We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.

Let X be a separable, complete metric space and P p (X) be the space of Borel probability measures with finite moment of order p > 1, metrized by the Wasserstein distance. In this paper we prove that every absolutely continuous curve with finite p-energy in the space P p (X) can be represented by a Borel probability measure on C([0, T]; X) concentrated on the set of absolutely continuous curves with finite p-energy in X. Moreover this measure satisfies a suitable property of minimality which entails an important relation on the energy of the curves. We apply this result to the geodesics of P p (X) and to the continuity equation in Banach spaces.

This paper focuses on the role of a government of a large population of interacting agents as a meanfield optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a Partial Differential Equation of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a Γ -convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn-Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution -non-negativity, conservation of the total mass and dissipation of the energy -are automatically guaranteed by the construction from minimizing movements in the energy landscape.

Abstract.We study existence and approximation of non-negative solutions of partial differential equations of the typewhere A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition,, we show that u is the "gradient flow" of φ with respect to the 2-Wasserstein distance between probability measures on the space R n , endowed with the Riemannian distance induced by A −1 . In the case of uniform convexity of V , long time asymptotic behaviour and decay rate to the stationary state for solutions of equation (0.1) are studied. A contraction property in Wasserstein distance for solutions of equation (0.1) is also studied in a particular case.Mathematics Subject Classification. 35K55, 35K15, 35B40.