Invariant Measures of Lévy-driven Stochastic Differential Equations

We study the structure and regularity of (infinitesimally) invariant measures of the solutions to stochastic differential equations $d{X}_t=b(X_t)dt+d{Z}_t$, where $(Z_t{)}_{t\ge 0}$ is a Lévy process. We show, in particular, that the invariant measure has to satisfy a Volterra-type convolution equation; since we can obtain the kernels explicitly, we are able to apply regularity methods from harmonic analysis. As an application, we get a very short proof – in any dimension – of a classic result due to Sato and Yamazato on the form of the invariant measure of a generalized Ornstein–Uhlenbeck process.