A Cyclically Catalytic Super-Brownian Motion

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