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BIRS Workshop Lecture Videos

Entropic optimal transport and nonlinear PDE's Savaré, Giuseppe


We discuss two examples of "dynamical optimal transport problems", whose formulations involve a relative entropy functional. The first case is related to the Hellinger-Kantorovich distance and induces an interesting geometric structure on the space of positive measures with finite (but possibly different) mass. In particular, contraction estimates of nonlinear flows are strongly related to geodesic convexity of the generating entropy functionals. In the second example an entropy functional penalizes the density of the connecting measures with respect to a given reference measure (typically the Lebesgue one) and leads to a first order "mean field planning" problem, which is classicaly formulated by a continuity equation and a Hamilton Jacobi equation with a nonlinear coupling. In this case, the variational approach and the displacement convexity of the entropy functionals (in the usual sense of optimal transport) provide crucial tools to give a precise meaning to the PDE system and to prove the existence of a solution.

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