### 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 language | English |
---|---|

Pages (from-to) | 2107-2143 |

Number of pages | 37 |

Journal | Journal of Functional Analysis |

Volume | 273 |

Issue number | 6 |

Early online date | 13 Jun 2017 |

DOIs | |

Publication status | Published - 15 Sep 2017 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*273*(6), 2107-2143. https://doi.org/10.1016/j.jfa.2017.06.006

**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.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 273, no. 6, pp. 2107-2143. https://doi.org/10.1016/j.jfa.2017.06.006

}

TY - JOUR

T1 - Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

AU - Berestycki, Julien

AU - Brunet, Éric

AU - Harris, Simon C.

AU - Miłoś, Piotr

PY - 2017/9/15

Y1 - 2017/9/15

N2 - 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.

AB - 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.

KW - Branching processes

KW - Dirichlet problem

KW - Parabolic partial differential equations

KW - Reaction-diffusion equations

UR - http://www.scopus.com/inward/record.url?scp=85021089315&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1016/j.jfa.2017.06.006

U2 - 10.1016/j.jfa.2017.06.006

DO - 10.1016/j.jfa.2017.06.006

M3 - Article

VL - 273

SP - 2107

EP - 2143

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 6

ER -