Singularities and their propagation in optimal transport
Piermarco Cannarsa, Wei Cheng, Tianqi Shi, Wenxue Wei
TL;DR
The paper addresses how singularities of potential energy functionals in the Wasserstein space propagate when the underlying dynamics are governed by Hamilton–Jacobi equations in measure spaces, using weak KAM theory and generalized characteristics. It develops random Lax–Oleinik operators and analyzes the regularity of dynamical cost functionals, establishing global propagation of singularities for weak KAM solutions and constructing evolutions along the measure cut locus via irregular Lagrangian semiflows. A variational movement framework is then introduced to realize a semiflow of generalized measure singular characteristics on the cut locus, connecting viscosity solutions, Aubry sets, and transport dynamics. These results enhance the understanding of singularity dynamics in optimal transport and offer a rigorous bridge between Hamilton–Jacobi theory in Wasserstein spaces and mass transport along calibrated curves.
Abstract
In this paper, we investigate the singularities of potential energy functionals \(φ(\cdot)\) associated with semiconcave functions \(φ\) in the Borel probability measure space and their propagation properties. Our study covers two cases: when \(φ\) 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 \(u\).