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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Branching Brownian Motion
Brownian Motion with Drift
Absorption
Real Line
Traveling Wave
Martingale Representation
Extremal Solutions
Tail Behavior
Cumulative distribution function
Series Expansion
Half line
Wave equation
If and only if
Converge

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift. / Berestycki, Julien; Brunet, Éric; Harris, Simon C.; Miłoś, Piotr.

In: Journal of Functional Analysis, Vol. 273, No. 6, 15.09.2017, p. 2107-2143.

Research output: Contribution to journalArticle

Berestycki, Julien ; Brunet, Éric ; Harris, Simon C. ; Miłoś, Piotr. / Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift. In: Journal of Functional Analysis. 2017 ; Vol. 273, No. 6. pp. 2107-2143.
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