On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

Let X be a n-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. $${\rm d}X_{t} = AX_{t} {\rm d}t + {\rm d}B_t$$ where A is a real n?× n matrix and B a Lévy process without Gaussian part. We show that when A is non-singular, the law of X ₁ is absolutely continuous in ${\mathbb{R}^n}$ if and only if the jumping measure of B fulfils a certain geometric condition with respect to A, which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process B, which might be singular and sometimes even have a one-dimensional discrete jumping measure. This improves on a result by Priola and Zabczyk.