  We prove existence and uniqueness of the invariant measure and exponential
mixing in the total-variation norm for a class of stochastic differential
equations driven by degenerate compound Poisson processes. In addition to mild
assumptions on the distribution of the jumps for the driving process, the
hypotheses for our main result are that the corresponding control system is
dissipative, approximately controllable and solidly controllable. The solid
controllability assumption is weaker than the well-known parabolic H\"ormander
condition and is only required from a single point to which the system is
approximately controllable. Our analysis applies to Galerkin projections of
stochastically forced parabolic partial differential equations with
asymptotically polynomial nonlinearities and to networks of quasi-harmonic
oscillators connected to different Poissonian baths.
