Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright-Fisher model

TL;DR

This paper investigates the spectral and differential properties of a class of two-dimensional Markov processes with both diffusion and discrete components, applying the findings to a Wright-Fisher model variant with mutation effects.

Contribution

It introduces a spectral analysis framework for Markov processes with matrix-valued differential operators, extending understanding of their probabilistic and spectral properties.

Findings

Derived backward and forward equations for the process

Established spectral representation of the probability density

Analyzed recurrence and invariant distribution properties

Abstract

The aim of this paper is to study differential and spectral properties of the infinitesimal operator of two dimensional Markov processes with diffusion and discrete components. The infinitesimal operator is now a second-order differential operator with matrix-valued coefficients, from which we can derive backward and forward equations, a spectral representation of the probability density, study recurrence of the process and the corresponding invariant distribution. All these results are applied to an example coming from group representation theory which can be viewed as a variant of the Wright-Fisher model involving only mutation effects.