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Proof of the Otto-Villani TheoremProof of the HWI inequality

Characterizations of T2 and PI in terms of dimension free concentration

Some applications of optimal transport infunctional inequalities

Nathaël Gozlan∗

∗LAMA Université Paris Est – Marne-la-Vallée

Some applications of optimal transport in functional inequalitiesToulouse, April 2014

Nathaël Gozlan Applications of optimal transport in functional inequalities

Functional approaches to dimension free concentration

Recall the following result from Lecture I:

Theorem.

Let µ ∈ P(Rd).1 [Gromov-Milman] If µ satisfies PI(C), then µ satisfies the dimension free

concentration property with the exponential profile

α(t) = b exp

(− a√

Ct

)where a, b are universal constant.

2 [Herbst/Marton/Talagrand] If µ satisfies LSI(C) or T2(C), then µsatisfies the dimension free concentration property with the Gaussianprofile

α(t) = exp

(− 1C

[t − to ]2+), with to =

√C log(2).

Functional approaches to dimension free concentration

Recall the following result from Lecture I:

Theorem.

Let µ ∈ P(Rd).1 [Gromov-Milman] If µ satisfies PI(C), then µ satisfies the dimension free

concentration property with the exponential profile

α(t) = b exp

(− a√

Ct

)where a, b are universal constant.

2 [Herbst/Marton/Talagrand] If µ satisfies LSI(C) or T2(C), then µsatisfies the dimension free concentration property with the Gaussianprofile

α(t) = exp

(− 1C

[t − to ]2+), with to =

√C log(2).

Characterization of T2 by concentration

Theorem. [G. ’09]

A probability µ satisfies the dimension free concentration property with a

Gaussian profile of the form α(t) = e−1C

[t−to ]2+ if and only if it satisfiesTalagrand’s transport entropy inequality

T2(ν, µ) ≤ C H(ν|µ), ∀ν ∈ P2(Rd).

(Proof at the end of Lecture II )

Remark.This result can be used to give an alternative proof of the implicationLSI⇒ T2, as follows

LSI(C)(Herbst)⇒ dim. free concentration with profile α(t) = e−

1C

[t−to ]2+ ⇒ T2(C)

Characterization of T2 by concentration

Theorem. [G. ’09]

A probability µ satisfies the dimension free concentration property with a

Gaussian profile of the form α(t) = e−1C

[t−to ]2+ if and only if it satisfiesTalagrand’s transport entropy inequality

T2(ν, µ) ≤ C H(ν|µ), ∀ν ∈ P2(Rd).

Remark.This result can be used to give an alternative proof of the implicationLSI⇒ T2, as follows

LSI(C)(Herbst)⇒ dim. free concentration with profile α(t) = e−

1C

[t−to ]2+ ⇒ T2(C)

Characterization of PI by concentration

Theorem. [G.-Roberto-Samson ’12]

Suppose that µ satisfies the dimension free concentration property with aconcentration profile α then µ satisfies PI(C) with the constant

C =

(inf

{t

Φ−1

(α(t)): t s.t. α(t) < 1/2

})2,

where

Φ(t) =1√2π

∫ +∞t

e−u2/2 du.

Remarks.• The constant C is optimal for the standard Gaussian measure.• The class of all probability measures enjoying some (non-trivial) dimensionfree concentration property thus coincides with the class of probabilitymeasures satisfying PI.

Characterization of PI in terms of dimension freeconcentration

Remarks.

Dimension free concentration can always be improved into an exponentialone (already observed by Talagrand ’91).

On the other hand, a too strong concentration inequality is not possible.If for instance, αp(t) = e

−a[t−to ]p+ for some p > 2, then the constantC = 0 and so µ is a Dirac.

Note however that concentration with a profile αp as above for p > 2 ispossible if the product space is equipped with the `p metric instead of `2metric. For instance µp(dx) =

1Zpe−|x|

p/p dx with p > 2 enjoys thisproperty.

For probability measures on the line, the convex - dimension freeconcentration was shown to be equivalent to Poincaré inequalityrestricted to convex functions in a paper by Bobkov and Götze ’99.

Remarks.

1Zpe−|x|

Remarks.

1Zpe−|x|

Remarks.

1Zpe−|x|

From concentration to PI and LSI for log-concave p.m.

The following result is a consequence of a general result by E. Milman.

(Particular cases of a) Theorem by E. Milman (’10)

Let µ ∈ P(Rd) be a log-concave probability measure.If µ satisfies a non-trivial concentration property : for some to > 0 andλo ∈ [0, 1/2)

µ(Ato ) ≥ 1− λo , ∀A ⊂ M, s.t. µ(A) ≥ 1/2,

then µ satisfies PI(C) with C = 4(

to1−2λo

)2.

If µ satisfies the concentration property with a Gaussian profile

α(t) = e−1a

[t−to ]2+ , then µ satisfies LSI(C) with a constant C dependingonly on a and to .

From concentration to PI and LSI for log-concave p.m.

Remarks.

