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

show all references

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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