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.

中文翻译：

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非均匀抛物线方程及其在随机电导模型中的应用

我们研究了与\(\mathbb Z^d\)上的随机电导模型相关的散度形式的线性非均匀抛物线有限差分算子的局部规律性。特别是，我们提供了一个振荡衰减，假设电导及其逆的某些可和性特性，从而改善了该方向的最新结果。作为一个应用程序，我们为随机退化和无界环境中的随机游走提供了一个局部极限定理。