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As we know, the only stationary Gaussian Markov process with continuous autocorrelation function is the stationary OU process (Doob's theorem). To show this, one shows that the autocorrelation function must be exponential. However, this only means that the only stationary Gaussian Markov process with continuous autocorrelation function has the same distribution as the stationary OU process.

Is it true that the stationary Gaussian Markov process with continuous autocorrelation function is equivalent pathwise to the stationary OU process, i.e. that the process must satisfy the SDE obeyed by the OU process?

More generally, if we know that the autocorrelation function of a Gaussian process is given by a (finite) sum of exponentials, can one write down SDEs whose pathwise solution is the above process? What happen if we have an infinite sum instead?

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