measure concentration, functional inequalities, and curvature ...gallay/taiwan/talks/...measure...

Measure concentration, functional inequalities,and curvature of metric measure spaces

M. Ledoux, Institut de Mathématiques de Toulouse, France

Luminy, March 2008

circle of ideas

between analysis, geometry and probability theory

• concentration of measure phenomenon

• geometric, measure/information theoretic, functionalinequalities

• curvature of metric measure spaces

theory of optimal transportation

circle of ideas

concentration of measure phenomenon

V. Milman 1970

Dvoretzky’s theorem on spherical sectionsof convex bodies in high dimension

for all ε > 0, there exists δ(ε) > 0 such that for every convex bodyK in Rn, there is a subspace F of Rn with dim (F ) ≥ δ(ε) log nand an ellipsoid E in F such that

Milman’s idea : find F random

using a concentration propertyof spherical measures in high dimension

V. Milman 1970

for all ε > 0, there exists δ(ε) > 0 such that for every convex bodyK in Rn,

there is a subspace F of Rn with dim (F ) ≥ δ(ε) log nand an ellipsoid E in F such that

V. Milman 1970

for all ε > 0, there exists δ(ε) > 0 such that for every convex bodyK in Rn, there is a subspace F of Rn with dim (F ) ≥ δ(ε) log nand an ellipsoid E in F

such that

V. Milman 1970

K ∩ F

V. Milman 1970

(1− ε) E ⊂ K ∩ F ⊂ (1 + ε) E

V. Milman 1970

(1− ε) E ⊂ K ∩ F ⊂ (1 + ε) E

V. Milman 1970

(1− ε) E ⊂ K ∩ F ⊂ (1 + ε) E

spherical concentration

isoperimetric inequality on Sn ⊂ Rn+1

geodesic balls are the extremal sets

if vol (A) ≥ vol (B), B geodesic ball

then vol (Ar ) ≥ vol (Br ), r > 0

normalized volume µ(·) = vol (·)vol (Sn)

if µ(A) ≥ 12

(B half-sphere)

then, for all r > 0,

µ(Ar ) ≥ µ(Br ) ≥ 1− e−(n−1)r2/2

r ∼ 1√n

(n →∞)

µ(Ar ) ≈ 1

if µ(A) ≥ 12

(B half-sphere)

µ(Ar ) ≥ µ(Br ) ≥ 1− e−(n−1)r2/2

r ∼ 1√n

(n →∞)

µ(Ar ) ≈ 1

if µ(A) ≥ 12

(B half-sphere)

µ(Ar ) ≥ µ(Br )

≥ 1− e−(n−1)r2/2

r ∼ 1√n

(n →∞)

µ(Ar ) ≈ 1

if µ(A) ≥ 12

(B half-sphere)

µ(Ar ) ≥ µ(Br ) ≥ 1− e−(n−1)r2/2

r ∼ 1√n

(n →∞)

µ(Ar ) ≈ 1

if µ(A) ≥ 12

(B half-sphere)

µ(Ar ) ≥ µ(Br ) ≥ 1− e−(n−1)r2/2

r ∼ 1√n

(n →∞)

µ(Ar ) ≈ 1

if µ(A) ≥ 12

(B half-sphere)

µ(Ar ) ≥ µ(Br ) ≥ 1− e−(n−1)r2/2

r ∼ 1√n

(n →∞)

µ(Ar ) ≈ 1

application by V. Milman

m (normalized) Haar measure on On+1

m({T ∈ On+1;Tx ∈ A}

)= µ(A), x ∈ Rn+1, A ⊂ Sn

‖ · ‖ gauge of K

show that for dim(E ) ≥ δ(ε) log n

m({

T ∈ On+1;|z |

1 + ε≤ ‖Tz‖ ≤ |z |

1− ε, z ∈ E

})> 0

F = ImT|E , E = T (BE )

application by V. Milman

m (normalized) Haar measure on On+1

m({T ∈ On+1;Tx ∈ A}

)= µ(A), x ∈ Rn+1, A ⊂ Sn

‖ · ‖ gauge of K

show that for dim(E ) ≥ δ(ε) log n

m({

T ∈ On+1;|z |

1 + ε≤ ‖Tz‖ ≤ |z |

1− ε, z ∈ E

})≈ 1

F = ImT|E , E = T (BE )

framework for measure concentration

metric measure space (X , d , µ)

