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On the space of probability densities, we extend the Wasserstein geodesics tothe case of higher-order interpolation such as cubic spline interpolation.After presenting the natural extension of cubic splines to the Wassersteinspace, we propose a simpler approach based on the relaxation of the variationalproblem on the path space. We explore two different numerical approaches, onebased on multi-marginal optimal transport and entropic regularization and theother based on semi-discrete optimal transport.

Second order models for optimal transport and cubic splines on the Wasserstein space