Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes.

This version: 09 - 02 -2005To appear in Ann. Prob., 2005Using Lamperti's relationship between Lévy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law $\p_x$ of a pssMp starting at $x>0$, in the Skorohod space of càdlàg paths, when $x$ tends to 0. To do so, we first give conditions which allow us to construct a càdlàg Markov process $X^{(0)}$, starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws $\p_x$ to converges weakly to the law of $X^{(0)}$ as $x$ goes to 0. In particular, this answers a question raised by Lamperti (1972) about the Feller property for pssMp at $x=0$