  We study Brownian motion on the space of distinct landmarks in
$\mathbb{R}^d$, considered as a homogeneous space with a Riemannian metric
inherited from a right-invariant metric on the diffeomorphism group. As of yet,
there is no proof of long-time existence of this process, despite its
fundamental importance in statistical shape analysis, where it is used to model
stochastic shape evolutions. We make some first progress in this direction by
providing a full classification of long-time existence for configurations of
exactly two landmarks, governed by a radial kernel. For low-order Sobolev
kernels, we show that the landmarks collide with positive probability in finite
time, whilst for higher-order Sobolev and Gaussian kernels, the landmark
Brownian motion exists for all times. We illustrate our theoretical results by
numerical simulations.
