Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Abstract

We establish the exponential convergence with respect to the $L^1$-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation (SDE) $$d{X}_t=d{Z}_t+b(X_t)dt,$$ where $(Z_t{)}_{t\ge 0}$ is a pure jump Lévy process whose Lévy measure $\nu$ fulfills $$\underset{x\in {\mathbb{R}}^d,|x|\le {\kappa }_0}{\inf }[\nu \wedge ({\delta }_x\ast \nu )]({\mathbb{R}}^d)>0$$ for some constant ${\kappa }_0>0$, and the drift term $b$ satisfies that for any $x,y\in {\mathbb{R}}^d$, $$⟨b(x)-b(y),x-y⟩\le \{\begin{array}{llc}{\Phi }_1(|x-y|)|x-y|,& |x-y|\le l_0;-K_2|x-y{|}^2,& |x-y|>l_0\end{array}$$ with some positive constants $K_2,l_0$ and positive measurable function ${\Phi }_1$. The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process $(X_t{)}_{t\ge 0}$.

44 pages

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• Gradient Estimates and Ergodicity for SDEs Driven by Multiplicative Lévy Noises via Coupling
• A Unified Approach to Coupling SDEs driven by Lévy Noise and Some Applications
• Spatial regularity of semigroups generated by Lévy type operators