Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities

TL;DR

This paper derives two-sided heat kernel estimates and establishes a local limit theorem for random conductance models with stable-like jumps on graphs, extending classical results to long-range, possibly degenerate settings.

Contribution

It introduces novel heat kernel estimates for stable-like jumps in conductance models where traditional Harnack inequalities fail.

Findings

Two-sided heat kernel estimates established

Local limit theorem proven for the models

Extension of classical Gaussian estimates to long-range jumps

Abstract

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases for nearest neighbor conductance models, the idea through parabolic Harnack inequalities does not work, since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.