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Convex Analysisの二,三の進展について

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

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Table Of Integrals Series And Products

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Methods Of Modern Mathematical Physics

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound

 $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$
 (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$
 and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincare inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

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On the geometry of metric measure spaces. II

We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) as well as in terms of transportation inequalities for volume measures, heat kernels, and Brownian motions and in terms of gradient estimates for the heat semigroup. © 2004 Wiley Periodicals, Inc.

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Transport inequalities, gradient estimates, entropy and Ricci curvature

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Emery (via energy and $$\Gamma _2$$
 -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.

On the equivalence of the entropic curvature-dimension condition and bochner's inequality on metric measure spaces

The basic object for the sequel is a fixed regul r Dirichlet form S with domain Q) ( } on a real Hubert space H = L(X, m). The underlying topological space X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[w] = X. The form S is always assumed to be strongly local (i.e. S (u, v) = 0 whenever u e Q) ((?) is constant on a neighborhood of the support of v €&(£)) and to be irreducible (i.e. we ^loc(<f) is constant on X whenever S (u, u) = 0). In other words, S has no killing measure and no jumping measure and X cannot be decomposed into (non-trivial) subsets which are invariant for S. For notions concerning Dirichlet forms we recommend the monograph [F] of M. Fukushima whose terminology we mostly follow. For a brief discussion of the notion or irreducibility we refer to the Appendix at the end of this article. Let us mention one crucial consequence of the regularity of S. Each function u€^(S) admits a quasi-continuous version (which is determined pointwise up to exceptional sets). For simplicity, we always write again u for and make the convention that whenever we use a pointwise version of u (e. g. in expressions like J φ or J M φ with a measure μ charging no exceptional sets) then {u>0} without restriction this version is always chosen quasi-continuous.

Karl-Theodor Sturm

Papers

On the geometry of metric measure spaces. II

Transport inequalities, gradient estimates, entropy and Ricci curvature

On the equivalence of the entropic curvature-dimension condition and bochner's inequality on metric measure spaces

Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties.

Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality