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Metric measure spaces with Riemannian Ricci curvature bounded from below. (English) Zbl 1304.35310
The authors introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces $$(X,d,m)$$ which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. Besides stability, it enjoys tensorization, global-to-local and local-to-global properties. They called these spaces $$\mathrm{RCD}(K,\infty)$$ spaces. They proved that the heat flow satisfied Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the $$L^{\infty}$$-Lip Feller regularization. They also proved that the distance induced by the Dirichlet form coincided with $$d$$, that the local energy measure has density given by the square of Cheeger’s relaxed slope. As a consequence the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact such as the infinite dimensional ones.

##### MSC:
 35K05 Heat equation 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 60J65 Brownian motion
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##### References:
 [1] L. Ambrosio and N. Gigli, “A user’s guide to optimal transport” in Modelling and Optimisation of Flows on Networks , Lecture Notes in Math. 2062 , Springer, Berlin, 2013. [2] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures , 2nd ed., Lectures Math. ETH Zürich, Birkhäuser, Basel, 2008. · Zbl 1145.35001 [3] L. Ambrosio, N. Gigli, and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below , Invent. Math. 195 (2014), 289-391. · Zbl 1312.53056 [4] L. Ambrosio, G. Savaré, and L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure , Probab. Theory Related Fields 145 (2009), 517-564. · Zbl 1235.60105 [5] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Sur les inégalités de Sobolev logarithmiques , Panor. Synthèses 10 , Soc. Math. France, Paris, 2000. · Zbl 0982.46026 [6] D. Bakry, “Functional inequalities for Markov semigroups” in Probability Measures on Groups: Recent Directions and Trends , Tata Inst. Fund. Res., Mumbai, 2006, 91-147. · Zbl 1148.60057 [7] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , North-Holland Math. Stud. 5 , North-Holland, Amsterdam, 1973. · Zbl 0252.47055 [8] D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry , Grad. Stud. Math. 33 , Amer. Math. Soc., Providence, 2001. · Zbl 0981.51016 [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces , Geom. Funct. Anal. 9 (1999), 428-517. · Zbl 0942.58018 [10] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below, I , J. Differential Geom. 46 (1997), 406-480. · Zbl 0902.53034 [11] P. Clément and W. Desch, Some remarks on the equivalence between metric formulations of gradient flows , Boll. Unione Mat. Ital. (9) 3 (2010), 583-588. · Zbl 1217.49009 [12] D. Cordero-Erausquin, R. J. McCann, and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb , Invent. Math. 146 (2001), 219-257. · Zbl 1026.58018 [13] G. Dal Maso, An Introduction to $${\Gamma}$$-Convergence , Progr. Nonlinear Differential Equations Appl. 8 , Birkhäuser, Boston, 1993. [14] S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance , SIAM J. Math. Anal. 40 (2008), 1104-1122. · Zbl 1166.58011 [15] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator , Invent. Math. 87 (1987), 517-547. · Zbl 0589.58034 [16] M. Fukushima, Dirichlet Forms and Markov Processes , North-Holland Math. Libr. 23 , North-Holland, Amsterdam, 1980. · Zbl 0422.31007 [17] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes , 2nd revised and extended ed., de Gruyter Stud. Math. 19 , Walter de Gruyter, Berlin, 2011. · Zbl 1227.31001 [18] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability , Calc. Var. Partial Differential Equations 39 (2010), 101-120. · Zbl 1200.35178 [19] N. Gigli, On the differential structure of metric measure spaces and applications , preprint, [math.MG]. 