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Stability of parabolic Harnack inequalities

Author(s): Martin T. Barlow; Richard F. Bass
Journal: Trans. Amer. Math. Soc. 356 (2004), 1501-1533.
MSC (2000): Primary 60J27; Secondary 60J35, 31B05
Posted: September 22, 2003
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Abstract: Let $(G,E)$ be a graph with weights $\{a_{xy}\}$ for which a parabolic Harnack inequality holds with space-time scaling exponent $\beta \ge 2$. Suppose $\{a'_{xy}\}$ is another set of weights that are comparable to $\{a_{xy}\}$. We prove that this parabolic Harnack inequality also holds for $(G,E)$ with the weights $\{a'_{xy}\}$. We also give stable necessary and sufficient conditions for this parabolic Harnack inequality to hold.

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Additional Information:

Martin T. Barlow
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2
Email: barlow@math.ubc.ca

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

DOI: 10.1090/S0002-9947-03-03414-7
PII: S 0002-9947(03)03414-7
Keywords: Harnack inequality, random walks on graphs, volume doubling, Green functions, Poincar\'{e} inequality, Sobolev inequality, anomalous diffusion
Received by editor(s): January 24, 2003
Posted: September 22, 2003
Additional Notes: The first author's research was partially supported by an NSERC (Canada) grant, and by CNRS (France)
The second author's research was partially supported by NSF Grant DMS 9988486
Copyright of article: Copyright 2003, American Mathematical Society

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