Dimensional contraction via Markov transportation distance

TL;DR

This paper introduces a new Markov transportation distance that captures dimensional contraction properties of Markov semigroups, extending classical Wasserstein distance results related to curvature conditions.

Contribution

It proposes a novel distance tailored to Markov semigroups, demonstrating its contraction properties and variational inequalities, thus generalizing Wasserstein-based curvature results.

Findings

Euclidean heat semigroup exhibits dimensional contraction.

New Markov transportation distance satisfies contraction properties.

The distance supports evolution variational inequalities.

Abstract

It is now well known that curvature conditions \`a la Bakry-Emery 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 strenghtening 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 that this Markov transportation distance satisfies the same properties for a general Markov semigroup as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolutional variational inequalities.