Self-improvement of the Bakry-Émery condition and  Wasserstein contraction of the heat flow in  $RCD (K, \infty)$ metric measure spaces

Giuseppe Savaré

doi:10.3934/dcds.2014.34.1641

Article Contents

2014, Volume 34, Issue 4: 1641-1661. Doi: 10.3934/dcds.2014.34.1641

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Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces

Giuseppe Savaré^1,

1.
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, I–27100 Pavia

Received: November 30, 2012

Revised: February 28, 2013

Published: March 2014

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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, $p\in [1,\infty]$. 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-Émery 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 Carré du Champ $\Gamma$ 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.

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Mathematics Subject Classification: Primary: 49Q20, 47D07; Secondary: 30L99.

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