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KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees
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  • Published: December 1988

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

  • Brigitte Chauvin1 &
  • Alain Rouault2 

Probability Theory and Related Fields volume 80, pages 299–314 (1988)Cite this article

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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:

$$\frac{{\partial u}}{{\partial t}} = \frac{1}{2}\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u)$$

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 → +∞.

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Author information

Authors and Affiliations

  1. Laboratoire de Probabilités, Université Paris VI., 4, place Jussieu, tour 56, 3ème étage, F-75230, Paris Cedex 05

    Brigitte Chauvin

  2. UA-CNRS 743, Statistique Appliquée, Université Paris Sud, Mathématiques, Bat. 425, F-91405, Orsay Cedex

    Alain Rouault

Authors
  1. Brigitte Chauvin
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  2. Alain Rouault
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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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  • Received: 26 February 1988

  • Revised: 06 June 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00356108

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Keywords

  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Point Measure
  • Mathematical Biology
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