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

@article{Aksoy2016RepresentationTO, title={Representation Theorems of \$\mathbb\{R\}\$-trees and Brownian Motions Indexed by \$\mathbb R\$-trees}, author={Asuman G{\"u}ven Aksoy and M. Alansari and Qidi Peng}, journal={arXiv: Metric Geometry}, year={2016} }

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.

## References

SHOWING 1-10 OF 16 REFERENCES

### Some Results on Metric Trees

- Mathematics, Computer Science
- 2010

This paper investigates barycenters, type and cotype, and various measures of compactness of metric trees, and the limit of the sequence of Kolmogorov widths of a metric tree is equal to its ball measure of non-compactness.

### Spherical and Hyperbolic Fractional Brownian Motion

- Mathematics
- 2005

We define a Fractional Brownian Motion indexed by a sphere, or more generally by a compact rank one symmetric space, and prove that it exists if, and only if, $0 < H \leq 1/2$. We then prove that…

### Probability and Real Trees

- Mathematics
- 2008

Around the Continuum Random Tree.- R-Trees and 0-Hyperbolic Spaces.- Hausdorff and Gromov-Hausdorff Distance.- Root Growth with Re-Grafting.- The Wild Chain and other Bipartite Chains.- Diffusions on…

### Multifractional brownian fields indexed by metric spaces with distances of negative type

- Mathematics
- 2013

We define multifractional Brownian fields indexed by a metric space, such as a manifold with its geodesic distance, when the distance is of negative type. This construction applies when the Brownian…

### Independence of the increments of Gaussian random fields

- MathematicsNagoya Mathematical Journal
- 1982

Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of…

### The Apple Doesn’t Fall Far From the (Metric) Tree: Equivalence of Definitions

- Mathematics
- 2014

In this paper we prove the equivalence of definitions for metric trees and for \delta-hyperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric…

### String Matching with Metric Trees Using an Approximate Distance

- Computer ScienceSPIRE
- 2002

This paper investigates the performance of metric trees, namely the M-tree, when they are extended using a cheap approximate distance function as a filter to quickly discard irrelevant strings, and shows an improvement in performance up to 90% with respect to the basic case.