Duality on gradient estimates and Wasserstein controls

We establish a duality between Lp-Wasserstein control and Lq-gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance.