Dobrushin ergodicity coefficient for Markov operators on cones, and beyond

Stéphane Gaubert 1, 2 Zheng Qu 1, 2
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert's projective metric or to a related family of seminorms (Hopf's oscillation or Hilbert's seminorm). We generalize these properties to abstract consensus operators over normal cones, which include the unital completely positive maps (Kraus operators) arising in quantum information theory. In particular, we show that the contraction rate of such operators, with respect to the Hopf oscillation seminorm, is given by an analogue of Dobrushin's ergodicity coefficient. We derive from this result a characterization of the contraction rate of a non-linear flow, with respect to Hopf's oscillation seminorm and to Hilbert's projective metric.
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https://hal.inria.fr/hal-00935272
Contributeur : Zheng Qu <>
Soumis le : jeudi 23 janvier 2014 - 12:12:26
Dernière modification le : jeudi 12 avril 2018 - 01:47:06

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  • HAL Id : hal-00935272, version 1
  • ARXIV : 1302.5226

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Stéphane Gaubert, Zheng Qu. Dobrushin ergodicity coefficient for Markov operators on cones, and beyond. Arxiv: 1302:5226. 2013. 〈hal-00935272〉

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