A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes

TL;DR

This paper introduces a new de Finetti type representation for exchangeable coalescent trees, enabling the definition of generalized tree-valued Fleming-Viot processes using various state spaces, including those with dust.

Contribution

It provides a novel representation for exchangeable coalescent trees and extends the construction of Fleming-Viot processes to include cases with dust and different state spaces.

Findings

Representation for exchangeable coalescent trees using iid samples from marked metric measure spaces

Definition of generalized tree-valued Fleming-Viot processes from a $\\Xi$-lookdown model

Inclusion of dust cases with isolated leaves in genealogical trees

Abstract

We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of tree-valued Fleming-Viot processes from a $\Xi$-lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.