Change of measure in the lookdown particle system

TL;DR

This paper explores measure changes in the lookdown particle system, providing new constructions and relationships for processes like the Fleming Viot and Dawson Watanabe, with applications in genetic modeling.

Contribution

It introduces novel measure change techniques, including pathwise constructions and intertwining relationships, for measure-valued processes in the lookdown particle system framework.

Findings

Pathwise construction of h-transform by 'forgetting' reproduction events

Intertwining relationship for Wright Fisher diffusion

Explicit additive h-transform for Dawson Watanabe process

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.