Optimal curves and applications with values in Wasserstein space.

Summary The Wasserstein space is the set of probability measures defined on a fixed domain and provided with the quadratic Wasserstein distance. In this work, we study variational problems in which the unknowns are Wasserstein-valued applications. When the starting space is a segment, i.e. when the unknowns are Wasserstein-valued curves, we are interested in models where, in addition to the action of the curves, terms penalizing congestion configurations are present. We develop techniques to extract regularity from the interaction between optimal density evolution (action minimization) and congestion penalization, and apply them to the study of mean-field games and the variational formulation of Euler equations.When the starting space is not only a segment but a domain of the Euclidean space, we consider only the Dirichlet problem, i.e. the minimization of the action (which can be called the Dirichlet energy) among all applications whose values on the edge of the starting domain are fixed. The solutions are called the harmonic applications with values in the Wasserstein space. We show that the different definitions of the Dirichlet energy present in the literature are in fact equivalent. that the Dirichlet problem is well posed under rather weak assumptions. that the superposition principle is defeated when the starting space is not a segment. that one can formulate a kind of maximum principle. and we propose a numerical method to compute these harmonic applications.