Symmetric Markov chains on 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

DOI:
https://doi.org/10.1090/S0002-9947-07-04281-X

Published electronically:
October 17, 2007

MathSciNet review:
2366974

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Abstract: We consider symmetric Markov chains on 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 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:
https://doi.org/10.1090/S0002-9947-07-04281-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.