(X , d) metric space

µ Borel measure on X , µ(X ) = 1

concentration function

α(r) = α(X ,d ,µ)(r) = sup{1− µ(Ar );µ(A) ≥ 1/2

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

}µ uniform on Sn ⊂ Rn+1 : α(r) ≤ e−(n−1)r2/2, r > 0

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

}

µ uniform on Sn ⊂ Rn+1 : α(r) ≤ e−(n−1)r2/2, r > 0

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

}µ uniform on Sn ⊂ Rn+1 :

α(r) ≤ e−(n−1)r2/2, r > 0

}, r > 0

Ar ={x ∈ X ; d(x ,A) < r

equivalent formulation on functions

F : X → R 1- Lipschitz

m = mF median of F for µ

µ(F ≤ m), µ(F ≥ m) ≥ 12

A = {F ≤ m} =⇒ Ar ⊂ {F < m + r}

µ({|F −m| < r

})≥ 1− 2 α(r), r > 0

µ(F ≤ m), µ(F ≥ m) ≥ 12

A = {F ≤ m} =⇒ Ar ⊂ {F < m + r}

µ({|F −m| < r

})≥ 1− 2 α(r), r > 0

µ(F ≤ m), µ(F ≥ m) ≥ 12

A = {F ≤ m} =⇒ Ar ⊂ {F < m + r}

µ({|F −m| < r

})≥ 1− 2 α(r), r > 0

µ(F ≤ m), µ(F ≥ m) ≥ 12

A = {F ≤ m} =⇒ Ar ⊂ {F < m + r}

µ({|F −m| < r

})≥ 1− 2 α(r), r > 0

µ(F ≤ m), µ(F ≥ m) ≥ 12

A = {F ≤ m} =⇒ Ar ⊂ {F < m + r}

µ({|F −m| < r

})≥ 1− 2 α(r), r > 0

dual description : observable diameter (M. Gromov)

κ > 0 (κ = 10−10)

PartDiamµ(X , d)

= inf{D ≥ 0,∃A ⊂ X ,Diam (A) ≤ D, µ(A) ≥ 1− κ

}

ObsDiamµ(X , d) = supF 1−Lip

PartDiamµF (R)

F : µ → µF

κ > 0 (κ = 10−10)

PartDiamµ(X , d)

}

PartDiamµF (R)

F : µ → µF

κ > 0 (κ = 10−10)

PartDiamµ(X , d)

}

PartDiamµF (R)

F : µ → µF

κ > 0 (κ = 10−10)

PartDiamµ(X , d)

}

PartDiamµF (R)

F : µ → µF

PartDiamµF (R)

µ state on the configuration space (X , d)

F : X → R observableyielding the tomographic image µF on R

µF(]m − r ,m + r [

)= µ

({|F −m| < r

})≥ 1− 2 α(r)

ObsDiamµ(X , d) ∼ α−1(κ)

ObsDiam(Sn) = O( 1√

n

)

PartDiamµF (R)

µF(]m − r ,m + r [

)= µ

({|F −m| < r

})≥ 1− 2 α(r)

n

)

PartDiamµF (R)

µF(]m − r ,m + r [

)= µ

({|F −m| < r

})≥ 1− 2 α(r)

n

)

PartDiamµF (R)

µF(]m − r ,m + r [

)= µ

({|F −m| < r

})≥ 1− 2 α(r)

n

)

PartDiamµF (R)

µF(]m − r ,m + r [

)= µ

({|F −m| < r

})≥ 1− 2 α(r)

n

)

PartDiamµF (R)

µF(]m − r ,m + r [

)= µ

({|F −m| < r

})≥ 1− 2 α(r)

n

)

measure concentration property

less restrictive than isoperimetry, widely shared

variety of examples and tools

• spectral methods

• combinatorial tools

• product measures (M. Talagrand 1995)

