ON THE STRUCTURE OF SPATIAL BRANCHING PROCESSES

The paper is a contribution to the theory of branching pro- cesses with discrete time and a general phase space in the sense of (2). We characterize the class of regular, i.e. in a sense sufficiently random, branch- ing processes (�k)k2Z by almost sure properties of their realizations with- out making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (�k) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the reg- ular ones are mixtures of left-tail trivial processes with a Poisson family structure.