Inserted: 27 nov 2010
This paper is meant to appear in the proceedings of the PDE Seminar of Ecole Polytechnique, where I gave a talk about gradient flow in Wasserstein spaces and applications to crowds. In particular, I detail(ed) the approach by ``vertical'' perturbations for optimality conditions that is complementary to the classical one. Here I only show easy examples where such an approach quickly gives an existence result without any convexity assumptions, and the intention is to undergo a deeper analysis later on, in a possible forthcoming paper
Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in Rn, 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 ``additive'' form. 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).
Keywords: Optimal transport, minimizing movements, Congestion, Necessary conditions for optimality