# zbMATH — the first resource for mathematics

On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process. (English) Zbl 1048.60060
Let $$X$$ be the Ornstein-Uhlenbeck process, solution of $$dX_t=-QX_t\,dt+dZ_t$$, where $$Z$$ is a Lévy process characterized by the vector $$b\in\mathbb R^d$$, the symmetric nonnegative definite $$d\times d$$ matrix $$C$$ and the $$\sigma$$-finite measure $$\nu$$, $$Q$$ is a $$d\times d$$ matrix, and the initial condition $$X_0$$ is independent of $$Z$$. The paper proves the following three results:
{1)} If $$\text{rank}(C)=d$$ or $$\nu(\mathbb R^d)=\infty$$ and $$\nu$$ is an absolutely continuous measure, then the Markov semigroup of $$X$$ is strong Feller.
{2)} If $$\text{rank}(C)=d$$ or there exists $$\alpha\in(0,2)$$ and $$c>0$$ such that $\int_{\{z:\;| v^Tz| \leq 1\}} | v^Tz| ^2\,\nu(dz)\geq c| v| ^{2-\alpha}$ for any $$v\in\mathbb R^d$$ with $$| v| \geq 1$$, then the transition probabilities have a smooth density.
{3)} If all eigenvalues of $$Q$$ have positive real parts, $$X$$ is strictly stationary, and its $$Q$$-decomposable limiting distribution has a finite absolute $$p$$-moment, for some $$p>0$$, then $$X$$ has an exponential $$\beta$$-mixing bound, and in particular is ergodic.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces
##### Keywords:
Ornstein-Uhlenbeck process; Lévy process
Full Text: