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

2006/06/27 by Hubert Hennion, Hennion, Hubert · 2 citations
Computer Science · Mathematics · #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #math.FA #math.PR #msc:47B07 #msc:47B65 #msc:60F15 #msc:60J05 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0606680

31 pages

arxiv created 2006/06/27 · arxiv updated 2009/12/01

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.

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• Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity
• Quasi-compactness for dominated kernels with application to quasi-stationary distribution theory

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