Abstract
We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on mathbb Z^d. In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.
Highlights
We continue our research [12,13] on regularity and stochastic homogenization of non-uniformly elliptic equations
In [12], we studied local regularity properties of weak solutions of elliptic equations in divergence form
Under relaxed ellipticity conditions compared to very recent contributions in the field. (B) Based on the regularity result in (A), we establish a local limit theorem for random walks among degenerate and unbounded random conductances
Summary
We continue our research [12,13] on regularity and stochastic homogenization of non-uniformly elliptic equations. In [12], we studied local regularity properties of weak solutions of elliptic equations in divergence form. ∇ · a∇u = 0 and proved local boundedness and the validity of Harnack inequality under essentially minimal integrability conditions on the ellipticity of the coefficients a. This generalizes the seminal theory of De Giorgi, Nash and Moser [23,32,34] and improves in an optimal way classic results due to Trudinger [36] (see [33]). Schäffner (A) (deterministic part) We establish local regularity properties in the sense of an oscillation decay (and Hölder-continuity) for solution to the discrete version of the parabolic equation. Under relaxed ellipticity conditions compared to very recent contributions in the field (see e.g. [3,5,7,21]). (B) (random part) Based on the regularity result in (A), we establish a local limit theorem for random walks among degenerate and unbounded random conductances
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