Actually, E. Milman’s result provides a general link between concentrationinequalities and isoperimetric type inequalities for probability measures onRiemannian manifolds such that Ric + HessV ≥ −K , with K ≥ 0.Milman’s result for LSI generalizes previous results by Wang or Bobkov

assuming (dimensional) integrability conditions∫eεd

2(x0,x) µ(dx)

Reduction of the KLS conjecture

Recall the KLS conjecture :There exists a universal constant C > 0 such that any isotropic log-concaveprobability measure satisfies PI(C).

The following result is a direct consequence of the preceding theorem.

Theorem (E. Milman)

The K.L.S conjecture is equivalent to the following statement :

There exist to > 0 and λo ∈ [0, 1/2) such that for any positive integer d , anyd-dimensional isotropic and log-concave probability measure µ satisfies

µ(Ato ) ≥ 1− λo , ∀A ⊂ Rd s.t. µ(A) ≥ 1/2.

Remark.There is an alternative proof using the characterization of Poincaré inequality interms of dimension free concentration. Indeed, the class of isotropiclog-concave probability measures is stable under tensor products.

Outline of Lecture 2

Part II -

Optimal transport as a tool to prove functionalinequalities

II.1 Proof of the Otto-Villani Theorem.

II.2 Proof of the HWI inequality and consequences.

II.3 Proof of the characterizations of T2 and PI in terms of dimensionfree concentration.

II.1 Proof of the Otto-Villani Theorem.

Otto-Villani theorem

Rd is equipped with the usual Euclidian norm ‖ · ‖.

(LSI(C)) Entµ(f2) ≤ C

∫‖∇f ‖2 dµ, ∀f

(PI(C)) Varµ(f ) ≤ C∫‖∇f ‖2 dµ, ∀f

(T2(C)) T2(ν, µ) ≤ C H(ν|µ), ∀ν

Theorem [Otto-Villani ’00].

LSI(C)⇒ T2(C)⇒ PI(C/2)

In the next slides, we sketch the proof by Bobkov-Gentil-Ledoux (’01).

The dual form of optimal transport

The following result is one of the basic tools in optimal transportation.

Theorem. [Kantorovich duality theorem]

Let µ, ν ∈ P2(Rd) ;

T2(ν, µ) = sup(f ,g)∈Φ

{∫f dν −

∫g dµ

},

where Φ is the class of pairs of functions (f , g) ∈ Cb(Rd)2 such that

f (x)− g(y) ≤ ‖x − y‖2, ∀x , y ∈ Rd .

Remark.This result is actually true on general complete separable metric spacesequipped with a lower semi-continuous cost function (x , y) 7→ c(x , y).

The dual form of optimal transport

Remark.For a given function g , the best function f such that f (x)− g(y) ≤ ‖x − y‖2 is

Qg(x) = infy∈Rd{g(y) + ‖x − y‖2}, x ∈ Rd .

Theorem. [Kantorovich duality theorem]

Let µ, ν ∈ P2(Rd) ;

T2(ν, µ) = supϕ∈Cb(Rd )

{∫Qϕ dν −

∫ϕ dµ

}.

The dual form of relative entropy

Theorem.

The following duality formulas hold

H(ν|µ) = supϕ∈Cb(Rd )

{∫ϕ dν − log

(∫eϕ dµ

)}

log

(∫eϕ dµ

)= supν∈P(Rd )

{∫ϕ dν − H(ν|µ)

}

The dual form of relative entropy

Theorem.

The following duality formulas hold

H(ν|µ) = supϕ∈Cb(Rd )

{∫ϕ dν − log

(∫eϕ dµ

)}

log

(∫eϕ dµ

)= supν∈P(Rd )

{∫ϕ dν − H(ν|µ)

}

Proof of the second identity

Define θ(u) = u log u, u > 0 and θ∗(v) = supu>0{uv − θ(u)} = ev−1, v ∈ R.Young’s inequality: uv ≤ θ(u) + θ∗(v).

Let ϕ ∈ Cb(Rd). For all ν � µ with h = dν/dµ it holds∫ϕ dν =

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

Replacing ϕ by ϕ+ a, a ∈ R, yields to

H(ν|µ) ≥∫ϕ dν + a− ea−1

∫eϕ dµ

=

∫ϕ dν − log

∫eϕ dµ

Formally, the other inequality sup{∫ϕ dν − log

∫eϕ dµ} ≥ H(ν|µ) follows by

taking ϕ = log dνdµ.

Let ϕ ∈ Cb(Rd). For all ν � µ with h = dν/dµ it holds

∫ϕ dν =

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

H(ν|µ) ≥∫ϕ dν + a− ea−1

∫eϕ dµ

=

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ

≤∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

H(ν|µ) ≥∫ϕ dν + a− ea−1

∫eϕ dµ

=

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

H(ν|µ) ≥∫ϕ dν + a− ea−1

∫eϕ dµ

=

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

H(ν|µ) ≥∫ϕ dν + a− ea−1

∫eϕ dµ

=

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

H(ν|µ) ≥∫ϕ dν + a− ea−1

∫eϕ dµ

=

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

H(ν|µ) ≥∫ϕ dν + sup

a∈R{a− ea−1

∫eϕ dµ}

=

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

a∈R{a− ea−1

∫eϕ dµ} =

∫ϕ dν − log

∫eϕ dµ

∫ϕh dµ ≤

∫θ(h) dµ+

∫θ∗(ϕ) dµ

= H(ν|µ) +∫

eϕ−1 dµ.

a∈R{a− ea−1

∫eϕ dµ} =

∫ϕ dν − log

∫eϕ dµ

Bobkov and Götze dual form of T2

Notation. For all t > 0, Qt f (x) = infy∈Rd

{f (y) +

1

t‖x − y‖2

}.

Theorem. [Bobkov-Götze ’99]

A probability µ satisfies T2(C) if and only if∫eQC f (x) µ(dx) ≤ e

∫f (x)µ(dx), ∀f ∈ Cb(Rd).

Proof.According to Kantorovich dual theorem, T2(C) holds iff for all g ∈ Cb(Rd) andν ∈ P(Rd) ∫

Q1g(x) ν(dx)−∫

g(x)µ(dx) ≤ C H(ν|µ).

This holds iff for all f ∈ Cb(Rd),

log

(∫eQC f (x) µ(dx)

)=

supν∈P(Rd )

{∫QC f (x) ν(dx)− H(ν|µ)

}≤∫

f (x)µ(dx)

{f (y) +

1

t‖x − y‖2

}.

Proof.According to Kantorovich dual theorem, T2(C) holds iff for all f ∈ Cb(Rd) andν ∈ P(Rd) ∫

QC f (x) ν(dx)−∫

f (x)µ(dx) ≤ H(ν|µ). g = Cf

log

)=

supν∈P(Rd )

}≤∫

f (x)µ(dx)

{f (y) +

1

t‖x − y‖2

}.

log

)=

supν∈P(Rd )

}≤∫

f (x)µ(dx)

{f (y) +

1

t‖x − y‖2

}.

log

)= sup

ν∈P(Rd )

}≤∫

f (x)µ(dx)

Hopf-Lax formula for Hamilton-Jacobi equations

Theorem.

For any function f ∈ Cb(Rd), the function

(t, x) 7→ Qt f (x) = infy∈Rd

{f (y) +

1

t‖x − y‖2

}is differentiable almost everywhere in (0,∞)× Rd and satisfies at every suchpoint

∂Qt f (x)

∂t= −1

4‖∇Qt f ‖2(x).

Remark.Ambrosio, Gigli and Savare (’12) have obtained a version of this result ingeodesic metric spaces (their formulation is free of measure theory).

Proof of the Otto-Villani Theorem

We follow the proof by Bobkov-Gentil-Ledoux (’01).

According to Bobkov and Götze dual criterion, µ satisfies T2(C) (C = 1) if andonly if

log

∫eQf dµ ≤

∫f dµ, ∀f ∈ Cb(Rd).

Consider

Z(t) =1

tlog

∫etQt f dµ

When t → 0+, Zt →∫f dµ.

So it is enough to show that under LSI the function t 7→ Z(t) is non-increasing.

A simple calculation proves that

dZ

dt=

1

t2∫etQt f dµ

[Entµ

(etQt f

)+ t2

∫d

dt(Qt f )e

tQt f dµ

]

(LSI)

≤ 1t2∫etQt f dµ

[t2∫

1

4|∇Qt f |2etQt f dµ+ t2

∫d

dt(Qt f )e

tQt f dµ

]

≤ 0,

because of the Hopf-Lax formula

d

dtQt f +

1

4|∇Qt f |2 = 0.

dZ

dt=

1

t2∫etQt f dµ

[Entµ

(etQt f

)+ t2

∫d

dt(Qt f )e

tQt f dµ

]

(LSI)

[t2∫

1

∫d

dt(Qt f )e

tQt f dµ

]

≤ 0,

d

dtQt f +

1

4|∇Qt f |2 = 0.

dZ

dt=

1

t2∫etQt f dµ

[Entµ

(etQt f

)+ t2

∫d

dt(Qt f )e

tQt f dµ

]

(LSI)

[t2∫

1

∫d

dt(Qt f )e

tQt f dµ

]

≤ 0,

d

dtQt f +

1

4|∇Qt f |2 = 0.

Refinement of Otto-Villani Theorem

Definition. Semi-convex functions

A function f : Rd → R is λ - (semi) convex if it satisfies

f ((1− t)x0 + tx1) ≤ (1− t)f (x0) + tf (x1)−λ

2t(1− t)‖x0 − x1‖2,

for all x0, x1 ∈ Rd and t ∈ [0, 1].

When f is C1 smooth, this is equivalent to

f (y) ≥ f (x) +∇f (x) · (y − x) + λ2‖x − y‖2, ∀x , y ∈ Rd .

