# Some explicit distributions for Brownian motion indexed by the Brownian tree

## TL;DR

This paper derives explicit distributions for functionals of Brownian motion indexed by the Brownian tree, providing new insights into the distribution of the density at zero of the integrated super-Brownian excursion.

## Contribution

It offers explicit distribution formulas for Brownian motion functionals indexed by the Brownian tree, including a direct proof of a key distribution result.

## Key findings

- Explicit distributions for Brownian motion functionals on the Brownian tree
- Direct proof of the distribution of the density at zero of the integrated super-Brownian excursion
- Enhanced understanding of Brownian motion indexed by complex structures

## Abstract

We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-M\'elou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion.