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

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on . 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 . particles and consider the scaled process Znt(.)=Znt(n....), where Z t is the measure.valued process representing the original particle system. We prove that Znt converges to 0 when .<14 and to a nondegenerate discrete distribution when .=14 . In addition, if 14<.<12 then n.(2..12)Znt converges to a random limit, while if .>12 then n..Znt converges to a deterministic limit.