Transport type metrics on the space of probability measures involving singular base measures
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We develop the theory of a metric, which we call the ν-based Wasserstein metric and denote by W_ν, on the set of probability measures 𝒫(X) on a domain X ⊆ℝ^m. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure ν and is relevant in particular for the case when ν is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The ν-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to ν; 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 ν, the ν-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when ν is sufficiently regular. When ν concentrates on a lower dimensional submanifold of ℝ^m, we prove that the variational problem in the definition of the ν-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 ν satisfies a strengthening of the generalized nestedness condition introduced in <cit.>. 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.
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