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GEOMETRICALASPECTS
OFOPTIMALTRANSPORT
Ischia, June 2010
C´edric Villani
ENS Lyon
& Institut Henri Poincar´eMAINTHEME
Interplay between hard/soft, and smooth/nonsmooth,
analysis/geometry
in relation with optimal transport
References
• Topics in Optimal Transportation [TOT] (AMS,
2003): Introduction
• Optimal transport, old and new [oldnew] (Springer,
2008): Reference text, more probabilistic & geometricPRELIMINARY: Push-forward, or change of variables
(dx), ν(dy) two (probability) measures
y =T(x) measurable
−1
Def: T =ν if ∀B, [T (B)] =ν[B]
#
Z Z
Equivalently: ∀ϕ, ϕ◦Td = ϕd(T )
#
Probabilistic formulation
law(U) =, law(V) =ν, V =T(U)
Analytic formulation
n
InR , T (f(x)dx) =g(y)dy, if T is 1-to-1, yields
#
f(x) =g(T(x))|det(dT)(x)|MONGE–KANTOROVICH PROBLEMI. BASIC THEORY OF OPTIMAL
TRANSPORT
• The modern core of the Monge–Kantorovich theory,
built from the eighties to now
• Simplified statements
• Reference: [oldnew, Chap. 4, 5, 10]The Kantorovich problem (Kantorovich, 1942)
• X,Y two Polish (= metric separable complete) spaces
• ∈P(X), ν ∈P(Y)
1 1
• c∈C(X ×Y;R), c≥ c∈L ()+L (ν)
n o
Π(;ν) = π∈P(X ×Y); marginals of π are and ν
(∀h,
R R R R
h(x)π(dxdy) = hd; h(y)π(dxdy) = hdν)
Z
(K) inf c(x,y)π(dxdy)
π∈Π(;ν)
Prop: Infimum achieved by compactness of Π(;ν)
In the sequel, assume infimum is finiteProbabilistic version
X and Y two given random variables (= with given laws)
(K’) inf Ec(X,Y)
(Infimum over all couplings of (X,Y))Engineer’s interpretation
π(dxdy)
T
y
x
y ν
x

remblais
d´eblais
Given the initial and final distributions, transport
matter at lowest possible costThe Monge problem (Monge, 1781)
Assume π = (Id,T) =(dx)δ
# y=T(x)
−→ belongs to Π(;ν) iff T =ν
#
=⇒ the Kantorovich problem becomes
Z
(M) inf c(x,T(x))(dx) = inf Ec(X,T(X))
T =ν
#
• Interpretation: Don’t split mass! Y =T(X)
• No compactness =⇒ not clear if infimum achieved