Abstract
In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued $\Xi $-Fleming-Viot processes and deduce path properties. For instance, these processes usually have a. s. cadlag paths with jumps at the times of large reproduction events. In the case of coming down from infinity, the construction on the lookdown space also allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a. s. cadlag paths with additional jumps at the extinction times of parts of the population.
Highlights
To the measure-valued Fleming-Viot process that is a model for the evolution of the type distribution in a large neutral haploid population, a tree-valued FlemingViot process models the evolution of the distribution of the genealogical distances between randomly sampled individuals
The tree-valued Fleming-Viot process is introduced in Greven, Pfaffelhuber, and Winter [19] and generalized in [22] to the setting with simultaneous multiple reproduction events
We identify each element (t, i) of R+ × N with the individual on level i at time t
Summary
To the measure-valued Fleming-Viot process that is a model for the evolution of the type distribution in a large neutral haploid population, a tree-valued FlemingViot process models the evolution of the distribution of the genealogical distances between randomly sampled individuals. Wakolbinger, we consider a process whose state space is endowed with a stronger topology, the GromovHausdorff-Prohorov topology, which highlights the overall structure of the population This process has jumps already in the Kingman case, namely at the times when the shape of the whole genealogical tree changes as all descendants of an ancestor die out (Theorem 3.5 and Proposition 4.2). Besides the space of isomorphy classes of marked metric measure spaces, another possible state space for tree-valued Fleming-Viot processes (in the case with or without dust) is a space of matrix distributions. Marked metric measure spaces are applied by Depperschmidt, Greven, and Pfaffelhuber [10, 11] to construct the tree-valued Fleming-Viot process with mutation and selection
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