Logarithmic Sobolev Inequalities Essentials: Probabilistic Side

213600

Université PSL (Paris Sciences & Lettres)

MasterMathématiques théoriques et appliquées

Logarithmic Sobolev Inequalities Essentials

Rough lecture notes

Djalil Chafaï & Joseph Lehec

Winter 2017, Université Paris-Dauphine, Paris

Revised Spring 2017, Universidad de Chile, Santiago de Chile

Revised Winter 2023, École normale supérieure, ParisThis course is a modern overview on logarithmic Sobolev inequalities, from the probabilistic

side. These inequalities have been the subject of intense activity in the recent decades in relation with the analysis and geometry of Markov processes and diffusion evolution equations. This course is designed to be accessible to a wide audience with a first year of master level in mathematics. It is divided into seven lectures of roughly three hours.

Short bibliography:

•Bakry, Gentil, and Ledoux -Analysis and Geometry of Markov Diffusion Operators •Villani -Optimal transport, old and new •Royer -An Initiation to Logarithmic Sobolev inequalities •Ané et al -Sur les inégalités de Sobolev logarithmiques 2

Contents

1 Introduction5

1.1 Generalities on Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Poincaré and logarithmic Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . 11

1.4 Convergence to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Amnesia and long time behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Tensorization and Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . 17

2 Hypercontractivity, spectral gap, information theory 21

2.1 Hypercontractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Spectral gap and Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Sub-Gaussian concentration and transportation 35

3.1 Concentration of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Transportation inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Isoperimetric inequalities 45

4.1 Proof of the second Bobkov inequality . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Bakry-Émery criterion 53

5.1 Gamma calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Poincaré inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Langevin semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.5 Log-Sobolev inequality for a diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.6 Kannan-Lovász-Simonovits conjecture . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Brenier and Caffarelli theorems 65

6.1 Brenier theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Monge-Ampère equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Caffarelli contraction theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.4 From Monge-Ampère to Gaussian log-Sobolev . . . . . . . . . . . . . . . . . . . . 69

7 Discrete space71

7.1 Bernoulli distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2 Poisson distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2.1 Poisson process andAmodified inequality . . . . . . . . . . . . . . . . . . . 76

7.2.2 M/M/∞queue andBmodified inequality . . . . . . . . . . . . . . . . . . . 76

7.2.3 Concentration andCmodified inequality . . . . . . . . . . . . . . . . . . . 78

3

4CONTENTS

Chapter 1

IntroductionThe first chapters focus on a Gaussian model formed with the product spaceRnequipped with the Gaussian probability measureγn=γ?n1. It allows explicit computations. It appears asympotically in other models due to the central limit phenomenon1, in particular from spheres and from cubes, both equipped with the uniform measure. In a wayγnplays the role of a uniform measure onRn. The standard Gaussian measureγnonRnhas expectation0, covariance the identity matrixIn, and density with respect to the Lebesgue measure given by x?→(2π)-n/2e-|x|2/2 where|x|=??

1/2denotes the Euclidean norm.

LetUnbe the uniform distribution on the unit sphere{s?Rn:|s|= 1}. Theorem 1.1(Polar factorization of spheres).IfX= (X1,...,Xn)≂γnthen |X|=?X

21+···+X2nandX|X|

are independent, with|X|2≂χ2(n)andX/|X| ≂ Un. Conversely ifRandUare indepen- dent withR2≂χ2(n)andU≂ Un, thenRU≂γn. Proof.Follows frome-|x|2/2dx= e-r2/2rn-1drduwherex=ru. The following result, known as the Borel or Poincaré observation, states that one can see the standard Gaussian as a the projection of the uniform distribution on high dimensional spheres with radius equal to the square root of the dimension. Theorem 1.2(Central Limit Theorem for spheres).IfUn≂ Unfor anyn≥1, then for any fixedk≥1,proj(⎷nU n,Rk)→γkin law asn→ ∞.

Proof.

Let(Xn)n≥1be independent and identically distributed random variables with law γ1. By the preceding theorem and the strong law of large numbers, proj( ⎷nU n,Rk)d=⎷n (X1,...,Xk)? X

21+···+X2na.s.

-→n→∞(X1,...,Xk)≂γk.1

IfX1,X2,...are i.i.d. real random variables with zero mean and unit variance thenX1+···+Xn⎷nconverges

in law asn→ ∞to the standard Gaussian distribution. 5

6CHAPTER 1. INTRODUCTION

Definition 1.3(Variance and entropy).Iffis a square integrable function with respect toγn, then itsvarianceis Var

γn(f) =?

R nf2dγn-? R nfdγn? 2 = Var(f(X))whereX≂γn. Iffis a non negative function, integrable with respect toγn, then itsentropyis Ent

γn(f) =?

