Heat kernel estimates for anomalous heavy-tailed random walks

TL;DR

This paper extends heat kernel estimates to anomalous heavy-tailed random walks with jump indices ≥ 2, using a robust adaptation of Davies' perturbation method to obtain sharp transition probability bounds.

Contribution

It develops a new approach to derive heat kernel bounds for heavy-tailed random walks with jump indices ≥ 2, overcoming limitations of existing methods.

Findings

Established global upper and lower bounds on transition probabilities

Demonstrated robustness of methods to small perturbations of the jump kernel

Extended Davies' perturbation technique to heavy-tailed jump processes

Abstract

Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature of our work is that the methods we develop are robust to small perturbations of the symmetric jump kernel.