On some generalisations of Optimal Transport problem

This manuscript is devoted to the study of some generalisation of Optimal Transport problem as well as some applications arising in Mathematical Finance, Game Theory and Quantum Physics. More specifically, we first consider the Multi-Marginal Optimal Transport (MMOT) problem by a careful refinement of the existing theory for the one dimensional case, we provide an explicit characterisation of spectral risk measures. We also study the entropic regularization of MMOT and in particular derive rate of convergence of the entropic cost to the multi-marginal cost as the regularization parameter vanishes. Moreover, in the framework of entropic transport we provide an ODE characterisation of the discrete problem. Concerning the application of Optimal Transport to Density Functional Theory (DFT) we propose a generalisation of MMOT to the grand canonical ensemble, that is we allow the number of marginals to vary. Furthermore, we also focus on a sparse approximation of the Lieb functional via moments constraints. Finally, we study unequal dimensional Optimal Transport and some related variational problems arising in Game Theory and Economics as, for instance, Cournot-Nash equilibria or the hedonic pricing problem. The last part of the manuscript is then devoted to develop the theory of some metrics involving singular measures.