Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

TL;DR

This paper introduces a new class of exchangeable coalescent processes with immigration, called M-coalescents, which model genealogies in populations with immigration and connect to generalized Fleming-Viot processes.

Contribution

It defines M-coalescents characterized by two measures, analyzes their properties, and links them to stochastic flows and generalized Fleming-Viot processes with immigration.

Findings

M-coalescents are characterized by measures  and .

Conditions for coming down from infinity match previous results for mbda-coalescents.

The superprocess relates to a generalized Fleming-Viot process with immigration.

Abstract

Coalescents with multiple collisions (also called Lambda-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Imagine an infinite population with immigration labelled at each generation by N:={1,2,...}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focussing on simple distinguished coalescents, i.e such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0,1] denoted by M=(\Lambda_{0},\Lambda_{1}). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the \Lambda-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures \Lambda_{0} and \Lambda_{1} specify respectively the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time, all individuals are immigrant children.