Gradient estimates and ergodicity for SDEs driven by multiplicative Levy noises via coupling

We consider SDEs driven by multiplicative pure jump Levy noises, where Levy processes are not necessarily comparable to alpha-stable-like processes. By assuming that the SDE has a unique strong solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Holder 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 Levy noises, which has been open for a long time in this area. (C) 2019 Elsevier B.V. All rights reserved.