Valeurs principales associées aux temps locaux Browniens. (Principal values associated to Brownian local times). (French) Zbl 0619.60072

Biane, Ph.; Yor, M.

Bull. Sci. Math., II. Sér. 111, 23-101 (1987).

This paper presents a number of results concerning linear Brownian motion and its local times, excursion theory and Bessel processes. The starting point of the paper is a deep study of the Cauchy principal value of Brownian local times, which leads to many remarkable identities in law, some of which are related to previous work of Chung, Pitman and Yor and other authors. These identities also provide a probabilistic interpretation of the functional equations satisfied by the Riemann \(\zeta\) function and the Jacobi \(\theta\) function.
The authors also obtain remarkable identities relating the Brownian excursion, the Brownian bridge and the Brownian meander. Some of these identities are interpreted via a trajectorial construction of the Brownian meander from the Brownian bridge.

Reviewer: J.Le Gall

MSC:

60J65	Brownian motion
60J55	Local time and additive functionals
60J60	Diffusion processes

Keywords:

Brownian motion and its local times; Bessel processes; Cauchy principal value; Brownian bridge; Brownian meander