Reversal property of the Brownian tree

TL;DR

This paper studies the Brownian tree, revealing a reversal symmetry where the tree's structure remains invariant when viewed from top to bottom, enhancing understanding of its probabilistic properties.

Contribution

It introduces a reversal procedure for the Brownian tree and proves its distributional invariance, offering new insights into its structural symmetries.

Findings

Reversal procedure preserves the distribution of the Brownian tree.

Provides a new perspective on the tree's structural symmetry.

Enhances understanding of the Brownian tree's probabilistic properties.

Abstract

We consider the Brownian tree introduced by Aldous and the associated Q-process which consists in an infinite spine on which are grafted independent Brownian trees. We present a reversal procedure on these trees that consists in looking at the tree downward from its top: the branching points becoming leaves and leaves becoming branching points. We prove that the distribution of the tree is invariant under this reversal procedure, which provides a better understanding of previous results from Bi and Delmas (2016).