On recurrence and transience of two-dimensional Lévy and Lévy-type processes

Abstract

In this paper, we study recurrence and transience of Lévy-type processes, that is, Feller processes associated with pseudo-differential operators. Since the recurrence property of Lévy-type processes in dimensions greater than two is vacuous and the recurrence and transience of onedimensional Lévy-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 Lévy-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 Lévy measure and we provide some comparison conditions for the recurrence and transience also in terms of the Lévy measures.

On transience of Lévy-type processes

“…By following the proof of [San15, Theorem 3.2] we get the following "perturbation" result. Let us remark here that if ν(x, dy) is the Lévy measure of a Lévy-type process {F t } t≥0 , then ν(Ox, dy) = ν(Ox, Ody) is a Lévy measure of the Lévy-type process {O −1 F t } t≥0 (see [San15,Proposition 3.1]). Further, observe that in the Lévy process case the condition in (4.2) will be satisfied if, and only if, R d |y| 2 ν(dy) = ∞.…”

An Introduction to Lévy and Feller Processes. Advanced Courses in Mathematics - CRM Barcelona 2014

“…With the machinery we have developed here, one can also study further path properties, such as invariant measures, ergodicity, transience and recurrence etc. For this we refer to the monograph [9] as well as recent developments by Behme & Schnurr [7] and Sandrić [49,50].…”