From flows of Lambda Fleming-Viot processes to lookdown processes via flows of partitions

TL;DR

This paper unifies two approaches to modeling the $mbda$-Fleming-Viot process by introducing a stochastic flow of partitions, analyzing its asymptotics, and constructing the lookdown representation from flows of bridges.

Contribution

It introduces a new stochastic flow of partitions that unifies the lookdown and flow of bridges representations of the $mbda$-Fleming-Viot process, and provides conditions for the existence of an infinite sequence of Eves.

Findings

The stochastic flow of partitions offers a new formulation of the lookdown representation.

Conditions are established for the existence of an infinite sequence of Eves.

The lookdown representation can be constructed pathwise from a flow of bridges under certain conditions.

Abstract

The goal of this paper is to unify the lookdown representation and the stochastic flow of bridges, which are two approaches to construct the $\Lambda$-Fleming-Viot process along with its genealogy. First we introduce the stochastic flow of partitions and show that it provides a new formulation of the lookdown representation. Second we study the asymptotic behaviour of the $\Lambda$-Fleming-Viot process and we provide sufficient conditions for the existence of an infinite sequence of Eves that generalise the primitive Eve of Bertoin and Le Gall. Finally under the condition that this infinite sequence of Eves does exist, we construct the lookdown representation pathwise from a flow of bridges.