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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This work is partly funded by the German Israeli Foundation with grant number G-807-227.6/2003.
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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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DOI: https://doi.org/10.1007/s00440-011-0376-1
Keywords
- Mutually catalytic branching
- Martingale problem
- Duality
- Stochastic differential equations
Mathematics Subject Classification (2000)
- 60K35
- 60K37
- 60J80
- 60J65
- 60H20