  In this paper, concerning SDEs with H\"older continuous drifts, which are
merely dissipative at infinity, and SDEs with piecewise continuous drifts, we
investigate the strong law of large numbers and the central limit theorem for
underlying additive functionals and reveal the corresponding rates of
convergence. To establish the limit theorems under consideration, the
exponentially contractive property of solution processes under the
(quasi-)Wasserstein distance plays an indispensable role. In order to achieve
such contractive property, which is new and interesting in its own right for
SDEs with H\"older continuous drifts or piecewise continuous drifts, the
reflection coupling method is employed and meanwhile a sophisticated test
function is built.
