Liouville theorem for elliptic systems with degenerate coefficients

I will discuss behavior of second order degenerate elliptic systems in divergence form with random ergodic coefficients. Assuming moment bounds on the coefficient field $a$ and its inverse, similar to the ones used by Chiarini and Deuschel [Arxiv preprint 1410.4483, 2014], we obtain $C^{1,\alpha }$ large scale regularity estimate for $a$-harmonic functions. As a consequence we obtain Liouville theorem for subquadratic $a$-harmonic functions, which in particular implies uniqueness of the correctors.