  We consider a stochastic process driven by a linear ordinary differential
equation whose right-hand side switches at exponential times between a
collection of different matrices. We construct planar examples that switch
between two matrices where the individual matrices and the average of the two
matrices are all Hurwitz (all eigenvalues have strictly negative real part),
but nonetheless the process goes to infinity at large time for certain values
of the switching rate. We further construct examples in higher dimensions where
again the two individual matrices and their averages are all Hurwitz, but the
process has arbitrarily many transitions between going to zero and going to
infinity at large time as the switching rate varies. In order to construct
these examples, we first prove in general that if each of the individual
matrices is Hurwitz, then the process goes to zero at large time for
sufficiently slow switching rate and if the average matrix is Hurwitz, then the
process goes to zero at large time for sufficiently fast switching rate. We
also give simple conditions that ensure the process goes to zero at large time
for all switching rates.
