Abstract
We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth coefficients for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive are given. In the Brownian motions case the domain of evolution can be non-smooth, including Lipschitz, with integrable Martin kernel. The results are valid for an arbitrary number of particles and non-uniform redistribution after branching. Additionally, with probability one, it is shown that only one ancestry line survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times.
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Grigorescu, I., Kang, M. Immortal particle for a catalytic branching process. Probab. Theory Relat. Fields 153, 333–361 (2012). https://doi.org/10.1007/s00440-011-0347-6
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DOI: https://doi.org/10.1007/s00440-011-0347-6