We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry-Émery (via energy and Γ2-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the L 2 -Wasserstein distance.

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐space produces the same evolution as the gradient flow of the relative entropy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐Wasserstein space. This means that the heat flow is well‐defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry‐Émery gradient estimates and the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\Gamma_2\end{align*} \end{document}‐condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift. © 2012 Wiley Periodicals, Inc.

We establish a duality between L p -Wasserstein control and L q -gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance.

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We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pathwise contraction result established by Arnaudon et al. (arXiv:0904.2762, to appear in Sém. Prob.). As an important tool which is of independent interest we derive an Itô formula for the distance from a fixed reference point, generalising a result of Kendall (Ann. Prob. 15:1491-1500, 1987.Keywords Ricci flow · Diffusion process · Non-explosion · Radial process Mathematics Subject Classification (2000) 53C21 · 53C44 · 58J35 · 60J60 Brownian motion with respect to time-changing Riemannian metricsLet M be a d-dimensional differentiable manifold with d ≥ 2, π : F (M) → M the frame bundle and (g(t)) t∈[0,T ] a family of Riemannian metrics on M depending smoothly on t such that (M, g(t)) is geodesically complete for all tβ=1 be the canonical vertical vector fields. Let (W t ) t≥0 be a standard R d -valued Brownian motion. In this situation Arnaudon et al. [1,5] defined horizontal Brownian motion on F (M) as the solution of the K. Kuwada was partially supported by the JSPS fellowship for research abroad.