The Optimal Transportation Problem

Abstract

Let X, Y be separable metric spaces such that any Borel probability measure in X, Y is tight (5.1.9), i.e. Radon spaces, according to Definition 5.1.4, and let c : X × Y → [0,+∞] be a Borel cost function. Given μ ∈ (X), ν ∈ (Y) the optimal transport problem, in Monge’s formulation, is given by

$$ \inf \left\{ {\smallint _X c\left( {x,t\left( x \right)} \right)d\mu \left( x \right):t_\# \mu = \nu } \right\}. $$

(6.0.1)

This problem can be ill posed because sometimes there is no transport map t such that t_#μ = ν (this happens for instance when μ is a Dirac mass and ν is not a Dirac mass). Kantorovich’s formulation

$$ \min \left\{ {\smallint _{X \times Y} c\left( {x,y} \right)d\gamma \left( {x,y} \right):\gamma \in \Gamma \left( {\mu ,\nu } \right)} \right\} $$

(6.0.2)

circumvents this problem (as μ× ν ∈ Г(μ, ν)). The existence of an optimal transpoplan, when c is l.s.c., is provided by (5.1.15) and by the tightness of Г(μ, ν) (this property is equivalent to the tightness of μ, ν, a property always guaranteed in Radon spaces).