Kantorovich problems and conditional measures depending on a parameter

On Sequential Properties of Spaces of Measures

The multistsochastic Monge--Kantorovich problem on the product $X = \prod_{i=1}^n X_i$ of $n$ spaces is a generalization of the multimarginal Monge--Kantorovich problem. For a given integer number $1 \le k<n$ we consider the minimization problem $\int c d \pi \to \inf$ of the space of measures with fixed projections onto every $X_{i_1} \times \dots \times X_{i_k}$ for arbitrary set of $k$ indices $\{i_1, \dots, i_k\} \subset \{1, \dots, n\}$. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual solution.

The multistochastic Monge-Kantorovich problem.

We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation We obtain broad sufficient conditions for the existence of conditional probabilities measurably depending on a parameter in the case of parametric families of measures and mappings A~particular emphasis is made on the Borel measurability (which cannot be always achieved) Our second main result gives sufficient conditions for the Borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter As a corollary we obtain the Borel measurability with respect to the parameter for conditional measures of optimal plans Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter

В работе дан обзор исследований последнего десятилетия и приведены новые результаты по различным новым модификациям классической задачи Канторовича оптимальной транспортировки мер. Подробно обсуждаются нелинейные задачи Канторовича, задачи с ограничениями на плотности транспортных планов, задачи оптимальной транспортировки с параметрами. Кроме того, рассмотрены связанные с этими новыми постановками вопросы геометрии и топологии пространств мер. 
Библиография: 134 названия.

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Задача Канторовича оптимальной транспортировки мер: новые направления исследований

The Kantorovich optimal transport problemwith a density constraint onmeasures on an infinite-dimensional space is considered. In this setting, the admissible transport plan is nonnegative and majorized by a given constraint function. The existence and the uniqueness of a solution of this problem are proved.

A. N. Doledenok

Papers

On a Kantorovich Problem with a Density Constraint