= Same-noise stationary Ornstein-Uhlenbeck processes
{title2=$Z_t-Y_t=e^{-\lambda t}(Z_0-Y_0)$}
Two <Ornstein-Uhlenbeck processes> with the same driving <Brownian motion> have the displayed difference. If $Y_0$ is standard normal and <independent> of future Brownian increments, replacing it by $-Y_0$ produces another <stationary process> with the same <covariance> and second <moment>, but a different pathwise <coupling>. Equal initial values are necessary for <indistinguishability of stochastic processes>.
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