Random conductance models with stable-like jumps: Quenched invariance principle

Abstract

We study the quenched invariance principle for random conductance models with long range jumps on ${\mathbb{Z}}^{\mathit{d}}$, where the transition probability from x to y is, on average, comparable to $|\mathit{x}-\mathit{y}{|}^{-(\mathit{d}+\mathit{\alpha })}$ with $\mathit{\alpha }\in (0,2)$ but is allowed to be degenerate. Under some moment conditions on the conductance, we prove that the scaling limit of the Markov process is a symmetric α-stable Lévy process on ${\mathbb{R}}^{\mathit{d}}$. The well-known corrector method in homogenization theory does not seem to work in this setting. Instead, we utilize probabilistic potential theory for the corresponding jump processes. Two essential ingredients of our proof are the tightness estimate and the Hölder regularity of caloric functions for nonelliptic α-stable-like processes on graphs. Our method is robust enough to apply not only for ${\mathbb{Z}}^{\mathit{d}}$ but also for more general graphs whose scaling limits are nice metric measure spaces.