Consider the linear stochastic differential equation (SDE) on $\mathbb{R}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\] where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a L��vy process with L��vy measure $��$ on $\mathbb{R}^d$. Assume that $��(\mathrm {d}{z})\ge ��_0(z)\,\mathrm{d}z$ for some $��_0\ge 0$. If $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}��_0(z)^{-1}\,\mathrm{d}z0$, then the associated Markov transition probability $P_t(x,\mathrm {d}{y})$ satisfies \[\|P_t(x,\cdot)-P_t(y,\cdot)\|_{\mathrm{var}}\le \frac{C(1+|x-y|)}{\sqrt{t}}, x,y\in \mathbb{R}^d,t>0,\] for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. The Harnack inequality, ultracontractivity and the strong Feller property are also investigated for the (conditional) transition semigroup.

Published in at http://dx.doi.org/10.3150/10-BEJ308 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)