Diffeomorphism-invariant metrics on densities for manifolds with boundary

Characterize all (weak) Riemannian metrics on the space of smooth probability densities \mathfrak{Dens}(M) on a compact manifold M with boundary that are invariant under the natural action of the diffeomorphism group \mathfrak{D}(M) (via pullbacks or pushforwards). In particular, determine whether an analogue of the Fisher–Rao uniqueness theorem holds in this setting and, if so, describe the complete class of diffeomorphism-invariant metrics on \mathfrak{Dens}(M) for manifolds with boundary.

Background

In the boundaryless case (compact manifolds without boundary, dimension n ≥ 2), the book states a uniqueness result (Theorem thm:invarFR): any (weak) right-invariant Riemannian metric on the diffeomorphism group \mathfrak{D}(M) that descends to the quotient of densities \mathfrak{Dens}(M) is a constant multiple of the Fisher–Rao metric. This places the Fisher–Rao metric as essentially the unique diffeomorphism-invariant metric on densities in that setting.

The remark points out that the analogous classification problem when the underlying manifold has a boundary remains unresolved. The open problem asks for a complete description of diffeomorphism-invariant metrics on \mathfrak{Dens}(M) in the presence of a boundary, clarifying whether Fisher–Rao retains uniqueness and, if not, identifying the full family of admissible invariant metrics.