The Monge problem in Banach spaces
type: Item
instance of: scholarly article
title: The Monge problem in Banach spaces
zbMATH Open document ID: 1120.49041
author: Henri Heinich
published in: Journal of Theoretical Probability
publication date: 2007-02-14T00:00:00Z
review text: 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.
reviewed by: Zagorka Lozanov-Crvenković
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60B11
zbMATH DE Number: 5126773
zbMATH Keywords: cost and profit functions
zbMATH Keywords: Monge-Kantorovich transportation problem
zbMATH Keywords: Monge problem
zbMATH Keywords: optimal coupling
zbMATH Keywords: Köthe space
zbMATH Keywords: Orlicz space
MaRDI profile type: Publication
full work available at URL: s10959-006-0017-2
OpenAlex ID: W1981847004
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DOI: 10.1007/S10959-006-0017-2
description: scientific article; zbMATH DE number 5126773