Abstract
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 Bertoin and Le Gall. Finally under the condition that this infinite sequence of Eves does exist, we construct the lookdown representation pathwise from a flow of bridges.
Highlights
The Λ-coalescent has been introduced by Pitman [25] and Sagitov [26]
If we assume that Π0 is the trivial partition O[∞] := {{1}, {2}, . . .} the process (Πt, t ≥ 0) can be interpreted as the genealogy of an infinite population: each individual is represented by an integer so that the coalescence of a collection of blocks corresponds to groups of individuals finding their most recent
First we propose a new formulation of the lookdown representation that relies on the introduction of an object called the stochastic flow of partitions
Summary
The Λ-coalescent has been introduced by Pitman [25] and Sagitov [26]. This is a Markov process (Πt, t ≥ 0) with values in the set P∞ of partitions of N := {1, 2 . . .} whose distribution is characterised by a finite measure Λ on [0, 1]. The Λ-coalescent is in duality (see Lemma 5 of Bertoin and Le Gall [6]) with the socalled Λ-Fleming-Viot process which, on the contrary, describes the evolution forward in time of an infinite population This Markov process has been introduced by Bertoin and Le Gall (see Theorem 3 in [6]), and implicitly by Donnelly and Kurtz [12]. We will call the underlying Poisson point process the lookdown graph for reasons that will be made clear later on Another - and seemingly different - approach to construct the Λ-Fleming-Viot process comes from the stochastic flow of bridges introduced by Bertoin and Le Gall [6]. We use the Eves and the stochastic flow of partitions to define the lookdown representation pathwise from the stochastic flow of bridges
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