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In this paper we introduce a synthetic notion of Riemannian Ricci bounds frombelow for metric measure spaces (X,d,m) which is stable under measuredGromov-Hausdorff convergence and rules out Finsler geometries. It can be givenin terms of an enforcement of the Lott, Sturm and Villani geodesic convexitycondition for the entropy coupled with the linearity of the heat flow. Besidesstability, it enjoys the same tensorization, global-to-local andlocal-to-global properties. In these spaces, that we call RCD(K,\infty) spaces,we prove that the heat flow (which can be equivalently characterized either asthe flow associated to the Dirichlet form, or as the Wasserstein gradient flowof the entropy) satisfies Wasserstein contraction estimates and severalregularity properties, in particular Bakry-Emery estimates and the L^\infty-LipFeller regularization. We also prove that the distance induced by the Dirichletform coincides with d, that the local energy measure has density given by thesquare of Cheeger's relaxed slope and, as a consequence, that the underlyingBrownian motion has continuous paths. All these results are obtainedindependently of Poincar\'e and doubling assumptions on the metric measurestructure and therefore apply also to spaces which are not locally compact, asthe infinite-dimensional ones

Metric measure spaces with Riemannian Ricci curvature bounded from below