 Mean-square contraction of synchronously coupled diffusions

If $2(x-y)(b(x)-b(y))+(\sigma (x)-\sigma (y){)}^2\le -k(x-y{)}^2$, the Itô formula for the squared difference and the Gronwall inequality give $\mathbb{E}|X_t-Y_t{|}^2\le e^{-kt}\mathbb{E}|X_0-Y_0{|}^2$. Localization of $e^{kt}|X_t-Y_t{|}^2$ as a nonnegative local supermartingale gives the same estimate under suitable existence hypotheses.