Acta Mathematica Sinica, English Series

, Volume 23, Issue 6, pp 997–1012

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

ORIGINAL ARTICLES

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 nβ particles and consider the scaled process \( Z^{n}_{t} {\left( \bullet \right)} = Z_{{nt}} {\left( {{\sqrt {n \bullet } }} \right)} \), where Zt 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.

Keywords

Renewal equation branching particle system scaling limit 

MR (2000) Subject Classification

60J80 60K25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  3. 3.Department of MathematicsHebei Normal UniversityShijiazhuang 050016P. R. China

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