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## The Monge-Kantorovich problem: Achievements, connections, and perspectives

This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse applications of these problems in non-linear analysis, probability theory, and differential geometry are discussed.

The aim of this paper is to show that the Kantorovich problem, well known in models of economics and very intensively studied in probability theory in recent years, can be viewed as the basis of some constructions in the theory of belief functions. We demonstrate this by analyzing specialization relation for finitely defined belief functions and belief functions defined on reals. In addition, for such belief functions, we consider the Wasserstein metric and study its connections to disjunctions of belief functions.

We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and

the cost function $c = \min\{l_1, \ldots, l_n\}$

that equals the minimum of a finite number $n$ of affine functions $l_i$

satisfying certain non-degeneracy assumptions. We prove that the problem

is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated

on the union of $n$ products $I_i \times J_i$, where $\{I_i\}$ and $\{J_i\}$

are partitions of the real line into unions of disjoint connected sets.

The families of sets $\{I_i\}$ and $\{J_i\}$ have the following properties: 1) $c=l_i$ on $I_i \times J_i$,

2) $\{I_i\}, \{J_i\}$ is a couple of partitions solving an auxiliary $n$-dimensional extremal problem.

The result is partially generalized to the case of more than two marginals.

We consider probability measures on R∞ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. We show in the latter case that existence problem for optimal transportation is closely related to ergodicity of the target measure. In particular, we prove existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.

Numerous applications of the optimal transportation theory in finite-dimensional spaces have been found during the last decade. They include differential equations, probability theory, and geometry. The situation in infinite-dimensional spaces has been much less studied. However, some partial results on existence, uniqueness, and regularity have been obtained in recent papers of D. Feyel, A.S. Ustunel, M. Zakai, and the authors.

In this paper we study regularity properties of the infinite-dimensional transportation of measures on the Wiener space, where the transportation cost is given by the integral squared Cameron-Martin norm, the target measure is the Wiener measure, and the source measure is absolutely continuous with respect to the Wiener measure. Assuming that the density has the finite Fisher information (belongs to a certain Sobolev class), we prove that the potential of the corresponding optimal transportation belongs to the second (weighted) Sobolev space W^{2,2}. Some estimates involving higher-order derivatives are given. This result settles a long-standing problem and is of principal importance for the whole area of infinite-dimensional optimal transport and its applications in stochastic analysis and measure theory in infinite dimensions.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.