The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation

TL;DR

This paper introduces a measure-valued construction of the spatial Lambda-Fleming-Viot process, fixing reproduction events to analyze local genetic diversity and genealogies, with a new lookdown particle system representation.

Contribution

It provides a fixed-event realization of the SLFV, clarifies its state-space, and develops a look-down construction for genealogical analysis.

Findings

Constructed a quenched measure-valued SLFV process.

Derived path properties and genealogical relations.

Established technical controls for fixed event configurations.

Abstract

We construct a measure-valued equivalent to the spatial Lambda-Fleming-Viot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the state-space of the SLFV and to derive several path properties of the measure-valued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a look-down construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.