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 (4)

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• Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
• Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises
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Cited by in corpus (4)

• On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks
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• Exponential ergodicity for general continuous-state nonlinear branching processes
• Spatial regularity of semigroups generated by Lévy type operators