Abstract: 
We develop a largescale regularity theory of higher order for divergenceform elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The largescale regularity of aharmonic functions is encoded by Liouville principles: The space of aharmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constantcoefficient case. This result can be seen as the qualitative side of a largescale Ck,αregularity theory, which in the present work is developed in the form of a corresponding Ck,α“excess decay” estimate: For a given aharmonic function u on a ball BR, its energy distance on some ball Br to the above space of aharmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r0. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kthorder correctors” and thereby the space of aharmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.
