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.

Classification : 60J25, 92D10, 60K37, 60J75, 60F15
Mots clés : random environment, generalised Fleming–Viot process, spatial coalescent, look-down construction

[1] O. Angel, N. Berestycki and V. Limic. Global divergence of spatial coalescents. Probab. Theory Related Fields 152 (2012) 625–679. | DOI | MR | Zbl

[2] N. H. Barton, A. M. Etheridge and A. Véber. A new model for evolution in a spatial continuum. Electron. J. Probab. 15 (2010) 162–216. | DOI | EuDML | MR | Zbl

[3] N. H. Barton, A. M. Etheridge and A. Véber. Modelling evolution in a spatial continuum. J. Stat. Mech. (2013) P01002. | MR | Zbl

[4] N. H. Barton, A. M. Etheridge, J. Kelleher and A. Véber. Inference in two dimensions: allele frequencies versus lengths of shared sequence blocks. Theor. Pop. Biol. 87 (2013) 105–119. | DOI | Zbl

[5] N. H. Barton, J. Kelleher and A. M. Etheridge. A new model for extinction and recolonization in two dimensions: Quantifying phylogeography. Evolution 64 (2010) 2701–2715. | DOI

[6] N. Berestycki, A. M. Etheridge and M. Hutzenthaler. Survival, extinction and ergodicity in a spatially continuous population model. Markov Process. Related Fields 15 (2009) 265–288. | MR | Zbl

[7] N. Berestycki, A. M. Etheridge and A. Véber. Large scale behaviour of the spatial $𝛬$-Fleming–Viot process. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013) 374–401. | DOI | EuDML | Numdam | MR | Zbl

[8] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes I. Probab. Theory Related Fields 126 (2003) 261–288. | DOI | MR | Zbl

[9] M. Birkner, J. Blath, M. Capaldo, A. M. Etheridge, M. Möhle, J. Schweinsberg and A. Wakolbinger. Alpha-stable branching and Beta-coalescents. Electron. J. Probab. 10 (2005) 303–325. | DOI | EuDML | MR | Zbl

[10] M. Birkner, J. Blath, M. Möhle, M. Steinrücken and J. Tams. A modified lookdown construction for the $𝛯$-Fleming–Viot process with mutation and populations with recurrent bottlenecks. Alea 6 (2009) 25–61. | MR | Zbl

[11] P. Donnelly, S. N. Evans, K. Fleischmann, T. G. Kurtz and X. Zhou. Continuum-sites stepping-stone models, coalescing exchangeable partitions, and random trees. Ann. Probab. 28 (2010) 1063–1110. | MR | Zbl

[12] P. Donnelly and T. G. Kurtz. A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Probab. 24 (1996) 698–742. | DOI | MR | Zbl

[13] P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166–205. | DOI | MR | Zbl

[14] A. M. Etheridge. Drift, draft and structure: Some mathematical models of evolution. Banach Center Publ. 80 (2008) 121–144. | DOI | MR | Zbl

[15] A. M. Etheridge and T. G. Kurtz. Genealogical constructions of population models. Preprint, 2012. | MR | Zbl

[16] A. M. Etheridge and A. Véber. The spatial $𝛬$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination. Ann. Appl. Probab. 22 (2012) 2165–2209. | DOI | MR | Zbl

[17] S. N. Evans. Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997) 339–358. | DOI | Numdam | MR | Zbl

[18] N. Freeman. The segregated Lambda-coalescent. Ann. Probab. 43 (2015) 435–467. | DOI | MR | Zbl

[19] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002. | DOI | MR | Zbl

[20] T. G. Kurtz and E. R. Rodrigues. Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 (2011) 939–984. | DOI | MR | Zbl

[21] V. Limic and A. Sturm. The spatial $𝛬$-coalescent. Electron. J. Probab. 11 (2006) 363–393. | DOI | MR | Zbl

[22] H. Liu and X. Zhou. The compact support property for the Lambda-Fleming–Viot process with underlying Brownian motion. Electron. J. Probab. 17 (2012) Article 73. | MR | Zbl

[23] P. Pfaffelhuber and A. Wakolbinger. The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 (2006) 1836–1859. | DOI | MR | Zbl

[24] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870–1902. | DOI | MR | Zbl

[25] J. Schweinsberg. Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 (2000) 1–50. | DOI | MR | Zbl

[26] J. Schweinsberg. A necessary and sufficient condition for the $𝛬$-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000) 1–11. | DOI | MR | Zbl