On the Markovian Similarity

  • Laurent Miclo
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)


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.


Markov generators Markov intertwinings Similarity relation Isospectrality Convergence to equilibrium φ-Entropies 

MSC 2010

Primary: 60J27 Secondary: 60J35 60J25 05C50 37A30 58J53 



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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse CedexFrance

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