TITLE: How was the Ornstein–Uhlenbeck process originally constructed?
QUESTION [5 upvotes]: It is natural to wonder about a Brownian motion with a drift toward $0$ whose rate is equal to the current value of the process. Unlike the standard Wiener process, which is null-recurrent, this is positive-recurrent and seldom wanders very far from $0.$
But the way I have seen it presented is this:

$$\tag 1 \text{Let } X(t) = e^{-t}W(e^{2t}),\qquad\qquad\qquad\qquad$$ where $W$ is the standard Wiener process. Now derive its covariance function and its drift.

Would any reasonable person think something like that is of interest other than as a routine exercise for undergraduates unless they knew the (to me) unexpected result?
What sort of thought process could possibly lead someone from wondering about the mean-reverting Brownian motion to coming up with line $(1)$ above as the answer?

REPLY [5 votes]: The answer by John Dawkins already covers most of the bases; I'll add two points:

The Langevin equation was considered independently by Ornstein in a 1919 paper 1; Ornstein attributes it to the PhD thesis of Haas-Lorentz, although further digging reveals that the Langevin equation was also considered by Einstein and Smoluchowski as well.

In fact, it was Doob in 1942 who first studied Brownian motion as a time-changed OU process (i.e. the inverse operation you are proposing). To see this, look at equation 1.2.1 of 2, set $\gamma = 2$ and let $u(t)$ be an OU process.



1 Ornstein, L. S. "On the Brownian motion." Proc. Amst. Vol. 21. 1919. Link
2 Doob, Joseph L. "The Brownian movement and stochastic equations." Annals of Mathematics (1942): 351-369. Link