From killed BBM with drift to BBM: the extremal process
Abstract
In this paper, we study the asymptotic behavior of the extremal process of a standard one-dimensional branching Brownian motion (BBM) with drift −ρ>−√2 and absorbing barrier at level −x. We prove that the two-dimensional point process, with first component being the extremal process of the BBM and the second component being the running minimum of the BBM with drift, converges weakly to a decorated Poisson point process (DPPP) on ℝ × [0, ∞). This framework allows us to supply a common limit coupling of the extremal processes for different barrier levels and to explicitly derive the limit, as t→∞, of the extremal process of the killed BBM killed at level −x, demonstrating that the double limit, when t→∞ first and then x→∞, of the extremal process of the BBM killed at level −x coincides with the limit of the extremal process of (un-killed) BBM.
Related
- A Functional Central Limit Theorem For A Rank-Killed Particle System With Persistent Copies
- A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements
- Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control
- Extremal projectors in the semi-classical case
- Learning extremal graphical structures in high dimensions
BibTeX
@misc{ren2026from,
title = {{From killed BBM with drift to BBM: the extremal process}},
author = {Yan-Xia Ren and Renming Song and Fan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28989},
eprint = {2609.28989},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28989},
}
RIS
TY - UNPB
TI - From killed BBM with drift to BBM: the extremal process
AU - Yan-Xia Ren
AU - Renming Song
AU - Fan Yang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.28989
UR - https://arxiv.org/abs/2609.28989
AB - In this paper, we study the asymptotic behavior of the extremal process of a standard one-dimensional branching Brownian motion (BBM) with drift −ρ>−√2 and absorbing barrier at level −x. We prove that the two-dimensional point process, with first component being the extremal process of the BBM and the second component being the running minimum of the BBM with drift, converges weakly to a decorated Poisson point process (DPPP) on ℝ × [0, ∞). This framework allows us to supply a common limit coupling of the extremal processes for different barrier levels and to explicitly derive the limit, as t→∞, of the extremal process of the killed BBM killed at level −x, demonstrating that the double limit, when t→∞ first and then x→∞, of the extremal process of the BBM killed at level −x coincides with the limit of the extremal process of (un-killed) BBM.
ER -