  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.
