Dunkl jump processes: relaxation and a phase transition

TL;DR

This paper investigates Dunkl jump processes, revealing a phase transition at critical parameters and characterizing their relaxation dynamics with power-law behavior, thus expanding understanding of these less-studied stochastic processes.

Contribution

It provides the first detailed analysis of Dunkl jump processes, deriving their master equation, identifying phase transitions, and describing their relaxation behavior.

Findings

Phase transition occurs as parameter β approaches one.

Relaxation follows a non-trivial power law.

Conjecture on jump rate asymptotics supported by simulations.

Abstract

Dunkl processes are multidimensional Markov processes defined through the use of Dunkl operators. These processes have discontinuities, and they can be separated into their continuous (radial) part, and their discontinuous (jump) part. While radial Dunkl processes have been studied thoroughly due to their relationship to families of stochastic particle systems such as the Dyson model and Wishart-Laguerre processes, Dunkl jump processes have gone largely unnoticed after the initial work of Gallardo, Yor and Chybiryakov. We study the dynamical properties of these processes, and we derive their master equation. By calculating the asymptotic behavior of their total jump rate, we find that the jump processes of types $A_{N-1}$ and $B_N$ undergo a phase transition when the parameter $\beta$ decreases toward one in the bulk scaling limit. In addition, we show that the relaxation behavior of these processes is given by a non-trivial power law, and formulate a conjecture for the jump rate asymptotics based on numerical simulations.