We consider a random dynamical system obtained by switching between the flows
generated by two smooth vector fields on the 2d-torus, with the random
switchings happening according to a Poisson process. Assuming that the driving
vector fields are transversal to each other at all points of the torus and that
each of them allows for a smooth invariant density and no periodic orbits, we
prove that the switched system also has a smooth invariant density, for every
switching rate. Our approach is based on an integration by parts formula
inspired by techniques from Malliavin calculus.Comment: 19 page
