Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

Julien Berestycki, Éric Brunet, Simon C. Harris, Piotr Miłoś

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Abstract

We study a dyadic branching Brownian motion on the real line with absorption at 0, drift μ∈R and started from a single particle at position x>0. With K(∞) the (possibly infinite) total number of individuals absorbed at 0 over all time, we consider the functions ωs(x):=Ex[sK(∞)] for s≥0. In the regime where μ is large enough so that K(∞)<∞ almost surely and that the process has a positive probability of survival, we show that ωs<∞ if and only if s∈[0,s0] for some s0>1 and we study the properties of these functions. Furthermore, ω(x):=ω0(x)=Px(K(∞)=0) is the cumulative distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption.We give descriptions of the family ωs,s∈[0,s0] through the single pair of functions ω0(x) and ωs0(x), as extremal solutions of the Kolmogorov-Petrovskii-Piskunov (KPP) travelling wave equation on the half-line, through a martingale representation, and as a single explicit series expansion. We also obtain a precise result concerning the tail behaviour of K(∞). In addition, in the regime where K(∞)>0 almost surely, we show that u(x,t):=Px(K(t)=0) suitably centred converges to the KPP critical travelling wave on the whole real line.

Original languageEnglish
Pages (from-to)2107-2143
Number of pages37
JournalJournal of Functional Analysis
Volume273
Issue number6
Early online date13 Jun 2017
DOIs
Publication statusPublished - 15 Sep 2017

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Keywords

  • Branching processes
  • Dirichlet problem
  • Parabolic partial differential equations
  • Reaction-diffusion equations

ASJC Scopus subject areas

  • Analysis

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