Long-time behavior for a class of Feller processes

Abstract

In this paper, as a main result, we derive a Chung-Fuchs type condition for the recurrence of Feller processes associated with pseudo-differential operators. In the L\'evy process case, this condition reduces to the classical and well-known Chung-Fuchs condition. Further, we also discuss the recurrence and transience of Feller processes with respect to the dimension of the state space and Pruitt indices and the recurrence and transience of Feller-Dynkin diffusions and stable-like processes. Finally, in the one-dimensional symmetric case, we study perturbations of Feller processes which do not affect their recurrence and transience properties, and we derive sufficient conditions for their recurrence and transience in terms of the corresponding L\'evy measure. In addition, some comparison conditions for recurrence and transience also in terms of the L\'evy measures are obtained.