Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${\mathbb{R}}^{n}$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step are obtained by looking at perturbation of the form ${\rho}_{\epsilon}=(1-\epsilon )\rho +\epsilon \tilde{\rho}$ instead of ${\rho}_{\epsilon}={(id+\epsilon \xi )}_{\#}\rho $. The ideas to make this approach rigorous are presented in the case of a Fokker-Planck equation, possibly with an interaction term, and then the paper is concluded by a section, where this method is applied to the original problem of crowd motion (referring to a recent paper in collaboration with B. Maury and A. Roudneff-Chupin for the details).

@article{SEDP_2009-2010____A27_0, author = {Santambrogio, Filippo}, title = {Gradient flows in Wasserstein spaces and applications to crowd movement}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2009-2010}, note = {talk:27}, language = {en}, url = {http://www.numdam.org/item/SEDP_2009-2010____A27_0} }

Santambrogio, Filippo. Gradient flows in Wasserstein spaces and applications to crowd movement. Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 27, 16 p. http://www.numdam.org/item/SEDP_2009-2010____A27_0/

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