Abstract
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.
Highlights
We recall that the Wasserstein distance W between two Borel probability measures μ0 and μ1 on Ω is defined by the following optimal transportation problem (Kantorovitch relaxed version)
After a brief review of some basic properties of the diffusion equation (1.6), in Section 4 we try to get some insight on the features of the generalized McCann condition (1.10a,b), we recall some basic facts on the metric characterization of contracting gradient flows and their relationships with geodesic convexity borrowed from [AGS05, DS08], and we state our main results Theorems 4.11 and 4.13
In a formal way, the conditions for the displacement convexity of the internal, the potential and the interaction energy, with respect to the geodesics corresponding to the distance (1.8)
Summary
Our aim is to prove rigorously the geodesic convexity of the integral functional U under conditions (1.10a,b) and the metric characterization of the nonlinear diffusion equation (1.9) as the gradient flow of U with respect to the distance Wm,Ω (1.8). Concerning the assumptions on m, its concavity is a necessary and sufficient condition to write the definition of Wm,Ω with a jointly convex integrand [DNS09], which is crucial in many properties of the distance, in particular for its lower semicontinuity with respect to the usual weak convergence of measures. After a brief review of some basic properties of the diffusion equation (1.6), in Section 4 we try to get some insight on the features of the generalized McCann condition (1.10a,b), we recall some basic facts on the metric characterization of contracting gradient flows and their relationships with geodesic convexity borrowed from [AGS05, DS08], and we state our main results Theorems 4.11 and 4.13. At the end of the paper we collect some final remarks and open problems
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
