KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees
type: Item
instance of: scholarly article
title: KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees
zbMATH Open document ID: 0653.60077
DOI: 10.1007/BF00356108
author: Brigitte Chauvin
author: Alain Rouault
publication date: 1988-01-01T00:00:00Z
review text: 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 the KPP equation: \[ \partial u/\partial t=2^{-1}\partial^ 2u/\partial x^ 2+f(u) \] where \(f(u)=\alpha (1-u-g(1-u))\), g is the generating function of the reproduction law and \(\alpha\) the inverse of the mean lifetime; if \(m=g'(1)>1\) and \(g(0)=0\), it is known that: \[ R_ t/t\to^{P}c_ o=\sqrt{2\alpha (m-1)},\quad when\quad t\to +\infty. \] For the general KPP equation, we show limit theorems for \(u(t,ct+\xi)\), \(c>c_ 0\), \(\xi\in {\mathbb{R}}\), \(t\to +\infty\). Large deviations for \(R_ t\) and probabilities of presence of particles for the branching process are deduced: \[ P(Z_ t(]ct+\xi,+\infty [)>0)\sim_{t\to +\infty}Const.E Z_ t(]ct+\xi,+\infty [)\quad (c>c_ 0) \] (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 \(\{Z_ t(]ct,+\infty [)>0\}\) is studied in the limit as \(t\to +\infty\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J85
zbMATH DE Number: 4066084
zbMATH Keywords: branching Brownian motion
zbMATH Keywords: generating function of the reproduction law
zbMATH Keywords: Large deviations
zbMATH Keywords: random point measure of particles
MaRDI profile type: Publication
cites work: Q4089232
cites work: Q3664178
cites work: Q5674726
cites work: Martingale convergence in the branching random walk
cites work: Chernoff's theorem in the branching random walk
cites work: Growth rates in the branching random walk
cites work: Convergence of solutions of the Kolmogorov equation to travelling waves
cites work: Arbres et processus de Bellman-Harris. (Trees and Bellman-Harris processes)
cites work: Sur la propriété de branchement. (On the property of branching)
cites work: Q3738360
cites work: Rate of expansion of an inhomogeneous branching process of brownian particles
cites work: Q4189096
cites work: Limit theorems for large deviations and reaction-diffusion equations
cites work: Location of Wave Fronts for the Multi-Dimensional K-P-P Equation and Brownian First Exit Densities
cites work: Q4127752
cites work: The growth and composition of branching populations
cites work: Branching Brownian motion with absorption
cites work: Subdiffusive behavior of random walk on a random cluster
cites work: Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov
cites work: BRANCHING DIFFUSION PROCESSES IN AN UNBOUNDED DOMAIN
cites work: Arbres et processus de Galton-Watson. (Trees and Galton-Watson processes)
cites work: A limit theorem for particle numbers in bounded domains of a branching diffusion process
cites work: Q3923366
cites work: Q3696260
cites work: Probabilités de présence dans un processus de branchement spatial markovien. (Presence probability in a spatial Markov branching process)
cites work: Q5184734
cites work: The behavior of solutions of some non-linear diffusion equations for large time
cites work: Brownian first exit from and sojourn over one sided moving boundary and application
full work available at URL: https://doi.org/10.1007/bf00356108
OpenAlex ID: W2009756903
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4066084