Positive self-similar Markov processes obtained by resurrection

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly alpha-stable process at its first exit time from (0, infinity). We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes. (c) 2022 Elsevier B.V. All rights reserved.