It is now well known that curvature conditions {\it a la} Bakry-Emery are equivalent to contraction properties of the heat semigroup with respect to the 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 its carre du champ, and adapted to them. We prove that this Markov transportation distance satisfies the same properties as the Wasserstein distance in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolutional variational inequalities.
['François Bolley', 'Ivan Gentil', 'Arnaud Guillin']
Dimensional contraction via Markov transportation distance