We study the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, where $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d)})^T$ and $Z_t^{(1)}, \ldots, Z_t^{(d)}$ are independent one-dimensional L{\'e}vy processes with characteristic exponents $\psi_1, \ldots, \psi_d$. We assume that each $\psi_i$ satisfies a weak lower scaling condition WLSC($\alpha,0,\underline{C}$), a weak upper scaling condition WUSC($\beta,1,\overline{C}$) (where $0 (2/3)\beta$. We also assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed $\gamma \in (0,\alpha) \cap (0,1]$ the semigroup $P_t$ of the process $X$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-\gamma/\alpha} |x - y|^{\gamma} ||f||_\infty$ for arbitrary bounded Borel function $f$. We also show the existence of a transition density of the process $X$.

中文翻译：

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由具有独立坐标的 Lévy 过程驱动的 SDE 解的半群性质

我们研究随机微分方程 $dX_t = A(X_{t-}) \, dZ_t$, $X_0 = x$, 其中 $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d) })^T$ 和 $Z_t^{(1)}, \ldots, Z_t^{(d)}$ 是独立的一维 L{\'e}vy 过程，具有特征指数 $\psi_1, \ldots, \ psi_d$。我们假设每个 $\psi_i$ 满足弱下缩放条件 WLSC($\alpha,0,\underline{C}$)，弱上缩放条件 WUSC($\beta,1,\overline{C}$) （其中 $0 (2/3)\beta$。我们还假设 $A(x) = (a_{ij}(x))$ 的行列式远离零，并且 $a_{ij}(x) $ 是有界且 Lipschitz 连续的。在两种情况 (i) 和 (ii) 中，我们证明对于任何固定的 $\gamma \in (0,\alpha) \cap (0, 1]$进程$X$的半群$P_t$满足$|P_t f(x) - P_t f(y)| \le ct^{-\gamma/\alpha} |x - y|^{\gamma} ||f||_\infty$ 用于任意有界 Borel 函数 $f$。我们还展示了过程 $X$ 的转移密度的存在。