Branching Random Walk with Catalysts

Abstract Shnerb et al. (2000), (2001) studied the following system of interacting particles on Z: There are two kinds of particles, called A-particles and B-particles. The A-particles perform continuous time simple random walks, independently of each other. The jumprate of each A-particle is DA. The B-particles perform continuous time simple random walks with jumprate DB , but in addition they die at rate δ and a B-particle at x at time s splits into two particles at x during the next ds time units with a probability βNA(x, s)ds+ o(ds), where NA(x, s) (NB(x, s)) denotes the number of A-particles (respectively B-particles) at x at time s. Conditionally on the A-system, the jumps, deaths and splittings of different B-particles are independent. Thus the B-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of A-particles which coincide with the appropriate B-particles. One starts the process with all the NA(x, 0), x ∈ Z, as independent Poisson variables with mean μA, and the NB(x, 0), x ∈ Z, independent of the A-system, translation invariant and with mean μB . Shnerb et al. (2000) made the interesting discovery that in dimension 1 and 2 the expectation E{NB(x, t)} tends to infinity, no matter what the values of δ, β,DA, DB , μA, μB ∈ (0,∞) are. We shall show here that nevertheless there is a phase transition in all dimensions, that is, the system becomes (locally) extinct for large δ but it survives for β large and δ small.