*preprint*

**Inserted:** 24 feb 2021

**Year:** 2013

**Abstract:**

We prove that the linear heat flow in a RCD(K,\infty) metric measure space
(X,d,m) satisfies a contraction property with respect to every
L^{p}-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a
precise estimate for the optimal W_{\infty}-coupling between two fundamental
solutions in terms of the distance of the initial points.
The result is a consequence of the equivalence between the RCD(K,\infty)
lower Ricci bound and the corresponding Bakry-\'Emery condition for the
canonical Cheeger-Dirichlet form in (X,d,m). The crucial tool is the extension
to the non-smooth metric measure setting of the Bakry's argument, that allows
to improve the commutation estimates between the Markov semigroup and the
Carr\'e du Champ associated to the Dirichlet form. This extension is based on a
new a priori estimate and a capacitary argument for regular and tight Dirichlet
forms that are of independent interest.