Caracterization of an optimal solution to Monge-Kantorovitch's problem

CARACTERIZATION OF AN OPTIMAL SOLUTION TO MONGE-KANTOROVITCH'S PROBLEM. - Let P and Q be two probabilities on a complete separable metric space M and c a continuous function on M x M. We consider d(c) (P, Q) = inf {E(c(X, Y)); X has the law P,. Y has the law Q}. We show that, when P is "c-nonatomic" and Q atomic, a pair (X, Y) verifies the relation d(c) (P, Q) = E(c(X, Y)) if and only if Y = f (X), where f is the unique optimal function for (P, Q). We also prove that the condition of c-cyclical monotonicity is necessary and sufficient for f to be optimal. For a Hilbert space we suppose that c(x, Y) = theta(X - Y), theta strictly convexe and P verifies a condition (*). We give the expression of the unique optimal function f for (P, Q).