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Essential spectral radius for Markov semigroups (I): discrete time case
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  • Published: 12 November 2003

Essential spectral radius for Markov semigroups (I): discrete time case

  • Liming Wu1,2 

Probability Theory and Related Fields volume 128, pages 255–321 (2004)Cite this article

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Abstract

Using two new measures of non-compactness βτ(P) and β w (P) for a positive kernel P on a Polish space E, we obtain a new formula of Nussbaum-Gelfand type for the essential spectral radius r ess (P) on bℬ. Using that formula we show that different known sufficient conditions for geometric ergodicity such as Doeblin’s condition, drift condition by means of Lyapunov function, geometric recurrence etc lead to variational formulas of the essential spectral radius. All those can be easily transported on the weighted space b u ℬ. Some related results on L 2(μ) are also obtained, especially in the symmetric case. Moreover we prove that for a strongly Feller and topologically transitive Markov kernel, the large deviation principle of Donsker-Varadhan for occupation measures of the associated Markov process holds if and only if the essential spectral radius is zero; this result allows us to show that the sufficient condition of Donsker-Varadhan for the large deviation principle is in fact necessary. The knowledge of r ess (P) allows us to estimate eigenvalues of P in L 2 in the symmetric case, and to estimate the geometric convergence rate by means of that in the metric of Wasserstein. Applications to different concrete models are provided for illustrating those general results.

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Authors and Affiliations

  1. Department of Mathematics, Wuhan University, 430072, Hubei, China

    Liming Wu

  2. and R. Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620, Université Blaise Pascal, 63177, Aubière, France

    Liming Wu

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  1. Liming Wu
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Correspondence to Liming Wu.

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Mathematics Subject Classification (2000): 60J05, 60F10, 47A10, 47D07

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Cite this article

Wu, L. Essential spectral radius for Markov semigroups (I): discrete time case. Probab. Theory Relat. Fields 128, 255–321 (2004). https://doi.org/10.1007/s00440-003-0304-0

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  • Received: 26 January 2003

  • Revised: 15 September 2003

  • Published: 12 November 2003

  • Issue Date: February 2004

  • DOI: https://doi.org/10.1007/s00440-003-0304-0

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Keywords

  •  Essential spectral radius
  • Markov processes
  • Large deviations
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