Regularity of invariant densities for random switching between two linear odes in Rd

In a paper entitled singularities of invariant densities for random switching between two linear odes in 2D, Bakhtin et al [5], consider a Markov process obtained by random switching between two stable linear vector fields in the plane and characterize the singularities of the invariant density in terms of the switching and contraction rates. This paper considers a generalization of this model obtained by random switching between two stable linear vector fields in Rd and provides sufficient conditions ensuring that the invariant distribution is absolutely continuous and has a Cr density. In dimension greater than 3 it provides, to the best of our knowledge, the first fully non-elliptic example of random switching for which quantitative conditions guaranteeing smoothness of the invariant density can be proved.