Optimal solution of a Monge-Kantorovitch transportation problem

✍ Scribed by Taoufiq Abdellaoui

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 631 KB
• Volume: 96
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(98)00094-6

✦ Synopsis

Suppose that P and Q are probabilities on a separable Banach space ~:. It is known that if (P, Q) satisfies certain regularity conditions and a random variable X has law P, then there exists a function f : ~: ~ IF, such that the function f(X) has the law Q and the random pair (X,f(X)) is an optimal coupling for the Monge--Kantorovitch problem. In this paper we provide an approximation of the function f when the law Q is discrete. Then we extend this main result to any law Q. The proofs are based on a relationship between optimal couplings and nonlinear equations. ¢~