Exponential functionals of Brownian motion and class-one Whittaker functions

TL;DR

This paper studies exponential functionals of multi-dimensional Brownian motion with drift, characterizing their joint law via PDEs and linking them to Whittaker functions, revealing new diffusion process properties.

Contribution

It provides a novel characterization of the joint law of exponential functionals using PDEs and connects these to class-one Whittaker functions for specific root systems.

Findings

Laplace transform characterized by a Schrödinger-type PDE

Explicit expression of the Laplace transform in terms of Whittaker functions

Identification of diffusion processes conditioned on exponential functionals

Abstract

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.