Transport type metrics on the space of probability measures involving singular base measures

We develop the theory of a metric, which we call the nu-based Wasserstein metric and denote by W_nu, on the set of probability measures mathcal P(X) on a domain X subseteq ℝm. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure nu and is relevant in particular for the case when nu is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The nu-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to nu; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure nu, the nu-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when nu is sufficiently regular. When nu concentrates on a lower dimensional submanifold of ℝm, we prove that the variational problem in the definition of the nu-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures μ such that optimal transport between μ and nu satisfies a strengthening of the generalized nestedness condition introduced in citeMcCannPass20.We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.