Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph

TL;DR

This paper develops a gradient flow formulation for diffusion equations on metric graphs within the Wasserstein space, establishing a Benamou-Brenier formula and linking McKean-Vlasov equations to gradient flows.

Contribution

It proves a Benamou-Brenier formula for Wasserstein distance on metric graphs and formulates McKean-Vlasov equations as gradient flows in this setting.

Findings

Established equivalence of static and dynamical optimal transport on metric graphs.

Formulated McKean-Vlasov equations as gradient flows of free energy.

Overcame challenges due to geodesic branching and entropy semi-convexity.

Abstract

This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan-Kinderlehrer-Otto, we show that McKean-Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.