Structured Continuity Equations in Fibred Wasserstein Spaces

TL;DR

This work extends mean-field theory to nonexchangeable, fibred particle systems by developing a full ODE theory for structured continuity equations on fibred probability spaces. It introduces fibred Wasserstein spaces with the distance $W_{\pi,p}$, establishes minimal regularity conditions for well-posedness (Cauchy–Lipschitz and Carathéodory–Peano), and connects fibred CE to both Lagrangian dynamics and classical CE via disintegration. The paper also develops rigorous particle-approximation results, including a two-level discretisation that yields quantitative convergence rates (notably $N^{-1/3}$ under mild regularity) for mean-field limits of nonexchangeable systems. These contributions provide a robust framework for modeling heterogeneous multi-agent dynamics with practical impact in neuroscience, biology, and networked systems where identity and structure matter.

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 Carathéodory-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.

Figures (2)

• Figure 1: We display a 3D illustration of a certain choice $\boldsymbol{\mu}^0 \in \mathcal{P}_{\pi,1}([0,1] \times [0,2])$ (left) and of the corresponding piecewise averaged measure $\boldsymbol{\mu}^{0,n} \in \mathcal{P}_{\pi,1}([0,1] \times [0,2])$ (right) for $n=20$. Their common marginal $\pi \in \mathcal{P}([0,1])$ is represented by the black curve in the plane $x=0$. Here $\boldsymbol{\mu}^0$ is taken to be absolutely continuous with respect to the Lebesgue measure on the product space, and the colour and height of the surfaces represent the densities of both measures with respect to the latter.
• Figure 2: 2D illustration of the same choice $\boldsymbol{\mu}^0$ (left) and the corresponding $\boldsymbol{\mu}^{0,n}$ (centre) with $n=20$, along with a sampling of the random empirical measure $\boldsymbol{\mu}^{0,n,m} \in \mathcal{P}_{\pi,1}([0,1] \times [0,2])$ (right) with $m=40$. In the first two plots, the colour represents the densities of $\boldsymbol{\mu}^0$ and $\boldsymbol{\mu}^{0,n}$ with respect to the Lebesgue measure, whereas in the third plot the horizontal black lines symbolise measures of the form $\pi_{\llcorner I_k^n} \times \delta_{x_{(k-1)m+\ell}^{0,{n,m}}} \in \mathcal{P}(I_k^n \times \mathbb{R})$.

Theorems & Definitions (95)

• Definition 2.1: Measurable and Carathéodory mappings
• Definition 2.2: Spaces of metric-valued and Bochner integrable functions
• Proposition 2.3: Some handy measurability results
• Theorem 2.4: A functional variant of Lusin's theorem
• Theorem 2.5: Disintegration theorem
• Theorem 2.6: Relative norm compactness in Bochner spaces
• Proposition 2.7: Miscellaneous facts about Wasserstein spaces
• Definition 2.8: Fibred narrow topology and stable topology
• Remark 2.9: Comparison between the stable, narrow and fibred narrow topologies
• Proposition 2.10: Topological properties of the fibred narrow topology