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On the Markovian Similarity

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Séminaire de Probabilités XLIX

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2215))

Abstract

Two finite Markov generators L and \(\widetilde L\) are said to be intertwined if there exists a Markov kernel Λ such that \(L\varLambda =\varLambda \widetilde L\). The goal of this paper is to investigate the equivalence relation between finite Markov generators obtained by imposing mutual intertwinings through invertible Markov kernels, in particular its links with the traditional similarity relation. Some consequences on the comparison of speeds of convergence to equilibrium for finite irreducible Markov processes are deduced. The situation of infinite state spaces is also quickly mentioned, by showing that the Laplacians of isospectral compact Riemannian manifolds are weakly Markov-similar.

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Acknowledgements

This paper was motivated by the presentation of Pierre Patie of his paper with Mladen Savov [16], I’m grateful to him for all the explanations he gave me. I’m also thankful to the ANR STAB (Stabilité du comportement asymptotique d’EDP, de processus stochastiques et de leurs discrètisations) for its support.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse Cedex, France

    Laurent Miclo

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  1. Laurent Miclo
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Correspondence to Laurent Miclo .

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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-les-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Alain Rouault

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Miclo, L. (2018). On the Markovian Similarity. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIX. Lecture Notes in Mathematics(), vol 2215. Springer, Cham. https://doi.org/10.1007/978-3-319-92420-5_10

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