Excursions of the integral of the Brownian motion

TL;DR

This paper develops a new approach to studying excursions of the Langevin process by constructing a stationary process and explicitly determining its excursion measure, offering fresh insights into Ito's excursion measure for reflected Langevin processes.

Contribution

It introduces a novel method for analyzing Langevin process excursions using stationarity and explicitly characterizes the excursion measure, extending recent work on inelastic boundary reflections.

Findings

Explicit stationary excursion measure for Langevin process

New descriptions of Ito's excursion measure for reflected Langevin process

Connection to recent developments by Bertoin

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.