Journal of Theoretical Probability

, Volume 9, Issue 2, pp 459–510 | Cite as

Nash inequalities for finite Markov chains

  • P. Diaconis
  • L. Saloff-Coste


This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) steps are necessary and suffice to reach stationarity. We consider local Poincaré inequalities and use them to prove Nash inequalities. These are bounds onl2-norms in terms of Dirichlet forms andl1-norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.

Key Words

Markov chains Dirichlet forms infinite graphs Nash inequalities 


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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • P. Diaconis
    • 1
  • L. Saloff-Coste
    • 2
  1. 1.Department of MathematicsHarvard UniversityCambridge
  2. 2.CNRSUniversité Paul Sabatier, Laboratoire Statistique et ProbabilitésToulouse CedexFrance

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