Bessel and Dunkl processes with drift

TL;DR

This paper extends the construction and analysis of Bessel and Dunkl processes with drift for arbitrary root systems, exploring their properties, transition densities, and probabilistic features.

Contribution

It introduces a general construction of Bessel and Dunkl processes with drift for all root systems and parameters, expanding their theoretical framework.

Findings

Constructed Bessel processes with drift for all root systems

Derived transition densities using multivariate Bessel functions

Analyzed properties like moments, martingales, and limit theorems

Abstract

For some discrete parameters $k\ge0$, multivariate (Dunkl-)Bessel processes on Weyl chambers $C$ associated with root systems appear as projections of Brownian motions without drift on Euclidean spaces $V$, and the associated transition densities can be described in terms of multivariate Bessel functions; the most prominent examples are Dyson Brownian motions. The projections of Brownian motions on $V$ with drifts are also Feller diffusions on $C$, and their transition densities and their generators can be again described via these Bessel functions. These processes are called Bessel processes with drifts. In this paper we construct these Bessel processes processes with drift for arbitrary root systems and parameters $k\ge 0$. Moreover, this construction works also for Dunkl processes. We study some features of these processes with drift like their radial parts, a Girsanov theorem, moments and associated martingales, strong laws of large numbers, and central limit theorems.