Generalized differentiation in Wasserstein space and application to multiagent control problem
Rossana Capuani, Antonio Marigonda, Marc Quincampoix
TL;DR
The paper addresses differentiability of functionals on the Wasserstein space $\mathscr P(\mathbb T^d)$, where the lack of a linear structure complicates classical derivatives. It develops a unified notion of admissible $q$-variations and defines derivatives along these variations, establishing comparisons with existing notions under smoothness and introducing sub-/superdifferentials for first-order Hamilton–Jacobi equations in Wasserstein space. The authors then apply this framework to a leader–follower mean-field control problem of Bolza type, formulating a fixed-point approach to the coupled dynamics and proving a Dynammical Programming Principle leading to a viscosity-solution treatment of the associated HJB equation. The results provide a rigorous bridge between various derivative concepts in Wasserstein spaces and yield a principled method for analyzing nonlocal mean-field control problems with viscosity-solutions techniques. This framework enhances the mathematical toolkit for mean-field games and control in spaces of measures and supports analysis of nonlocal, high-dimensional agent systems.
Abstract
Several concepts of generalized differentiation in Wasserstein space have been proposed in order to deal with the intrinsic nonsmoothness arising in the context of optimization problems in Wasserstein spaces. In this paper we introduce a concept of admissible variation encompassing some of the most popular definitions as special cases, and using it to derive a comparison principle for viscosity solutions of an Hamilton Jacobi Bellman equation following from an optimal control of a multiagent systems.