%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 }