  We study stochastically perturbed non-holonomic systems from a geometric
point of view. In this setting, it turns out that the probabilistic properties
of the perturbed system are intimately linked to the geometry of the constraint
distribution. For $G$-Chaplygin systems, this yields a stochastic criterion for
the existence of a smooth preserved measure. As an application of our results
we consider the motion planning problem for the noisy two-wheeled robot and the
noisy snakeboard.
