Nonlinear Kantorovich problems with a parameter
type: Item
instance of: scholarly article
title: Nonlinear Kantorovich problems with a parameter
zbMATH Open document ID: 1504.49063
DOI: 10.26516/1997-7670.2022.41.96
publication date: 2022-11-11T00:00:00Z
full work available at URL: https://mathizv.isu.ru/en/article/file
review text: The paper contains two results concerning the existence of optimal transportation plans for nonlinear Kantorovich problems depending on a parameter. The parameter set is a general Souslin space \(T\), while the base spaces \(X\) and \(Y\) are completely regular Luzin spaces. In the first result the cost function that induce the Kantorovich problem is of the form \(h:T\times X\times Y\times \mathcal{P}(X\times Y)\rightarrow [0,+\infty)\) and the existence of a Borel mapping \(t\mapsto \sigma_t\) from \(T\) to \(\mathcal{P}(X\times Y)\) such that \(\sigma_t\) is optimal for every \(t\) is ensured under natural measurability and regularity assumptions of the involved maps (mainly a lower semicontinuity property of the cost is required). In the second result, the analogous conclusion is reached for cost functions of the form \(H:T\times X\times \mathcal{P}(Y)\) requiring in addition that the function \(p\mapsto H(t,x,p)\) is convex for all \(t,x\).
reviewed by: Nicolò De Ponti
Mathematics Subject Classification ID: 49Q22
Mathematics Subject Classification ID: 28C15
zbMATH DE Number: 7615828
zbMATH Keywords: Kantorovich problem
zbMATH Keywords: optimal plan
zbMATH Keywords: measurability with respect to parameter
author: Vladimir I. Bogachev
author: Ilya I. Malofeev
MaRDI profile type: Publication
OpenAlex ID: W4296051492
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published in: The Bulletin of Irkutsk State University. Series Mathematics
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description: scientific article; zbMATH DE number 7615828