A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows

In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions, and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.Comment: Final version, to appear in SIAM SIM

Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces

In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context of SCE Density Functional Theory introduced in "Strong-interaction limit of density-functional theory" by M. Seidl.Comment: 18 page

Riemannian Ricci curvature lower bounds in metric measure spaces with σ\sigma-finite measure

Using techniques of optimal transportation and gradient flows in metric spaces, we extend the notion of Riemannian Curvature Dimension condition $RCD(K,\infty)$ introduced (in case the reference measure is finite) by Giuseppe Savare', the first and the second author, to the case the reference measure is $\sigma$-finite; in this way the theory includes natural examples as the euclidean $n$-dimensional space endowed with the Lebesgue measure, and noncompact manifolds with bounded geometry endowed with the Riemannian volume measure. Another major goal of the paper is to simplify the axiomatization of $RCD(K,\infty)$ (even in case of finite reference measure) replacing the assumption of strict $CD(K,\infty)$ with the classic notion of $CD(K,\infty)$.Comment: 42 pages; final version (minor changes to the old one, in particular we added some more preliminaries and explanations) to be published in Transactions of the AM

Electrostatic energy calculation for the interpretation of scanning probe microscopy experiments

We discuss the correct expression for the classical electrostatic energy used while analysing scanning probe microscopy (SPM) experiments if either a conducting tip or a substrate or both are used in the experiment. For this purpose a general system consisting of an arbitrary arrangement of finite metallic conductors at fixed potentials (maintained by external sources) and a distribution of point charges in free space are considered using classical electrostatics. We stress the crucial importance of incorporating into the energy the contribution coming from the external sources (the `battery'). Using the Green function of the Laplace equation, we show in a very general case that the potential energy of point charges which are far away from metals is equally shared by their direct interaction and the polarization interaction due to charge induced in metals by the remote charges (the image interaction). When the charges are located close to the metals, there is an additional negative term in the energy entirely due to image interaction. The exact Hamiltonian of a quantum system interacting classically with polarized metal conductors is derived and its application in the Hartree-Fock and the density functional theories is briefly discussed. As an illustration of the theory, we consider an interaction of several point charges with a metal plane and a spherical tip, based on the set-up of a real SPM experiment. We show the significance of the image interaction for the force imposed on the tip

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4, Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop. 4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6 simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients, still equivalent to all other ones, has been propose