Exponential functionals of brownian motion and class-one Whittaker functions
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1096-1120.

Nous étudions certaines fonctionelles d'un mouvement Brownien avec dérive dans ℝn qui sont définies par une collection de fonctionnelles linéaires. Nous donnons une caractérisation de la transformée de Laplace de leur loi jointe comme l'unique solution bornée, à une constante près d'une équation aux dérivées partielles de type Schrödinger. Nous déduisons une équation similaire pour la densité. Nous caractérisons ensuite toutes les diffusions qui peuvent être interprétées comme ayant la loi d'un mouvement Brownien avec dérive conditionné par la loi de ses fonctionelles exponentielles. Dans le cas où la famille des fonctionelles est un ensemble de racines simples, la transformée de Laplace de la densité jointe des fonctionnelles exponentielles correspondantes peut être exprimée en termes d'une fonction de Whittaker de classe 1 associée au système. Dans ce cadre, nous établissons quelques propriétés du processus de diffusion correspondant.

We consider exponential functionals of a brownian motion with drift in ℝn, defined via a collection of linear functionals. We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrödinger-type partial differential equation. We derive a similar equation for the probability density. We then characterise all diffusions which can be interpreted as having the law of the brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (class-one) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes.

DOI : 10.1214/10-AIHP401
Classification : 60J65, 60J55, 37K10, 22E27
Mots-clés : conditioned brownian motion, quantum Toda lattice 
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Baudoin, Fabrice; O’Connell, Neil. Exponential functionals of brownian motion and class-one Whittaker functions. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1096-1120. doi : 10.1214/10-AIHP401. http://www.numdam.org/articles/10.1214/10-AIHP401/

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