Dunkl jump processes: relaxation and a phase transition

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.