Exponential functionals of Brownian motion and class-one Whittaker functions

Fabrice Baudoin*, Neil O'Connell

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

18 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1096-1120
Number of pages25
JournalAnnales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Volume47
Issue number4
DOIs
Publication statusPublished - Nov 2011

Keywords

  • Conditioned Brownian motion
  • Quantum Toda lattice

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