Sample genealogies within a Brownian excursion

Abstract

In this article, we apply It\^o'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 $a \in (0,1]$. 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.