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

1 LAREMA - Laboratoire Angevin de REcherche en MAthématiques 2 Department of Mathematical Sciences, University of Bath 3 CIMAT - Centro de Investigación en Matemáticas

Abstract : 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.

Keywords : self-similar Markov processes Entrance laws exit systems excursion theory ladder processes Lamperti-s transformation Lévy processes self-similar Markov processes.

Author: Loïc Chaumont - Andreas Kyprianou - Juan Carlos Pardo - Victor Rivero -

Source: https://hal.archives-ouvertes.fr/