Metric measure spaces with Riemannian Ricci curvature bounded from below

Abstract

In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces $(X,\mathsf{d},\mathfrak{m})$ which is stable under measured Gromov– Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm, and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local, and local-to-global properties. In these spaces, which we call $\mathrm{RCD}(K,\infty )$ spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry–Émery estimates and the $L^{\infty }\text{-}\mathrm{Lip}$ Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with $\mathsf{d}$, that the local energy measure has density given by the square of Cheeger’s relaxed slope, and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.