• product measures

(M. Talagrand 1995)

dimension free inequalities

model : Gaussian measure

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

concentration property

R ∼√

n

µ uniform on Sn√n → γ Gaussian

Gauss space : curvature 1 dimension ∞

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

R ∼√

n

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

R ∼√

n

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

Sn : α(r) ≤ e−(n−1)r2/2

R ∼√

n

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

SnR : α(r) ≤ e−(n−1)r2/2R2

R ∼√

n

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

SnR : α(r) ≤ e−(n−1)r2/2R2

R ∼√

n

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

SnR : α(r) ≤ e−(n−1)r2/2R2

R ∼√

n

µ uniform on Sn√n

→ γ Gaussian

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

SnR : α(r) ≤ e−(n−1)r2/2R2

R ∼√

n

dγ(x) = e−|x |2/2 dx

(2π)k/2on (Rk , | · |)

SnR : α(r) ≤ e−(n−1)r2/2R2

R ∼√

n

Gaussian concentration α(r) ≤ e−r2/2

if A ⊂ Rk , γ(A) ≥ 12

then γ(Ar ) ≥ 1− e−r2/2, r > 0

ObsDiamγ(Rk) = O(1)

independent of the dimension

infinite dimensional analysis

if A ⊂ Rk , γ(A) ≥ 12

then γ(Ar ) ≥ 1− e−r2/2, r > 0

if A ⊂ Rk , γ(A) ≥ 12

then γ(Ar ) ≥ 1− e−r2/2, r > 0

if A ⊂ Rk , γ(A) ≥ 12

then γ(Ar ) ≥ 1− e−r2/2, r > 0

if A ⊂ Rk , γ(A) ≥ 12

then γ(Ar ) ≥ 1− e−r2/2, r > 0

if A ⊂ Rk , γ(A) ≥ 12

then γ(Ar ) ≥ 1− e−r2/2, r > 0

triple description of Gaussian concentration α(r) ≤ e−r2/2

• geometric

• measure/information theoretic

• functional

notion of curvature in metric measure spaces

(PDE and calculus of variations viewpoint)

• geometric

• functional

• geometric

• functional

• geometric

• functional

• geometric

• functional

• geometric

• functional

geometric description : Brunn-Minkowski inequality

Prékopa-Leindler theorem

θ ∈ [0, 1], u, v ,w : Rn → R+

if w(θx + (1− θ)y

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

u = χA, v = χB , multiplicative form of Brunn-Minkowski

voln(θA + (1− θ)B

)≥ voln(A)θvoln(B)1−θ

θ ∈ [0, 1], u, v ,w : Rn → R+

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

θ ∈ [0, 1], u, v ,w : Rn → R+

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

θ ∈ [0, 1], u, v ,w : Rn → R+

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

θ ∈ [0, 1], u, v ,w : Rn → R+

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

θ ∈ [0, 1], u, v ,w : Rn → R+

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

u = χA, v = χB ,

multiplicative form of Brunn-Minkowski

θ ∈ [0, 1], u, v ,w : Rn → R+

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dx ≥

( ∫u dx

)θ( ∫v dx

)1−θ

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

choose θ = 1/2, w ≡ 1, v = χA, u = ed(·,A)2/4

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

extension : dµ = e−V dx , V ′′ ≥ c > 0

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

)≥ u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

)≥ e−θ(1−θ)|x−y |2/2u(x)θv(y)1−θ, x , y ∈ Rn

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

choose θ = 1/2, w ≡ 1, v = χA,

u = ed(·,A)2/4

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

dx → dγ(x) = e−|x |2/2 dx(2π)n/2

then

∫w dγ ≥

( ∫u dγ

)θ( ∫v dγ

)1−θ

∫ed(·,A)

2/4dγ ≤ 1γ(A)

γ(Ar ) ≥ 1− 2 e−r2/4, γ(A) ≥ 1

2

measure/information theoretic description :

transportation cost inequality

ν probability measure on Rn, ν

concentration via the transportation cost inequality (K. Marton)

A,B ⊂ Rn, d(A,B) ≥ r > 0

γA = γ(· |A), γB = γ(· |B)

W2(γA, γB) ≤(

log1

γ(A)

