  We consider exponential functionals of a multi-dimensional Brownian motion
with drift, defined via a collection of linear functionals. We give a
characterization of the Laplace transform of their joint law as the unique
bounded solution, up to a constant factor, to a Schrodinger-type partial
differential equation. We derive a similar equation for the probability
density. We then characterize all diffusion processes 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.
