KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees

Abstract

If Rt is the position of the rightmost particle at time t in a one dimensional branching brownian motion, u(t, x)=P(Rt>x) is a solution of KPP equation: {Mathematical expression} where f(u)=α(1-u-g(1-u))g is the generating function of the reproduction law and α the inverse of the mean lifetime; if m=g′(1)>1 and g(0)=0, it is known that: {Mathematical expression} For the general KPP equation, we show limit theorems for u(t, ct+ζ), c>c0, ξ ∈ ℝ, t → +∞. Large deviations for Rt and probabilities of presence of particles for the branching process are deduced: [Figure not available: see fulltext.] (where Zt 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 {Zt(]ct, +∞[)>0}, is studied in the limit as t → +∞. © 1988 Springer-Verlag.