On geodesic completeness for Riemannian metrics on smooth probability densities

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 Otto metric, yielding the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-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.