Brownian Excursion Theory: A First Approach

Abstract

Brownian excursions away from 0 may be labeled by the inverse local time, which also allows to define Ito’s excursion process. This process is a Poisson Point Process. Descriptions of its intensity measure n shall be the subject of next chapters. Two master formulae, the additive one and the multiplicative one, are proven. They allow to compute expectations of sums or products of excursion functionals in terms of n. The Levy measures of Brownian additive functionals, considered at inverse local time are shown to be expressible in terms of n. The distributions of the lifetime and the maximum of the generic excursion under n are computed.