)1/2+

(log

1

γ(B)

)1/2r√2≤ W2(γA, γB) ≤

(log

1

γ(A)

)1/2+

(log

1

γ(B)

)1/2γ(A) ≥ 1/2, B = complement of Ar

γ(Ar ) ≥ 1− e−r2/4, r ≥ r0

functional description : logarithmic Sobolev inequality

f ≥ 0 smooth,∫

f dγ = 1

∫f log f dγ ≤ 1

2

∫|∇f |2

fdγ

∫f log f dγ = H(fdγ | dγ) entropy∫|∇f |2

fdγ Fisher information

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

A. Stam 1959, L. Gross 1975

f ≥ 0 smooth,∫

f dγ = 1

2

∫|∇f |2

fdγ

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

f ≥ 0 smooth,∫

f dγ = 1

2

∫|∇f |2

fdγ

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

f ≥ 0 smooth,∫

f dγ = 1

2

∫|∇f |2

fdγ

∫f log f dγ = H(fdγ | dγ) entropy

∫|∇f |2

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

f ≥ 0 smooth,∫

f dγ = 1

2

∫|∇f |2

fdγ

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

f ≥ 0 smooth,∫

f dγ = 1

2

∫|∇f |2

fdγ

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

f ≥ 0 smooth,∫

f dγ = 1

2

∫|∇f |2

fdγ

f → f 2 :∫

f 2 log f 2 dγ ≤ 2∫|∇f |2 dγ

concentration via the logarithmic Sobolev inequality (I. Herbst)

F : Rn → R 1-Lipschitz,∫

F dγ = 0

f = eλF/∫

eλFdγ, λ ∈ R

differential inequality on∫

eλFdγ

∫eλFdγ ≤ eλ2/2, λ ∈ R

γ({F < r}

)≥ 1− e−r2/2, r > 0

Brunn-Minkowski inequality

logarithmic Sobolev inequality

hierarchy

⇓

dimension free measure concentration

tools to establish these inequalities

parametrisation methods

hierarchy

⇓

hierarchy

⇓

hierarchy

⇓

heat kernel parametrisation

logarithmic Sobolev inequality∫

f log f dγ ≤ 12∫ |∇f |2

f dγ

generator Lf = ∆f − x · ∇f , Pt = etL semigroup

γ invariant measure

d

dt

∫Pt f log Pt f dγ = −

1

2

∫|∇Pt f |2

Pt fdγ

commutation |∇Pt f | ≤ e−tPt(|∇f |)

equivalent to a curvature condition

f log f dγ ≤ 12∫ |∇f |2

f dγ

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

f log f dγ ≤ 12∫ |∇f |2

f dγ

generator Lf = ∆f − x · ∇f ,

Pt = etL semigroup

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

f log f dγ ≤ 12∫ |∇f |2

f dγ

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

f log f dγ ≤ 12∫ |∇f |2

f dγ

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

f log f dγ ≤ 12∫ |∇f |2

f dγ

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

f log f dγ ≤ 12∫ |∇f |2

f dγ

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

f log f dγ ≤ 12∫ |∇f |2

f dγ

d

dt

1

2

∫|∇Pt f |2

Pt fdγ

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

� Riemannian manifolds Ric ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

equivalent to |∇Pt f | ≤ e−ctPt(|∇f |)

(Gauss space : curvature 1 )

� manifolds with weights

dµ = e−V dx , Ric + Hess(V ) ≥ c > 0

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f )

≥ c |∇f |2

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

extensions

� dµ = e−V dx on Rn, V ′′ ≥ c > 0

Bochner’s formula

1

2∆

(|∇f |2

)−∇f · ∇(∆f ) = ‖Hess (f )‖2 + Ric(∇f ,∇f ) ≥ c |∇f |2

parametrisation by optimal transportation

Riemannian geometry of (P2,W2) F. Otto (2001), C. Villani (2005)

(Brunn-Minkowski, transportation cost inequalities)

ν probability measure on Rn

T : γ → ν

optimal : W2(γ, ν)2 =

∫1

2

∣∣x − T (x)∣∣2dγ(x)Y. Brenier, S. T. Rachev - L. Rüschendorf 1990 T = ∇φ, φ convex

