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

Véber, A.; Wakolbinger, A.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 570-598.

Nous construisons un processus à valeurs mesures équivalent au processus $𝛬$-Fleming–Viot spatial (SLFV) introduit dans (Banach Center Publ. 80 (2008) 121–144). Contrairement à la construction effectuée dans (Banach Center Publ. 80 (2008) 121–144), nous fixons une réalisation de la suite d’événements de reproduction et obtenons une évolution quenched des diversités génétiques locales. Pour ce faire, nous utilisons une représentation particulaire qui met en avant le rôle des généalogies dans l’attribution des types (ou allèles) aux individus de la population. Cette construction nous permet également de clarifier l’espace d’états du SLFV et d’obtenir plusieurs propriétés trajectorielles du processus à valeurs mesures, ainsi que des arbres étiquetés qui décrivent les relations généalogiques liant un échantillon d’individus. Nous complétons ces résultats avec une construction look-down fournissant un système de particules dont la mesure empirique au temps $t$, vue comme un processus en $t$, a même loi que le SLFV quenched. Dans tous ces résultats, le fait que nous travaillions à configuration d’événements fixée et que les reproductions ne se produisent que localement (en espace) introduisent de sérieuses difficultés techniques qui sont surmontées en contrôlant le nombre d’événements et de particules présentes dans une zone donnée de l’espace pendant un laps de temps macroscopique.

We construct a measure-valued equivalent to the spatial $𝛬$-Fleming–Viot process (SLFV) introduced in (Banach Center Publ. 80 (2008) 121–144). In contrast with the construction carried out there, 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.

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