Gradient Estimates and Ergodicity for SDEs Driven by Multiplicative L\'{e}vy Noises via Coupling

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.