A Liouville principle for the random conductance model under degenerate conditions

We consider a random conductance model on the $d$-dimensional lattice, $d\in[2,\infty)\cap\mathbb{N}$, where the conductances take values in $(0,\infty)$ and are however not assumed to be bounded from above and below. We assume that the law of the random conductances is stationary and ergodic with respect to translations on $\mathbb{Z}^d$ and invariant with respect to reflections on $\mathbb{Z}^d$ and satisfies a similar moment bound as that by Andres, Deuschel, and Slowik (2015), under which a quenched FCLT holds. We prove a first-order Liouville theorem. In the proof we construct the sublinear correctors in the discrete and adapt boundary estimates for harmonic extensions from the work in the continuum done by Bella, Fehrman, and Otto (2018) to the discrete.