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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
Klaus Schürger*
Affiliation:
German Cancer Research Centre
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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).

Keywords

SPATIAL SPREAD PROCESSBRANCHING PROCESS WITH INTERACTIONSTUMOUR GROWTHEDEN PROCESSSTRONG LAW OF LARGE NUMBERSCORRELATION INEQUALITY
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 1 , March 1981 , pp. 14 - 39
Copyright
Copyright © Applied Probability Trust 1981 

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