Fluctuation theory and exit systems for positive self-similar Markov processes

For a positive self-similar Markov process, X, we construct a local time for the random set, Θ, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, viz. a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set Θ and the process X sampled on the local time scale. The process (R,H) is described in terms of ladder process associated to the Lévy process associated to X via Lamperti’s transformation. In the case where X never hits 0 and the upward ladder height process is not arithmetic and has finite mean we prove the finite dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in [8], namely to remove a redundant hypothesis and to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates