A survey on weak optimal transport

This talk will present the framework of weak optimal transport which allows to incorporate more general penalizations on elementary mass transports. After recalling general duality results and different optimality criteria, we will focus on recent applications of weak optimal transport. We will see in particular how a weak variant of the squared Wasserstein distance can be used to characterize the Gaussian concentration of measure phenomenon for convex functions or to study the contraction properties of the Brenier map. If time permits we will also discuss a new variant of the weak transport problem which has applications in economy. Based on joint works with P. Choné, M. Fathi, N. Juillet, F. Kramarz, M. Prodhomme, C. Roberto, P-M Samson, Y. Shu and P. Tetali.