PreprintCC BY 4.0arXiv
Sample genealogies within a Brownian excursion
Abstract
In this article, we apply Itô's excursion theory to find the joint law of minima of a Brownian excursion conditioned to go above level $1$ over the intervals defined by k independent identically distributed (i.i.d.) points of intersection of the excursion with level a ∈ (0,1]. This approach offers an intuitive way to find the joint law of coalescent times of an i.i.d. sample of k particles alive at time a in Aldous' continuum random tree of height $1$. This can be thought of as ``sampling of the limit" of critical Galton-Watson trees conditioned to survive a large time, agreeing with some special cases obtained in [4] and [5] which instead consider ``the limit of sampling" from critical Galton-Watson trees.
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BibTeX
@misc{bocharov2026sample,
title = {{Sample genealogies within a Brownian excursion}},
author = {Sergey Bocharov and Simon C. Harris},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06291},
eprint = {2609.06291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06291},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sample genealogies within a Brownian excursion
AU - Sergey Bocharov
AU - Simon C. Harris
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06291
UR - https://arxiv.org/abs/2609.06291
AB - In this article, we apply Itô's excursion theory to find the joint law of minima of a Brownian excursion conditioned to go above level $1$ over the intervals defined by k independent identically distributed (i.i.d.) points of intersection of the excursion with level a ∈ (0,1]. This approach offers an intuitive way to find the joint law of coalescent times of an i.i.d. sample of k particles alive at time a in Aldous' continuum random tree of height $1$. This can be thought of as ``sampling of the limit" of critical Galton-Watson trees conditioned to survive a large time, agreeing with some special cases obtained in [4] and [5] which instead consider ``the limit of sampling" from critical Galton-Watson trees.
ER -