Optimal solution of a Monge-Kantorovitch transportation problem

Suppose that P and Q are probabilities on a separable Banach space E. 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 : E → E, 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. Thenwe extend this main result to any law Q. The proofs are based on a relationship between optimal couplings and nonlinear equations.