Some bivariate stochastic models arising from group representation theory

TL;DR

This paper explores new bivariate Markov processes derived from group representation theory, analyzing their spectral properties and probabilistic behaviors, with applications to queueing and genetic models.

Contribution

It introduces matrix-valued infinitesimal operators and spherical functions for these processes, providing a novel spectral analysis framework.

Findings

Spectral analysis of matrix-valued operators reveals eigenfunctions from spherical functions.

Models include rational extensions of queue and Wright-Fisher processes.

Probabilistic properties are studied through the spectral framework.

Abstract

The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued second-order differential operator). The matrix-valued spherical functions associated to the compact symmetric pair $(\mathrm{SU}(2)\times \mathrm{SU}(2), \mathrm{diag} \, \mathrm{SU}(2))$ will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright-Fisher models involving only mutation effects.