Heterogeneous gradient flows in the topology of fibered optimal transport

TL;DR

This paper introduces a new optimal transport topology on fibered probability measures, enabling the analysis of gradient flows and PDEs with heterogeneities, and demonstrates its applications in long-time behavior and mean-field limits.

Contribution

It develops a novel fibered optimal transport framework with a weak Riemannian structure, addressing gradient flows and applications to complex PDE models.

Findings

The space of measures with fixed marginal becomes a Polish space under the new topology.

A gradient flow theory is established for the fibered transport distance.

Applications include long-time behavior analysis and mean-field limits in PDE models.

Abstract

We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case $\mathbb{R}^d\times \mathbb{R}^d$, where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures $(\mathcal{P}_{2,\nu}(\mathbb{R}^{2d}),W_{2,\nu})$ with fixed marginal $\nu\in \mathcal{P}(\mathbb{R}^d)$ in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.