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

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

作者姓名：	Vladimir VATUTIN Jie XIONG

作者单位：	[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

摘要：	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^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.
关键词：	补充方程分歧颗粒系统标定极限随机通道
收稿时间：	21 April 2004
修稿时间：	2004-04-212005-01-19
Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

Vladimir VATUTIN Jie XIONG.Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks[J].Acta Mathematica Sinica,2007,23(6):997-1012.

Authors:	Vladimir?Vatutin Jie?Xiong

Institution:	(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 $n^{{ - {\left( {2\beta - \frac{1} {2}} \right)}}} Z^{n}_{t}$ converges to a random limit, while if $\beta > \frac{1} {2}$ then $n^{{ - \beta }} Z^{n}_{t}$ converges to a deterministic limit. * 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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