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 and Continuous Dynamical Systems, 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, arXiv:1207.4924, (2012). To appear in Transactions of the AMS.

[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, Birkhäuser Verlag, Basel, 2008.

[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, arXiv:1106.2090, (2011), 1-74. To appear in Invent. Math. doi: 10.1007/s00222-013-0456-1.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, preprint, arXiv:1109.0222, (2011), 1-60. To appear in Duke Math. J.

[5]

L. Ambrosio, N. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, preprint, arXiv:1209.5786, (2012), 1-61.

[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), 517-564. doi: 10.1007/s00440-008-0177-3.

[7]

D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition $\Gamma_2\geq 0$, Séminaire de probabilités, XIX, 1983/84, 145-174, Lecture Notes in Math., 1123, Springer, Berlin, 1985. doi: 10.1007/BFb0075844.

[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), 1-114, Lecture Notes in Math., 1581, Springer, Berlin, 1994. doi: 10.1007/BFb0073872.

[9]

D. Bakry, Functional inequalities for Markov semigroups, in "Probability Measures on Groups: Recent Directions and Trends," 91-147, Tata Inst. Fund. Res., Mumbai, 2006.

[10]

D. Bakry and M. Émery, Diffusions hypercontractives, (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, Springer, Berlin, 1123 (1985), 177-206. doi: 10.1007/BFb0075847.

[11]

D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Rev. Mat. Iberoamericana, 22 (2006), 683-702. doi: 10.4171/RMI/470.

[12]

M. Biroli and U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl., 169 (1995), 125-181. doi: 10.1007/BF01759352.

[13]

V. I. Bogachev, "Measure Theory," Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[14]

N. Bouleau and F. Hirsch, "Dirichlet Forms and Analysis on Wiener Spaces," De Gruyter studies in Mathematics, 14. Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389.

[15]

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094.

[16]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 46 (1997), 406-480.

[17]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom., 54 (2000), 13-35.

[18]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom., 54 (2000), 37-74.

[19]

Z.-Q. Chen and M. Fukushima, "Symmetric Markov Processes, Time Change, and Boundary Theory," London Mathematical Society Monographs Series, 35. Princeton University Press, Princeton, NJ, 2012.

[20]

D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257. doi: 10.1007/s002220100160.

[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), 1-23. doi: 10.1214/08-AIHP306.

[22]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9.

[23]

N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Comm. Pure Appl. Math., 66 (2013), 307-331. doi: 10.1002/cpa.21431.

[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).

[25]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[26]

K. Kuwada, Duality on gradient estimates and Wasserstein controls, Journal of Functional Analysis, 258 (2010), 3758-3774. doi: 10.1016/j.jfa.2010.01.010.

[27]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.

[28]

Z.-M. Ma and M. Röckner, "Introduction to The Theory of (Non-symmetric) Dirichlet Forms," Universitext. Springer-Verlag, Berlin, 1992. vi+209 pp doi: 10.1007/978-3-642-77739-4.

[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), 18-35. doi: 10.1016/j.matpur.2010.07.003.

[30]

S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math., 62 (2009), 1386-1433. doi: 10.1002/cpa.20273.

[31]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

[32]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures," Tata Institute of Fundamental Research Studies in Mathematics, No. 6. Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. xii+393 pp.

[34]

K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.

[35]

K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7.

[36]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. doi: 10.1007/978-3-540-71050-9.

[37]

F.-Y. Wang, Equivalent semigroup properties for the curvature-dimension condition, Bull. Sci. Math., 135 (2011), 803-815. doi: 10.1016/j.bulsci.2011.07.005.

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, arXiv:1207.4924, (2012). To appear in Transactions of the AMS.

[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, Birkhäuser Verlag, Basel, 2008.

[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, arXiv:1106.2090, (2011), 1-74. To appear in Invent. Math. doi: 10.1007/s00222-013-0456-1.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, preprint, arXiv:1109.0222, (2011), 1-60. To appear in Duke Math. J.

[5]

L. Ambrosio, N. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, preprint, arXiv:1209.5786, (2012), 1-61.

[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), 517-564. doi: 10.1007/s00440-008-0177-3.

[7]

D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition $\Gamma_2\geq 0$, Séminaire de probabilités, XIX, 1983/84, 145-174, Lecture Notes in Math., 1123, Springer, Berlin, 1985. doi: 10.1007/BFb0075844.

[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), 1-114, Lecture Notes in Math., 1581, Springer, Berlin, 1994. doi: 10.1007/BFb0073872.

[9]

D. Bakry, Functional inequalities for Markov semigroups, in "Probability Measures on Groups: Recent Directions and Trends," 91-147, Tata Inst. Fund. Res., Mumbai, 2006.

[10]

D. Bakry and M. Émery, Diffusions hypercontractives, (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, Springer, Berlin, 1123 (1985), 177-206. doi: 10.1007/BFb0075847.

[11]

D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Rev. Mat. Iberoamericana, 22 (2006), 683-702. doi: 10.4171/RMI/470.

[12]

M. Biroli and U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl., 169 (1995), 125-181. doi: 10.1007/BF01759352.

[13]

V. I. Bogachev, "Measure Theory," Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[14]

N. Bouleau and F. Hirsch, "Dirichlet Forms and Analysis on Wiener Spaces," De Gruyter studies in Mathematics, 14. Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389.

[15]

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094.

[16]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 46 (1997), 406-480.

[17]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom., 54 (2000), 13-35.

[18]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom., 54 (2000), 37-74.

[19]

Z.-Q. Chen and M. Fukushima, "Symmetric Markov Processes, Time Change, and Boundary Theory," London Mathematical Society Monographs Series, 35. Princeton University Press, Princeton, NJ, 2012.

[20]

D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257. doi: 10.1007/s002220100160.

[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), 1-23. doi: 10.1214/08-AIHP306.

[22]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9.

[23]

N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Comm. Pure Appl. Math., 66 (2013), 307-331. doi: 10.1002/cpa.21431.

[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).

[25]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[26]

K. Kuwada, Duality on gradient estimates and Wasserstein controls, Journal of Functional Analysis, 258 (2010), 3758-3774. doi: 10.1016/j.jfa.2010.01.010.

[27]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.

[28]

Z.-M. Ma and M. Röckner, "Introduction to The Theory of (Non-symmetric) Dirichlet Forms," Universitext. Springer-Verlag, Berlin, 1992. vi+209 pp doi: 10.1007/978-3-642-77739-4.

[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), 18-35. doi: 10.1016/j.matpur.2010.07.003.

[30]

S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math., 62 (2009), 1386-1433. doi: 10.1002/cpa.20273.

[31]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

[32]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures," Tata Institute of Fundamental Research Studies in Mathematics, No. 6. Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. xii+393 pp.

[34]

K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.

[35]

K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7.

[36]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. doi: 10.1007/978-3-540-71050-9.

[37]

F.-Y. Wang, Equivalent semigroup properties for the curvature-dimension condition, Bull. Sci. Math., 135 (2011), 803-815. doi: 10.1016/j.bulsci.2011.07.005.

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