SMOOTHNESS OF OPTIMAL TRANSPORT

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SMOOTHNESS OF OPTIMAL TRANSPORT IN CURVED GEOMETRY Videoseminaire, October 9, 2008 Cedric Villani ENS Lyon & Institut Universitaire de France

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fluid mechanics

finslerian geometry

optimal transport

deblais remblais

monge–kantorovich problem

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English

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.