# American Institute of Mathematical Sciences

April  2014, 34(4): 1641-1661. doi: 10.3934/dcds.2014.34.1641

## Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces

 1 Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, I–27100 Pavia

Received  December 2012 Revised  March 2013 Published  October 2013

We prove that the linear heat'' flow in a $RCD (K, \infty)$ metric measure space $(X, d, m)$ satisfies a contraction property with respect to every $L^p$-Kantorovich-Rubinstein-Wasserstein distance, $p\in [1,\infty]$. In particular, we obtain a precise estimate for the optimal $W_\infty$-coupling between two fundamental solutions in terms of the distance of the initial points.
The result is a consequence of the equivalence between the $RCD (K, \infty)$ lower Ricci bound and the corresponding Bakry-Émery condition for the canonical Cheeger-Dirichlet form in $(X, d, m)$. The crucial tool is the extension to the non-smooth metric measure setting of the Bakry's argument, that allows to improve the commutation estimates between the Markov semigroup and the Carré du Champ $\Gamma$ associated to the Dirichlet form.
This extension is based on a new a priori estimate and a capacitary argument for regular and tight Dirichlet forms that are of independent interest.
Citation: Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641
##### References:
 [1] L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure,, preprint, (2012). Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures,", Second edition. Lectures in Mathematics ETH Zürich, (2008). Google Scholar [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,, preprint, (2011), 1. doi: 10.1007/s00222-013-0456-1. Google Scholar [4] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below,, preprint, (2011), 1. Google Scholar [5] L. Ambrosio, N. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds,, preprint, (2012), 1. Google Scholar [6] L. Ambrosio, G. Savaré and L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure,, Probab. Theory Relat. Fields, 145 (2009), 517. doi: 10.1007/s00440-008-0177-3. Google Scholar [7] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition $\Gamma_2\geq 0$,, Séminaire de probabilités, (1983), 145. doi: 10.1007/BFb0075844. Google Scholar [8] D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes,, (French) [Hypercontractivity and its use in semigroup theory] Lectures on probability theory (Saint-Flour, (1992), 1. doi: 10.1007/BFb0073872. Google Scholar [9] D. Bakry, Functional inequalities for Markov semigroups,, in, (2006), 91. Google Scholar [10] D. Bakry and M. Émery, Diffusions hypercontractives,, (French) [Hypercontractive diffusions] Séminaire de probabilités, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar [11] D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality,, Rev. Mat. Iberoamericana, 22 (2006), 683. doi: 10.4171/RMI/470. Google Scholar [12] M. Biroli and U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media,, Ann. Mat. Pura Appl., 169 (1995), 125. doi: 10.1007/BF01759352. Google Scholar [13] V. I. Bogachev, "Measure Theory," Vol. I, II,, Springer-Verlag, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar [14] N. Bouleau and F. Hirsch, "Dirichlet Forms and Analysis on Wiener Spaces,", De Gruyter studies in Mathematics, (1991). doi: 10.1515/9783110858389. Google Scholar [15] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces,, Geom. Funct. Anal., 9 (1999), 428. doi: 10.1007/s000390050094. Google Scholar [16] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I,, J. Differential Geom., 46 (1997), 406. Google Scholar [17] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. II,, J. Differential Geom., 54 (2000), 13. Google Scholar [18] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. III,, J. Differential Geom., 54 (2000), 37. Google Scholar [19] Z.-Q. Chen and M. Fukushima, "Symmetric Markov Processes, Time Change, and Boundary Theory,", London Mathematical Society Monographs Series, (2012). Google Scholar [20] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb,, Invent. Math., 146 (2001), 219. doi: 10.1007/s002220100160. Google Scholar [21] M. Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space,, Annales de l'Institut Henri Poincaré - Probabilités et Statistiques, 46 (2010), 1. doi: 10.1214/08-AIHP306. Google Scholar [22] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calc. Var. Partial Differential Equations, 39 (2010), 101. doi: 10.1007/s00526-009-0303-9. Google Scholar [23] N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces,, Comm. Pure Appl. Math., 66 (2013), 307. doi: 10.1002/cpa.21431. Google Scholar [24] N. Gigli, A. Mondino and G. Savaré, A notion of pointed convergence of non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows,, In preparation, (2013). Google Scholar [25] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar [26] K. Kuwada, Duality on gradient estimates and Wasserstein controls,, Journal of Functional Analysis, 258 (2010), 3758. doi: 10.1016/j.jfa.2010.01.010. Google Scholar [27] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math. (2), 169 (2009), 903. doi: 10.4007/annals.2009.169.903. Google Scholar [28] Z.-M. Ma and M. Röckner, "Introduction to The Theory of (Non-symmetric) Dirichlet Forms,", Universitext. Springer-Verlag, (1992). doi: 10.1007/978-3-642-77739-4. Google Scholar [29] L. Natile, M. A. Peletier and G. Savaré, Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts,, Journal de Mathématiques Pures et Appliqués, 95 (2011), 18. doi: 10.1016/j.matpur.2010.07.003. Google Scholar [30] S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds,, Comm. Pure Appl. Math., 62 (2009), 1386. doi: 10.1002/cpa.20273. Google Scholar [31] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,, J. Funct. Anal., 173 (2000), 361. doi: 10.1006/jfan.1999.3557. Google Scholar [32] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [33] L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,", Tata Institute of Fundamental Research Studies in Mathematics, (1973). Google Scholar [34] K.-T. Sturm, On the geometry of metric measure spaces. I,, Acta Math., 196 (2006), 65. doi: 10.1007/s11511-006-0002-8. Google Scholar [35] K.-T. Sturm, On the geometry of metric measure spaces. II,, Acta Math., 196 (2006), 133. doi: 10.1007/s11511-006-0003-7. Google Scholar [36] C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar [37] F.-Y. Wang, Equivalent semigroup properties for the curvature-dimension condition,, Bull. Sci. Math., 135 (2011), 803. doi: 10.1016/j.bulsci.2011.07.005. Google Scholar

