Transience of symmetric non-local Dirichlet forms

TL;DR

This paper develops criteria to determine when symmetric non-local Dirichlet forms are transient or recurrent based on coefficient growth, impacting the understanding of associated jump processes.

Contribution

It provides new transience and recurrence criteria for symmetric non-local Dirichlet forms with unbounded or degenerate coefficients.

Findings

Established transience criteria based on coefficient growth rates.

Derived a necessary and sufficient condition for recurrence of stable-like Dirichlet forms.

Showed that jump part coefficients influence sample path properties.

Abstract

We establish transience criteria for symmetric non-local Dirichlet forms on $L^2({\mathbb R}^d)$ in terms of the coefficient growth rates at infinity. Applying these criteria, we find a necessary and sufficient condition for recurrence of Dirichlet forms of symmetric stable-like with unbounded/degenerate coefficients. This condition indicates that both of the coefficient growth rates of small and big jump parts affect the sample path properties of the associated symmetric jump processes.