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. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale C-k,C-alpha-regularity theory, which in the present work is developed in the form of a corresponding C-k,C-alpha-excess decay estimate: For a given a-harmonic function u on a ball BR, its energy distance on some ball B-r to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r(0). 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 (phi, sigma) of scalar and vector potentials of the harmonic coordinates, where phi is the usual corrector, grows sublinearly in a mildly quantified way. We then construct kth-order correctors and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.