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

Authors

    • Steklov Mathematical Institute
  • Jie Xiong**
    • Department of MathematicsUniversity of Tennessee
    • Department of MathematicsHebei Normal University
ORIGINAL ARTICLES

DOI: 10.1007/s10114-005-0757-4

Cite this article as:
Vatutin*, V. & Xiong**, J. Acta Math Sinica (2007) 23: 997. doi:10.1007/s10114-005-0757-4

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 equationbranching particle systemscaling limit

MR (2000) Subject Classification

60J8060K25

Copyright information

© Springer-Verlag Berlin Heidelberg 2006