Djordje Nikolic: Vector Valued Optimal Transport

Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (à la Benamou-Brenier) to  static formulations (à la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product of Euclidean space and graph G, where G is a weighted graph over a finite set of nodes, and the graph geometry strongly influences the associated dynamic and static distances. In this talk, we will present sharp inequalities relating four notions of vector valued optimal transport metrics, which show that the metrics are mutually bi-Hölder equivalent. We will also discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis.