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

TL;DR

This paper develops new gradient estimates and ergodicity results for SDEs driven by multiplicative pure jump Lévy noises, introducing a novel explicit Markov coupling method for such processes.

Contribution

It introduces the first explicit Markov coupling for SDEs with multiplicative pure jump Lévy noises and derives gradient estimates and ergodicity under less restrictive conditions.

Findings

Established gradient estimates for the semigroup of the SDEs.

Proved ergodicity in both Wasserstein and total variation distances.

Developed a new explicit coupling method for multiplicative Lévy noises.

Abstract

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.