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

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