Mean-square contraction of synchronously coupled diffusions
= Mean-square contraction of synchronously coupled diffusions
{title2=$\mathbb E|X_t-Y_t|^2\leq e^{-kt}\mathbb E|X_0-Y_0|^2$}
If $2(x-y)(b(x)-b(y))+(\sigma(x)-\sigma(y))^2\leq-k(x-y)^2$, the <Itô formula> for the squared difference and the <Gronwall inequality> give $\mathbb E|X_t-Y_t|^2\leq e^{-kt}\mathbb E|X_0-Y_0|^2$. Localization of $e^{kt}|X_t-Y_t|^2$ as a nonnegative local supermartingale gives the same estimate under suitable existence hypotheses.