Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps

TL;DR

This paper explores the connections between parabolic Harnack inequalities, heat kernel estimates, and geometric and analytic conditions for long-range jump random walks, highlighting differences from diffusion processes.

Contribution

It clarifies the relationship between key inequalities and estimates for long-range jump processes, showing that Harnack inequality does not always imply heat kernel bounds.

Findings

Parabolic Harnack inequality does not necessarily imply heat kernel estimates for long-range jumps.

The paper establishes conditions under which heat kernel estimates hold.

Differences between diffusion processes and jump processes are highlighted.

Abstract

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.