On the conservativeness and the recurrence of symmetric jump-diffusions
  Sufficient conditions for a symmetric jump-diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process. A number of examples are presented to illustrate the optimality of these conditions; in particular, the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components. 
Jun Masamune, Toshihiro Uemura, Jian Wang
math.PR