Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On the Convex Hull of a Brownian Excursion with Parabolic Drift

Auteur(s):

Code(s) de Classification MSC:

• 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
• 70Fxx Dynamics of a system of particles, including celestial mechanics

Résumé: The solutions of Burgers equation with white noise initial velocity are closely connected to the convex hull $\h_a$ of a Brownian excursion with parabolic drift $s\mapsto e_s+{a\over 2}\,s^2$. We derive from the law of the minimum $\sigma$ and the location $\eta$ of the minimum of $s \mapsto 2 e_s/s(1-s)$ a complete description of the convex hull $\h_a$. As an application, we determine the statistical properties of the Burgers turbulence on the circle.

Mots Clés: Brownian excursion ; convex hull ; Burgers turbulence

Date: 2002-05-02

Prépublication numéro: PMA-722

Postscript file : PMA-722.ps

Pdf file : PMA-722.pdf