A Liouville property for the random conductance model

In the talk, I will give an introduction to my research based on a work by Bella, Fehrman, and Otto on stochastic homogenization. Consider the random differential operator $\mathrm{\nabla }\cdot a\mathrm{\nabla }$ where the random matrix (coefficient field) $a$ is assumed to be stationary and ergodic. By making use of the extended correctors $(\varphi ,\sigma )$ and choosing a reasonable homogenization error, they can obtain a regularity estimate, namely the excess decay, which implies a Liouville principle for $a$-harmonic functions, i.e. functions satisfying $\mathrm{\nabla }\cdot a\mathrm{\nabla }u=0$. It is interesting to know whether their ideas work in the discrete case (the random conductance model on the lattice). The answer is positive: By using several analytic and numerical methods, it is possible to implement their ideas in the discrete case.