The aim of this paper is to prove a second order differentiation formula for H^{2,2} functions along geodesics in RCD* (K,N) spaces with K ıin \mathbb R and N < \infty . This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that W_2 -geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolations. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: Finally, the techniques adopted in this paper can be used to show that in the RCD setting the viscous solution of the Hamilton–Jacobi equation can be obtained via a vanishing viscosity method, as in the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality.

We develop the theory of tamed spaces which are Dirichlet spaces with distributionvalued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with either interior singularities or singular boundary behavior. MATTHIAS ERBAR, CHIARA RIGONI, KARL-THEODOR STURM, AND LUCA TAMANINI 4.2. A manifold which is tamed but not 2-tamed 36 4.3. Manifolds with boundary and potentially singular curvature 36 4.4. A tamed domain with boundary that is not semiconvex 40 4.5. A Tamed Manifold with Highly Irregular Boundary 41 5. Functional Inequalities for Tamed Spaces 45 6. Self-Improvement of the Taming Condition 47 6.1. Measure-Valued Taming Operator and Bochner Inequality 47 6.2. Self-Improvement of the L 2 -Taming Condition 51 7. Sub-tamed Spaces 55 7.1. Reflected Dirichlet Spaces and Sub-taming 56 7.2. Doubling of Dirichlet Spaces and Sub-taming 58 7.3. Doubling of Riemannian Surfaces 64 References 66

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