Synchronous coupling

A Synchronous coupling drives two processes with the same fresh random inputs. For example, recursions $X_{n+1}=F(X_n,{\xi }_{n+1})$ and $Y_{n+1}=F(Y_n,{\xi }_{n+1})$ use the same input ${\xi }_{n+1}$ at each step. This can reveal synchronous contraction of a random affine recursion even when independently simulated trajectories would conceal the relationship.

A Synchronous coupling drives two solutions of a stochastic differential equation by the same Brownian motion. Their difference has noise coefficient $\sigma (X)-\sigma (Y)$. A dissipativity inequality for the drift and this coefficient gives the mean-square contraction of synchronously coupled diffusions.

For $k>0$, 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.

If $|b|\le B$, then $2(x-y)(b(x)-b(y))+(\sigma (x)-\sigma (y){)}^2\ge -4B|x-y|$. It cannot also be at most $-k|x-y{|}^2$ for every pair on the real line when $k>0$. Global strict dissipativity thus requires relaxing global boundedness of the drift; linear confining drifts give natural compatible examples.

If two coupled processes approach each other in L2 and one family of marginals is uniformly tight, then their expectations of any bounded continuous test function approach each other. On a large compact set use uniform continuity; outside it use boundedness; control excessive coupling separation by the second-moment Markov bound. A constant expectation for one process then identifies the other limit. No global derivative or Lipschitz bound is necessary.