Gradient Estimates and Ergodicity for SDEs Driven by Multiplicative Lévy Noises via Coupling

Abstract

We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy 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 Hölder 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 Lévy noises, which is derived for the first time in this paper.

34 pages

References in corpus (2)

• Ergodicity of stochastic differential equations with jumps and singular coefficients
• Uniform Hölder Estimates on Semigroups Generated by Non-Local Operators of Variable Order