Persistence of integrable stable processes

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Zhan Shi (2003). Along the way, we investigate the law of the stable process evaluated at the first time its integral hits zero, extending classical formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion. This is joint work with Christophe Profeta (Evry-Val d’Essonne).