2015/06/03 by Julien Berestycki, Éric Brunet-Gouet, Berestycki, Julien +5 · 2 citations
Economics, Econometrics and Finance · Mathematics · Medicine · #60J25 (Secondary) #60J65 (Primary) #60J80 #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1506.01429
openalex publication_date 2015/06/03 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28
We study a dyadic branching Brownian motion on the real line with absorption\nat 0, drift \μ \∈ \ℝ and started from a single particle at position\nx>0. When \μ is large enough so that the process has a positive\nprobability of survival, we consider K(t), the number of individuals absorbed\nat 0 by time t and for s\≥ 0 the functions \ωs(x):=\n\𝔼x[sK(\∞)]. We show that \ωs<\∞ if and only of\ns\∈[0,s0] for some s0>1 and we study the properties of these functions.\nFurthermore, for s=0, \ω(x) := \ω0(x) =\ℙx(K(\∞)=0) is\nthe cumulative distribution function of the all time minimum of the branching\nBrownian motion with drift started at 0 without absorption.\n We give three descriptions of the family \ωs, s\∈ [0,s0] through a\nsingle pair of functions, as the two extremal solutions of the\nKolmogorov-Petrovskii-Piskunov (KPP) traveling wave equation on the half-line,\nthrough a martingale representation and as an explicit series expansion. We\nalso obtain a precise result concerning the tail behavior of K(\∞). In\naddition, in the regime where K(\∞)>0 almost surely, we show that u(x,t)\n:= \ℙx(K(t)=0) suitably centered converges to the KPP critical\ntravelling wave on the whole real line.\n