1205.6622v2 [20] N. Gigli, K. Kuwada, and S.-I. Ohta, Heat flow on Alexandrov spaces , Comm. Pure Appl. Math. 66 (2013), 307-331. · Zbl 1267.58014 [21] N. Gigli and S.-I. Ohta, First variation formula in Wasserstein spaces over compact Alexandrov spaces , Canad. Math. Bull. 55 (2012), 723-735. · Zbl 1264.53050 [22] A. Joulin, A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature , Bernoulli 15 (2009), 532-549. · Zbl 1202.60136 [23] P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces , Studia Math. 131 (1998), 1-17. · Zbl 0918.30011 [24] P. Koskela and Y. Zhou, Geometry and analysis of Dirichlet forms , Adv. Math. 231 (2012), 2755-2801. · Zbl 1253.53035 [25] K. Kuwada, Duality on gradient estimates and Wasserstein controls , J. Funct. Anal. 258 (2010), 3758-3774. · Zbl 1194.53032 [26] B. Levi, Sul principio di dirichlet , Rend. Circ. Mat. Palermo 22 (1906), 293-359. · JFM 37.0414.04 [27] S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces , Calc. Var. Partial Differential Equations 28 (2007), 85-120. · Zbl 1132.60004 [28] J. Lott and C. Villani, Weak curvature conditions and functional inequalities , J. Funct. Anal. 245 (2007), 311-333. · Zbl 1119.53028 [29] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport , Ann. of Math. (2) 169 (2009), 903-991. · Zbl 1178.53038 [30] Z.-M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms , Universitext, Springer, Berlin, 1992. [31] S.-I. Ohta, Finsler interpolation inequalities , Calc. Var. Partial Differential Equations 36 (2009), 211-249. · Zbl 1175.49044 [32] S.-I. Ohta, Gradient flows on Wasserstein spaces over compact Alexandrov spaces , Amer. J. Math. 131 (2009), 475-516. · Zbl 1169.53053 [33] S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds , Comm. Pure Appl. Math. 62 (2009), 1386-1433. · Zbl 1176.58012 [34] S.-I. Ohta and K.-T. Sturm, Non-contraction of heat flow on Minkowski spaces , Arch. Ration. Mech. Anal. 204 (2012), 917-944. · Zbl 1257.53098 [35] S.-I. Ohta and K.-T. Sturm, Adv. Math. 252 (2014), 429-448. · Zbl 1321.53089 [36] Y. Ollivier, Ricci curvature of Markov chains on metric spaces , J. Funct. Anal. 256 (2009), 810-864. · Zbl 1181.53015 [37] A. Petrunin, Alexandrov meets Lott-Villani-Sturm , Münster J. Math. 4 (2011), 53-64. · Zbl 1247.53038 [38] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces , Calc. Var. Partial Differential Equations 44 (2012), 477-494. · Zbl 1250.53040 [39] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds , C. R. Math. Acad. Sci. Paris 345 (2007), 151-154. · Zbl 1125.53064 [40] G. Savaré, Gradient flows and evolution variational inequalities in metric spaces , in preparation. [41] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces , Rev. Mat. Iberoam. 16 (2000), 243-279. · Zbl 0974.46038 [42] K.-T. Sturm, Analysis on local Dirichlet spaces, III: The parabolic Harnack inequality , J. Math. Pures Appl. (9) 75 (1996), 273-297. · Zbl 0854.35016 [43] K.-T. Sturm, Is a diffusion process determined by its intrinsic metric? Chaos Solitons Fractals 8 (1997), 1855-1860. · Zbl 0939.58032 [44] K.-T. Sturm, Acta Math. 196 (2006), 65-131. · Zbl 1105.53035 [45] K.-T. Sturm, Acta Math. 196 (2006), 133-177. · Zbl 1106.53032 [46] M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature , Comm. Pure Appl. Math. 58 (2005), 923-940. · Zbl 1078.53028 [47] C. Villani, Optimal Transport: Old and New , Grundlehren Math. Wiss. 338 , Springer, Berlin, 2009. · Zbl 1156.53003 [48] H.-C. Zhang and X.-P. Zhu, Ricci curvature on Alexandrov spaces and rigidity theorems , Comm. Anal. Geom. 18 (2010), 503-553. · Zbl 1230.53064 [49] H.-C. Zhang and X.-P. Zhu, Yau’s gradient estimates on Alexandrov spaces , J. Differential Geom. 91 (2012), 445-522. · Zbl 1258.53075
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