Subordination of trees and the Brownian map

TL;DR

This paper explores the subordination of the Brownian tree using the maximum process of Brownian motion, revealing its connection to stable Levy trees and applying findings to properties of the Brownian map.

Contribution

It introduces a novel subordination framework for the Brownian tree and identifies the subordinate tree as a stable Levy tree with index 3/2, linking it to the Brownian map.

Findings

The subordinate tree is a stable Levy tree with index 3/2.

The maximum process of the Brownian snake is a time change of the height process.

Revealed properties of the Brownian map's metric net.

Abstract

We discuss subordination of random compact R-trees. We focus on the case of the Brownian tree, where the subordination function is given by the past maximum process of Brownian motion indexed by the tree. In that particular case, the subordinate tree is identified as a stable Levy tree with index 3/2. As a more precise alternative formulation, we show that the maximum process of the Brownian snake is a time change of the height process coding the Levy tree. We then apply our results to properties of the Brownian map. In particular, we recover, in a more precise form, a recent result of Miller and Sheffield identifying the metric net associated with the Brownian map.