Dunkl jump processes: relaxation and a phase transition

Abstract

Dunkl processes are multidimensional Markov processes defined through the use of Dunkl operators. Their paths show discontinuities, and so 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 with 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 the latter 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 AN−1 and BN undergo a phase transition when the parameter 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 we derive an asymptotic relation for the relaxation exponent in order to discuss its -dependence.