Hess f ≥ λ

f ((1− t)x0 + tx1) ≤ (1− t)f (x0) + tf (x1)−λ

2t(1− t)‖x0 − x1‖2,

for all x0, x1 ∈ Rd and t ∈ [0, 1].When f is C1 smooth, this is equivalent to

Hess f ≥ λ

f ((1− t)x0 + tx1) ≤ (1− t)f (x0) + tf (x1)−λ

2t(1− t)‖x0 − x1‖2,

for all x0, x1 ∈ Rd and t ∈ [0, 1].When f is C1 smooth, this is equivalent to

Hess f ≥ λ

Theorem (G. Roberto-Samson ’11- ’12)

Let µ be a probability on a Rd . The following are equivalent.

1 There is C > 0 such that µ satisfies T2(C).

2 There is D > 0 such that µ satisfies the following restricted log-Sobolevinequality : for all 0 < K < 1/D and all function f : Rd → R which is−K semi-convex, it holds

Entµ(ef ) ≤ D

(1− DK)2

∫|∇f |2e f dµ

Moreover Dopt ≤ Copt ≤ e2Dopt .

Entµ(ef ) ≤ D

(1− DK)2

∫|∇f |2e f dµ

Entµ(ef ) ≤ D

(1− DK)2

∫|∇f |2e f dµ

Entµ(ef ) ≤ D

(1− DK)2

∫|∇f |2e f dµ

II.2 Proof of the HWI inequality.

The HWI inequality of Otto and Villani

Theorem. [Otto-Villani ’00]

Let µ(dx) = e−V (x) dx be an absolutely continuous probability measure with asmooth density on Rd such that

HessV ≥ K ,

for some K ∈ R.Then for all ν0, ν1 ∈ P(Rd) absolutely continuous with respect to µ, it holds

H(ν0|µ) ≤ H(ν1|µ) + W2(ν0, ν1)√

I(ν0|µ)−K

2W 22 (ν0, ν1),

where I(ν0|µ) is the (relative) Fisher information of ν0 with respect to µdefined by

I(ν0|µ) =∫|∇h0|2

h0dµ, with ν0 = h0 µ

Consequences

H(ν0|µ) ≤ H(ν1|µ) + W2(ν0, ν1)√

I(ν0|µ)−K

2W 22 (ν0, ν1), ∀ν0, ν1

Consequence 1 : Talagrand’s inequality when K > 0.Take ν0 = µ, it remains

0 ≤ H(ν1|µ)−K

2W 22 (ν1, µ)

and so µ satisfies T2(2/K)

Consequence 2 : Log-Sobolev inequality when K > 0.Take ν1 = µ and use the inequality ab ≤ K2 a

2 + 12K

b2, with a = W2(ν0, ν1) and

b =√

I(ν0|µ).It remains

H(ν0|µ) ≤1

2KI(ν0|µ)

and so µ satisfies LSI(2/K).

Consequences

H(ν0|µ) ≤ H(ν1|µ) + W2(ν0, ν1)√

I(ν0|µ)−K

2W 22 (ν0, ν1), ∀ν0, ν1

0 ≤ H(ν1|µ)−K

2W 22 (ν1, µ)

2 + 12K

b =√

H(ν0|µ) ≤1

2KI(ν0|µ)

Consequences

H(ν0|µ) ≤ H(ν1|µ) + W2(ν0, ν1)√

I(ν0|µ)−K

2W 22 (ν0, ν1), ∀ν0, ν1

0 ≤ H(ν1|µ)−K

2W 22 (ν1, µ)

2 + 12K

b =√

H(ν0|µ) ≤1

2KI(ν0|µ)

Consequences

H(ν0|µ) ≤W2(ν0, µ)√

I(ν0|µ)−K

2W 22 (ν0, µ), ∀ν0

Consequence 3 : Sometimes T2 implies LSI.Assume K ≤ 0 and that µ satisfies T2(C) for some C > 0.

Inserting the inequality W 22 ≤ C H into HWI, yields to

H(ν0|µ) ≤√

C H(ν0|µ)√

I(ν0|µ)−KC

2H(ν0|µ), ∀ν0

and so (1 +

KC

2

)√H(ν0|µ) ≤

√C I(ν0|µ).

Therefore, if 1 + KC2> 0, then µ satisfies LSI(C/(1 + KC/2)2).

Consequences

H(ν0|µ) ≤W2(ν0, µ)√

I(ν0|µ)−K

2W 22 (ν0, µ), ∀ν0

H(ν0|µ) ≤√

C H(ν0|µ)√

I(ν0|µ)−KC

2H(ν0|µ), ∀ν0

and so (1 +

KC

2

)√H(ν0|µ) ≤

√C I(ν0|µ).

Consequences

H(ν0|µ) ≤W2(ν0, µ)√

I(ν0|µ)−K

2W 22 (ν0, µ), ∀ν0

H(ν0|µ) ≤√

C H(ν0|µ)√

I(ν0|µ)−KC

2H(ν0|µ), ∀ν0

and so (1 +

KC

2

)√H(ν0|µ) ≤

√C I(ν0|µ).

Consequences

H(ν0|µ) ≤W2(ν0, µ)√

I(ν0|µ)−K

2W 22 (ν0, µ), ∀ν0

H(ν0|µ) ≤√

C H(ν0|µ)√

I(ν0|µ)−KC

2H(ν0|µ), ∀ν0

and so (1 +

KC

2

)√H(ν0|µ) ≤

√C I(ν0|µ).

Consequences

Consequence 4 : HWI can be used to bound the deficit in the Log-Sobolevinequality for the standard Gaussian γd .Joint work with S. Bobkov, C. Roberto, P-M Samson. (see Lecture III )

Consequence 5 : HWI can also be used to prove a cheap version of E. Milman’sresult mentioned in Lecture I:

If µ is a log-concave probability measure enjoying the concentration property

with a Gaussian profile α(t) = e1C

[t−to ]2+ , then it satisfies LSI(D) with aconstant D = D(C , to).(see Lecture III )

Brenier Theorem

Theorem [Brenier ’91]

If ν0, ν1 ∈ P2(Rd) are two probability measures on Rd and ν0 is absolutelycontinuous with respect to Lebesgue, then there exists a unique optimaltransport plan π∗ from ν0 to ν1 and this plan is induced by a map T .

In other words, π∗(dxdy) = δT (x)(dy) ν0(dx), ν1 = T#ν0 and

T2(ν0, ν1) =∫‖T (x)− x‖2 ν0(dx).

Moreover, there exists a convex function φ : Rd → R such that T (x) = ∇φ(x)for ν0 almost all x ∈ Rd .

Brenier Theorem

T2(ν0, ν1) =∫‖T (x)− x‖2 ν0(dx).

Brenier Theorem

T2(ν0, ν1) =∫‖T (x)− x‖2 ν0(dx).

Monge-Ampère equation

Let ν0, ν1 be two probability measures on Rd absolutely continuous with

respect to Lebesgue and let f0, f1 be their densities.

Formal derivation: assume T : Rd → Rd is a C1 diffeomorphism.

For any test function φ : Rd → R, since ν1 = T#ν0, it holds∫φ(y) ν1(dy) =

∫φ ◦ T (x) ν0(dx) =

∫φ ◦ T (x)f0(x) dx .

On the other hand,∫φ(y) ν1(dy) =

∫φ(y)f1(y) dy =

∫φ ◦ T (x)f1(T (x))| detDT (x)| dx .

Identifying the densities, one obtains

f0(x) = f1(T (x))| detDT (x)|

= f1(∇φ(x)) detHessφ(x)

.

∫φ ◦ T (x) ν0(dx) =

∫φ ◦ T (x)f0(x) dx .

∫φ(y)f1(y) dy =

∫φ ◦ T (x)f1(T (x))| detDT (x)| dx .

f0(x) = f1(T (x))| detDT (x)|

.

∫φ ◦ T (x) ν0(dx) =

∫φ ◦ T (x)f0(x) dx .

∫φ(y)f1(y) dy =

∫φ ◦ T (x)f1(T (x))| detDT (x)| dx .

f0(x) = f1(T (x))| detDT (x)|

.

∫φ ◦ T (x) ν0(dx) =

∫φ ◦ T (x)f0(x) dx .

∫φ(y)f1(y) dy =

∫φ ◦ T (x)f1(T (x))| detDT (x)| dx .

f0(x) = f1(T (x))| detDT (x)|

.

∫φ ◦ T (x) ν0(dx) =

∫φ ◦ T (x)f0(x) dx .

∫φ(y)f1(y) dy =

∫φ ◦ T (x)f1(T (x))| detDT (x)| dx .

f0(x) = f1(T (x))| detDT (x)|

.

∫φ ◦ T (x) ν0(dx) =

∫φ ◦ T (x)f0(x) dx .

∫φ(y)f1(y) dy =

∫φ ◦ T (x)f1(T (x))| detDT (x)| dx .

f0(x) = f1(T (x))| detDT (x)| = f1(∇φ(x)) detHessφ(x).

Definition. Hessian in the sense of Aleksandrov

A function φ : Rd → R differentiable at a point x ∈ Rd is said to have aHessian in the sense of Aleksandrov at x if there exists a symmetric linear mapA such that

φ(x + t) = φ(x) +∇φ · t + 12At · t + o(‖t‖2).

When it exists, A is unique and denoted by A = Hessφ(x).

Theorem. [Aleksandrov]

A convex function φ : Rd → R admits a Hessian in the sense above almosteverywhere. At every point where it is well defined Hessφ(x) is non-negative.

Definition. Hessian in the sense of Aleksandrov

A function φ : Rd → R differentiable at a point x ∈ Rd is said to have aHessian in the sense of Aleksandrov at x if there exists a symmetric linear mapA such that

φ(x + t) = φ(x) +∇φ · t + 12At · t + o(‖t‖2).

When it exists, A is unique and denoted by A = Hessφ(x).

