= 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 <diffusion coefficient>[noise coefficient] $\sigma(X)-\sigma(Y)$. A dissipativity inequality for the drift and this coefficient gives the <mean-square contraction of synchronously coupled diffusions>.
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