The Brownian disk viewed from a boundary point

TL;DR

This paper introduces a new construction of Brownian disks using forests of random trees with labels indicating distances, revealing how boundary distances evolve as a Bessel bridge and connecting boundary measures with volume measures.

Contribution

It provides a novel construction of Brownian disks via random trees and clarifies the relationship between boundary measures and volume measures, also simplifying the proof of the Brownian half-plane equivalence.

Findings

Distances from a distinguished boundary point follow a five-dimensional Bessel bridge.

Boundary measures are limits of volume measures on neighborhoods.

New construction simplifies understanding of Brownian half-plane.

Abstract

We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This construction shows in particular that distances from the distinguished point evolve along the boundary as a five-dimensional Bessel bridge. As an important ingredient of our proofs, we show that the uniform measure on the boundary, as defined in the earlier work of Bettinelli and Miermont, is the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. Our construction also yields a simple proof of the equivalence between the two definitions of the Brownian half-plane.