  Vector fields that are discontinuous on codimension-one surfaces are known as
Filippov systems and can have attracting periodic orbits involving segments
that are contained on a discontinuity surface of the vector field. In this
paper we consider the addition of small noise to a general Filippov system and
study the resulting stochastic dynamics near such a periodic orbit. Since a
straight-forward asymptotic expansion in terms of the noise amplitude is not
possible due to the presence of discontinuity surfaces, in order to
quantitatively determine the basic statistical properties of the dynamics, we
treat different parts of the periodic orbit separately. Dynamics distant from
discontinuity surfaces is analyzed by the use of a series expansion of the
transitional probability density function. Stochastically perturbed sliding
motion is analyzed through stochastic averaging methods. The influence of noise
on points at which the periodic orbit escapes a discontinuity surface is
determined by zooming into the transition point. We combine the results to
quantitatively determine the effect of noise on the oscillation time for a
three-dimensional canonical model of relay control. For some parameter values
of this model, small noise induces a significantly large reduction in the
average oscillation time. By interpreting our results geometrically, we are
able to identify four features of the relay control system that contribute to
this phenomenon.
