DISCRETE SELF-SIMILAR AND ERGODIC MARKOV CHAINS

• 成果类型: Article
• 署名作者: Miclo, Laurent; Patie, Pierre; Sarkar, Rohan
• 署名单位: Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics; Cornell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1577
• 发表日期: 2022
• 页码: 2085-2132
• 关键词: scaling limits; Duality; hypocoercivity

摘要

The first aim of this paper is to introduce a class of Markov chains on Z(+) which are discrete self-similar in the sense that their semigroups satisfy an in-variance 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 pro-cesses on R+. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric to the associ-ated 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 expan-sion of their semigroups and the study of their convergence to equilibrium in the Phi-entropy sense as well as their hypercontractivity property.