Abstract. 
Let D be a finite collection of smooth vector fields on R^n.
Given an initial vector field u from D, we flow along u for a random time.
Then, we switch to a new vector field that is randomly selected from the
remaining vector fields in D.  We flow along this new vector field for a
random time until another switch occurs.  Reiterating this procedure, we
obtain a Markov process on R^n x D.  If the associated Markov semigroup has
an absolutely continuous invariant measure with density rho, we can study
the projections (rho_u)_{u in D} defined by rho_u(x) = rho(x,u).  In
particular, we ask whether these projections have smooth representatives
and at which points singularities may appear.  If the dimension n equals 1,
under mild assumptions on D, singularities may only occur at critical
points of the vector fields and the projections are smooth at noncritical
points. In dimension 2, singularities may also form away from critical
points. Whether and where singularities occur depends critically on the
switching rate.  The talk is based on work with Yuri Bakhtin, Sean Lawley
and Jonathan Mattingly.