Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479–497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.
"Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior." Ann. Probab. 40 (1) 103 - 129, January 2012. https://doi.org/10.1214/10-AOP621