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References
Alexandrov, A.D.: A theorem on triangles in a metric space and some applications (Russian). Trudy Math. Inst. Steklov, 38, 5–23 (1951)
Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Société Mathématique de France, Paris (2000)
Ballmann, W.: Lectures on spaces of nonpositive curvature. DMV Seminar Band 25, Birkhäuser Verlag, Basel (1995)
Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature. Progress in Mathematics, 61, Birkhäuser Boston Inc., Boston, MA (1985)
Bakry, D., Emery, M.: Hypercontractivite de semi-groupes de diffusion. C. R. Acad. Sci., Paris, Ser. I, 299, 775–778 (1984)
Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin (1999)
Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI (2001)
Cartan, H.: Lecons sur lá geometrie des espaces de Riemann. Gauthiers-Villars, Paris (1928)
Cordero-Erausquin, D., McCann, R., Schnuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamb and Lieb. Invent. Math., 146, 219–257 (2001)
Da Prato, G., Röckner, M.: Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Relat. Fields, 124, 261–303 (2002)
Doss, S.: Sur la moyenne d'un élément aléatoire dans un espace distancié. Bull. Sci. Math., 73, 48–72 (1949)
Dudley, R.M.: Real analysis and probability. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. (1989)
Eells, J., Fuglede, B.: Harmonic maps between Riemannian polyhedra. Cambridge Tracts in Mathematics, 142, Cambridge University Press, Cambridge (2001)
Emery, D., Mokobodzki, G.: Sur le barycentre d'une probabilité dans une variété. Séminaire de Probabilités XXV, 220–233. Lecture Notes in Math. 1485, Springer, Berlin (1991)
Es-Sahib, A., Heinich, H.: Barycentre canonique pour un espace métrique à courbure négative. Séminaire de Probabilités, XXXIII, 355–370. Lecture Notes in Math., 1709, Springer, Berlin (1999)
Fréchet, M.: Les éléments alétoires de nature quelconque dans un espace distancié, Ann. Inst. H. Poincaré, 10, 215–310 (1948)
Carl Friedrich Gauß,: Theoria Motus Corporum Celestium (1809)
Gromov, M.: Structures métriques pour les variétés Riemanniennes. Rédigé par J. Lafontaine et P. Pansu, Cedic/Fernand Nathan (1981), (1999)
Herer, W.: Espérance mathématique au sens Doss d'une variable aléatoire dans un espace métrique. C. R. Acad. Sci. Paris Sér. I, 302, 131–134 (1991)
Jost, J.: Equilibrium maps between metric spaces, Calc. Var. Partial Differential Equations, 2, 173–204 (1994)
Jost, J.: Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1997)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math., 30, 509–541 (1977)
Kendall, W.S.: Probability, convexity, and harmonic maps with small images. I. Uniqueness and fine existence. Proc. London Math. Soc., 61, 371–406 (1990)
Kendall, W.S.: From stochastic parallel transport to harmonic maps. In: Jost, J., Kendall, W.S., Mosco, U., Röckner, M., Sturm, K.T. (ed) New directions in Dirichlet forms. AMS and International Press (1998)
Korevaar, N., Schoen, R.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom., 1, 561–569 (1993)
McCann, R.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal., 11, 589–608 (2001)
Otto, F.: The geometry of dissipative evolution equation: the porous medium equation. Comm. PDE, 26, 101–174 (2001)
Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173, 361–400 (2000)
Picard, J.: Barycentres et martingales sur une variéte. Ann. Inst. Henri Poincaré, Prob. Stat., 30, 647–702 (1994)
Rachev, S.T., Rüschendorf, L.: Mass transportation problems. Vol. I. Theory. Probability and its Applications. Springer, Berlin Heidelberg New York (1998)
von Renesse, M.-K.: Intrinsic Coupling on Riemannian Manifolds and Polyhedra. Preprint, University of Bonn, Bonn (2003)
von Renesse, M.-K., Sturm, K.T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. SFB 611 Preprint, 80, University of Bonn, Bonn (2003)
Stroock, D.W.: Markov processes from K. Ito's perspective. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (2003)
Sturm, K.T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab., 26, 1–55 (1998)
Sturm, K.T.: Metric spaces of lower bounded curvature. Expo. Math., 17, 35–47 (1999)
Sturm, K.T.: Stochastics and analysis on metric spaces. Lecture Notes, University of Bonn, Bonn (2001)
Sturm, K.T.: Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. Ann. Probab., 30, 1195–1222 (2002)
Sturm, K.T.: A Semigroup Approach to Harmonic Maps. SFB 611 Preprint, 39, University of Bonn, Bonn (2003)
Sturm, K.T.: Nonlinear diffusions as gradient flows: convexity and contraction properties. Preprint, University of Bonn, Bonn (2004)
Villani, C.: Topics in Mass Transportation. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003)
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Sturm, KT. (2005). Coupling, Regularity and Curvature. In: Deuschel, JD., Greven, A. (eds) Interacting Stochastic Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27110-4_14
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