Multi-scale Clustering for a Non-Markovian Spatial Branching Process |
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Authors: | Email author" target="_blank">Klaus?FleischmannEmail author Vladimir?A?Vatutin |
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Institution: | (1) Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr 39, D–10117 Berlin, Germany;(2) Department of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 117 966 Moscow, GSP-1, Russia |
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Abstract: | Consider a system of particles which move in Rd according to a symmetric α-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+β. In case of the critical dimension d=α/β the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start
with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform,
but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+β. Our result generalizes the case α=2 of Brownian particles of Klenke (1998), where p.d.e. methods had been used which are not available in the present setting.
Supported in part by the DFG.
Supported in part by the grants RFBR 02-01-00266 and Russian Scientific School 1758.2003.1. |
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Keywords: | Age-dependent process branching particle system critical dimension continuous-state branching scaling limit theorem |
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