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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Brownian motion on $ \mathbb{R}$-trees


Authors: Siva Athreya, Michael Eckhoff and Anita Winter
Journal: Trans. Amer. Math. Soc. 365 (2013), 3115-3150
MSC (2010): Primary 60B05, 60J60; Secondary 60J25, 60B99
Published electronically: December 26, 2012
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Siva Athreya
Affiliation: Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India
Email: athreya@isibang.ac.in

Anita Winter
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Universitätsstrasse 2, 45141 Essen, Germany
Email: anita.winter@uni-due.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05752-7
PII: S 0002-9947(2012)05752-7
Keywords: $\mathbb{R}$-trees, Brownian motion, diffusions on metric measure trees, Dirichlet forms, spectral gap, mixing times, recurrence
Received by editor(s): October 13, 2011
Published electronically: December 26, 2012
Additional Notes: The first author was supported in part by a CSIR Grant in Aid scheme and Homi Bhaba Fellowship.
The third author was supported in part at the Technion by a fellowship from the Aly Kaufman Foundation
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.