Singularities and their propagation in optimal transport

TL;DR

This paper studies the behavior and propagation of singularities in potential energy functionals related to semiconcave functions and weak KAM solutions within the Wasserstein space, revealing global propagation and solutions along the cut locus.

Contribution

It establishes the global propagation of singularities for weak KAM solutions and introduces solutions evolving along the cut locus governed by an irregular Lagrangian semiflow.

Findings

Singularities of weak KAM solutions propagate globally.

Existence of solutions evolving along the cut locus.

Propagation properties are linked to Hamilton-Jacobi equations in Wasserstein space.

Abstract

In this paper, we investigate the singularities of potential energy functionals \(\phi(\cdot)\) associated with semiconcave functions \(\phi\) in the Borel probability measure space and their propagation properties. Our study covers two cases: when \(\phi\) is a semiconcave function and when \(u\) is a weak KAM solution of the Hamilton-Jacobi equation \(H(x, Du(x)) = c[0]\) on a smooth closed manifold. By applying previous work on Hamilton-Jacobi equations in the Wasserstein space, we prove that the singularities of \(u(\cdot)\) will propagate globally when \(u\) is a weak KAM solution, and the dynamical cost function \(C^t\) is the associated fundamental solution. We also demonstrate the existence of solutions evolving along the cut locus, governed by an irregular Lagrangian semiflow on the cut locus of \(u\).