Brownian motion on ℝ-trees

Athreya, Siva; Eckhoff, Michael; Winter, Anita

doi:10.1090/S0002-9947-2012-05752-7

Brownian motion on $\mathbb {R}$-trees
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by Siva Athreya, Michael Eckhoff and Anita Winter PDF

Trans. Amer. Math. Soc. 365 (2013), 3115-3150 Request permission

Abstract:

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as locally infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact $\mathbb {R}$-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 $\mathbb {R}$-trees we provide bounds on the mixing time.

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