field. Key concepts of periodic and stochastic homogenization will also be introduced at the beginning.