On the comparison between jump processes and subordinated diffusions

TL;DR

This paper investigates the conditions under which jump processes can be represented as subordinated diffusions on the same space, establishing criteria and applying results to prove parabolic Harnack inequalities for such jump processes.

Contribution

It provides necessary and sufficient conditions for jump processes to be comparable to subordinated diffusions under certain heat kernel and jump kernel bounds.

Findings

Established criteria for the existence of a subordinator linking jump processes and diffusions.

Proved parabolic Harnack inequality for jump processes with polynomial jump kernel bounds.

Extended stability results to a broad class of jump processes.

Abstract

Given a symmetric diffusion process and a jump process on the same underlying space, is there a subordinator such that the jump process and the subordinated diffusion processes are comparable? We address this question when the diffusion satisfies a sub-Gaussian heat kernel estimate and the jump process satisfies a polynomial-type jump kernel bounds. Under these assumptions, we obtain necessary and sufficient conditions on the jump kernel estimate for such a subordinator to exist. As an application of our results and the recent stability results of Chen, Kumagai, and Wang, we obtain parabolic Harnack inequality for a large family of jump processes. In particular, we show that any jump process with polynomial-type jump kernel bounds on such a space satisfies the parabolic Harnack inequality.