Invariant Measures of Lévy-driven Stochastic Differential Equations

Abstract

We study the structure and regularity of (infinitesimally) invariant measures of the solutions to stochastic differential equations $dX_t = b(X_t)\,dt + dZ_t$, where $(Z_t)_{t\geq 0}$ 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.