Quantitative logarithmic Sobolev inequalities and stability estimates

Previous Article
Global stability of a price model with multiple delays
DCDS Home
This Issue
Next Article
Gradient flow structure for McKean-Vlasov equations on discrete spaces

December 2016, 36(12): 6835-6853. doi: 10.3934/dcds.2016097

Quantitative logarithmic Sobolev inequalities and stability estimates

Max Fathi ^1, , Emanuel Indrei ^2, and Michel Ledoux ^3,

1.	Université Pierre et Marie Curie, Paris, France
2.	Carnegie Mellon University, Pittsburgh, United States
3.	University of Toulouse and Institut Universitaire de France, Toulouse, France

Received December 2015 Revised June 2016 Published October 2016

Abstract
Related Papers

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${ L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.

Keywords: deficit estimates, optimal transport theory, semigroup theory., transportation inequalities, Logarithmic Sobolev inequalities.

Mathematics Subject Classification: Primary: 58J35, 60J6.

Citation: Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6835-6853. doi: 10.3934/dcds.2016097

References:

[1]	D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, École d'Été de Probabilités de Saint-Flour, Lecture Notes in Math., Springer, 1581 (1994), 1-114. doi: 10.1007/BFb0073872.
[2]	D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality in a large class of probability measures including log-concave cases, Elec. Comm. Prob., 13 (2008), 60-66. doi: 10.1214/ECP.v13-1352.
[3]	D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de Probabilités XIX, Lecture Notes in Math., Springe, 1123 (1985), 177-206. doi: 10.1007/BFb0075847.
[4]	M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, to appear in Ann. Probab., 2015.
[5]	G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.
[6]	D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der mathematischen Wissenschaften, 348, Springer, Berlin, 2014. doi: 10.1007/978-3-319-00227-9.
[7]	F. Barthe and A. V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal., 18 (2008), 921-979. doi: 10.1007/s12220-008-9039-6.
[8]	S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28. doi: 10.1006/jfan.1998.3326.
[9]	S. Bobkov, N. Gozlan, C. Roberto and P.-M. Samson, Bounds on the deficit in the logarithmic Sobolev inequality, J. Funct. Anal., 267 (2014), 4110-4138. doi: 10.1016/j.jfa.2014.09.016.
[10]	L. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134. doi: 10.2307/1971509.
[11]	L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. doi: 10.1090/S0894-0347-1992-1124980-8.
[12]	E. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., 97 (1991), 231-249. doi: 10.1016/0022-1236(91)90022-W.
[13]	E. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211. doi: 10.1016/0022-1236(91)90155-X.
[14]	M. Christ, A sharpened Hausdorff-Young inequality, arXiv:1406.1210.
[15]	A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105-1139. doi: 10.4171/JEMS/176.
[16]	D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., 161 (2002), 257-269. doi: 10.1007/s002050100185.
[17]	G. De Philipis and A. Figalli, $W^{2,1}$ regularity of solutions to the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69. doi: 10.1007/s00222-012-0405-4.
[18]	R. Eldan, A two-sided estimate for the Gaussian noise stability deficit, Invent. Math., 201 (2015), 561-624. doi: 10.1007/s00222-014-0556-6.
[19]	A. Figalli and E. Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones, J. Geom. Anal., 23 (2013), 938-969. doi: 10.1007/s12220-011-9270-4.
[20]	A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality, J. Eur. Math. Soc., to appear.
[21]	A. Figalli, F. Maggi and A. Pratelli, A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2511-2519. doi: 10.1016/j.anihpc.2009.07.004.
[22]	A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211. doi: 10.1007/s00222-010-0261-z.
[23]	A. Figalli, F. Maggi and A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, Adv. Math., 242 (2013), 80-101. doi: 10.1016/j.aim.2013.04.007.
[24]	J. Fontbona, N. Gozlan and J.-F. Jabir, A variational approach to some transport inequalities, preprint (2015).
[25]	N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math., 168 (2008), 941-980. doi: 10.4007/annals.2008.168.941.
[26]	A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$, preprint, 2015.
[27]	L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.
[28]	E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit, Proc. Amer. Math. Soc., 144 (2016), 3115-3122. doi: 10.1090/proc/12947.
[29]	E. Indrei and D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not. IMRN, (2014), 5563-5580.
[30]	E. Indrei and L. Nurbekyan, On the stability of the polygonal isoperimetric inequality, Advances in Mathematics, 276 (2015), 62-86. doi: 10.1016/j.aim.2015.02.013.
[31]	M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, Séminaire de Probabilités XXXV, Lecture Notes in Math., Springer, 1755 (2001), 167-194. doi: 10.1007/978-3-540-44671-2_13.
[32]	J. Lehec, Representation formula for the entropy and functional inequalities, Ann. IHP: Probab. Stat., 49 (2013), 885-899. doi: 10.1214/11-AIHP464.
[33]	E. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys., 62 (1978), 35-41. doi: 10.1007/BF01940328.
[34]	E. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53 (1981), 603-641. doi: 10.1103/RevModPhys.53.603.
[35]	R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.
[36]	C. Mooney, Partial regularity for singular solutions to the Monge-Ampere equation, Comm. Pure Appl. Math., 68 (2015), 1066-1084. doi: 10.1002/cpa.21534.
[37]	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.
[38]	I. Segal, Mathematical characterization of the physical vacuum of the physical vacuum for a linear Bose-Einstein field, Illinois J. Math., 6 (1962), 500-523.
[39]	I. Segal, Mathematical Problems in Relativistic Quantum Mechanics, American Mathematical Society, Providence, 1963.
[40]	I. Segal, Construction of non-linear local quantum processes I, Ann. of Math., 92 (1970), 462-481. doi: 10.2307/1970628.
[41]	M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600. doi: 10.1007/BF02249265.
[42]	C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematic,s 58, American Mathematical Society, Providence, 2003. doi: 10.1007/b12016.
[43]	C. Villani, Optimal transport. Old and new, Grundlehren der mathematischen Wissenschaften, 338, Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
[44]	A. Wehrl, On the relation between classical and quantum mechanical entropy, Rep. Mat. Phys., 16 (1979), 353-358. doi: 10.1016/0034-4877(79)90070-3.

