The Monge problem in Banach spaces (Q867073)

Let \(E\) be Köthe space, \((M, d)\) be a Polish space, and \(P\), \(Q\) two Borel probabilities on \(M\). Let \((P|Q)\) denote the family of probabilities on \(M^2\) whose marginals are respectively \(P\) and \(Q\) and \(E[X]\) and \({\mathcal L}[X]\) be the expectation and probability distribution of a random variable \(X\). Let \(c: M\times M\to\mathbb{R}^+\) be a cost function. The functional \[ (P, Q)\to{\mathcal W}_c(P, Q)= \inf\{E[c(X, Y)],{\mathcal L}[(X,Y)]\in (P|Q)\} \] is Kantorovich functional, and represents a cost of the transport from \(P\) to \(Q\). In this paper the Kantorovich functional is generalized to Köthe spaces. The Köthe functional \({\mathcal I}_{c,E}\) is defined by \[ {\mathcal I}_{c,E}(P, Q)= \inf\{\| c(X, Y)\|,{\mathcal L}[(X, Y)]\in (P|Q)\}. \] The Monge problem is in finding conditions assuring the existence of a function \(\phi: M\to M\) such that if \({\mathcal L}[X]= P\), then \({\mathcal L}[\phi(X)]= P\) and \({\mathcal W}(P, Q)= E[c(X,\phi(X))]\). The main aim of the paper is to solve the Monge problem for the Köthe functional.