We consider SDEs driven by multiplicative pure jump L\'{e}vy noises, where L\'evy processes are not necessarily comparable to $\alpha$-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally H\"{o}lder continuous, and we establish the ergodicity of the process both in the $L^1$-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump L\'{e}vy noises, which is derived for the first time in this paper.

中文翻译：

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由乘法 Lévy 噪声通过耦合驱动的 sdes 的梯度估计和遍历性

我们考虑由乘法纯跳跃 L\'{e}vy 噪声驱动的 SDE，其中 L\'evy 过程不一定与 $\alpha$-stable-like 过程相当。通过假设 SDE 有唯一解，当漂移项局部 H\"{o}lder 连续时，我们获得相关半群的梯度估计，并且我们在 $L^1$- 中建立过程的遍历性Wasserstein 距离和总变异，当系数在大距离上是耗散的。证明是基于由乘法纯跳跃 L\'{e}vy 噪声驱动的 SDE 的新显式马尔可夫耦合，这是首次在这篇报告。