Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift
Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift
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带吸收的分支布朗运动和带漂移的分支布朗运动的历史最小值
DOI：
10.1016/j.jfa.2017.06.006
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Piotr Milo's
中科院分区：
文献类型：
--
作者：
J. Berestycki;'Eric Brunet;S. Harris;Piotr Milo's
本文研究了真实的直线上的并矢分支布朗运动，其中吸收在0，漂移μ∈ R，从位置x> 0的单个粒子出发。用K（∞）表示在所有时间内在0点吸收的个体总数（可能是无限的），我们考虑函数ω s（x）：= E x [s K（∞）]，其中s≥ 0。在μ大到几乎必然使K（∞）<∞且过程有正生存概率的情况下，我们证明了ω s<∞当且仅当对某些s 0> 1，s∈[0，s 0]，并研究了这些函数的性质.此外，ω（x）：= ω 0（x）= Px（K（∞）= 0）是漂移起始于0而无吸收的分支布朗运动的全时极小值的累积分布函数.通过函数ω 0（x）和ω s 0（x）的单对，作为Kolmogorov-Petrovskii-Piskunov（KPP）行波方程在半直线上的极值解，通过鞅表示，作为一个显式级数展开，给出了族ω s，s∈[0，s 0]的描述.我们还得到了关于K（∞）的尾部行为的一个精确结果。此外，在K（∞）> 0的区域中，我们证明了u（x，t）：= Px（K（t）= 0）在整条真实的直线上收敛于KPP临界行波.
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 ω s (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 ω s<∞ if and only if s∈[0, s 0] for some s 0> 1 and we study the properties of these functions. Furthermore, ω (x):= ω 0 (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 ω s, s∈[0, s 0] through the single pair of functions ω 0 (x) and ω s 0 (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):= P x (K (t)= 0) suitably centred converges to the KPP critical travelling wave on the whole real line.
影响因子：
2.3
作者：
Berestycki, Julien;Berestycki, Nathanael;Schweinsberg, Jason
通讯作者：
Schweinsberg, Jason