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KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. (English) Zbl 0653.60077

If \(R_ t\) is the position of the rightmost particle at time t in a one- dimensional branching Brownian motion, \(u(t,x)=P(R_ t>x)\) is a solution of the KPP equation: \[ \partial u/\partial t=2^{-1}\partial^ 2u/\partial x^ 2+f(u) \] where \(f(u)=\alpha (1-u-g(1-u))\), g is the generating function of the reproduction law and \(\alpha\) the inverse of the mean lifetime; if \(m=g'(1)>1\) and \(g(0)=0\), it is known that: \[ R_ t/t\to^{P}c_ o=\sqrt{2\alpha (m-1)},\quad when\quad t\to +\infty. \] For the general KPP equation, we show limit theorems for \(u(t,ct+\xi)\), \(c>c_ 0\), \(\xi\in {\mathbb{R}}\), \(t\to +\infty\). Large deviations for \(R_ t\) and probabilities of presence of particles for the branching process are deduced: \[ P(Z_ t(]ct+\xi,+\infty [)>0)\sim_{t\to +\infty}Const.E Z_ t(]ct+\xi,+\infty [)\quad (c>c_ 0) \] (where \(Z_ t\) denotes the random point measure of particles living at time t), and a Yaglom type theorem is proved. The conditional distribution of the spatial tree, given \(\{Z_ t(]ct,+\infty [)>0\}\) is studied in the limit as \(t\to +\infty\).

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 Applications of branching processes
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