Refined basic couplings and Wasserstein-type distances for SDEs with L\'{e}vy noises

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 L'{e}vy process whose L'{e}vy measure $\nu$ fulfills $$\underset{x\in {\mathbb{R}}^d,\mathrm{\mid }x\mathrm{\mid }\le {\kappa }_0}{\inf }[\nu \wedge ({\delta }_x\ast \nu )]({\mathbb{R}}^d)>0$$ for some constant ${\kappa }_0>0$, and the drift term $b$ satisfies that for any $x,y\in {\mathbb{R}}^d$, $$⟨b(x)-b(y),x-y⟩\le \{\begin{array}{lll}{\mathrm{\Phi }}_1(\mathrm{\mid }x-y\mathrm{\mid })\mathrm{\mid }x-y\mathrm{\mid },& \mathrm{\mid }x-y\mathrm{\mid }\le l_0;-K_2\mathrm{\mid }x-y{\mathrm{\mid }}^2,& \mathrm{\mid }x-y\mathrm{\mid }>l_0\end{array}$$ with some positive constants $K_2,l_0$ and positive measurable function ${\mathrm{\Phi }}_1$. The method is based on the refined basic coupling for L'{e}vy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process $(X_t{)}_{t\ge 0}$.