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.
"Sub-exponential convergence to equilibrium for Gaussian driven Stochastic Differential Equations with semi-contractive drift." Electron. J. Probab. 25 1 - 43, 2020. https://doi.org/10.1214/20-EJP464