2013/04/30 by Giuseppe Savaré · 1 citation
Mathematics · #math.FA #math.AP #math.MG #math.PR #msc:49Q20 #msc:47D07 #msc:30L99
paper · pdf · doi:10.3934/dcds.2014.34.1641
published as Discrete Contin. Dyn. Syst. (A) 34 (2014) 1641-1661 · (v2) Minor corrections. A discussion of quasi-regular Dirichlet forms has been added (Section 2.3) to cover the case of a sigma-finite reference measure. The proof of the quasi regularity of the Cheeger energy has been added (Thm. 4.1)
arxiv created 2013/05/17 · arxiv updated 2013/11/22
We prove that the linear heat flow in a RCD(K,∞) metric measure space (X,d,m) satisfies a contraction property with respect to every Lp-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a precise estimate for the optimal W_∞-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,∞) 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 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.