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Stability of parabolic Harnack inequalities


Authors: Martin T. Barlow and Richard F. Bass
Journal: Trans. Amer. Math. Soc. 356 (2004), 1501-1533
MSC (2000): Primary 60J27; Secondary 60J35, 31B05
DOI: https://doi.org/10.1090/S0002-9947-03-03414-7
Published electronically: September 22, 2003
MathSciNet review: 2034316
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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: https://doi.org/10.1090/S0002-9947-03-03414-7
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
Published electronically: 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
Article copyright: © Copyright 2003 American Mathematical Society