show all references

References:

[1]	D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, École d'Été de Probabilités de Saint-Flour, Lecture Notes in Math., Springer, 1581 (1994), 1-114. doi: 10.1007/BFb0073872.
[2]	D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality in a large class of probability measures including log-concave cases, Elec. Comm. Prob., 13 (2008), 60-66. doi: 10.1214/ECP.v13-1352.
[3]	D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de Probabilités XIX, Lecture Notes in Math., Springe, 1123 (1985), 177-206. doi: 10.1007/BFb0075847.
[4]	M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, to appear in Ann. Probab., 2015.
[5]	G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.
[6]	D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der mathematischen Wissenschaften, 348, Springer, Berlin, 2014. doi: 10.1007/978-3-319-00227-9.
[7]	F. Barthe and A. V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal., 18 (2008), 921-979. doi: 10.1007/s12220-008-9039-6.
[8]	S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28. doi: 10.1006/jfan.1998.3326.
[9]	S. Bobkov, N. Gozlan, C. Roberto and P.-M. Samson, Bounds on the deficit in the logarithmic Sobolev inequality, J. Funct. Anal., 267 (2014), 4110-4138. doi: 10.1016/j.jfa.2014.09.016.
[10]	L. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134. doi: 10.2307/1971509.
[11]	L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. doi: 10.1090/S0894-0347-1992-1124980-8.
[12]	E. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., 97 (1991), 231-249. doi: 10.1016/0022-1236(91)90022-W.
[13]	E. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211. doi: 10.1016/0022-1236(91)90155-X.
[14]	M. Christ, A sharpened Hausdorff-Young inequality, arXiv:1406.1210.
[15]	A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105-1139. doi: 10.4171/JEMS/176.
[16]	D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., 161 (2002), 257-269. doi: 10.1007/s002050100185.
[17]	G. De Philipis and A. Figalli, $W^{2,1}$ regularity of solutions to the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69. doi: 10.1007/s00222-012-0405-4.
[18]	R. Eldan, A two-sided estimate for the Gaussian noise stability deficit, Invent. Math., 201 (2015), 561-624. doi: 10.1007/s00222-014-0556-6.
[19]	A. Figalli and E. Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones, J. Geom. Anal., 23 (2013), 938-969. doi: 10.1007/s12220-011-9270-4.
[20]	A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality, J. Eur. Math. Soc., to appear.
[21]	A. Figalli, F. Maggi and A. Pratelli, A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2511-2519. doi: 10.1016/j.anihpc.2009.07.004.
[22]	A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211. doi: 10.1007/s00222-010-0261-z.
[23]	A. Figalli, F. Maggi and A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, Adv. Math., 242 (2013), 80-101. doi: 10.1016/j.aim.2013.04.007.
[24]	J. Fontbona, N. Gozlan and J.-F. Jabir, A variational approach to some transport inequalities, preprint (2015).
[25]	N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math., 168 (2008), 941-980. doi: 10.4007/annals.2008.168.941.
[26]	A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$, preprint, 2015.
[27]	L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.
[28]	E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit, Proc. Amer. Math. Soc., 144 (2016), 3115-3122. doi: 10.1090/proc/12947.
[29]	E. Indrei and D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not. IMRN, (2014), 5563-5580.
[30]	E. Indrei and L. Nurbekyan, On the stability of the polygonal isoperimetric inequality, Advances in Mathematics, 276 (2015), 62-86. doi: 10.1016/j.aim.2015.02.013.
[31]	M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, Séminaire de Probabilités XXXV, Lecture Notes in Math., Springer, 1755 (2001), 167-194. doi: 10.1007/978-3-540-44671-2_13.