Theorem. [Aleksandrov]

A convex function φ : Rd → R admits a Hessian in the sense above almosteverywhere. At every point where it is well defined Hessφ(x) is non-negative.

Theorem. [McCann ’97]

Let ν0, ν1 ∈ P2(Rd) be two absolutely continuous probability measures, andT = ∇φ be the Brenier map from ν0 to ν1.There exists a set Ω of measure 1 for ν0 such that for all x ∈ Ω, the map φadmits a Hessian in the sense of Aleksandrov and the Monge-Ampère equationis satisfied

f0(x) = f1(∇φ(x)) detHessφ(x),where f0, f1 are the densities of ν0, ν1 with respect to Lebesgue.

Proof of the HWI inequality

We follow the proof of Cordero-Erausquin (’02).

In fact we shall establish the following useful lemma first.

Above tangent lemma

Let ν0, ν1 ∈ P2(Rd) be two compactly supported probability measuresabsolutely continuous with respect to Lebesgue and µ(dx) = e−V (x) dx be anabsolutely continuous probability measure with V : Rd → R a C2 smoothfunction such that HessV ≥ K , for some K ∈ R.Suppose that there is a C1 diffeomorphism T : Rd → Rd such that ν1 = T#ν0and DT (x) has a non-negative spectrum at every point x ∈ Rd , then thefollowing inequality holds

H(ν1|µ) ≥ H(ν0|µ) +∫∇h0(x) · (T (x)− x)µ(dx) +

K

2

∫‖T (x)− x‖2 dν0

We admit that the conclusion of the lemma is still valid when T is the Breniermap. Then the HWI inequality follows by applying Cauchy Schwarz inequality.

Above tangent lemma

H(ν1|µ) ≥ H(ν0|µ) +∫∇h0(x) · (T (x)− x)µ(dx) +

K

2

∫‖T (x)− x‖2 dν0

We admit that the conclusion of the lemma is still valid when T is the Breniermap.

Then the HWI inequality follows by applying Cauchy Schwarz inequality.

Above tangent lemma

H(ν1|µ) ≥ H(ν0|µ) +∫∇h0(x) · (T (x)− x)µ(dx) +

K

2

∫‖T (x)− x‖2 dν0

We admit that the conclusion of the lemma is still valid when T is the Breniermap. Then the HWI inequality follows by applying Cauchy Schwarz inequality.

Proof of the above tangent lemma I/III

Let h0, h1 be the densities of ν0, ν1 with respect to µ.We assume that h0, h1 are smooth and compactly supported.

By the change of variable formula, the densities f0, f1 with respect to Lebesguesatisfy

f0(x) = f1(T (x)) det(DT (x))

and soh0(x)e

−V (x) = h1(T (x))−V (T (x)) det(DT (x)).

Taking the log,

log h0 = log h1(T (x)) + V (x)− V (T (x)) + log detDT (x),

and integrating with respect to ν0,

H(ν0|µ) = H(ν1|µ) +∫

V (x)− V (T (x)) ν0(dx) +∫

log detDT (x) ν0(dx).

f0(x) = f1(T (x)) det(DT (x))

and soh0(x)e

−V (x) = h1(T (x))−V (T (x)) det(DT (x)).

Taking the log,

H(ν0|µ) = H(ν1|µ) +∫

V (x)− V (T (x)) ν0(dx) +∫

f0(x) = f1(T (x)) det(DT (x))

and soh0(x)e

−V (x) = h1(T (x))−V (T (x)) det(DT (x)).

Taking the log,

H(ν0|µ) = H(ν1|µ) +∫

V (x)− V (T (x)) ν0(dx) +∫

f0(x) = f1(T (x)) det(DT (x))

and soh0(x)e

−V (x) = h1(T (x))−V (T (x)) det(DT (x)).

Taking the log,

H(ν0|µ) = H(ν1|µ) +∫

V (x)− V (T (x)) ν0(dx) +∫

f0(x) = f1(T (x)) det(DT (x))

and soh0(x)e

−V (x) = h1(T (x))−V (T (x)) det(DT (x)).

Taking the log,

H(ν0|µ) = H(ν1|µ) +∫

V (x)− V (T (x)) ν0(dx) +∫

f0(x) = f1(T (x)) det(DT (x))

and soh0(x)e

−V (x) = h1(T (x))−V (T (x)) det(DT (x)).

Taking the log,

H(ν1|µ) = H(ν0|µ) +∫

V (T (x))− V (x) ν0(dx)−∫

Proof of the above tangent lemma II/III

(∗) H(ν1|µ) = H(ν0|µ)+∫

V (T (x))−V (x) ν0(dx)−∫

HessV ≥ K means that V is K semi-convex which translates into

V (y) ≥ V (x) +∇V (x) · (y − x) + K2‖y − x‖2, ∀x , y ∈ Rd .

Inserting into (∗) gives

H(ν1|µ) ≥ H(ν0|µ) +∫∇V (x) · (T (x)−x) ν0(dx)−

∫log detDT (x) ν0(dx)

+K

2

∫‖T (x)− x‖2 ν0(dx).

