Shape structure preservation for Riemannian Brownian motion on landmark manifolds

Determine whether, on the landmark configuration manifold S_n (the space of n distinct points in R^d endowed with the LDDMM-induced Riemannian metric), the Riemannian Brownian motion preserves shape structure for n > 2; specifically, prove or refute that starting from an initial configuration in S_n, the diffusion avoids landmark collisions and remains within S_n for all t ≥ 0.

Background

The paper formalizes desirable properties for stochastic shape processes, including a shape structure preservation property requiring that a process starting from a shape remains within the shape space for all finite times. On landmark manifolds S_n, this amounts to preventing collisions of landmarks under the dynamics.

Riemannian Brownian motion is the canonical diffusion on a Riemannian manifold with generator 1/2 Δ. In the landmark setting with the metric induced by LDDMM-type kernels, the process has been studied for parameter estimation and long-time existence, but it remains unclear whether collisions can be ruled out when n > 2, which is necessary to ensure that the process stays in S_n. This uncertainty is explicitly flagged by the authors as an open question.