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 performs random walks on \ZZ 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^{\be} particles and consider the scaled process $Z^n_t(\bullet)=Z_{nt}(\sqrt{n}\, \bullet)$ where $Z_t$ is the measure-valued process representing the original particle system. We prove that $Z^n_t$ converges to 0 when \be<\frac{1}{4} and to a nondegenerate discrete distribution when \be=\frac{1}{4}. In addition, if \frac{1}{4}<\be<\frac{1}{2} then n^{-(2\be-\frac{1}{2})}Z^n_t converges to a random limit while if \be>\frac{1}{2} then n^{-\be}Z^n_t converges to a deterministic limit. Note that according to Kaj and Sagitov \cite{KS} $n^{-\frac{1}{2}}Z^n_t$ converges to a random limit if $\be=\frac{1}{2}.