The intrinsic dynamics of optimal transport

Robert J Mccann,Ludovic Rifford

doi:10.5802/jep.29

Abstract

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a suciently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probility densities is unique.

Highlights

Let M and N be smooth closed manifolds of dimensions m and n 1 respectively, and c : M × N → R a continuous cost function
Given two probability measures μ and ν respectively on M and N, the Monge problem consists in minimizing the transportation cost
That problem is a linear optimization problem under convex constraints, it consists in minimizing the transportation cost c(x, y) dγ(x, y), among all transport plans between μ and ν, meaning γ belongs to the set Π(μ, ν) of non-negative measures having marginals μ and ν

Summary

Introduction

Let M and N be smooth closed manifolds (meaning compact, without boundary) of dimensions m and n 1 respectively, and c : M × N → R a continuous cost function. If there is a uniform bound K on the length of all chains in M × N , our theorem applies a fortiori with S = M × N and E0 = ∅, since E∞ = ∅ We shall see this occurs in many cases of interest, including for the smooth costs that we construct on compact manifolds with arbitrary topology. Elaborating on a celebrated result by Mañé [18] in the framework of Aubry-Mather theory, we are able to prove that uniqueness of optimal transport plans holds for generic costs in Ck if the marginals are fixed. Theorem 1.8 (Optimal transport between given marginals is generically unique) Fix Borel probability measures on compact manifolds M and N. We give the proofs of Theorem 1.4 in Section 4, of Theorem 1.7 in Section 6, and of Theorem 1.8 in the appendix

Examples and applications

Costs on manifolds

Preliminaries on numbered limb systems

Generic costs in smooth topology

The set V is closed and stratified by the smooth submanifolds