Abstract. 
By the Krylov-Bogoliubov method, a Markov process that is Feller
and evolves on a compact state space M admits at least one invariant
probability measure.  If M contains an invariant closed set M_0, one can ask 
whether there are invariant measures that assign full measure to the complement
M \ M_0.  We present conditions, due to Benaïm, under which this is the case.
Then, we apply these conditions to Markov processes characterized by
Poissonian switching between deterministic vector fields, and discuss
sufficient conditions for exponential convergence to equilibrium in 
total variation.  The last part is based on work with Michel Benaïm and 
Edouard Strickler.