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

Journal article

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) := E x [s K(∞) ] 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 of s ∈ [0, s0] for some s0 > 1 and we study the properties of these functions. Furthermore, ω(x) := ω0(x) = P x (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) traveling 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 behavior of K(∞). In addition, in the regime where K(∞) > 0 almost surely, we show that u(x, t) := P x (K(t) = 0) suitably centered converges to the KPP critical travelling wave on the whole real line.

Authors: Berestycki, J; Brunet, É; Harris, S; Milos, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Functional Analysis
• Volume: 273
• Issue: 6
• Pages: 2107-2143
• Publication date: 2017-06-13
• Acceptance date: 2017-04-07
• DOI: 10.1016/j.jfa.2017.06.006
• ISSN: 0022-1236
• Keywords: parabolic partial differential equation; Dirichlet problem; branching processes; reaction–diffusion equations