Regularity of the stationary density for systems with fast random switching

We consider the piecewise-deterministic Markov process obtained by randomly switching between the flows generated by a finite set of smooth vector fields on a compact set. We obtain Hörmander-type conditions on the vector fields guaranteeing that the stationary density is: C k whenever the jump rates are sufficiently fast, for any k < ∞ ; unbounded whenever the jump rates are sufficiently slow and lower semicontinuous regardless of the jump rates. Our proofs are probabilistic, relying on a novel application of stopping times.