On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

Abstract

Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$ matrix and $B$ a L\'evy process without Gaussian part. We show that when $A$ is non-singular, the law of $X_1$ is absolutely continuous in $\r^n$ 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 L\'evy process $B$, which might be very singular and sometimes even have a one-dimensional discrete jumping measure. It also solves a difficult problem for a certain class of multivariate Non-Gaussian infinitely divisible distributions.