Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

Abstract

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on ℤ and can branch at 0 only. Varying the initial (finite) number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n ^β particles and consider the scaled process $Z^{n}_{t} {\left( \bullet \right)} = Z_{{nt}} {\left( {{\sqrt {n \bullet } }} \right)}$, where Z _t is the measure–valued process representing the original particle system. We prove that $Z^{n}_{t}$ converges to 0 when $\beta < \frac{1} {4}$ and to a nondegenerate discrete distribution when $\beta = \frac{1} {4}$ . In addition, if $\frac{1} {4} < \beta < \frac{1} {2}$ then $n^{{ - {\left( {2\beta - \frac{1} {2}} \right)}}} Z^{n}_{t}$ converges to a random limit, while if $\beta > \frac{1} {2}$ then $n^{{ - \beta }} Z^{n}_{t}$ converges to a deterministic limit.