December  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.  Google Scholar

[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.  Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives,, Séminaire de Probabilités XIX, 1123 (1985), 177.  doi: 10.1007/BFb0075847.  Google Scholar

[4]

M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality,, to appear in Ann. Probab., (2015).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities,, Arch. Rational Mech. Anal., 161 (2002), 257.  doi: 10.1007/s002050100185.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Fontbona, N. Gozlan and J.-F. Jabir, A variational approach to some transport inequalities,, preprint (2015)., (2015).   Google Scholar

[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.  Google Scholar

[26]

A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$,, preprint, (2015).   Google Scholar

[27]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.  doi: 10.2307/2373688.  Google Scholar

[28]

E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit,, Proc. Amer. Math. Soc., 144 (2016), 3115.  doi: 10.1090/proc/12947.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. Lehec, Representation formula for the entropy and functional inequalities,, Ann. IHP: Probab. Stat., 49 (2013), 885.  doi: 10.1214/11-AIHP464.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

[43]

C. Villani, Optimal transport. Old and new,, Grundlehren der mathematischen Wissenschaften, (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives,, Séminaire de Probabilités XIX, 1123 (1985), 177.  doi: 10.1007/BFb0075847.  Google Scholar

[4]

M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality,, to appear in Ann. Probab., (2015).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities,, Arch. Rational Mech. Anal., 161 (2002), 257.  doi: 10.1007/s002050100185.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Fontbona, N. Gozlan and J.-F. Jabir, A variational approach to some transport inequalities,, preprint (2015)., (2015).   Google Scholar

[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.  Google Scholar

[26]

A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$,, preprint, (2015).   Google Scholar

[27]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.  doi: 10.2307/2373688.  Google Scholar

[28]

E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit,, Proc. Amer. Math. Soc., 144 (2016), 3115.  doi: 10.1090/proc/12947.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. Lehec, Representation formula for the entropy and functional inequalities,, Ann. IHP: Probab. Stat., 49 (2013), 885.  doi: 10.1214/11-AIHP464.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

[43]

C. Villani, Optimal transport. Old and new,, Grundlehren der mathematischen Wissenschaften, (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[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.  Google Scholar

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