Regularity of the stationary density for systems with fast random switching

Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> ; 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.