Discrete self-similar and ergodic Markov chains

Abstract

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 intertwining relation 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 nonreversible. By means of intertwining relations and their strengthened interweaving versions, we derive several deep analytical properties of such ergodic chains, including the description of the spectrum, the spectral expansion of their semigroups and the study of their convergence to equilibrium in the Φ-entropy sense as well as their hypercontractivity property.