Exponential functionals of brownian motion and class-one Whittaker functions

Fabrice Baudoin; Neil O’Connell

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 4, page 1096-1120
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Baudoin, Fabrice, and O’Connell, Neil. "Exponential functionals of brownian motion and class-one Whittaker functions." Annales de l'I.H.P. Probabilités et statistiques 47.4 (2011): 1096-1120. <http://eudml.org/doc/242110>.

@article{Baudoin2011,
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.},
author = {Baudoin, Fabrice, O’Connell, Neil},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {conditioned brownian motion; quantum Toda lattice; conditioned Brownian motion},
language = {eng},
number = {4},
pages = {1096-1120},
publisher = {Gauthier-Villars},
title = {Exponential functionals of brownian motion and class-one Whittaker functions},
url = {http://eudml.org/doc/242110},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Baudoin, Fabrice
AU - O’Connell, Neil
TI - Exponential functionals of brownian motion and class-one Whittaker functions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 4
SP - 1096
EP - 1120
AB - 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.
LA - eng
KW - conditioned brownian motion; quantum Toda lattice; conditioned Brownian motion
UR - http://eudml.org/doc/242110
ER -

References

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