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

We introduce a new metric ($W_\nu$ $\nu$-based Wasserstein metric) on the set of probability measures on $X \subseteq \mathbb{R}^m$, based on a slight refinement of the notion of generalized geodesics with respect to a base measure $\nu$, relevant in particular for the case when $\nu$ is singular with respect to $m$-dimensional Lebesgue measure. $W_\nu$ 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 with respect to $\nu$, and through limits of certain multi-marginal optimal transport problems. We also introduce a class of metrics which are dual in a certain sense to $W_\nu$ on the set of measures which are absolutely continuous with respect to a second fixed based measure $\sigma$.As we vary the base measure $\nu$, $W_\nu$ 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 $\mathbb{R}^m$, 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 $\mu$ such that optimal transport between $\mu$ and $\nu$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}. We also present two applications of the ideas introduced here. First, our dual metric is used to prove convergence of an iterative scheme to solve a variational problem arising in game theory. We also use the multi-marginal formulation to characterize solutions to the multi-marginal problem by an ordinary differential equation, yielding a new numerical method for it.