Mathematische Annalen

, Volume 324, Issue 3, pp 521–556 | Cite as

Harnack inequalities and sub-Gaussian estimates for random walks

  • A. Grigor'yan
  • A. Telcs


We show that the \(\beta \)-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to the sub-Gaussian estimate for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order \(R^{\beta }\). The latter condition can be replaced by a certain estimate of the resistance of annuli.


Exit Time Harnack Inequality Volume Property Doubling Volume Parabolic Harnack Inequality 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • A. Grigor'yan
    • 1
  • A. Telcs
    • 2
  1. 1.Department of Mathematics, London SW7 2BZ, UK 662 (e-mail: GB
  2. 2. IMC, Graduate School of Business, Zrinyi u. 14, Budapest, 1051, Hungary (e-mail: HU

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