Quasi-compactness and absolutely continuous kernels, applications to Markov chains

TL;DR

This paper explores the spectral properties of Markov kernels, linking their essential spectral radius to absolutely continuous kernels, and provides new characterizations of quasi-compactness without requiring irreducibility or aperiodicity.

Contribution

It introduces a novel connection between the spectral radius of kernels and their approximation by absolutely continuous kernels, extending quasi-compactness characterization.

Findings

Linked spectral radius to absolutely continuous kernel approximation

Reformulated Doeblin's condition for spectral analysis

Characterized quasi-compactness without irreducibility oraperiodicity

Abstract

We show how the essential spectral radius of a bounded positive kernel, acting on bounded functions, is linked to its lower approximation by certain absolutely continuous kernels. The standart Doeblin's condition can be interpreted in this context, and, when suitably reformulated, it leads to a formula for the essential spectral radius. This results may be used to characterize the Markov kernels having a quasi-compact action on a space of measurable functions bounded with respect to some test function, when no irreducibilty and aperiodicity are assumed.