The shape of the front of multidimensional branching Brownian motion

• 成果类型: Article; Early Access
• 署名作者: Kim, Yujin H.; Zeitouni, Ofer
• 署名单位: New York University; Weizmann Institute of Science; New York University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01396-3
• 发表日期: 2025
• 关键词: extremal process; equation; CONVERGENCE

摘要

We study the shape of the outer envelope of a branching Brownian motion (BBM) in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>d$$\end{document}, d >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document}. We focus on the extremal particles: those whose norm is within O(1) of the maximal norm amongst the particles alive at time t. Our main result is a scaling limit, with exponent 3/2, for the outer-envelope of the BBM around each extremal particle (the front); the scaling limit is a continuous random surface given explicitly in terms of a Bessel(3) process. Towards this end, we introduce a point process that captures the full landscape around each extremal particle and show convergence in distribution to an explicit point process. This complements the global description of the extremal process given in Berestycki et. al. (Ann. Probab. 52 (2024), no. 3, 955-982), where the local behavior at directions transversal to the radial component of the extremal particles is not addressed.