Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Slowdown in branching Brownian motion with inhomogeneous variance

Pascal Maillard and Ofer Zeitouni

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Abstract

We consider the distribution of the maximum $M_{T}$ of branching Brownian motion with time-inhomogeneous variance of the form $\sigma^{2}(t/T)$, where $\sigma(\cdot)$ is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) equation $F_{t}(x,t)=\sigma^{2}(1-t/T)F_{xx}(x,t)/2+g(F(x,t))$, for appropriate nonlinearities $g(\cdot)$. Fang and Zeitouni (J. Stat. Phys. 149 (2012) 1–9) showed that $M_{T}-v_{\sigma}T$ is negative of order $T^{1/3}$, where $v_{\sigma}=\int_{0}^{1}\sigma(s)\,\mathrm{d}s$. In this paper, we show the existence of a function $m'_{T}$, such that $M_{T}-m'_{T}$ converges in law, as $T\rightarrow\infty$. Furthermore, $m'_{T}=v_{\sigma}T-w_{\sigma}T^{1/3}-\sigma(1)\log T+O(1)$ with $w_{\sigma}=2^{-1/3}\alpha_{1}\int_{0}^{1}\sigma(s)^{1/3}|\sigma'(s)|^{2/3}\,\mathrm{d}s$. Here, $-\alpha_{1}=-2.33811\ldots$ is the largest zero of the Airy function $\operatorname{Ai}$. The proof uses a mixture of probabilistic and analytic arguments.

Résumé

Nous étudions la loi du maximum $M_{T}$ d’un mouvement brownien branchant avec une variance inhomogène en temps de la form $\sigma^{2}(t/T)$, où $\sigma(\cdot)$ est une fonction strictement décroissante. Ceci correspond à étudier l’équation Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) inhomogène en temps, $F_{t}(x,t)=\sigma^{2}(1-t/T)F_{xx}(x,t)/2+g(F(x,t))$, pour des nonlinéarités $g(\cdot)$ appropriées. Fang et Zeitouni (J. Stat. Phys. 149 (2012) 1–9) ont montré que $M_{T}-v_{\sigma}T$ est negatif de l’ordre $T^{1/3}$, où $v_{\sigma}=\int_{0}^{1}\sigma(s)\,\mathrm{d}s$. Dans cet article, nous montrons l’existence d’une fonction $m'_{T}$ telle que $M_{T}-m'_{T}$ converge en loi quand $T\rightarrow\infty$. De plus, $m'_{T}=v_{\sigma}T-w_{\sigma}T^{1/3}-\sigma(1)\log T+O(1)$ avec $w_{\sigma}=2^{-1/3}\alpha_{1}\int_{0}^{1}\sigma(s)^{1/3}|\sigma'(s)|^{2/3}\,\mathrm{d}s$. Ici, $-\alpha_{1}=-2.33811\ldots$ est la plus grande racine de la fonction d’Airy $\operatorname{Ai}$. La démonstration repose sur un mélange d’arguments probabilistes et analytiques.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1144-1160.

Dates
Received: 3 February 2014
Revised: 6 March 2015
Accepted: 6 March 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723514

Digital Object Identifier
doi:10.1214/15-AIHP675

Mathematical Reviews number (MathSciNet)
MR3531703

Zentralblatt MATH identifier
1366.60105

Citation

Maillard, Pascal; Zeitouni, Ofer. Slowdown in branching Brownian motion with inhomogeneous variance. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1144--1160. doi:10.1214/15-AIHP675. https://projecteuclid.org/euclid.aihp/1469723514


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