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

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.
Comment: 17 pages, 805C05, 05C62, 60J65, 54E35 figures