# Ornstein-Uhlenbeck and Stationarity

I'm taking a Statistical Mechanics course and we just started discussing the Ornstein-Uhlenbeck process. I'm confused by what seems to be two contradictory facts about this process. For one, it is Gaussian, Markov, and Stationary. Stationary, as I currently understand it, means the moments of its distribution do not vary with time and, specifically, the autocorrelation function doesn't vary with time. On the other hand, Doob's theorem shows that any Markov and Gaussian process has a correlation function which is proportional to $e^{-t}$. But this would suggest the correlation function does vary with time, meaning it isn't stationary.

What false assumption am I making that leads to this contradiction?

• I think stationary in this context means that the stochastic component of the process ($W_t$) has constant moments – citronrose Sep 21 '17 at 15:11

## 1 Answer

The correlation function is proportional to $e^{-\gamma\tau} = e^{-\gamma|t-t'|}$, see e.g. this paper for a more verbose definition than the MathWorld page. And this is indeed stationary under a translation $t \mapsto t + dt$ of the time.