Integral Equations and Operator Theory

, Volume 81, Issue 1, pp 127–150 | Cite as

Dobrushin’s Ergodicity Coefficient for Markov Operators on Cones

  • Stéphane Gaubert
  • Zheng QuEmail author


Doeblin and Dobrushin characterized the contraction rate of Markov operators with respect the total variation norm. We generalize their results by giving an explicit formula for the contraction rate of a Markov operator over a cone in terms of pairs of extreme points with disjoint support in a set of abstract probability measures. By duality, we derive a characterization of the contraction rate of consensus dynamics over a cone with respect to Hopf’s oscillation seminorm (the infinitesimal seminorm associated with Hilbert’s projective metric). We apply these results to Kraus maps (noncommutative Markov chains, representing quantum channels), and characterize the ultimate contraction of the map in terms of the existence of a rank one matrix in a certain subspace.


Markov operator Dobrushin’s ergodicity coefficient ordered linear space invariant measure contraction ratio consensus noncommutative Markov chain quantum channel zero error capacity rank one matrix 


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© Springer Basel 2014

Authors and Affiliations

  1. 1.École PolytechniqueINRIA and CMAP UMR 7641 CNRSPalaiseau CedexFrance
  2. 2.School of MathematicsUniversity of EdinburghEdinburghUK

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