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Coupling for Ornstein--Uhlenbeck processes with jumps

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 Lévy process with Lévy measure $ν$ on $\mathbb{R}^d$. Assume that $ν(\mathrm {d}{z})\ge ρ_0(z)\,\mathrm{d}z$ for some $ρ_0\ge 0$. If $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}ρ_0(z)^{-1}\,\mathrm{d}z<\infty$ holds for some $z_0\in \mathbb{R}^d$ and some $\varepsilon>0$, 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.