Optimal transport, Euler equations, Mather and DiPerna-Lions theories

Chapter 1
The optimal transport problem
The optimal transport problem (whose origin goes back to Monge [68]) is nowadays formulated
in the following general form: given two probability measures µ and ν, defined on the measurable
spaces X and Y , find a measurable map T : X → Y with T♯µ = ν, i.e.
ν(A) = µ T −1 (A)
∀A ⊂ Y measurable,
and in such a way that T minimizes the transportation cost. This last condition means
c(x, T (x)) dµ(x) = min
S♯µ=ν
c(x, S(x)) dµ(x) ,
X
where c : X × Y → R is some given cost function, and the minimum is taken over all measurable
maps S : X → Y with S♯µ = ν. When the transport condition T♯µ = ν is satisfied, we say that
T is a transport map, and if T minimizes also the cost we call it an optimal transport map.
Even in Euclidean spaces, with the cost c equal to the Euclidean distance or its square,
the problem of the existence of an optimal transport map is far from being trivial. Moreover,
it is easy to build examples where the Monge problem is ill-posed simply because there is no
transport map: this happens for instance when µ is a Dirac mass while ν is not. This means
that one needs some restrictions on the measures µ and ν.
The major advance on this problem is due to Kantorovitch, who proposed in [55], [56] a notion
of weak solution of the optimal transport problem. He suggested to look for plans instead of
transport maps, that is probability measures γ in X × Y whose marginals are µ and ν, i.e.
(πX)♯γ = µ and (πY )♯γ = ν,
where πX : X ×Y → X and πY : X ×Y → Y are the canonical projections. Denoting by Π(µ, ν)
the set of plans, the new minimization problem becomes
C(µ, ν) = min
γ∈Π(µ,ν)
c(x, y) dγ(x, y) . (1.0.1)
M×M
If γ is a minimizer for the Kantorovich formulation, we say that it is an optimal plan. Due to
the linearity of the constraint γ ∈ Π(µ, ν), it turns out that weak topologies can be used to
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