The multistochastic Monge-Kantorovich problem

TL;DR

This paper explores the multistochastic Monge-Kantorovich problem, a generalization involving multiple spaces and projections, analyzing its fundamental properties such as well-posedness and dual solutions.

Contribution

It introduces and studies the basic properties of the multistochastic Monge-Kantorovich problem, including existence, boundedness, and continuity of solutions.

Findings

Established well-posedness of the problem

Proved existence of dual solutions

Analyzed boundedness and continuity of dual solutions

Abstract

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.