Exponential functionals of Brownian motion and class-one Whittaker functions

TY - JOUR

T1 - Exponential functionals of Brownian motion and class-one Whittaker functions

AU - Baudoin, Fabrice

AU - O'Connell, Neil

PY - 2011/11

Y1 - 2011/11

N2 - 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.

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.

KW - Conditioned Brownian motion

KW - Quantum Toda lattice

UR - http://www.scopus.com/inward/record.url?scp=80054106617&partnerID=8YFLogxK

U2 - 10.1214/10-AIHP401

DO - 10.1214/10-AIHP401

M3 - Journal article

AN - SCOPUS:80054106617

SN - 0246-0203

VL - 47

SP - 1096

EP - 1120

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 4

ER -