A class of branching processes on a lattice with interactions

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Abstract
In this paper, a very general class of branching processes on thed-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(·) onRdsuch that, for all 0 <∊< 1, we have that almost surely for all sufficiently larget, all sites in theN-ball of radius (1 –∊)tare contained in(the set of sites occupied at timet) andis contained in the set of all sites in theN-ball of radius (1 +∊)t(given that the process starts with finitely many particles).

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