Summary
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:
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:
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 → +∞.
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Chauvin, B., Rouault, A. KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees. Probab. Th. Rel. Fields 80, 299–314 (1988). https://doi.org/10.1007/BF00356108
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DOI: https://doi.org/10.1007/BF00356108
Keywords
- Stochastic Process
- Brownian Motion
- Probability Theory
- Point Measure
- Mathematical Biology