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

 

Symmetric Markov chains on $ \mathbb{Z}^d$ with unbounded range


Authors: Richard F. Bass and Takashi Kumagai
Journal: Trans. Amer. Math. Soc. 360 (2008), 2041-2075
MSC (2000): Primary 60J10; Secondary 60F05, 60J27
Published electronically: October 17, 2007
MathSciNet review: 2366974
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Abstract: We consider symmetric Markov chains on $ \mathbb{Z}^d$ where we do not assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of normalized symmetric Markov chains to converge to a diffusion on $ \mathbb{R}^d$ corresponding to an elliptic operator in divergence form.


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Additional Information

Richard F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: bass@math.uconn.edu

Takashi Kumagai
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Email: kumagai@math.kyoto-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04281-X
PII: S 0002-9947(07)04281-X
Received by editor(s): August 30, 2005
Received by editor(s) in revised form: February 16, 2006
Published electronically: October 17, 2007
Additional Notes: The first author’s research was partially supported by NSF grant DMS0244737.
The second author’s research was partially supported by Ministry of Education, Japan, Grant-in-Aid for Scientific Research for Young Scientists (B) 16740052.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.