A Spectral decomposition for the Bolthausen-Sznitman coalescent and the Kingman coalescent

TL;DR

This paper derives spectral decompositions for the generators of the Bolthausen-Sznitman and Kingman coalescents, providing new formulas for Green's functions and transition probabilities, which enhance understanding of these stochastic processes.

Contribution

It introduces spectral decompositions for the generators of both coalescents, offering new analytical tools and simplified derivations of key formulas.

Findings

Spectral decompositions of the generators are obtained.

New formulas for Green's functions are derived.

Transition probabilities for the Bolthausen-Sznitman coalescent are simplified.

Abstract

We consider both the Bolthausen-Sznitman and the Kingman coalescent restricted to the partitions of $\{1, \ldots, n\}.$ Spectral decompositions of the corresponding generators are derived. As an application we obtain a formula for the Green's functions and a short derivation of the well-known formula for the transition probabilities of the Bolthausen-Sznitman coalescent.