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 - Article (Academic Journal)
AN - SCOPUS:80054106617
SN - 0246-0203
VL - 47
SP - 1096
EP - 1120
JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
IS - 4
ER -