This question continues the discussion from an earlier post on this website found here : Why the operator is termed as Ornstein–Uhlenbeck operator? I am interested in the relationship between the Ornstein-Uhlenbeck operator $L f(x)=\Delta f(x) - x\cdot\nabla f(x)$ and the Ornstein-Uhlenbeck stochastic process. The earlier post states that the operator is the infinitesimal generator of the [stochastic] process. When I select the provided link which routes the reader to a Wikipedia post, the infinitesimal generator for the Ornstein- Uhlenbeck process on $\mathbb{R}$ which this post lists is this one : $$ \mathcal{A}f(x)=\theta(\mu-x)f'(x)+\frac{\sigma^2}{2}f''(x)\ .$$ Specifically, my questions are the following.
(1) How can the generator $\mathcal{A}f(x)$ be written in the form given by $L f(x)$ ? If possible, I am interested in seeing the steps by which $\mathcal{A}f(x)$ can be rewritten as the form $L f(x)$.
(2) How did Ornstein and Uhlenbeck use the operator $L f(x) $ in their research ? If possible, I am intersted in any source references.
