Coupling for Ornstein–Uhlenbeck processes with jumps
Coupling for Ornstein–Uhlenbeck processes with jumps
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DOI：
10.3150/10-bej308
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发表时间：
2010-02
期刊：
影响因子：
1.5
通讯作者：
Feng-Yu Wang
中科院分区：
文献类型：
--
作者：
Feng-Yu Wang
考虑$\mathbb{R}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\]上的线性随机微分方程（SDE），其中$A$是一个实矩阵$n\times n$, $B$是一个实矩阵$n\times d$, $L_t$是一个lsamvy过程，lsamvy测量$\nu$和$\mathbb{R}^d$。假设是$\nu(\mathrm {d}{z})\ge \rho_0(z)\,\mathrm{d}z$对于一些$\rho_0\ge 0$。如果$A\le 0,\operatorname {Rank}(B)=n$和$\int_{\{|z-z_0|\le\varepsilon\}}\rho_0(z)^{-1}\,\mathrm{d}z 0$，则相关的马尔可夫转移概率$P_t(x,\mathrm {d}{y})$对于某个常数$C>0$满足\[\|P_t(x,\cdot)-P_t(y,\cdot)\|_{\mathrm{var}}\le \frac{C(1+|x-y|)}{\sqrt{t}}, x,y\in \mathbb{R}^d,t>0,\]，对于较大的$t$是锐利的，这意味着该过程具有成功的耦合。研究了（条件）跃迁半群的harack不等式、超收缩性和强Feller性质。
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\'{e}vy process with L\'{e}vy measure $\nu$ on $\mathbb{R}^d$. Assume that $\nu(\mathrm {d}{z})\ge \rho_0(z)\,\mathrm{d}z$ for some $\rho_0\ge 0$. If $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}\rho_0(z)^{-1}\,\mathrm{d}z 0$, 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.