  The convergence to the stationary regime is studied for Stochastic
Differential Equations driven by an additive Gaussian noise and evolving in a
semi-contractive environment, i.e. when the drift is only contractive out of a
compact set but does not have repulsive regions. In this setting, we develop a
synchronous coupling strategy to obtain sub-exponential bounds on the rate of
convergence to equilibrium in Wasserstein distance. Then by a coalescent
coupling close to terminal time, we derive a similar bound in total variation
distance.
