Exponential convergence in the Wasserstein metric \begin{document}$ W_1 $\end{document} for one dimensional diffusions

In this paper, we find some general and efficient sufficient conditions for the exponential convergence \begin{document}$ W_{1,d}(P_t(x,\cdot), P_t(y,\cdot) )\le Ke^{-\delta t}d(x,y) $\end{document} for the semigroup \begin{document}$ (P_t) $\end{document} of one-dimensional diffusion. Moreover, some sharp estimates of the involved constants \begin{document}$ K\ge 1, \delta>0 $\end{document} are provided. Those general results are illustrated by a series of examples.