A higher-order large-scale regularity theory for random elliptic operators

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. Under the assumptions of stationarity and slightly quantified ergodicity of the ensemble, we derive a $C^{2,\alpha }$-"excess decay" estimate on large scales and a $C^{2,\alpha }$-Liouville principle: For a given $a$-harmonic function $u$ on a ball $B_R$, we show that its energy distance to the space of $a$-harmonic "corrected quadratic polynomials" on some ball $B_r$ has the natural decay in the radius $r$ above some minimal (random) radius $r_0$. Our Liouville principle states that the space of $a$-harmonic functions growing at most quadratically has (almost surely) the same dimension as in the constant-coefficient case. The existence of $a$-harmonic "corrected quadratic polynomials" - and therefore our regularity theory - relies on the existence of second-order correctors for the homogenization problem. By an iterative construction, we are able to establish existence of subquadratically growing second-order correctors.