  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
(peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived
using this method. These numerical simulations show that peakon soliton
solutions of the stochastically perturbed CH equation persist and provide an
interesting laboratory for investigating the sensitivity and accuracy of adding
stochasticity to finite dimensional solutions of stochastic partial
differential equations (SPDE). In particular, some choices of stochastic
perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian
stochastic deformations, or CH-SD) allow peakons to interpenetrate and exchange
order on the real line in overtaking collisions, although this behaviour does
not occur for other choices of stochastic perturbations which preserve the
Euler-Poincar\'e structure of the CH equation (parametric stochastic
deformations, or P-SD), and it also does not occur for peakon solutions of the
unperturbed deterministic CH equation. The discussion raises issues about the
science of stochastic deformations of finite-dimensional approximations of
evolutionary PDE and the sensitivity of the resulting solutions to the choices
made in stochastic modelling.
