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Tree-valued resampling dynamics Martingale problems and applications
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  • Published: 03 February 2012

Tree-valued resampling dynamics Martingale problems and applications

  • Andreas Greven1,
  • Peter Pfaffelhuber2 &
  • Anita Winter3 

Probability Theory and Related Fields volume 155, pages 789–838 (2013)Cite this article

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  • 28 Citations

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Abstract

The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming–Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled “individuals”. We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming–Viot dynamics.

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Author information

Authors and Affiliations

  1. Department Mathematik, University of Erlangen, Cauerstrasse 11, 91058, Erlangen, Germany

    Andreas Greven

  2. Abteilung für Mathematische Stochastik, Albert-Ludwigs University of Freiburg, Eckerstrasse 1, 79104, Freiburg, Germany

    Peter Pfaffelhuber

  3. Fakultät für Mathematik, Universität Duisburg-Essen, Universitätsstr. 2, 45141, Essen, Germany

    Anita Winter

Authors
  1. Andreas Greven
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  2. Peter Pfaffelhuber
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  3. Anita Winter
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Corresponding author

Correspondence to Peter Pfaffelhuber.

Additional information

All authors were supported in part by the DFG-Forschergruppe 498 through grant GR 876/13-1,2,3.

P. Pfaffelhuber was supported in part by the BMBF, Germany, through FRISYS (Freiburg Initiative for Systems biology), Kennzeichen 0313921.

A. Winter was supported in part at the Technion by a fellowship from the Aly Kaufman Foundation.

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Greven, A., Pfaffelhuber, P. & Winter, A. Tree-valued resampling dynamics Martingale problems and applications. Probab. Theory Relat. Fields 155, 789–838 (2013). https://doi.org/10.1007/s00440-012-0413-8

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  • Received: 20 May 2010

  • Revised: 14 December 2011

  • Published: 03 February 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-012-0413-8

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Keywords

  • Tree-valued Markov process
  • Fleming–Viot process
  • Moran model
  • Genealogical tree
  • Martingale problem
  • Duality
  • (ultra-)Metric measure space
  • Gromov-weak topology

Mathematics Subject Classification (2000)

  • Primary: 60K35
  • 60J25
  • Secondary: 60J70
  • 92D10
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