## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 43, Number 2 (2011), 348-374.

### Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

#### Abstract

Coalescents with multiple collisions (also called Λ-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. Consider 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, focusing 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* = (Λ_{0}, Λ_{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 Λ-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 Λ_{0} and
Λ_{1} respectively specify 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.

#### Article information

**Source**

Adv. in Appl. Probab. Volume 43, Number 2 (2011), 348-374.

**Dates**

First available in Project Euclid: 21 June 2011

**Permanent link to this document**

http://projecteuclid.org/euclid.aap/1308662483

**Mathematical Reviews number (MathSciNet)**

MR2848380

**Zentralblatt MATH identifier**

1300.60086

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability

Secondary: 92D25: Population dynamics (general)

**Keywords**

Exchangeable partition coalescent theory genealogy for a population with immigration stochastic flow coming down from infinity

#### Citation

Foucart, Clément. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration. Adv. in Appl. Probab. 43 (2011), no. 2, 348--374. http://projecteuclid.org/euclid.aap/1308662483.