Heterogeneous gradient flows in the topology of fibered optimal transport

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 R-d x 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 (P-2,P-nu (R-2d), W-2,W-nu) with fixed marginal nu is an element of P(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.

Automated summary

In this article, we introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost between different fibers. We illustrate this construction in the Euclidean case and show that even though the cost is infinitely valued and discontinuous, the space of probability measures with fixed marginal conditions is still a Polish space with a weak Riemannian structure. The article also explores the abstract theory of gradient flows with respect to the new topology, applications to evolution PDEs with heterogeneities, and the long-time behavior and global-in-time mean-field limits in a specific alignment model.