On Recurrence and Transience of Two-Dimensional L\'evy and L\'evy-Type Processes

TL;DR

This paper investigates the recurrence and transience properties of two-dimensional Le9vy-type processes, providing conditions based on Le9vy measures and analyzing perturbations that do not alter these properties.

Contribution

It offers new criteria for recurrence and transience of two-dimensional Le9vy-type processes, focusing on the impact of Le9vy measures and perturbations.

Findings

Derived sufficient conditions for recurrence and transience.

Provided comparison conditions based on Le9vy measures.

Analyzed the effect of perturbations on process properties.

Abstract

In this paper, we study recurrence and transience of L\'evy-type processes, that is, Feller processes associated with pseudo-differential operators. Since the recurrence property of L\'evy-type processes in dimensions greater than two is vacuous and the recurrence and transience of one-dimensional L\'evy-type processes have been very well investigated, in this paper we are focused on the two-dimensional case only. In particular, we study perturbations of two-dimensional L\'evy-type processes which do not affect their recurrence and transience properties, we derive sufficient conditions for their recurrence and transience in terms of the corresponding L\'evy measure and we provide some comparison conditions for the recurrence and transience also in terms of the L\'evy measures.