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The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are the Wasserstein metric of optimal mass transport, and the Fisher--Rao metric. On the space of smooth probability densities, none of these Riemannian metrics are geodesically completes. Here we study geodesic equations for higher-order Sobolev metrics. We give order conditions for global existence and uniqueness, thereby providing geodesic completeness.
This presentation is part of Minisymposium “MS63 - Geometric methods for shape analysis with applications to biomedical imaging and computational anatomy”
organized by: Martin Bauer (Florida State University) , Nicolas Charon (Johns Hopkins University) .