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November, 1982 Spatial Growth of a Branching Process of Particles Living in $\mathbb{R}^d$
Kohei Uchiyama
Ann. Probab. 10(4): 896-918 (November, 1982). DOI: 10.1214/aop/1176993712


Consider a branching process in which particles are located in $\mathbb{R}^d$, do not move during their life times, die according to the exponential holding law, and at their deaths give birth to random number of particles which are located at distances from their parents. The total number process is supposed supercritical. We are interested in the number of particles living in a shifted region $D + tc$, denoted by $Z_t(D + tc)$, where $c \in \mathbb{R}^d$ and $D$ is a bounded set of $\mathbb{R}^d$, and observe a.s. convergences of $Z_t(D + tc)/E\lbrack Z_t(D + tc)\rbrack$ as $t \rightarrow \infty$. The result is applied to an associated non-linear evolution equation, which reduces, in a special case, to the equation of a deterministic model of simple epidemics.


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Kohei Uchiyama. "Spatial Growth of a Branching Process of Particles Living in $\mathbb{R}^d$." Ann. Probab. 10 (4) 896 - 918, November, 1982.


Published: November, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0499.60088
MathSciNet: MR672291
Digital Object Identifier: 10.1214/aop/1176993712

Primary: 60J80
Secondary: 60J85

Keywords: model of simple epidemics , Spatial growth of brancing process , super-critical

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 4 • November, 1982
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