(∗) H(ν1|µ) = H(ν0|µ)+∫

V (T (x))−V (x) ν0(dx)−∫

HessV ≥ K means that V is K semi-convex

which translates into

H(ν1|µ) ≥ H(ν0|µ) +∫∇V (x) · (T (x)−x) ν0(dx)−

+K

2

∫‖T (x)− x‖2 ν0(dx).

(∗) H(ν1|µ) = H(ν0|µ)+∫

V (T (x))−V (x) ν0(dx)−∫

H(ν1|µ) ≥ H(ν0|µ) +∫∇V (x) · (T (x)−x) ν0(dx)−

+K

2

∫‖T (x)− x‖2 ν0(dx).

(∗) H(ν1|µ) = H(ν0|µ)+∫

V (T (x))−V (x) ν0(dx)−∫

H(ν1|µ) ≥ H(ν0|µ) +∫∇V (x) · (T (x)−x) ν0(dx)−

+K

2

∫‖T (x)− x‖2 ν0(dx).

(∗) H(ν1|µ) = H(ν0|µ)+∫

V (T (x))−V (x) ν0(dx)−∫

H(ν1|µ) ≥ H(ν0|µ) +∫∇V (x) · (T (x)−x) ν0(dx)−

+K

2

∫‖T (x)− x‖2 ν0(dx).

Proof of the above tangent lemma III/III

To conclude one needs to show that

θ(x) = T (x)− x

∫∇V (x)·(T (x)−x) ν0(dx)−

∫log detDT (x) ν0(dx) ≥

∫∇h0(x)·(T (x)−x)µ(dx)

∫∇V (x) · θ(x) ν0(dx) =

d∑i=1

∫θi (x)h0(x)∂iV (x)e

−V (x) dx

Int. Part=

d∑i=1

∫∂iθi (x)h0(x)e

−V (x)dx +d∑

i=1

∫θi∂ih0(x)e

−V (x) dx

=

∫Tr(Dθ) ν0(dx) +

∫∇h0(x) · θ(x)µ(dx).

Using the inequality log(1 + u) ≤ u, u > −1, one gets

log det(I + Dθ(x)) ≤ Tr(Dθ(x))

which completes the proof.

To conclude one needs to show that θ(x) = T (x)− x∫∇V (x)·θ(x) ν0(dx)−

∫log det (I + Dθ(x)) ν0(dx) ≥

∫∇h0(x)·θ(x)µ(dx).

∫∇V (x) · θ(x) ν0(dx) =

d∑i=1

−V (x) dx

Int. Part=

d∑i=1

−V (x)dx +d∑

i=1

∫θi∂ih0(x)e

−V (x) dx

=

∫∇h0(x) · θ(x)µ(dx).

∫∇h0(x)·θ(x)µ(dx).

∫∇V (x) · θ(x) ν0(dx) =

d∑i=1

−V (x) dx

Int. Part=

d∑i=1

−V (x)dx +d∑

i=1

∫θi∂ih0(x)e

−V (x) dx

=

∫∇h0(x) · θ(x)µ(dx).

∫∇h0(x)·θ(x)µ(dx).

∫∇V (x) · θ(x) ν0(dx) =

d∑i=1

−V (x) dx

Int. Part=

d∑i=1

−V (x)dx +d∑

i=1

∫θi∂ih0(x)e

−V (x) dx

=

∫∇h0(x) · θ(x)µ(dx).

∫∇h0(x)·θ(x)µ(dx).

∫∇V (x) · θ(x) ν0(dx) =

d∑i=1

−V (x) dx

Int. Part=

d∑i=1

−V (x)dx +d∑

i=1

∫θi∂ih0(x)e

−V (x) dx

=

∫∇h0(x) · θ(x)µ(dx).

∫∇h0(x)·θ(x)µ(dx).

∫∇V (x) · θ(x) ν0(dx) =

d∑i=1

−V (x) dx

Int. Part=

d∑i=1

−V (x)dx +d∑

i=1

∫θi∂ih0(x)e

−V (x) dx

=

∫∇h0(x) · θ(x)µ(dx).

Displacement convexity

Definition. [McCann]

Let ν0, ν1 ∈ P2(Rd) be two absolutely continuous probability measures and letT be the Brenier map from ν0 to ν1. The displacement interpolation betweenν0 and ν1 is defined by

νt = [(1− t)Id + tT ]# ν0, ∀t ∈ [0, 1].

A functional F is displacement convex if

F(νt) ≤ (1− t)F(ν0) + tF(ν1), ∀t ∈ [0, 1].

Remark.The path (νt)t∈[0,1] is a constant speed geodesic for W2 :

W2(νs , νt) = |s − t|W2(ν0, ν1), ∀s, t ∈ [0, 1].

Displacement convexity

The following result follows from the successive works by McCann,Otto-Villani, Cordero-McCann-Schmuckenschläger, Von Renesse-Sturm, Sturm,Lott-Villani,. . .

