Using Lamperti’s relationship between Lévy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law ℙx 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 ℙx 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.
M. E. Caballero. L. Chaumont. "Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes." Ann. Probab. 34 (3) 1012 - 1034, May 2006. https://doi.org/10.1214/009117905000000611