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Geodesic convexity of the relative entropy in reversible Markov chains

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Abstract

We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some discretizations of one-dimensional Fokker–Planck equations.

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Authors and Affiliations

  1. WIAS Berlin, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Alexander Mielke

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  1. Alexander Mielke
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Correspondence to Alexander Mielke.

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Communicated by L. Ambrosio.

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Mielke, A. Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. 48, 1–31 (2013). https://doi.org/10.1007/s00526-012-0538-8

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  • DOI: https://doi.org/10.1007/s00526-012-0538-8

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