  We mimic the stochastic Hamiltonian reduction of Lazaro-Cami and Ortega [17,
18] for the case of certain non-holonomic systems with symmetries.
  Using the non-holonomic connection it is shown that the drift of the
stochastically perturbed $n$-dimensional Chaplygin ball is a certain gradient
of the density of the preserved measure of the deterministic system.
