Dobrushin’s Ergodicity Coefficient for Markov Operators on Cones

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.