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**Monge-Kantorovich problem on mass transition and its applications in stochastics.**
*(Russian)*
Zbl 0565.60010

This survey paper considers extreme functionals on the space of probability measures with fixed marginals. Such functionals were first used by Kantorovich in connection with solving the economic problem on the most gainful translocation of masses. Owing to a number of their important applications these functionals have become recognized in probability theory and have been studied as independent objects by many authors.

The main purpose of the paper is to present a variety of problems connected with Monge-Kantorovich extreme problems in stochastics. The main result is the solution of the multivariate Kantorovich duality problem.

The paper includes: Monge optimization problem, Kantorovich problem on the most gainful translocation of masses, Gini dissimilarity index, Wasserstein and Ornstein distances, multivariate Kantorovich problem, duality representations for extreme functionals with fixed marginals (or with fixed difference of marginals), explicit representations of extreme functionals of Kantorovich and Kantorovich-Rubinstein types, inequalities between extreme functionals with given marginals, convergence, applications (uniform convergence of measures, convergence of empirical measures, functional limit theorem, stability of queueing system). The bibliography numbers almost one hundred authors.

For additional acquaintance with marginals problems in probability theory the author recommends the new papers of H. G. Kellerer, Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 399-432 (1984; Zbl 0535.60002) and L. Rüschendorf, ibid. 70, 117-129 (1985; Zbl 0554.60024).

The main purpose of the paper is to present a variety of problems connected with Monge-Kantorovich extreme problems in stochastics. The main result is the solution of the multivariate Kantorovich duality problem.

The paper includes: Monge optimization problem, Kantorovich problem on the most gainful translocation of masses, Gini dissimilarity index, Wasserstein and Ornstein distances, multivariate Kantorovich problem, duality representations for extreme functionals with fixed marginals (or with fixed difference of marginals), explicit representations of extreme functionals of Kantorovich and Kantorovich-Rubinstein types, inequalities between extreme functionals with given marginals, convergence, applications (uniform convergence of measures, convergence of empirical measures, functional limit theorem, stability of queueing system). The bibliography numbers almost one hundred authors.

For additional acquaintance with marginals problems in probability theory the author recommends the new papers of H. G. Kellerer, Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 399-432 (1984; Zbl 0535.60002) and L. Rüschendorf, ibid. 70, 117-129 (1985; Zbl 0554.60024).

### MSC:

60E05 | Probability distributions: general theory |

60F17 | Functional limit theorems; invariance principles |