Supercritical spatially homogeneous branching in \(\mathbb{R}^n\) admits no equilibria (Q2747572)
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scientific article; zbMATH DE number 1658048
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Supercritical spatially homogeneous branching in \(\mathbb{R}^n\) admits no equilibria |
scientific article; zbMATH DE number 1658048 |
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9 April 2002
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branching particle system
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spatially homogeneous equilibria
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Abelian locally compact Hausdorff topological group
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Supercritical spatially homogeneous branching in \(\mathbb{R}^n\) admits no equilibria (English)
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At the beginning of investigations in spatially homogeneous branching processes in Euclidean space [see \textit{A. Liemant}, Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-naturw. R. 18, 361-372 (1969; Zbl 0265.60082)] it seemed to be obvious that the existence of equilibria implies criticality of branching. This prejudice was disproved by the example of \textit{Ra. Siegmund-Schultze} [Math. Nachr. 151, 101-120 (1991; Zbl 0739.60077)] of a subcritical spatially homogeneous branching equilibrium in dimension one. Let \((G,\rho_G)\) be an Abelian locally compact Hausdorff topological group \(G\) with a countable base and with a metric \(\rho_G\). It is proved that supercritical spatially homogeneous branching processes with the phase space \((G,\rho_G)\) (in particular in Euclidean space \(\mathbb R^n\)) have no (non-void, homogeneous) equilibria.
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