In a prior work of the first two authors with Savaré, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces (X, d, m) was introduced, and the corresponding class of spaces was denoted by RCD (K, ∞). This notion relates the CD(K, N ) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In this prior work the RCD(K, ∞) property is defined in three equivalent ways and several properties of RCD(K, ∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In the above-mentioned work only finite reference measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, and on the other we remove a technical assumption that appeared in the earlier work concerning a strengthening of the CD(K, ∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds. enforced by adding the assumption that the so-called Cheeger energy (playing here the role of the classical Dirichlet energy) is quadratic.More precisely, the class of RCD(K, ∞) spaces of [4] can be defined in three equivalent ways thanks to this equivalence result (see §2.3 for the precise formulation of gradient flows involved here, in the metric sense and in the EV I K sense):Theorem 1.1 ([4]). Let (X, d, m) be a metric measure space with (X, d) complete and separable, m(X) ∈ (0, ∞) and supp m = X. Then the following are equivalent:m) is a strict CD(K, ∞) space and Ch is a quadratic form on L 2 (X, m). (iii) (X, d, m) is a length space and any μ ∈ P 2 (X) is the starting point of an EV I K gradient flow of Ent m . This equivalence is crucial for the study of the spaces RCD(K, ∞): for instance the fine properties of the heat flow and the Bakry-Emery condition obtained in [4] need (ii), while the stability of RCD(K, ∞) spaces under Sturm's convergence [33] of metric measure spaces (a variant of measured Gromov-Hausdorff convergence) depends in a crucial way on (iii) and on the stability properties of EV

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak C D(K , N ) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a flow satisfying an evolution variational inequality for either the Rényi entropy functional E N (ρm) = − X ρ 1−1/N dm or the Shannon entropy functional E ∞ (ρm) = X ρ log ρdm. We also prove that if the Rényi entropy functional is strongly displacement convex in the Wasserstein space, then at every point of the space we have unique geodesics to almost all points of the space.

We prove that in metric measure spaces where the entropy functional is Kconvex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map.The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant.

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincaré inequality is implied by the weak displacement convexity of the functional.