On the conservativeness and the recurrence of symmetric jump-diffusions

TL;DR

This paper establishes conditions based on volume and jump kernel for symmetric jump-diffusions to be conservative and recurrent, including cases with disconnected state spaces where jumps connect components.

Contribution

It provides new sufficient conditions for recurrence and conservativeness of symmetric jump-diffusions, extending to disconnected spaces with jump connectivity.

Findings

Conditions relate volume and jump kernel to process behavior

Examples demonstrate optimality of conditions

Includes cases with disconnected state spaces connected by jumps

Abstract

Sufficient conditions for a symmetric jump-diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process. A number of examples are presented to illustrate the optimality of these conditions; in particular, the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components.