Characterization of subordinate symmetric Markov processes

TL;DR

This paper studies subordinate symmetric Markov processes on metric measure spaces, providing estimates of their jump kernels and conditions for their behavior, which enhances understanding of their non-diffusive characteristics.

Contribution

It introduces new estimates for jump kernels of subordinate processes and clarifies their scale, extending results to non-subordinate processes via Dirichlet form stability.

Findings

Derived estimates for jump kernels of subordinate processes

Established equivalent conditions for jump kernel behavior

Clarified the scale of jump kernels distinct from diffusion types

Abstract

In this paper, we consider subordinate symmetric Markov processes which correspond to non-killing Dirichlet forms enjoying heat kernel estimates on a metric measure space with the volume doubling property. We obtain estimates of the jump kernel of the subordinate process and establish equivalent conditions for the jump kernel following Liu-Murugan. In particular, we clarify the scale of the jump kernel, which is different from the diffusion type. This result is appliable to non-subordinate processes by the transferring method, which uses stability of Dirichlet forms.