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The geometric approach to optimal transport and information theory hastriggered the interpretation of probability densities as aninfinite-dimensional Riemannian manifold. The most studied Riemannianstructures are Otto's metric, yielding the $L^2$-Wasserstein distance ofoptimal mass transport, and the Fisher--Rao metric, predominant in the theoryof information geometry. On the space of smooth probability densities, none ofthese Riemannian metrics are geodesically complete---a property desirable forexample in imaging applications. That is, the existence interval for solutionsto the geodesic flow equations cannot be extended to the whole real line. Herewe study a class of Hamilton--Jacobi-like partial differential equationsarising as geodesic flow equations for higher-order Sobolev type metrics on thespace of smooth probability densities. We give order conditions for globalexistence and uniqueness, thereby providing geodesic completeness. The systemwe study is an interesting example of a flow equation with loss of derivatives,which is well-posed in the smooth category, yet non-parabolic and fullynon-linear. On a more general note, the paper establishes a link betweengeometric analysis on the space of probability densities and analysis ofEuler-Arnold equations in topological hydrodynamics.

On Geodesic Completeness for Riemannian Metrics on Smooth Probability Densities