On the Markovian similarity

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Two finite Markov generators $L$ and $\widetilde L$ are said to be intertwined if there exists a Markov kernel $\Lambda$ such that $L\Lambda=\Lambda\widetilde L$. 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.

Mots clés: Markov generators; Markov intertwinings; similarity relation; isospectrality; convergence to equilibrium; con-vergence to equilibrium; ϕ-entropies

Domaines: Probabilités [math.PR]

Laurent Miclo. On the Markovian similarity. Séminaire de Probabilités XLIX, pp.375-403, 2018, ⟨10.1007/978-3-319-92420-5_10⟩. ⟨hal-01281029v2⟩