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

Abstract

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 KPP equation: $$\frac{{\partial u}}{{\partial t}} = \frac{1}{2}\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u)$$ 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: $$\frac{{R_t }}{t}\xrightarrow{P}c_0 = \sqrt {2a(m - 1)} ,{\text{ when }}t \to + \infty .$$ For the general KPP equation, we show limit theorems for u(t, ct+ζ), c>c 0 , ξ ∈ ℝ, t → +∞. Large deviations for R t and probabilities of presence of particles for the branching process are deduced: (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, +∞[)>0}, is studied in the limit as t → +∞.