Stochastic EPDiff Landmark Dynamics

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension we numerically simulate singular solutions (landmarks) of the stochastically perturbed EPDiff equation derived using this method. These numerical simulations show that singular solutions of the stochastically perturbed EPDiff equation persist, and some choices of stochastic perturbations allow landmarks to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for singular solutions of the unperturbed deterministic EPDiff equation. This solution behaviour introduces the possibility of a topological change and may be of importance in registration of noisy images in computational anatomy.