The longtime behavior of branching random walkin a catalytic

Consider a countable collection (t) of particles located on a countable group, performing a critical branching random walk where the branching rate of a particle is given by a random medium uctuating both in space and time. Here we study the case where the time{space random medium (t) (called catalyst) is also a critical branching random walk evolving autonomously while the local branching rate of the reactant process (t) is proportional to the number of catalytic particles present at a site. The process (t) and the process (t) typically have diierent underlying motions. Our main interest is to analyze the behavior of the bivariate system (t; t) as t ! 1 and to exhibit features of (t) t0 which are diierent from those of classical branching random walk. Some of these features have already been noticed for super processes. First we show rst that if the symmetrized motion of the catalytic particles is transient then (t) behaves similarly as a classical branching random walk: depending on whether the symmetrized motion of the reactant is transient or recurrent we have an equilibrium with the original intensity, or we have local extinction. Next, we consider the case where the symmetrized motion of the catalyst is recurrent. It is well known that in this case the catalyst goes locally to extinction; however, we discover new features of the reactant depending on the mobility both of catalyst and reactant. Now three regimes are possible. In one regime the reactant (t) behaves like a system of independent random walks in the limit t ! 1, that is it converges to a Poisson system. In a second regime the reactant (t) approaches also a nontrivial stable state which is now a mixture of Poisson systems. Thirdly the reactant (t) can become locally extinct even in this case of a catalyst going to local extinction. We give examples for all these regimes and set the framework to develop a complete theory.