Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior

TL;DR

This paper studies the long-term behavior of the infinite rate mutually catalytic branching process in infinitely many colonies, showing conditions for survival and coexistence of types based on the interaction kernel's recurrence or transience.

Contribution

It provides a rigorous analysis of the longtime behavior of IMUB, establishing criteria for survival and coexistence depending on the interaction kernel's properties.

Findings

Only one type survives under recurrent kernels.

Both types coexist under certain transient conditions.

Results extend understanding of IMUB dynamics in infinite settings.

Abstract

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.