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January  2021, 14(1): 273-297. doi: 10.3934/dcdss.2020327

Contraction and regularizing properties of heat flows in metric measure spaces

1. 

Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK

2. 

Dipartimento di Matematica 'Felice Casorati', University of Pavia, Via Ferrata 1, 27100 Pavia, Italy

* Corresponding author: Giuseppe Savaré

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  April 2019 Revised  October 2019 Published  January 2021 Early access  April 2020

Fund Project: The second author is partially supported by PRIN2015 grant from MIUR for the project Calculus of Variations and by IMATI-CNR

We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorovich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow.

When weaker transport distances are involved, we will show that contraction and regularizing effects rely on the dual formulations of the distances and are strictly related to lower Ricci curvature bounds in the setting of $ \mathrm{RCD}(K, \infty) $ metric measure spaces. As a byproduct, when $ K\ge0 $ we will also find new estimates for the asymptotic decay of the solution.

Citation: Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
References:
[1]

L. Ambrosio, A. Mondino and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces,, Amer. Math. Soc., 262 (2019), v+121 pp. doi: 10.1090/memo/1270.

[2]

L. AmbrosioM. Erbar and G. Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces, Nonlinear Anal., 137 (2016), 77-134.  doi: 10.1016/j.na.2015.12.006.

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[4]

L. AmbrosioN. 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.  doi: 10.1007/s00222-013-0456-1.

[5]

L. AmbrosioN. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490.  doi: 10.1215/00127094-2681605.

[6]

L. AmbrosioN. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab., 43 (2015), 339-404.  doi: 10.1214/14-AOP907.

[7]

L. AmbrosioG. 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.

[8]

D. Bakry and M. Émery, Diffusions hypercontractives,, Séminaire de Probabilités de Strasbourg, 19 (1985), 177–206. doi: 10.1007/BFb0075847.

[9]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[10]

D. BakryI. Gentil and M. Ledoux, On Harnack inequalities and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14 (2015), 705-727. 

[11]

A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, vol. 17 of EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/099.

[12]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).

[13]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[14]

L. ChizatG. PeyréB. Schmitzer and F.-X. Vialard, An interpolating distance between optimal transport and Fisher-Rao metrics, Found. Comput. Math., 18 (2018), 1-44.  doi: 10.1007/s10208-016-9331-y.

[15]

L. ChizatG. PeyréB. Schmitzer and F.-X. Vialard, Unbalanced optimal transport: Dynamic and Kantorovich formulations, J. Funct. Anal., 274 (2018), 3090-3123.  doi: 10.1016/j.jfa.2018.03.008.

[16]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318. 

[17]

S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122.  doi: 10.1137/08071346X.

[18]

C. Dellacherie and P.-A. Meyer, Probabilities and Potential. C, vol. 151 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1988, Potential theory for discrete and continuous semigroups, Translated from the French by J. Norris.

[19]

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.

[20]

M. ErbarK. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces, Invent. Math., 201 (2015), 993-1071.  doi: 10.1007/s00222-014-0563-7.

[21]

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

[22]

J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), 1-61.  doi: 10.1007/BF02392747.

[23]

J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, vol. 27 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2015, An approach based on upper gradients. doi: 10.1017/CBO9781316135914.

[24]

E. Hellinger, Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen, J. Reine Angew. Math., 136 (1909), 210-271.  doi: 10.1515/crll.1909.136.210.

[25]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[26]

S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2), 49 (1948), 214-224.  doi: 10.2307/1969123.

[27]

S. Kondratyev, L. Monsaingeon and D. Vorotnikov, A new optimal transport distance on the space of finite Radon measures, Adv. Differential Equations, 21 (2016), 1117–1164, URL http://projecteuclid.org/euclid.ade/1476369298.

[28]

P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math., 131 (1998), 1-17. 

[29]

S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics, 22 (1951), 79-86.  doi: 10.1214/aoms/1177729694.

[30]

K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal., 258 (2010), 3758-3774.  doi: 10.1016/j.jfa.2010.01.010.

[31]

M. LieroA. Mielke and G. Savaré, Optimal transport in competition with reaction: The Hellinger-Kantorovich distance and geodesic curves, SIAM J. Math. Anal., 48 (2016), 2869-2911.  doi: 10.1137/15M1041420.

[32]

M. LieroA. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Invent. Math., 211 (2018), 969-1117.  doi: 10.1007/s00222-017-0759-8.

[33]

F. Liese and I. Vajda, On divergences and informations in statistics and information theory, IEEE Trans. Inform. Theory, 52 (2006), 4394-4412.  doi: 10.1109/TIT.2006.881731.

[34]

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.

[35]

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.

[36]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[37]

F. Otto and M. Westdickenberg, Eulerian calculus for the contraction in the Wasserstein distance,, SIAM J. Math. Anal., 37 (2005), 1227–1255 (electronic). doi: 10.1137/050622420.

