Mokameeting, Federico Stra (Pisa) & Hugo Lavenant (Orsay), May 24th 2017

Mokameeting, May 24th 2017, 10:30-12:00, room A415

Federico Stra (SNS Pisa)
Title: Finiteness and continuity of multimarginal optimal transport cost
Abstract: In the talk I will present some new results regarding the finiteness and continuity of the multimarginal optimal transport for a repulsive cost. We exhibit also counterexamples showing the sharpness of the finiteness condition. These results are the product of an ongoing collaboration with Maria Colombo and Simone Di Marino. If time permits, I will mention some results about the structure of the optimal plans for the 3-marginal case with Coulomb interaction.

Hugo Lavenant (Orsay):
Title: L^inf bounds in optimal density evolution with congestion via flow interchange techniques
Abstract: We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games with a variational structure. We prove L^inf regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent’s trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar L^inf results have already been found by P.-L. Lions in his course on Mean Field Games, using a proof based on the use of a (very degenerate) elliptic equation on the dual potential (the value function) phi, in the case where the initial and final density were prescribed (planning problem). Here the strategy is highly different, and allows for instance to prove local-in-time estimates without assumptions on the initial and final data, and to insert a potential in the dynamics. (see http://cvgmt.sns.it/paper/3385/)

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