(manifold case R. McCann 1995)

T : γ → ν

∫1

2

T : γ → ν

∫1

2

T : γ → ν

∫1

2

T : γ → ν

∫1

2

T : γ → ν

∫1

2

∣∣x − T (x)∣∣2dγ(x)

Y. Brenier, S. T. Rachev - L. Rüschendorf 1990 T = ∇φ, φ convex

T : γ → ν

∫1

2

T : γ → ν

∫1

2

parametrisation Tθ = (1− θ) Id + θT , θ ∈ [0, 1]

(T0γ = γ, T1γ = ν)

Tθ : γ → fθ dγ

Monge-Ampère equation

e−|x |2/2 = fθ ◦ Tθ e−|Tθ|

2/2det((1− θ) Id + θφ′′

)φ′′ symmetric positive definite

non-smooth analysis, PDE methods

(T0γ = γ, T1γ = ν)

e−|x |2/2 = fθ ◦ Tθ e−|Tθ|

2/2det((1− θ) Id + θφ′′

(T0γ = γ, T1γ = ν)

e−|x |2/2 = fθ ◦ Tθ e−|Tθ|

2/2det((1− θ) Id + θφ′′

)

φ′′ symmetric positive definite

(T0γ = γ, T1γ = ν)

e−|x |2/2 = fθ ◦ Tθ e−|Tθ|

2/2det((1− θ) Id + θφ′′

(T0γ = γ, T1γ = ν)

e−|x |2/2 = fθ ◦ Tθ e−|Tθ|

2/2det((1− θ) Id + θφ′′

mass transportation method

• F. Barthe (1998) : geometric Brascamp-Lieb inequalities,inverse forms

• D. Cordero-Erausquin, R. McCann, M. Schmuckenschläger(2001, 2006) : extension of Brunn-Minkowski inequality tomanifolds, J. Lott - C. Villani, K. Th. Sturm (2006) : notionof Ricci curvature in metric measure spaces

• D. Cordero- Erausquin (2002) : transportation cost andfunctional inequalities (logarithmic Sobolev...),D. Cordero-Erausquin, B. Nazaret, C. Villani (2004) : optimalclassical Sobolev inequalities

• D. Cordero-Erausquin, R. McCann, M. Schmuckenschläger(2001, 2006) : extension of Brunn-Minkowski inequality tomanifolds,

J. Lott - C. Villani, K. Th. Sturm (2006) : notionof Ricci curvature in metric measure spaces

• D. Cordero- Erausquin (2002) : transportation cost andfunctional inequalities (logarithmic Sobolev...),

D. Cordero-Erausquin, B. Nazaret, C. Villani (2004) : optimalclassical Sobolev inequalities

optimal parametrisation and entropy

J. Lott - C. Villani, K. Th. Sturm

µ0, µ1 probability measures on Rn, T : µ0 → µ1 optimal

geodesic Tθ = (1− θ) Id + θT , θ ∈ [0, 1]

reference measure dµ = e−V dx on Rn, V ′′ ≥ c

c-convexity property of entropy along geodesic Tθ

H(µθ |µ) ≤ (1− θ)H(µ0 |µ) + θH(µ1 |µ)− cθ(1− θ)W2(µ0, µ1)2

H relative entropy, W2 Wasserstein distance

characterizes V ′′ ≥ c

extends to (weighted) manifolds

characterizes Ric + Hess(V ) ≥ c

notion of Ricci curvature

in a metric measure space (length space) (X , d , µ)

Tθ : [0, 1] → P2 probability measures on X

connecting µ0, µ1, µθ = Tθ(µ0)

definition of curvature : postulate that entropy is c-convex along Tθ

� generalizes Ricci curvature in manifolds

� allows for geometric and functional inequalities

� main result : stability of the definition by Gromov-Hausdorff limit

J. Lott - C. Villani, K.-Th. Sturm 2006

definition of curvature :

postulate that entropy is c-convex along Tθ

Intro

measure concentration, functional inequalities, and curvature ...gallay/taiwan/talks/...measure...

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