The Multistochastic Monge-Kantorovich Problem

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 thesis, we study the basic properties of the multistochastic problem and its connection to other types of optimal transportation problems. We show that the multistochastic problem is a particular case of more general optimal transportation problem with additional linear constraints. Using this inclusion, we prove the absence of duality gap without the assumption of compactness of spaces. Finally, we consider some explicit solutions to multistochastic problems which show that the properties of the optimal transport plans in the multistochastic case can be much more complicated than in the classical case.