Excursions of the integral of the Brownian motion

Abstract

The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Ito's excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.

In this second version, some consequent changes of notations and presentation. The space we work on for Proposition 2 and Theorem 2 changed a bit (the proofs are unchanged)