R nflogfdγn-? R nfdγn? log? R nfdγn? .The functionflogfmay not be integrable, but since the functionxlogxis bounded from below, the integral offlogfalways makes sense inR? {+∞}. Remark(φentropies).One can define a more general object: Given a convex functionφ on an intervalI?Rtheφ-entropy off:R→Iis E n(f) =? R nφ(f)dγn-φ? R nfdγn? We recover the variance forφ(x) =x2andI=Rand the entropy forφ(x) =xlog(x)and

Université PSL (Paris Sciences & Lettres)

MasterMathématiques théoriques et appliquées

Logarithmic Sobolev Inequalities Essentials

Rough lecture notes

Djalil Chafaï & Joseph Lehec

Winter 2017, Université Paris-Dauphine, Paris

Revised Spring 2017, Universidad de Chile, Santiago de Chile

Revised Winter 2023, École normale supérieure, ParisThis course is a modern overview on logarithmic Sobolev inequalities, from the probabilistic

side. These inequalities have been the subject of intense activity in the recent decades in relation with the analysis and geometry of Markov processes and diffusion evolution equations. This course is designed to be accessible to a wide audience with a first year of master level in mathematics. It is divided into seven lectures of roughly three hours.

Short bibliography:

•Bakry, Gentil, and Ledoux -Analysis and Geometry of Markov Diffusion Operators •Villani -Optimal transport, old and new •Royer -An Initiation to Logarithmic Sobolev inequalities •Ané et al -Sur les inégalités de Sobolev logarithmiques 2

Contents

1 Introduction5

1.1 Generalities on Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Poincaré and logarithmic Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . 11

1.4 Convergence to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Amnesia and long time behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Tensorization and Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . 17

2 Hypercontractivity, spectral gap, information theory 21

2.1 Hypercontractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Spectral gap and Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Sub-Gaussian concentration and transportation 35

3.1 Concentration of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Transportation inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Isoperimetric inequalities 45

4.1 Proof of the second Bobkov inequality . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Bakry-Émery criterion 53

5.1 Gamma calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Poincaré inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Langevin semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.5 Log-Sobolev inequality for a diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.6 Kannan-Lovász-Simonovits conjecture . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Brenier and Caffarelli theorems 65

6.1 Brenier theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Monge-Ampère equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Caffarelli contraction theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.4 From Monge-Ampère to Gaussian log-Sobolev . . . . . . . . . . . . . . . . . . . . 69

7 Discrete space71

7.1 Bernoulli distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2 Poisson distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2.1 Poisson process andAmodified inequality . . . . . . . . . . . . . . . . . . . 76

7.2.2 M/M/∞queue andBmodified inequality . . . . . . . . . . . . . . . . . . . 76

7.2.3 Concentration andCmodified inequality . . . . . . . . . . . . . . . . . . . 78

3

4CONTENTS

Chapter 1

IntroductionThe first chapters focus on a Gaussian model formed with the product spaceRnequipped with the Gaussian probability measureγn=γ?n1. It allows explicit computations. It appears asympotically in other models due to the central limit phenomenon1, in particular from spheres and from cubes, both equipped with the uniform measure. In a wayγnplays the role of a uniform measure onRn. The standard Gaussian measureγnonRnhas expectation0, covariance the identity matrixIn, and density with respect to the Lebesgue measure given by x?→(2π)-n/2e-|x|2/2 where|x|=??

1/2denotes the Euclidean norm.

LetUnbe the uniform distribution on the unit sphere{s?Rn:|s|= 1}. Theorem 1.1(Polar factorization of spheres).IfX= (X1,...,Xn)≂γnthen |X|=?X

21+···+X2nandX|X|

are independent, with|X|2≂χ2(n)andX/|X| ≂ Un. Conversely ifRandUare indepen- dent withR2≂χ2(n)andU≂ Un, thenRU≂γn. Proof.Follows frome-|x|2/2dx= e-r2/2rn-1drduwherex=ru. The following result, known as the Borel or Poincaré observation, states that one can see the standard Gaussian as a the projection of the uniform distribution on high dimensional spheres with radius equal to the square root of the dimension. Theorem 1.2(Central Limit Theorem for spheres).IfUn≂ Unfor anyn≥1, then for any fixedk≥1,proj(⎷nU n,Rk)→γkin law asn→ ∞.

Proof.

Let(Xn)n≥1be independent and identically distributed random variables with law γ1. By the preceding theorem and the strong law of large numbers, proj( ⎷nU n,Rk)d=⎷n (X1,...,Xk)? X

21+···+X2na.s.

-→n→∞(X1,...,Xk)≂γk.1

IfX1,X2,...are i.i.d. real random variables with zero mean and unit variance thenX1+···+Xn⎷nconverges

in law asn→ ∞to the standard Gaussian distribution. 5

6CHAPTER 1. INTRODUCTION

Definition 1.3(Variance and entropy).Iffis a square integrable function with respect toγn, then itsvarianceis Var

γn(f) =?

R nf2dγn-? R nfdγn? 2 = Var(f(X))whereX≂γn. Iffis a non negative function, integrable with respect toγn, then itsentropyis Ent

γn(f) =?

R nflogfdγn-? R nfdγn? log? R nfdγn? .The functionflogfmay not be integrable, but since the functionxlogxis bounded from below, the integral offlogfalways makes sense inR? {+∞}. Remark(φentropies).One can define a more general object: Given a convex functionφ on an intervalI?Rtheφ-entropy off:R→Iis E n(f) =? R nφ(f)dγn-φ? R nfdγn? We recover the variance forφ(x) =x2andI=Rand the entropy forφ(x) =xlog(x)and