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##### References:
 [1] L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure,, preprint, (2012). Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures,", Second edition. Lectures in Mathematics ETH Zürich, (2008). Google Scholar [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,, preprint, (2011), 1. doi: 10.1007/s00222-013-0456-1. Google Scholar [4] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below,, preprint, (2011), 1. Google Scholar [5] L. Ambrosio, N. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds,, preprint, (2012), 1. Google Scholar [6] L. Ambrosio, G. Savaré and L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure,, Probab. Theory Relat. Fields, 145 (2009), 517. doi: 10.1007/s00440-008-0177-3. Google Scholar [7] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition $\Gamma_2\geq 0$,, Séminaire de probabilités, (1983), 145. doi: 10.1007/BFb0075844. Google Scholar [8] D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes,, (French) [Hypercontractivity and its use in semigroup theory] Lectures on probability theory (Saint-Flour, (1992), 1. doi: 10.1007/BFb0073872. Google Scholar [9] D. Bakry, Functional inequalities for Markov semigroups,, in, (2006), 91. Google Scholar [10] D. Bakry and M. Émery, Diffusions hypercontractives,, (French) [Hypercontractive diffusions] Séminaire de probabilités, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar [11] D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality,, Rev. Mat. Iberoamericana, 22 (2006), 683. doi: 10.4171/RMI/470. Google Scholar [12] M. Biroli and U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media,, Ann. Mat. Pura Appl., 169 (1995), 125. doi: 10.1007/BF01759352. Google Scholar [13] V. I. Bogachev, "Measure Theory," Vol. I, II,, Springer-Verlag, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar [14] N. Bouleau and F. Hirsch, "Dirichlet Forms and Analysis on Wiener Spaces,", De Gruyter studies in Mathematics, (1991). doi: 10.1515/9783110858389. Google Scholar [15] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces,, Geom. Funct. Anal., 9 (1999), 428. doi: 10.1007/s000390050094. Google Scholar [16] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I,, J. Differential Geom., 46 (1997), 406. Google Scholar [17] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. II,, J. Differential Geom., 54 (2000), 13. Google Scholar [18] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. III,, J. Differential Geom., 54 (2000), 37. Google Scholar [19] Z.-Q. Chen and M. Fukushima, "Symmetric Markov Processes, Time Change, and Boundary Theory,", London Mathematical Society Monographs Series, (2012). Google Scholar [20] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb,, Invent. Math., 146 (2001), 219. doi: 10.1007/s002220100160. Google Scholar [21] M. Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space,, Annales de l'Institut Henri Poincaré - Probabilités et Statistiques, 46 (2010), 1. doi: 10.1214/08-AIHP306. Google Scholar [22] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calc. Var. Partial Differential Equations, 39 (2010), 101. doi: 10.1007/s00526-009-0303-9. Google Scholar [23] N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces,, Comm. Pure Appl. Math., 66 (2013), 307. doi: 10.1002/cpa.21431. Google Scholar [24] N. Gigli, A. Mondino and G. Savaré, A notion of pointed convergence of non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows,, In preparation, (2013). Google Scholar [25] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar [26] K. Kuwada, Duality on gradient estimates and Wasserstein controls,, Journal of Functional Analysis, 258 (2010), 3758. doi: 10.1016/j.jfa.2010.01.010. Google Scholar [27] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math. (2), 169 (2009), 903. doi: 10.4007/annals.2009.169.903. Google Scholar [28] Z.-M. Ma and M. Röckner, "Introduction to The Theory of (Non-symmetric) Dirichlet Forms,", Universitext. Springer-Verlag, (1992). doi: 10.1007/978-3-642-77739-4. Google Scholar [29] L. Natile, M. A. Peletier and G. Savaré, Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts,, Journal de Mathématiques Pures et Appliqués, 95 (2011), 18. doi: 10.1016/j.matpur.2010.07.003. Google Scholar [30] S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds,, Comm. Pure Appl. Math., 62 (2009), 1386. doi: 10.1002/cpa.20273. Google Scholar [31] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,, J. Funct. Anal., 173 (2000), 361. doi: 10.1006/jfan.1999.3557. Google Scholar [32] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [33] L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,", Tata Institute of Fundamental Research Studies in Mathematics, (1973). Google Scholar [34] K.-T. Sturm, On the geometry of metric measure spaces. I,, Acta Math., 196 (2006), 65. doi: 10.1007/s11511-006-0002-8. Google Scholar [35] K.-T. Sturm, On the geometry of metric measure spaces. II,, Acta Math., 196 (2006), 133. doi: 10.1007/s11511-006-0003-7. Google Scholar [36] C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar [37] F.-Y. Wang, Equivalent semigroup properties for the curvature-dimension condition,, Bull. Sci. Math., 135 (2011), 803. doi: 10.1016/j.bulsci.2011.07.005. Google Scholar
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