We describe a dissipation principle/variational principle which may be useful in modeling motion in small viscous systems and provide brief illustrations to brownian motor or molecular rachet situations which are found in intracellular transport. Monge-Kantorovich mass transport and Wasserstein metric play an interesting role in these developments. Some properties of the system that ensure the ...

Fabien Silly,*,†,‡ Adam Q. Shaw,† Martin R. Castell,† G. A. D. Briggs,† Manuela Mura,§ Natalia Martsinovich,§ and Lev Kantorovich§,⊥ Department of Materials, UniVersity of Oxford, Parks Road, Oxford OX1 3PH, U.K., Zernike Institute for AdVanced Materials, RijksuniVersiteit Groningen, Nijenborgh 4, NL-9747 AG, Groningen, The Netherlands, and Department of Physics, King’s College London, The Stra...

We study the two-sample matching problem and its connections with the Monge-Kantorovich problem of optimal transportation of mass. We exploit this connection to obtain moderate and large deviation principles. For the classical problem on the unit square we present a conjecture which, if true, yields an explicit formula for the rate function.

We unify Kantorovich and Young’s theory by formulating the Monge mass transfer problem as a variational problem involving Young measures. This is done thanks to the disintegration theorem and a density result on the set of all Borel measures on R ×R with fixed marginals. We mention applications, such as Bernoulli convolution.

A new technique is used instead of the classical majorant principle to analyze the R-order of convergence of the Newton process when more general conditions than the Kantorovich ones are considered. © 2005 Elsevier B.V. All rights reserved. MSC: 47H17; 65J15

1 Primal and dual problems 2 1.1 Kantorovich and Monge problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The case c(x, y) = |x− y| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 c(x, y) = h(x− y) with h strictly convex and the existence of an optima...

The well-known Kantorovich technique based on majorizing sequences is used to analyse the convergence of Newton’s method when it is used to solve nonlinear Fredholm integral equations. In addition, we obtain information about the domains of existence and uniqueness of a solution for these equations. Finally, we illustrate the above with two particular Fredholm integral equations.