Theorem.

Let µ(dx) = e−V (x) dx be an absolutely continuous probability measure on acomplete connected Riemannian manifold with V : Rd → R a smooth functionof class C2. The following are equivalent :

1 Ric + HessV ≥ K , for some K ∈ R2 For any ν0, ν1 ∈ P(Rd) compactly supported and absolutely continuous

with respect to µ

H(νt |µ) ≤ (1−t) H(ν0|µ)+t H(ν1|µ)−K

2t(1−t)W 22 (ν0, ν1), ∀t ∈ [0, 1].

Remark.Differentiating at t = 0 gives back HWI.

II.3 Proof of the characterization of T2 in terms ofdimension free concentration

Characterization of T2 by dimension free Gaussianconcentration

Theorem. [G. ’09]

If µ satisfies the dimension free concentration property with the profile

α(t) = e−1C

[t−to ]2+ ,

then it satisfies the inequality T2(C).

Remark.• Conversely, we have seen that T2(C) implies the dimension freeconcentration property with the profile α as above (and with to =

√C log 2).

• The optimal constant in T2 is equal to the optimal constant in the dimensionfree Gaussian concentration property.

Large deviations of empirical measures

Xi an i.i.d sequence of law µ.

Ln =1n

∑ni=1 δXi .

P(Rd) is equipped with the topology of narrow convergence.

Theorem (Sanov)

Roughly speaking, for all A ⊂ P(Rd),

P(Ln ∈ A) ' e−n H(A|µ), with H(A|µ) = inf{H(ν|µ) : ν ∈ A}.

More precisely,

− inf {H(ν|µ), ν ∈ int(A)} ≤ lim infn→+∞

1

nlog P(Ln ∈ A)

lim supn→+∞

1

nlog P(Ln ∈ A) ≤ − inf {H(ν|µ), ν ∈ cl(A)}

Xi an i.i.d sequence of law µ.Ln =

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

1n

∑ni=1 δXi .

Theorem (Sanov)

More precisely,

1

nlog P(Ln ∈ A)

lim supn→+∞

1

A key lemma

Lemma

The function F : (Rd)n → R defined by

F (x) = W2(Lxn, µ), x = (x1, . . . , xn) ∈ (Rd)n, where Lxn =

1

n

n∑i=1

δxi ,

is 1/√n-Lipschitz with respect to the Euclidean norm on (Rd)n.

Proof.Take x , y ∈ (Rd)n ; since W2 is a distance,

|W2(Lxn, µ)−W2(Lyn , µ)| ≤W2(Lxn, Lyn) ≤

√√√√1n

n∑i=1

‖xi − yi‖2,

where the second inequality comes from the fact that Lyn is the image of Lxn by

the application xi 7→ yi .

Proof of the theorem

• For all x = (x1, . . . , xn) ∈ (Rd)n, one defines

Lxn =1

n

n∑i=1

δxi and Fn(x) = W2(Lxn, µ).

• The function Fn is 1/√n-Lipschitz with respect to the Euclidean norm.

• For all 1-Lipschitz G : (Rd)n → R with median m, it holds

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn(√

nFn >√nmn + t

)≤ exp

(− 1C

(t − to)2), ∀t ≥ to , ∀n ≥ 1

where mn is a median of Fn.• Sanov Theorem lower bound:

− inf {H(ν|µ) : ν s.t W2(ν, µ) > u} ≤ lim infn→+∞

1

nlog P (W2(Ln, µ) > u) .

So,1

Cu2 ≤ inf {H(ν|µ) : ν s.t W2(ν, µ) > u} , ∀u > 0.

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn(√

nFn >√nmn + t

)≤ exp

(− 1C

(t − to)2), ∀t ≥ to , ∀n ≥ 1

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn(√

nFn >√nmn + t

)≤ exp

(− 1C

(t − to)2), ∀t ≥ to , ∀n ≥ 1

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn(√

nFn >√nmn + t

)≤ exp

(− 1C

(t − to)2), ∀t ≥ to , ∀n ≥ 1

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn(√

nFn >√nmn + t

)≤ exp

(− 1C

(t − to)2), ∀t ≥ to , ∀n ≥ 1

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn(√

nFn >√nmn + t

)≤ exp

(− 1C

(t − to)2), ∀t ≥ to , ∀n ≥ 1

where mn is a median of Fn.

• Sanov Theorem lower bound:

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

µn (Fn > u) ≤ exp(− 1Cn(u −mn − to/

√n)2), ∀u > mn + to/

√n, ∀n ≥ 1

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

lim supn→+∞

1

nlog P (W2(Ln, µ) > u) ≤ −

1

Cu2, ∀u > 0.

1

So,1

Lxn =1

n

n∑i=1

µn (G > m + t) ≤ exp(− 1C

(t − to)2), ∀t ≥ to .

So,

lim supn→+∞

1

nlog P (W2(Ln, µ) > u) ≤ −

1

Cu2, ∀u > 0.

1

nlog P (W2

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