Nonlinear Kantorovich problems with a parameter
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 \(t,x\).
scientific article; zbMATH DE number 7615828
scientific article