Brownian motion on R trees

Abstract

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact $R$-tree {$(T,r)$} equipped with a Radon measure $\nu$ {on $(T,\mathcal{B}(T))$}. We specify a criterion under which the Brownian motion is recurrent or transient. For compact recurrent $R$-trees we provide bounds on the mixing time. In this revised version, assumption (A3) for an $R$-tree has been removed.

47 pages, 1 figure

Cited by in corpus (5)

• Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
• Scaling limits of the three-dimensional uniform spanning tree and associated random walk
• The harmonic measure of balls in random trees
• Scaling limits of stochastic processes associated with resistance forms
• Representation Theorems of $\mathbb{R}$-trees and Brownian Motions Indexed by $\mathbb{R}$-trees