  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 $(0,\infty)$. 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.

Probability