Change of measure in the lookdown particle system

Abstract

We perform various changes of measure in the lookdown particle system of Donnelly and Kurtz. The first example is a product type h-transform related to conditioning a Generalized Fleming Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this h-transform by just "forgetting" some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright Fisher diffusion and explicit the associated pathwise decomposition. The second example, called the linear or additive h-transform, concerns a wider class of measure valued processes with spatial motion. Applications include: -a simple description of the additive h-transform of the Generalized Fleming Viot process, which confirms a suggestion of Overbeck for the usual Fleming Viot process -an immortal particle representation for the additive h-transform of the Dawson Watanabe process.

The fixation line in the ${\Lambda}$-coalescent

“…This is a review of the literature: the origin of the fixation line may be traced back to Pfaffelhuber and Wakolbinger [26] in the Kingman case Λ(dx) = δ 0 (dx). For general Λ(dx), it appeared in Labbé [19] and in [17]. The coalescent we will focus on is the Λ-coalescent, that was introduced independently and simultaneously by Donnelly and Kurtz [10], Pitman [27] and Sagitov [29].…”

On the block counting process and the fixation line of exchangeable coalescents

“…This process can be traced back to Pfaffelhuber and Wakolbinger [22] for the Kingman coalescent. For the Λ-coalescent the fixation line appears in Labbé [15] and was further studied by Hénard [8,9]. We close the introduction by a brief summary of the organization of the article.…”

Probabilités et biologie

“…L'étude [34] présentée par Olivier Hénard concerne une représentation des processus de Wright-Fisher général-isés pour lesquels on impose la coexistence de deux types neutres dans la population. Il faut remarquer que ce dernier évènement est de probabilité nulle, car dans ces modèles la diversité disparaît.…”

“…Our interest in [34] is in a microscopic understanding of the macroscopic conditioning on non-absorption. That is, we are interested in an individual-based explanation of the form of the generatorG (Λ) f (x).…”