Dimensional contraction via Markov transportation distance

Abstract

It is now well known that curvature conditions à la Bakry–Émery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strengthening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove, in the setting of a compact Riemannian manifold, that this Markov transportation distance satisfies the same properties as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and evolution variational inequalities.