Discrete self-similar and ergodic Markov chains

Laurent Miclo , Pierre Patie (1) , Rohan Sarkar (2)

1 Cornell University [New York]
2 Department of Mathematics [Cornell]

The first aim of this paper is to introduce a class of Markov chains on $\mathbb{Z}_+$ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway relation, a concept introduced in [26], between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on $\mathbb{R}_+$. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric, to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains, which are non-reversible. By means of intertwining and interweaving relations, where the latter was recently introduced in [27], we derive several deep analytical properties of such ergodic chains including the description of the spectrum, the spectral expansion of their semigroups, the study of their convergence to equilibrium in the $\Phi$-entropy sense as well as their hypercontractivity property.

Mots clés: Discrete self-similarity; Markov chains; generalized Meixner polynomials; intertwining; non-reversible; spectral theory; ergodicity constants; convergence to equilibrium; hypercontractivity 2010 Mathematical Subject Classification: 41A60; 47G20; 33C45; 47D07; 37A30; 60J27; 60J60

Domaines: Probabilités [math.PR]

Laurent Miclo, Pierre Patie, Rohan Sarkar. Discrete self-similar and ergodic Markov chains. Annals of Probability, In press. ⟨hal-03777200⟩