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, $\Theta,$ 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 \textit{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 $\Theta$ 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 \cite{CCh}, 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