Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Vladimir?Vatutin Jie?Xiong

Affiliation:	(1) Steklov Mathematical Institute, Gubkin street, 8, 119991 Moscow, Russia;(2) Department of Mathematics, University of Tennessee, Knoxville, TN 37996–1300, USA;(3) Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, P. R. China

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 n ^βparticles and consider the scaled process $Z^{n}_{t} {left( bullet right)} = Z_{{nt}} {left( {{sqrt {n bullet } }} right)}$ , where Z _t 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 $n^{{ - {left( {2beta - frac{1} {2}} right)}}} Z^{n}_{t}$ converges to a random limit, while if $beta > frac{1} {2}$ then $n^{{ - beta }} Z^{n}_{t}$ converges to a deterministic limit. * Research supported partially by DFG and grants RFBR 02–01–00266 and Russian Scientific School 1758.2003.1 ** Research supported partially by NSA and by Alexander von Humboldt Foundation

Keywords:	Renewal equation branching particle system scaling limit
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