First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes

TL;DR

This paper presents two methods for calculating the probability distribution of the first time a radial Dunkl process hits the boundary of the Weyl chamber, using spectral analysis and Dunkl-Hermite polynomials, with applications to Brownian motion.

Contribution

It introduces two equivalent approaches for tail distribution computation of the first hitting time in radial Dunkl processes, connecting spectral problems and Dunkl-Hermite polynomials.

Findings

Derived explicit formulas for types A, B, D root systems

Connected the problem to spectral analysis and polynomial expansions

Obtained determinantal formulas for Brownian motion case

Abstract

We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the $W$-invariant Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible root systems of types $A$, $B$, $D$. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms.