Structured Continuity Equations in Fibred Wasserstein Spaces

TL;DR

This paper develops a new ODE theory for structured continuity equations in fibred Wasserstein spaces, linking particle systems, PDEs, and meanfield limits in heterogeneous settings.

Contribution

It introduces a comprehensive ODE framework for structured continuity equations in fibred probability spaces, extending meanfield approximation results to nonexchangeable particle systems.

Findings

Established well-posedness results for structured continuity equations.

Derived quantitative meanfield approximation for nonexchangeable particles.

Analyzed topologies induced by fibred and classical Wasserstein metrics.

Abstract

In this article, we develop a comprehensive ODE-theory for structured continuity equations in fibred probability spaces, which represent a class of heterogeneous PDEs arising as the meanfield limit nonexchangeable particle systems. After investigating in depth the topologies induced by the so-called fibred and classical Wasserstein metrics on such probability spaces, we establish quantitative Cauchy-Lipschitz and qualitative Carath\'eodory-Peano well-posedness results for structured continuity equations, along with precise correspondences between this class of evolutions, classical Lagrangian dynamics, and continuity equations. In keeping with what has long been known for exchangeable dynamics, we derive a general meanfield approximation result by solutions of nonexchangeable particle systems, along with a quantitative variant thereof under practically reasonable regularity assumptions on the driving field and initial data.