Large scale behaviour of the spatial $𝛬$-Fleming–Viot process

Berestycki, N., Etheridge, A. M., and Véber, A.. "Large scale behaviour of the spatial $\varLambda $-Fleming–Viot process." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 374-401. <http://eudml.org/doc/271991>.

@article{Berestycki2013,
abstract = {We consider the spatial $\varLambda $-Fleming–Viot process model (Electron. J. Probab.15(2010) 162–216) for frequencies of genetic types in a population living in $\mathbb \{R\}^\{d\}$, in the special case in which there are just two types of individuals, labelled $0$ and $1$. At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$. We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the dynamics of the process are driven by purely ‘local’ events and one incorporating large-scale extinction recolonisation events. We choose the frequency of these events in such a way that, under a suitable rescaling of space and time, the ancestry of a single individual in the population converges to a symmetric stable process of index $\alpha \in (1,2]$ (with $\alpha =2$ corresponding to Brownian motion). We consider the behaviour of the process of allele frequencies under the same space and time rescaling. For $\alpha =2$, and $d\ge 2$ it converges to a deterministic limit. In all other cases the limit is random and we identify it as the indicator function of a random set. In particular, there is no local coexistence of types in the limit. We characterise the set in terms of a dual process of coalescing symmetric stable processes, which is of interest in its own right. The complex geometry of the random set is illustrated through simulations.},
author = {Berestycki, N., Etheridge, A. M., Véber, A.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {generalised Fleming–Viot process; limit theorems; duality; symmetric stable processes; population genetics; generalised Fleming-Viot process},
language = {eng},
number = {2},
pages = {374-401},
publisher = {Gauthier-Villars},
title = {Large scale behaviour of the spatial $\varLambda $-Fleming–Viot process},
url = {http://eudml.org/doc/271991},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Berestycki, N.
AU - Etheridge, A. M.
AU - Véber, A.
TI - Large scale behaviour of the spatial $\varLambda $-Fleming–Viot process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 374
EP - 401
AB - We consider the spatial $\varLambda $-Fleming–Viot process model (Electron. J. Probab.15(2010) 162–216) for frequencies of genetic types in a population living in $\mathbb {R}^{d}$, in the special case in which there are just two types of individuals, labelled $0$ and $1$. At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$. We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the dynamics of the process are driven by purely ‘local’ events and one incorporating large-scale extinction recolonisation events. We choose the frequency of these events in such a way that, under a suitable rescaling of space and time, the ancestry of a single individual in the population converges to a symmetric stable process of index $\alpha \in (1,2]$ (with $\alpha =2$ corresponding to Brownian motion). We consider the behaviour of the process of allele frequencies under the same space and time rescaling. For $\alpha =2$, and $d\ge 2$ it converges to a deterministic limit. In all other cases the limit is random and we identify it as the indicator function of a random set. In particular, there is no local coexistence of types in the limit. We characterise the set in terms of a dual process of coalescing symmetric stable processes, which is of interest in its own right. The complex geometry of the random set is illustrated through simulations.
LA - eng
KW - generalised Fleming–Viot process; limit theorems; duality; symmetric stable processes; population genetics; generalised Fleming-Viot process
UR - http://eudml.org/doc/271991
ER -