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Multiplicative martingales for spatial branching processes. (English) Zbl 0652.60089

Stochastic processes, Semin., Princeton/New Jersey 1987, Prog. Probab. Stat. 15, 223-242 (1988).
[For the entire collection see Zbl 0635.00011.]
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 \(N_ t\) be the point process of the particles present at time t. Then \[ \exp (\int \log (\Phi (x+y- \lambda t))N_ t(dy)) \] form martingales, provided that \(\Phi\) solves \[ \Phi ''/2-\lambda \Phi '+\Phi (1-\Phi)=0\quad with\quad \Phi (x)\in (0,1) \] for all x. Such solutions exist for \(\lambda^ 2\geq 2\). It is shown that the limit of these martingales is the same as that of the additive martingales \[ \int \exp (ay-v(a)t)N_ t(dy),\quad where\quad a=\lambda -\sqrt{\lambda^ 2-2},\quad provided\quad \lambda^ 2>2. \] Similar additive martingales have been discussed, for example, by the reviewer [J. Appl. Probab. 14, 25-37 (1977; Zbl 0356.60053)] and K. Uchiyama [Ann. Probab 10, 896-918 (1982; Zbl 0499.60088)]. When \(\lambda^ 2=2\) 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 \[ \int (\sqrt{2}t-y)\exp (\sqrt{2y}-2t)N_ t(dy). \] There are close connections here with Theorem 1 of S. P. Lalley and T. Sellke, ibid. 15, 1052-1061 (1987; Zbl 0622.60085).
Finally, the number, \(Z_ s^{\lambda}\), 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 \(x=\lambda t-s\) is considered. It is indicated that \(\{Z_ s^{\lambda}:\) \(s\geq 0\}\) forms a Markov branching process and that its asymptotics are related to those of the multiplicative martingales.
Reviewer: J.Biggins

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G44 Martingales with continuous parameter
60J60 Diffusion processes