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A class of branching processes on a lattice with interactions

Published online by Cambridge University Press:  01 July 2016

Klaus Schürger*
Affiliation:
German Cancer Research Centre

Abstract

In this paper, a very general class of branching processes on the d-dimensional square lattice is studied. It is assumed that the division rates as well as the spatial distribution of offspring are configuration-dependent. The main interest of this paper is in the asymptotic geometrical behaviour of such processes. Utilizing techniques mainly due to Richardson [28], we derive conditions which are necessary and sufficient for such branching processes to have the following property: there exists a norm N(·) on Rd such that, for all 0 < < 1, we have that almost surely for all sufficiently large t, all sites in the N-ball of radius (1 – )t are contained in (the set of sites occupied at time t) and is contained in the set of all sites in the N-ball of radius (1 + )t (given that the process starts with finitely many particles).

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1981 

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