Weak convergence of positive self-similar Markov processes and overshoots of L\'{e}vy processes

Using Lamperti's relationship between L\'{e}vy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law $\mathbb{P}_x$ of a pssMp starting at $x>0$, in the Skorohod space of c\`{a}dl\`{a}g paths, when $x$ tends to 0. To do so, we first give conditions which allow us to construct a c\`{a}dl\`{a}g 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 $\mathbb{P}_x$ to converge weakly to the law of $X^{(0)}$ as $x$ goes to 0. In particular, this answers a question raised by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205--225] about the Feller property for pssMp at $x=0$.