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

Quantitative logarithmic Sobolev inequalities and stability estimates

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

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.
Citation: Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete & Continuous Dynamical Systems - A, 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, 1581 (1994), 1. 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. doi: 10.1214/ECP.v13-1352.

[3]

D. Bakry and M. Émery, Diffusions hypercontractives,, Séminaire de Probabilités XIX, 1123 (1985), 177. 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. 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, (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. 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. 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. 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. doi: 10.2307/1971509.

[11]

L. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. 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. 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. doi: 10.1016/0022-1236(91)90155-X.

[14]

M. Christ, A sharpened Hausdorff-Young inequality,, , ().

[15]

A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form,, J. Eur. Math. Soc., 11 (2009), 1105. 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. 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. 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. 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. doi: 10.1007/s12220-011-9270-4.

[20]

A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality,, J. Eur. Math. Soc., ().

[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. 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. 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. 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)., (2015).

[25]

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality,, Ann. of Math., 168 (2008), 941. 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. doi: 10.2307/2373688.

[28]

E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit,, Proc. Amer. Math. Soc., 144 (2016), 3115. 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.

[30]

E. Indrei and L. Nurbekyan, On the stability of the polygonal isoperimetric inequality,, Advances in Mathematics, 276 (2015), 62. doi: 10.1016/j.aim.2015.02.013.

[31]

M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited,, Séminaire de Probabilités XXXV, 1755 (2001), 167. 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. doi: 10.1214/11-AIHP464.

[33]

E. Lieb, Proof of an entropy conjecture of Wehrl,, Comm. Math. Phys., 62 (1978), 35. doi: 10.1007/BF01940328.

[34]

E. Lieb, Thomas-Fermi and related theories of atoms and molecules,, Rev. Mod. Phys., 53 (1981), 603. doi: 10.1103/RevModPhys.53.603.

[35]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps,, Duke Math. J., 80 (1995), 309. 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. 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. 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.

[39]

I. Segal, Mathematical Problems in Relativistic Quantum Mechanics,, American Mathematical Society, (1963).

[40]

I. Segal, Construction of non-linear local quantum processes I,, Ann. of Math., 92 (1970), 462. doi: 10.2307/1970628.

[41]

M. Talagrand, Transportation cost for Gaussian and other product measures,, Geom. Funct. Anal., 6 (1996), 587. doi: 10.1007/BF02249265.

[42]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematic, (2003). doi: 10.1007/b12016.

[43]

C. Villani, Optimal transport. Old and new,, Grundlehren der mathematischen Wissenschaften, (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. 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, 1581 (1994), 1. 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. doi: 10.1214/ECP.v13-1352.

[3]

D. Bakry and M. Émery, Diffusions hypercontractives,, Séminaire de Probabilités XIX, 1123 (1985), 177. 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. 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, (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. 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. 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. 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. doi: 10.2307/1971509.

[11]

L. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. 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. 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. doi: 10.1016/0022-1236(91)90155-X.

[14]

M. Christ, A sharpened Hausdorff-Young inequality,, , ().

[15]

A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form,, J. Eur. Math. Soc., 11 (2009), 1105. 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. 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. 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. 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. doi: 10.1007/s12220-011-9270-4.

[20]

A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality,, J. Eur. Math. Soc., ().

[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. 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. 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. 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)., (2015).

[25]

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality,, Ann. of Math., 168 (2008), 941. 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. doi: 10.2307/2373688.

[28]

E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit,, Proc. Amer. Math. Soc., 144 (2016), 3115. 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.

[30]

E. Indrei and L. Nurbekyan, On the stability of the polygonal isoperimetric inequality,, Advances in Mathematics, 276 (2015), 62. doi: 10.1016/j.aim.2015.02.013.

[31]

M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited,, Séminaire de Probabilités XXXV, 1755 (2001), 167. 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. doi: 10.1214/11-AIHP464.

[33]

E. Lieb, Proof of an entropy conjecture of Wehrl,, Comm. Math. Phys., 62 (1978), 35. doi: 10.1007/BF01940328.

[34]

E. Lieb, Thomas-Fermi and related theories of atoms and molecules,, Rev. Mod. Phys., 53 (1981), 603. doi: 10.1103/RevModPhys.53.603.

[35]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps,, Duke Math. J., 80 (1995), 309. 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. 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. 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.

[39]

I. Segal, Mathematical Problems in Relativistic Quantum Mechanics,, American Mathematical Society, (1963).

[40]

I. Segal, Construction of non-linear local quantum processes I,, Ann. of Math., 92 (1970), 462. doi: 10.2307/1970628.

[41]

M. Talagrand, Transportation cost for Gaussian and other product measures,, Geom. Funct. Anal., 6 (1996), 587. doi: 10.1007/BF02249265.

[42]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematic, (2003). doi: 10.1007/b12016.

[43]

C. Villani, Optimal transport. Old and new,, Grundlehren der mathematischen Wissenschaften, (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. doi: 10.1016/0034-4877(79)90070-3.

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