Representation Theorems of $\mathbb{R}$-trees and Brownian Motions Indexed by $\mathbb R$-trees

TL;DR

This paper introduces a novel way to represent -trees using metric rays, establishing equivalences with certain metrics and facilitating the study of Brownian motions indexed by these trees.

Contribution

It provides a new representation of -trees via metric rays and links this to the characterization of Brownian motions indexed by -trees.

Findings

Captured the four-point condition from metric rays

Established equivalence between -trees with specific metrics and sets of metric rays

Facilitated identification of Brownian motions indexed by -trees

Abstract

We provide a new representation of an $\mathbb R$-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the $\mathbb R$-trees with radial and river metrics, and these sets of metric rays. In stochastic analysis, these graphical representation theorems are of particular interest in identifying Brownian motions indexed by $\mathbb R$-trees.