Long-time behavior and coexistence in a mutually catalytic branching model

Abstract

We study a system of two interacting populations which undergo random migration and mutually catalytic branching. The branching rate of one population at a site is proportional to the mass of the other population at the site. The system is modelled by an infinite system of stochastic differential equations, allowing symmetric Markov migration, if the set of sites is discrete $(\mathbb{Z}^d)$, or by a stochastic partial differential equation with Brownian migration if the set of sites is the real line. A duality technique of Leonid Mytnik, which gives uniqueness in law, is used to examine the long-time behavior of the solutions. For example, with uniform initial conditions, the process converges to an equilibrium distribution as $t \to \infty$, and there is coexistence of types in the equilibrium “iff ” the random migration is transient.