WIAS Preprint No. 919, (2004)

Some limit theorems for a particle system of single point catalytic branching random walks


  • Vatutin, Vladimir
  • Xiong, Jie

2010 Mathematics Subject Classification

  • 60J80 60K25


  • Renewal equation, branching particle system, scaling limit


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 $Z^n_t(bullet)=Z_nt(sqrtn, bullet)$ where $Z_t$ is the measure-valued process representing the original particle system. We prove that $Z^n_t$ converges to 0 when $befrac12$ then $n^-beZ^n_t$ converges to a deterministic limit. Note that according to Kaj and Sagitov citeKS $n^-frac12Z^n_t$ converges to a random limit if $be=frac12.$

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