Bessel and Dunkl processes with drift

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.

💡 Research Summary

This paper by Michael Voit presents a systematic generalization of multivariate Bessel and Dunkl processes to include a “drift” parameter, constructing these processes for arbitrary root systems and multiplicity parameters k≥0.

The work is motivated by known geometric examples where Bessel processes without drift arise as projections of driftless Brownian motions on certain Euclidean spaces onto associated Weyl chambers (e.g., Dyson Brownian motion from Hermitian matrix spaces). The paper observes that projecting Brownian motions with constant drift vectors similarly yields diffusion processes on the Weyl chambers, whose generators and transition densities can be expressed via multivariate Bessel functions. This inspires the general definition.

The core construction in Section 3 defines “Bessel processes with drift” on a Weyl chamber C by a conjugation method. Specifically, the generator and transition densities of the standard (driftless) Bessel process are transformed using multiplication operators involving the Bessel function J_k(x, λ), where λ∈C represents the drift vector. This yields a well-defined Feller diffusion for all root systems and k≥0.

The author then extends this idea to define analogous processes on the full space R^N. Two distinct constructions emerge:

1. Dunkl processes with drift: Using the Dunkl kernel E_k(x, λ) in the conjugation. These processes have algebraic properties that make them natural from a Dunkl-theoretic perspective.
2. Hybrid Dunkl-Bessel processes with drift: Using the Bessel function J_k(x, λ) in the conjugation. These processes behave more like their Bessel counterparts on the chamber, inheriting the jump structure of classical Dunkl processes.

A key result shows that the radial parts (norms) of all these new processes with drift are themselves one-dimensional Bessel processes with drift.

Section 4 provides a simple Girsanov-type theorem linking the laws of processes with and without drift.

Section 5 introduces “modified moments” tailored for Dunkl processes with drift. These moments lead to associated martingales and form the basis for deriving limit theorems. However, a significant challenge is highlighted: the lack of explicit asymptotic information for the Dunkl kernels E_k for N>1 prevents fully explicit strong laws of large numbers (SLLN) and central limit theorems (CLT) in the general multivariate Dunkl case. Satisfactory explicit results are achieved only for the one-dimensional (N=1) case.

Finally, Section 6 establishes SLLN and CLT results for Bessel processes with drift on the chamber C. For the hybrid Dunkl-Bessel processes, no new limit theorems beyond those covered by the Bessel case are found.

In summary, this paper successfully lays the foundation for a theory of Bessel and Dunkl processes with drift, providing general constructions, studying fundamental properties like radial parts and Girsanov transformations, and initiating the analysis of their long-time behavior through moment methods and limit theorems, while also clearly delineating the current limitations due to the analytical complexity of Dunkl kernels.