Wang, Feng-Yu Coupling for Ornstein-Uhlenbeck processes with jumps. (English) Zbl 1238.60090 Bernoulli 17, No. 4, 1136-1158 (2011). Summary: Consider the linear stochastic differential equation (SDE) on \(\mathbb{R}^n\): \[ dX_t=AX_tdt+BdL_t, \] 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 \(\nu\) on \(\mathbb{R}^d\). Assume that \(\nu(dz)\geq\rho_0(z)dz\) for some \(\rho_0\geq 0\). If \(A\leq 0\), Rank\((B)=n\) and \(\int_{\{|z-z_0|\leq\varepsilon\}}\rho_0(z)^{-1}dz<\infty\) holds for some \(z_0\in\mathbb{R}^d\) and some \(\varepsilon>0\), then the associated Markov transition probability \(P_t(x,dy)\) satisfies \[ \|P_t(x,\cdot)-P_t(y,\cdot)\|_{\text{var}}\leq\frac{C(1+|x-y|)}{\sqrt t},\quad x,y\in\mathbb{R}^d,t>0, \] 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. Cited in 2 ReviewsCited in 26 Documents MSC: 60J75 Jump processes (MSC2010) 60G51 Processes with independent increments; Lévy processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J35 Transition functions, generators and resolvents Keywords:coupling; Harnack inequality; Lévy process; quasi-invariance; strong Feller × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Aldous, D. and Thorisson, H. (1993). Shift-coupling. Stochastic Proc. Appl. 44 1-14. · Zbl 0769.60062 · doi:10.1016/0304-4149(93)90034-2 [2] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus . Cambridge Univ. Press. · Zbl 1073.60002 [3] Bakry, D. and Qian, Z. (1999). Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Mat. 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