Expressivity of autonomous Riemannian Hamiltonian transport

Characterize the pairs of initial and target densities that can be connected exactly, or to prescribed accuracy, by an autonomous scalar potential with an isotropic non-Dirac initial momentum distribution.

Background

The paper establishes exact finite-time transport for isotropic Gaussian targets with a matched quadratic potential and develops a local harmonic approximation around non-degenerate modes of general targets. However, these results do not determine the global expressive power of autonomous Hamiltonian flows driven by a scalar potential and isotropic random momenta. In particular, it remains unresolved which pairs of Riemannian densities can be transported exactly and which can only be approximated, as well as how the attainable accuracy depends on the potential, momentum distribution, integration time, and underlying geometry.