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:
∂u∂t=12∂2u∂x2+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:
Rtt−→Pc0=2a(m−1)‾‾‾‾‾‾‾‾‾√, when t→+∞.
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:
http://static-content.springer.com.libproxy1.usc.edu/image/art%3A10.1007%2FBF00356108/MediaObjects/440_2006_BF00356108_f1.jpg
(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 → +∞.
Description
KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees - Springer
%0 Journal Article
%1 chauvin1988equation
%A Chauvin, Brigitte
%A Rouault, Alain
%D 1988
%I Springer-Verlag
%J Probability Theory and Related Fields
%K Fisher-KPP branching_Brownian_motion branching_processes edge_of_the_wave large_deviations point_processes wave_of_advance
%N 2
%P 299-314
%R 10.1007/BF00356108
%T KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees
%U http://dx.doi.org/10.1007/BF00356108
%V 80
%X 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:
∂u∂t=12∂2u∂x2+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:
Rtt−→Pc0=2a(m−1)‾‾‾‾‾‾‾‾‾√, when t→+∞.
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:
http://static-content.springer.com.libproxy1.usc.edu/image/art%3A10.1007%2FBF00356108/MediaObjects/440_2006_BF00356108_f1.jpg
(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 → +∞.
@article{chauvin1988equation,
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:
∂u∂t=12∂2u∂x2+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:
Rtt−→Pc0=2a(m−1)‾‾‾‾‾‾‾‾‾√, when t→+∞.
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:
http://static-content.springer.com.libproxy1.usc.edu/image/art%3A10.1007%2FBF00356108/MediaObjects/440_2006_BF00356108_f1.jpg
(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 → +∞.},
added-at = {2015-08-30T08:16:15.000+0200},
author = {Chauvin, Brigitte and Rouault, Alain},
biburl = {https://www.bibsonomy.org/bibtex/2bed576c5be1d77deeba1458a7c2c7a43/peter.ralph},
description = {KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees - Springer},
doi = {10.1007/BF00356108},
interhash = {4249f35f2b6f59bcb6e24dac184a49ed},
intrahash = {bed576c5be1d77deeba1458a7c2c7a43},
issn = {0178-8051},
journal = {Probability Theory and Related Fields},
keywords = {Fisher-KPP branching_Brownian_motion branching_processes edge_of_the_wave large_deviations point_processes wave_of_advance},
language = {English},
number = 2,
pages = {299-314},
publisher = {Springer-Verlag},
timestamp = {2015-08-30T08:16:15.000+0200},
title = {KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees},
url = {http://dx.doi.org/10.1007/BF00356108},
volume = 80,
year = 1988
}