Dimensional contraction via Markov transportation distance (Q2874661)

The curvature condition, i.e., the Ricci curvature being bounded from below, is well known to be equivalent to the contraction property of the heat semigroup in the Wasserstein distance. In the present paper, it is shown that, when the dimension is considered, it brings a negative correction term in the contraction property. While in the main part of this paper, larger than the Wasserstein distance, a Markov transportation distance \(T_2\) based on the squared field operator of the generator and its reversible measure is defined, which is a generalized form of the modification of a dynamical formulation of Wasserstein distance provided by \textit{J.-D. Benamou} and \textit{Y. Brenier} [Numer. Math. 84, No. 3, 375--393 (2000; Zbl 0968.76069)]. Many inequalities in Wasserstein distance are established in \(T_2\) distance as well. Under a curvature dimension condition, the contraction property and the evolution variational inequality are refined.