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Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence
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  • Published: 18 June 2011

Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence

  • Achim Klenke1 &
  • Leonid Mytnik2 

Probability Theory and Related Fields volume 154, pages 533–584 (2012)Cite this article

  • 159 Accesses

  • 8 Citations

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Abstract

We construct a mutually catalytic branching process on a countable site space with infinite “branching rate”. The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins (Ann Probab 26(3):1088–1138, 1998). We show that our model is the limit for a class of models and in particular for the Dawson–Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as a weak solution of an infinite system of stochastic integral equations driven by a Poisson noise.

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099, Mainz, Germany

    Achim Klenke

  2. Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Leonid Mytnik

Authors
  1. Achim Klenke
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  2. Leonid Mytnik
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Corresponding author

Correspondence to Achim Klenke.

Additional information

This work is partly funded by the German Israeli Foundation with grant number G-807-227.6/2003.

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Cite this article

Klenke, A., Mytnik, L. Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence. Probab. Theory Relat. Fields 154, 533–584 (2012). https://doi.org/10.1007/s00440-011-0376-1

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  • Received: 10 March 2011

  • Revised: 30 May 2011

  • Published: 18 June 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0376-1

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Keywords

  • Mutually catalytic branching
  • Martingale problem
  • Duality
  • Stochastic differential equations

Mathematics Subject Classification (2000)

  • 60K35
  • 60K37
  • 60J80
  • 60J65
  • 60H20
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