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

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 one-dimensional 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.