Article,
Limiting distributions for branching random fields
Trans. Amer. Math. Soc., (1978)Branching Brownian motion, the subject of this article, is a continuous time Galton-Watson process in which particles also have positions. On splitting, each particle is replaced by its daughter particles and they then follow independent Brownian motions until they split. The branching process is assumed to be critical. Let $N_A(t)$ be the number of particles in the bounded set $A$, of Lebesgue measure $m(A)$, at the time $t$. It is shown that if there is only one particle initially and the movement of the particles occurs in the plane then $c_2(tlogt)PN_A(t)>c_1m(A)łambdate^-łambda$ for $łambda>0$ as $t\rightarrowınfty$, where $c_1$ and $c_2$ are specified constants. The method of proof is to obtain good estimates of $EN_A(t)^k$ for all $k$ and hence of the moment generating function of $N_A(t)/(m(A)t)$; from this the result is derived. If the set of initial particles forms a homogeneous Poisson process of unit rate in the plane (in fact, a slightly weaker assumption, as made by the author, is sufficient), then $c_3logtPN_A(t)>c_1m(A)łambdate^-łambda$ for $łambda>0$ as $t\rightarrowınfty$, where $c_1$ and $c_3$ are specified constants..
Abstract
In this paper we derive limiting distributions for branching Brownian motion. The cases considered are where the state space is (1) the line and (2) the plane where (a) initially there's but one particle and (b) initially there's a random number of independent particles. In all cases the branching process is critical and we obtain results for the growth of selectively neutral mutant types. We use generating functions to derive these results.
