A class of branching processes on a lattice with interactions

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).