WIAS Preprint No. 528, (1999)

A cyclically catalytic super-Brownian motion



Authors

  • Fleischmann, Klaus
  • Xiong, Jie

2010 Mathematics Subject Classification

  • 60K35 60G57 60J80

Keywords

  • Catalyst, reactant, superprocess, duality, martingale problem, cyclic reaction, global segregation of neighboring types, finite time survival, extinction, strong Markov selection, stochastic equation

Abstract

In generalization of the mutually catalytic super-Brownian motion in R of Dawson/Perkins (1998) and Mytnik (1998), a function-valued cyclically catalytic model X is constructed as a strong Markov solution to a martingale problem. Starting with a finite population X0, each pair of neighboring types will globally segregate in the long-term limit (non-coexistence of neighboring types). Also finer extinction/survival properties in dependence on X0 are studied in the spirit of Mueller and Perkins (1999). In fact, X0 can be chosen in such a way that all types survive all finite times. On the other hand, sufficient conditions on X0 are stated for the following situation: Given a type k and a positive time t, the kth subpopulation Xk dies by time t with a large probability, provided that its initial value Xk0 was sufficiently small.

Appeared in

  • Ann. Prob. 29 (2001), p-p. 820-861

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