Relations entre pont et excursion du mouvement brownien réel. (Relations between bridge and excursion of real Brownian motion) (Q1078927)

Following from a result of \textit{W. Verwaat} [Ann. Probab. 7, 143-149 (1979; Zbl 0392.60058)], it is possible to construct a normalized excursion of a real Brownian motion by taking a Brownian bridge and cutting its trajectory at the moment at which it attains its minimum and then exchanging the end points of the resulting trajectory. A representation of Itô's measure of the excursions of the Brownian motion permits a new proof of this theorem and moreover it is possible to show the reverse proposition: Given a normalized excursion of the Brownian motion it is possible to construct a Brownian bridge, such that Verwaats method returns the given excursion. Furthermore this representation allows the derivation of some results concerning the processes of local times of Brownian bridges and Brownian excursions.