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

 

Volume growth and stochastic completeness of graphs


Author: Matthew Folz
Journal: Trans. Amer. Math. Soc. 366 (2014), 2089-2119
MSC (2010): Primary 60G50; Secondary 60J60, 31C25
Published electronically: September 4, 2013
MathSciNet review: 3152724
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Abstract: Given the variable-speed random walk on a weighted graph and a metric adapted to the structure of the random walk, we construct a Brownian motion on a closely related metric graph which behaves similarly to the VSRW and for which the associated intrinsic metric has certain desirable properties. Jump probabilities and moments of jump times for Brownian motion on metric graphs with varying edge lengths, jump conductances, and edge densities are computed. We use these results together with a theorem of Sturm for stochastic completeness, or non-explosiveness, on local Dirichlet spaces to prove sharp volume growth criteria in adapted metrics for stochastic completeness of graphs.


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Additional Information

Matthew Folz
Affiliation: Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
Email: mfolz@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05930-2
PII: S 0002-9947(2013)05930-2
Keywords: Random walks, stochastic completeness, volume growth, intrinsic metrics, Brownian motion on metric graphs, local Dirichlet spaces
Received by editor(s): March 16, 2012
Received by editor(s) in revised form: June 20, 2012, and August 2, 2012
Published electronically: September 4, 2013
Additional Notes: This research was supported by an NSERC Alexander Graham Bell Canada Graduate Scholarship
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.