Note on radial Dunkl processes

TL;DR

This paper provides shorter, more informative proofs of key properties of radial Dunkl processes, including their uniqueness, behavior at boundaries, and jump characteristics, enhancing understanding of their stochastic dynamics.

Contribution

It offers simplified proofs of known results on radial Dunkl processes, revealing new insights into their boundary hitting times and jump behavior, and addresses a conjecture on jump lengths.

Findings

Radial Dunkl process is the unique strong solution to a stochastic differential equation with singular drift.

The process hits the Weyl chamber boundary almost surely for small multiplicities.

The total jump length during finite time is finite, confirming a conjecture.

Abstract

We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times of a stochastic differential equation with a singular drift, the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. Compared to the original proofs, ours give more information on the behavior of the process. More precisely, the first proof allows to give a positive answer to a conjecture announced by Gallardo and Yor for the finiteness of the total length of the jumps performed by the Dunkl process during a finite amount of time, while the second one shows that the process hits almost surely the wall corresponding to the simple root with a small multiplicity value. Further results on the jumps of the Dunkl process (communicated by D. L\'epingle) and and additional references are included.