Multiplicative Martingales for Spatial Branching Processes

Abstract

For the entire collection see Zbl 0635.00011. par This is a very interesting paper about spatial branching processes or, more specifically, binary splitting branching Brownian motion on the real line. Particles perform independent Brownian motions. The initial ancestor splits into two particles and after an exponential time (mean one) these behave similarly and so on. Let Nsb t be the point process of the particles present at time t. Then exp (int log (Φ(x+y- λt))Nsb t(dy)) form martingales, provided that Φ solves Φ''/2-λΦ'+Φ(1-Φ)=0quad withquad Φ(x)in (0,1) for all x. Such solutions exist for λsp 2ge 2. It is shown that the limit of these martingales is the same as that of the additive martingales int exp (ay-v(a)t)Nsb t(dy),quad wherequad a=λ-sqrtlambdasp 2-2,quad providedquad λsp 2>2. Similar additive martingales have been discussed, for example, by the reviewer J. Appl. Probab. 14, 25-37 (1977; Zbl 0356.60053) and it K. Uchiyama Ann. Probab 10, 896-918 (1982; Zbl 0499.60088). When λsp 2=2 the multiplicative martingale has a non-trivial limit, whilst the additive one does not, and it is shown that this non-trivial limit is related to that of the martingale int (sqrt2t-y)exp (sqrt2y-2t)Nsb t(dy). There are close connections here with Theorem 1 of it S. P. Lalley and it T. Sellke, ibid. 15, 1052-1061 (1987; Zbl 0622.60085). par Finally, the number, Zsb ssplambda, of points, where the space- time tree formed by the process' first crosses (in the sense that no `ancestor' has already done so) of the line x=λt-s is considered. It is indicated that Zsb ssplambda: sge 0 forms a Markov branching process and that its asymptotics are related to those of the multiplicative martingales.