Mappings valued in the Wasserstein space and their links with Q-valued functions

Lundi 19 novembre 2018 14:00-15:00 - Hugo Lavenant - LMO

Résumé : The Wasserstein space, which is the space of probability measures endowed with the so-called (quadratic) Wasserstein distance coming from optimal transport, can formally be seen as a Riemannian manifold of infinite dimension. In a first part, we propose, through a variational approach, a definition of harmonic mappings defined over a domain of R^n and valued in the Wasserstein space. As the latter has nonnegative curvature, we cannot rely on the theory of Koorevaar, Schoen and Jost of harmonic mappings valued in metric spaces and we use arguments based on optimal transport instead. In a second part, we will explain why the object we introduced cannot be seen as the limit Q \to + \infty of Q-valued functions, the latter being introduced by Almgren in a completely different context. The obstruction will reveal the absence of a Lagrangian point of view for mappings valued in the Wasserstein space.

Lieu : IMO ; salle 3L8.

Mappings valued in the Wasserstein space and their links with Q-valued functions  Version PDF