Title: On Geodesic Completeness for Riemannian Metrics on Smooth Probability Densities,
Abstract: 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 Otto's metric, yielding the $L^2$-Wasserstein distance of
optimal mass transport, and the Fisher--Rao metric, predominant in the theory
of information geometry. On the space of smooth probability densities, none of
these Riemannian metrics are geodesically complete---a property desirable for
example in imaging applications. That is, the existence interval for solutions
to the geodesic flow equations cannot be extended to the whole real line. Here
we study a class of Hamilton--Jacobi-like partial differential equations
arising as geodesic flow equations for higher-order Sobolev type metrics on the
space of smooth probability densities. We give order conditions for global
existence and uniqueness, thereby providing geodesic completeness. The system
we 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 fully
non-linear. On a more general note, the paper establishes a link between
geometric analysis on the space of probability densities and analysis of
Euler-Arnold equations in topological hydrodynamics.