[38]

G. Savaré, Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD$ (K,\infty) $ metric measure spaces, Discrete Contin. Dyn. Syst., 34 (2014), 1641-1661.  doi: 10.3934/dcds.2014.34.1641.

[39]

N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana, 16 (2000), 243-279.  doi: 10.4171/RMI/275.

[40]

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

[41]

K.-T. Sturm and M.-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

[42]

C. Villani, Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

L. Ambrosio, A. Mondino and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces,, Amer. Math. Soc., 262 (2019), v+121 pp. doi: 10.1090/memo/1270.

[2]

L. AmbrosioM. Erbar and G. Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces, Nonlinear Anal., 137 (2016), 77-134.  doi: 10.1016/j.na.2015.12.006.

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[4]

L. AmbrosioN. 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.  doi: 10.1007/s00222-013-0456-1.

[5]

L. AmbrosioN. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490.  doi: 10.1215/00127094-2681605.

[6]

L. AmbrosioN. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab., 43 (2015), 339-404.  doi: 10.1214/14-AOP907.

[7]

L. AmbrosioG. 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.

[8]

D. Bakry and M. Émery, Diffusions hypercontractives,, Séminaire de Probabilités de Strasbourg, 19 (1985), 177–206. doi: 10.1007/BFb0075847.

[9]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[10]

D. BakryI. Gentil and M. Ledoux, On Harnack inequalities and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14 (2015), 705-727. 

[11]

A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, vol. 17 of EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/099.

[12]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).

[13]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[14]

L. ChizatG. PeyréB. Schmitzer and F.-X. Vialard, An interpolating distance between optimal transport and Fisher-Rao metrics, Found. Comput. Math., 18 (2018), 1-44.  doi: 10.1007/s10208-016-9331-y.

[15]

L. ChizatG. PeyréB. Schmitzer and F.-X. Vialard, Unbalanced optimal transport: Dynamic and Kantorovich formulations, J. Funct. Anal., 274 (2018), 3090-3123.  doi: 10.1016/j.jfa.2018.03.008.

[16]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318. 

[17]

S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122.  doi: 10.1137/08071346X.

[18]

C. Dellacherie and P.-A. Meyer, Probabilities and Potential. C, vol. 151 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1988, Potential theory for discrete and continuous semigroups, Translated from the French by J. Norris.

[19]

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.

[20]

M. ErbarK. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces, Invent. Math., 201 (2015), 993-1071.  doi: 10.1007/s00222-014-0563-7.

[21]

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

[22]

J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), 1-61.  doi: 10.1007/BF02392747.

[23]

J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, vol. 27 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2015, An approach based on upper gradients. doi: 10.1017/CBO9781316135914.

[24]

E. Hellinger, Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen, J. Reine Angew. Math., 136 (1909), 210-271.  doi: 10.1515/crll.1909.136.210.

[25]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[26]

S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2), 49 (1948), 214-224.  doi: 10.2307/1969123.

[27]

S. Kondratyev, L. Monsaingeon and D. Vorotnikov, A new optimal transport distance on the space of finite Radon measures, Adv. Differential Equations, 21 (2016), 1117–1164, URL http://projecteuclid.org/euclid.ade/1476369298.

[28]

P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math., 131 (1998), 1-17. 

[29]

S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics, 22 (1951), 79-86.  doi: 10.1214/aoms/1177729694.

[30]

K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal., 258 (2010), 3758-3774.  doi: 10.1016/j.jfa.2010.01.010.

[31]

M. LieroA. Mielke and G. Savaré, Optimal transport in competition with reaction: The Hellinger-Kantorovich distance and geodesic curves, SIAM J. Math. Anal., 48 (2016), 2869-2911.  doi: 10.1137/15M1041420.

[32]

M. LieroA. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Invent. Math., 211 (2018), 969-1117.  doi: 10.1007/s00222-017-0759-8.

[33]

F. Liese and I. Vajda, On divergences and informations in statistics and information theory, IEEE Trans. Inform. Theory, 52 (2006), 4394-4412.  doi: 10.1109/TIT.2006.881731.

[34]

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.

[35]

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.

[36]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[37]

F. Otto and M. Westdickenberg, Eulerian calculus for the contraction in the Wasserstein distance,, SIAM J. Math. Anal., 37 (2005), 1227–1255 (electronic). doi: 10.1137/050622420.

[38]

G. Savaré, Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD$ (K,\infty) $ metric measure spaces, Discrete Contin. Dyn. Syst., 34 (2014), 1641-1661.  doi: 10.3934/dcds.2014.34.1641.

[39]

N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana, 16 (2000), 243-279.  doi: 10.4171/RMI/275.

[40]

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

[41]

K.-T. Sturm and M.-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

[42]

C. Villani, Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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