Kantorovich problems and conditional measures depending on a parameter (Q2304880)

Let us recall the classical Kantorovich optimal transportation problem. Given two probability spaces \((X,\mathcal{B}_X,\mu)\) and \((Y,\mathcal{B}_Y,\nu)\) and a nonnegative cost function \(h:X \times Y\rightarrow \mathbb{R}^+\), the Kantorovich problem consists to find the value of \(K_h(\mu, \nu) = \inf_{\pi \in \Pi(\mu, \nu)} \int h d\pi\), where \(\Pi\) denotes the set of couplings whose marginals are \(\mu\) and \(\nu\). The authors consider the Kantorovich problem with a parameter; namely, given \((T,\mathcal{T})\) a measurable space and for all \(t\in T\) let us consider marginal probability measures \(\mu_t\) and \(\nu_t\) which depend on t measurably. The question is whether the infimum in the Kantorovich problem depends measurably on a parameter \(t\) and whether there are optimal measures \(\pi_t\) depending measurably on \(t\). Among other results, the authors are able to prove that under suitable conditions, the function \(t \mapsto K_{h_t}(\mu_t, \nu_t)\) is Borel and there are optimal measures \(\pi_t\in \Pi(\mu_t,\nu_t)\) for \(h_t\) such that the mapping \(t \mapsto \pi_t\) is Borel measurable.