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Branching-coalescing particle systems
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  • Published: 20 August 2004

Branching-coalescing particle systems

  • Siva R. Arthreya1 &
  • Jan M Swart2 

Probability Theory and Related Fields volume 131, pages 376–414 (2005)Cite this article

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An Erratum to this article was published on 10 June 2009

Abstract.

We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.

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Authors and Affiliations

  1. Indian Statistical Institute, 8th mile Mysore Road, RV College PO, Bangalore, -560059, India

    Siva R. Arthreya

  2. Mathematical Institute, University Erlangen-Nuremberg, Bismarckstraße 1 1/2, 91054, Erlangen, Germany

    Jan M Swart

Authors
  1. Siva R. Arthreya
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  2. Jan M Swart
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Corresponding author

Correspondence to Jan M Swart.

Additional information

Mathematics Subject Classification (2000):Primary: 60K35, 92D25; Secondary: 60J80, 60J60

Research supported in part by the German Science Foundation.

AcknowledgementWe thank Klaus Fleischmann who played a stimulating role during the early stages of this project and answered a question about Laplace functionals, Claudia Neuhauser for answering questions about branching-coalescing processes, Olle Häggström for answering questions on nonamenable groups, and Tokuzo Shiga for answering our questions about his work. We thank the referee for drawing our attention to the reference [SU86]. Part of this work was carried out during the visits of Siva Athreya to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin and to the Friedrich-Alexander University Erlangen-Nuremberg, and of Jan Swart to the Indian Statistical Institute, Delhi. We thank all these places for their kind hospitality.

An erratum to this article is available at http://dx.doi.org/10.1007/s00440-009-0232-8.

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Arthreya, S., Swart, J. Branching-coalescing particle systems. Probab. Theory Relat. Fields 131, 376–414 (2005). https://doi.org/10.1007/s00440-004-0377-4

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  • Received: 12 August 2003

  • Revised: 14 June 2004

  • Published: 20 August 2004

  • Issue Date: March 2005

  • DOI: https://doi.org/10.1007/s00440-004-0377-4

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Key words or phrases:

  • First Schlögl Model
  • Reaction-diffusion process
  • Autocatalytic reaction
  • Branching
  • Coalescence
  • Resampling
  • Selection
  • Mutation
  • Contact process
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