Application of Space-time Harmonic Functions to Branching Processes

Abstract

In attempting to determine the growth properties of a branching process (b.p.), a standard method of attack is to look for the appropriate martingale. Here we show that for many b.p., this really corresponds to looking for the harmonic functions associated with the space-time process. As a particular application of the above we show that in the case of the classical Galton-Watson continuous time process with $m < \infty$ there exists constants $c(t)$ such that $Z_t/c(t)$ converges w.p. 1 to a nontrivial random variable.