  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.
