Dobrushin’s Ergodicity Coefficient for Markov Operators on Cones

被引：0
作者：
Stéphane Gaubert
Zheng Qu
机构：
[1] INRIA and CMAP UMR 7641 CNRS,École Polytechnique
[2] University of Edinburgh,School of Mathematics
来源：
Integral Equations and Operator Theory | 2015年 / 81卷
关键词：
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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暂无
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学科分类号：
摘要：
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.
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页码：127 / 150
页数：23
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