SMOOTHNESS

OFOPTIMALTRANSPORT

INCURVEDGEOMETRY

Vid´eos´eminaire, October 9, 2008

C´edric Villani

ENS Lyon

& Institut Universitaire de FranceMAINTHEMES

• In the optimal transport problem, smoothness and

geometry are mixed in an inextricable way

• Gives rise to progress in “nonsmooth” geometry

• Provides a “soft” tool to attack “hard problems”Optimal transport (Monge–Kantorovich problem)

μ(dx), ν(dy), c(x,y) given

Z

inf c(x,T(x))μ(dx)

T ¹=ν

#

−1

T μ[A] = μ[T (A)]

#

T

x

y ν

μ

remblais

d´eblais

Given the initial and ﬁnal distributions, transport

matter at lowest possible costO.T. has been exploding in the past 20 years:

• modelling in economics, meteorology, ...

• ﬂuid mechanics

• linear and nonlinear diﬀusion equations

• concentration of measure

• (optimal) Sobolev inequalities

• Riemannian and Finslerian geometry

• ...

Ref: Optimal Transport, old and new

(Grundlehren to appear)Which cost function?

It depends.!

Some noticeable cost functions 973

cost setting what use where quoted

2 3|x! y| R orR Monge’s original cost function [15, 606]

shape optimization, sandpile growth, compression molding [315]

d(x,y) Polish space Kantorovich’s cost function [482]

deﬁnition of Kantorovich distance/norm Chapter 6

1 Polish space representation of total variation [783, Appendix 1.4]x=y

1 Polish space Strassen’s duality theorem [783, Appendix 1.4]d(x,y)"!

pd(x,y) Polish space p-Wasserstein distances Chapter 6

2 n|x! y| R Tanaka’s study of Boltzmann equation [746] [783, Section 7.5]

Brenier’s study of incompressible ﬂuids [18, 145, 147, 150] [783, Section 3.2]

nmost useful for geometric applications inR [783, Chapter 6]

di!usion equations of Fokker–Planck type [474, 639, 641]

semi-geostrophic equations [256]

2!log(1! x · y) S Far-ﬁeld reﬂector antenna design [402, 801]

2!log(1!!(x) · y) x" surface, y " S Near-ﬁeld reﬂector antenna design [630]

nlog#x,y$ S prescribed integral curvature problem [629]+

n!log |x! y| R (ﬂat) conformal geometry [499, 757]

!

321! 1! |x! y| R relativistic theory [155]

2R Rubinstein–Wolansky’s design of lens [681]

!

321 + |x! y| R relativistic heat equation [32, 33, 155]

2 2(x ! y ) + (x ! y )1 1 2 2 3R semi-geostrophic equations [255]

f(x ! y )3 3

erf("|x! y|) R Hsu–Sturm’s maximal coupling of Brownian paths [466]

" 2|x! y| ,0 < # < 1 R orR modelling in economy [383]

2d(x,y) Riemannian manifold Riemannian geometry [235, 587] and Part II

"

2min(d(x ,y ),d(x ,y ) ) product metric space Talagrand’s study of exponential concentration [743] and Chapter 22i i i ii

#

inf L(x,v,t)dt Riemannian manifold Mather’s theory of Lagrangian mechanics [95, 96, 574] and Chapter 8

# 2|v| ninf ( ! p(t,x))dt subset ofR incompressible Euler equation [18, 19, 146]

2#

1 2inf (|v| + Scal(t,x))dt Riemannian manifold Study of Ricci ﬂow [550, 751]2

2d(x,y) geodesic space Lott–Sturm–Villani’s nonsmooth Ricci curvature bounds [551, 732, 733] and Part IIIWhich cost function?

It depends.

In many cases, the quadratic cost function

2

d(x,y)

c(x,y) =

2

d = geodesic distance

Use in Riemannian geometry goes back to Otto–V (2000)

In the sequel, stick to that cost.Illustration: Two “external” results

• (M ,g ) compact manifolds with Ric≥ 0, converging

k k

in weak sense [measured Gromov–Hausdorﬀ] to (M,g); then

(M,g) also has Ric≥ 0

(Lott–Sturm–V.)

n

• k¢k a norm on R ,

Z

n−1

|A| = λ [A], |∂A| = kσ k H (dx), then

n x ∗

∂A

( Ã !)

µ ¶

2

|ΩΔB|

|∂Ω|≥ inf |∂B| 1+const.

|B|=|Ω| |Ω|

(Figalli–Fusco–Maggi–Pratelli)Regularity mystery

Up to recently, nothing was known about regularity of

optimal transport, apart from Euclidean case (Caﬀarelli,

Urbas)

=⇒ Whole theory built with nonsmooth analysis, relying

only on regularity of (semi)convex functions.Example: O.T. on Riemannian manifold

(M,g) a compact Riemannian manifold

exp the Riemannian exponential: Let x∈ M, v∈ T M,

x

start a constant-speed geodesic from x with velocity v,

then exp v = position at time 1

x

2

Let μ(dx) = f(x)vol(dx), ν(dy), c(x,y) = d(x,y)

McCann: T = exp(∇ψ)

opt

ψ is c-convex, i.e. ∃ζ s.t.

³ ´

ψ(x) = sup ζ(y)−c(x,y) .

y∈M

In particular, ψ is semiconvex.