[32]	J. Lehec, Representation formula for the entropy and functional inequalities, Ann. IHP: Probab. Stat., 49 (2013), 885-899. doi: 10.1214/11-AIHP464.
[33]	E. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys., 62 (1978), 35-41. doi: 10.1007/BF01940328.
[34]	E. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53 (1981), 603-641. doi: 10.1103/RevModPhys.53.603.
[35]	R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.
[36]	C. Mooney, Partial regularity for singular solutions to the Monge-Ampere equation, Comm. Pure Appl. Math., 68 (2015), 1066-1084. doi: 10.1002/cpa.21534.
[37]	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.
[38]	I. Segal, Mathematical characterization of the physical vacuum of the physical vacuum for a linear Bose-Einstein field, Illinois J. Math., 6 (1962), 500-523.
[39]	I. Segal, Mathematical Problems in Relativistic Quantum Mechanics, American Mathematical Society, Providence, 1963.
[40]	I. Segal, Construction of non-linear local quantum processes I, Ann. of Math., 92 (1970), 462-481. doi: 10.2307/1970628.
[41]	M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600. doi: 10.1007/BF02249265.
[42]	C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematic,s 58, American Mathematical Society, Providence, 2003. doi: 10.1007/b12016.
[43]	C. Villani, Optimal transport. Old and new, Grundlehren der mathematischen Wissenschaften, 338, Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
[44]	A. Wehrl, On the relation between classical and quantum mechanical entropy, Rep. Mat. Phys., 16 (1979), 353-358. doi: 10.1016/0034-4877(79)90070-3.

[1]	Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
[2]	Takayoshi Ogawa, Kento Seraku. Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079
[3]	Daesung Kim. Instability results for the logarithmic Sobolev inequality and its application to related inequalities. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4297-4320. doi: 10.3934/dcds.2022053
[4]	Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237
[5]	Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems and Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003
[6]	Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[7]	Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[8]	Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure and Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533
[9]	Xiaoli Chen, Jianfu Yang. Improved Sobolev inequalities and critical problems. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3673-3695. doi: 10.3934/cpaa.2020162
[10]	G. Idone, A. Maugeri. Variational inequalities and a transport planning for an elastic and continuum model. Journal of Industrial and Management Optimization, 2005, 1 (1) : 81-86. doi: 10.3934/jimo.2005.1.81
[11]	Marita Thomas. Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2741-2761. doi: 10.3934/dcds.2015.35.2741
[12]	T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105
[13]	Maria J. Esteban. Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2101-2114. doi: 10.3934/cpaa.2022051
[14]	Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201
[15]	Friedemann Brock, Francesco Chiacchio, Giuseppina di Blasio. Optimal Szegö-Weinberger type inequalities. Communications on Pure and Applied Analysis, 2016, 15 (2) : 367-383. doi: 10.3934/cpaa.2016.15.367
[16]	José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure and Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271
[17]	Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[18]	David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311
[19]	Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure and Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[20]	Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Max Fathi Emanuel Indrei Michel Ledoux
on Google Scholar
Max Fathi Emanuel Indrei Michel Ledoux

[Back to Top]