Optimal transportation, topology and uniqueness

Abstract

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

Multi‐to One‐Dimensional Optimal Transport

“…At the end of the preceding section we identified a natural candidate for this iso-husband set: namely the potential indifference set which divides the mass of µ in the same ratio as y divides ν; whether or not these natural candidates actually fit together to form the level sets of a function F or not depends on a subtle interaction between µ, ν and s. When they do, we say the model is nested, and in that case we show that the resulting function F : X −→ Y produces the unique optimizer γ = (id × F ) # µ for (1). Note that except in the Lorentz/Becker/Mirrlees/Spence case m = 1 = n, this nestedness depends not only on s, but also on µ and ν.…”

“…For example, is there a map F : X −→ Y such that γ vanishes outside Graph(F ), and if so, what can be said about its analytical and geometric properties? Such a map is called a Monge or pure solution, deterministic coupling, matching function or optimal map, and we have γ = (id×F ) # µ where id denotes the identity map on X in that case [1]. More generally, if F : X −→ Y is any µ-measurable map, we define the push-forward F # µ of µ through F by…”

“…In this case the potential indifference set X(y, k) is said to split the population proportionately at y, making it a natural candidate for being the iso-husband set F −1 (y) to be matched with y. 1 In the next sections, we go on to describe and contrast situations in which this expectation is born out and leads to a complete solution from those in which it does not.…”

“…Since k ± are both constant on any interval I ⊂ R with ν[I] = 0, we conclude they are continuous functions except perhaps at the countably many points in spt ν where they disagree (k − being left continuous and k + right continuous at such points). We shall show γ := (id × F ) # µ is well-defined and maximizes (1).…”

Multivariate shuffles and approximation of copulas

“…In particular, 2-fold stochastic measures (better known as doubly stochastic measures) represent an infinitedimensional generalization of doubly stochastic matrices and originated from an idea by Birkhoff (1948, Problem 111). Notably, such measures also appear in several problems connected with optimal transportation (Ahmad et al, 2009;Gangbo and McCann, 2000;Rachev and Rüschendorf, 1998).…”