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.

60J25 Continuous-time Markov processes on general state spaces
Full Text: DOI