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 Hörmander 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.

我们证明了由退化复合泊松过程驱动的一类随机微分方程在总变分范数中不变测度和指数混合的存在性和唯一性。除了对驾驶过程中跳跃分布的温和假设外，我们主要结果的假设是，相应的控制系统是耗散的，近似可控制的和可稳定控制的。固体可控性假设比众所周知的抛物线Hörmander条件要弱，并且仅从系统可控制的单个点开始才需要。