 Same-noise stationary Ornstein-Uhlenbeck processes

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.