Constant curvature metrics for Markov chains

Abstract

We consider metrics which are preserved under a $p$-Wasserstein transport map, up to a possible contraction. In the case $p=1$ this corresponds to a metric which is uniformly curved in the sense of coarse Ricci curvature. We investigate the existence of such metrics in the more general sense of pseudo-metrics, where the distance between distinct points is allowed to be 0, and show the existence for general Markov chains on compact Polish spaces. Further we discuss a notion of algebraic reducibility and its relation to the existence of multiple true pseudo-metrics with constant curvature. Conversely, when the Markov chain is irreducible and the state space finite we obtain effective uniqueness of a metric with uniform curvature up to scalar multiplication and taking the $p$th root, making this a natural intrinsic distance of the Markov chain. An application is given in the form of concentration inequalities for the Markov chain.