Quantitative logarithmic Sobolev inequalities and stability estimates

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e 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 ${\rm L}^1$-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-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.

[1]  N. Gozlan,et al.  A variational approach to some transport inequalities , 2015, 1508.07642.

[2]  E. Indrei.  A sharp lower bound on the polygonal isoperimetric deficit , 2015, 1502.05978.

[3]  Ronen Eldan.  A two-sided estimate for the Gaussian noise stability deficit , 2013, 1307.2781.

[4]  A. Figalli,et al.  Quantitative stability for the Brunn-Minkowski inequality , 2014, 1502.06513.

[5]  A. Brancolini,et al.  Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality , 2014, 1409.2106.

[6]  Cyril Roberto,et al.  Bounds on the deficit in the logarithmic Sobolev inequality , 2014, 1408.2115.

[7]  M. Christ.  A sharpened Hausdorff-Young inequality , 2014, 1406.1210.

[8]  L. Nurbekyan,et al.  On the stability of the polygonal isoperimetric inequality , 2014, 1402.4460.

[9]  M. Ledoux,et al.  Logarithmic Sobolev Inequalities , 2014 .

[10]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[11]  A. Figalli,et al.  Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation , 2013 .

[12]  Connor Mooney.  Partial Regularity for Singular Solutions to the Monge‐Ampère Equation , 2013, 1304.2706.

[13]  Diego Marcon,et al.  A quantitative log-Sobolev inequality for a two parameter family of functions , 2013, 1302.4910.

[14]  A. Figalli,et al.  W2,1 regularity for solutions of the Monge–Ampère equation , 2013 .

[15]  A. Figalli,et al.  A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones , 2012, 1210.3113.

[16]  J. Lehec,et al.  Representation formula for the entropy and functional inequalities , 2010, 1006.3028.

[17]  A. Figalli,et al.  A mass transportation approach to quantitative isoperimetric inequalities , 2010 .

[18]  A. Figalli,et al.  A refined Brunn-Minkowski inequality for convex sets , 2009 .

[19]  Aldo Pratelli,et al.  The sharp Sobolev inequality in quantitative form , 2009 .

[20]  C. Villani.  Optimal Transport: Old and New , 2008 .

[21]  N. Fusco,et al.  The sharp quantitative isoperimetric inequality , 2008 .

[22]  D. Bakry,et al.  A simple proof of the Poincaré inequality for a large class of probability measures , 2008 .

[23]  F. Barthe,et al.  Mass Transport and Variants of the Logarithmic Sobolev Inequality , 2007, 0709.3890.

[24]  C. Villani.  Topics in Optimal Transportation , 2003 .

[25]  C. Villani,et al.  Optimal Transportation and Applications , 2003 .

[26]  Dario Cordero-Erausquin,et al.  Some Applications of Mass Transport to Gaussian-Type Inequalities , 2002 .

[27]  Michel Ledoux,et al.  Logarithmic Sobolev Inequalities for Unbounded Spin Systems Revisited , 2001 .

[28]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[29]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[30]  M. Talagrand.  Transportation cost for Gaussian and other product measures , 1996 .

[31]  R. McCann.  Existence and uniqueness of monotone measure-preserving maps , 1995 .

[32]  D. Bakry.  L'hypercontractivité et son utilisation en théorie des semigroupes , 1994 .

[33]  L. Caffarelli.  The regularity of mappings with a convex potential , 1992 .

[34]  E. Carlen.  Superadditivity of Fisher's information and logarithmic Sobolev inequalities , 1991 .

[35]  G. Bianchi,et al.  A note on the Sobolev inequality , 1991 .

[36]  E. Carlen.  Some integral identities and inequalities for entire functions and their application to the coherent state transform , 1991 .

[37]  Luis A. Caffarelli,et al.  A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity , 1990 .

[38]  E. Lieb.  Thomas-fermi and related theories of atoms and molecules , 1981 .

[39]  A. Wehrl.  On the relation between classical and quantum-mechanical entropy , 1979 .

[40]  E. Lieb.  Proof of an entropy conjecture of Wehrl , 1978 .

[41]  I. Segal.  Construction of Non-Linear Local Quantum Processes: I , 1970 .

[42]  I. Segal.  Mathematical characterization of the physical vacuum for a linear Bose-Einstein field , 1962 .

[43]  A. Figalli.  GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY : THE CASE p ≥ 2 , 2022 .