From flows of \(\Lambda\)-Fleming-Viot processes to lookdown processes via flows of partitions. (English) Zbl 1334.60152

Labbé, Cyril

Electron. J. Probab. 19, Paper No. 55, 49 p. (2014).

Summary: The goal of this paper is to unify the lookdown representation and the stochastic flow of bridges, which are two approaches to construct the \(\Lambda\)-Fleming-Viot process along with its genealogy. First, we introduce the stochastic flow of partitions and show that it provides a new formulation of the lookdown representation. Second, we study the asymptotic behaviour of the \(\Lambda\)-Fleming-Viot process and we provide sufficient conditions for the existence of an infinite sequence of Eves that generalise the primitive Eve of J. Bertoin and J.-François Le Gall [Probab. Theory Relat. Fields 126, No. 2, 261–288 (2003; Zbl 1023.92018)]. Finally, under the condition that this infinite sequence of Eves does exist, we construct the lookdown representation pathwise from a flow of bridges.

MSC:

60J25	Continuous-time Markov processes on general state spaces
60G09	Exchangeability for stochastic processes
92D25	Population dynamics (general)

Keywords:

Fleming-Viot process; lookdown process; stochastic flow; partition; coalescent; exchangeable bridge

Citations:

Zbl 1023.92018