Electronic Journal of Probability

Pathwise construction of tree-valued Fleming-Viot processes

Stephan Gufler

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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. càdlàg 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. càdlàg paths with additional jumps at the extinction times of parts of the population.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 42, 58 pp.

Dates
Received: 2 November 2016
Accepted: 6 April 2018
First available in Project Euclid: 9 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1525852819

Digital Object Identifier
doi:10.1214/18-EJP166

Mathematical Reviews number (MathSciNet)
MR3806410

Zentralblatt MATH identifier
1390.60350

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
lookdown model tree-valued Fleming-Viot process evolving coalescent $\Xi $-coalescent (marked) metric measure space (marked) Gromov-weak topology Gromov-Hausdorff-Prohorov topology

Rights
Creative Commons Attribution 4.0 International License.

Citation

Gufler, Stephan. Pathwise construction of tree-valued Fleming-Viot processes. Electron. J. Probab. 23 (2018), paper no. 42, 58 pp. doi:10.1214/18-EJP166. https://projecteuclid.org/euclid.ejp/1525852819


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