I woud like to show that the generator of the OU process is given by
$\displaystyle Af(x)=\frac{1}{2} \sigma^2f''(x)-\alpha x f'(x)$

i know generator of brownian motion Af=1/2f''(x) and it should be something like this:

first proof that
$\displaystyle P_{t}^{X}f(x)=P^{W}_{g(t)}f(e^{-\alpha t}x)$ with $\displaystyle P_{t}^{X}, P_{t}^{W}$ transitions functions of OU process and BM, $\displaystyle g(t)=\sigma^2(1-e^{-2\alpha t})/2\alpha$
and then (i guess) something like
$\displaystyle \frac{d}{dt}P_{t}f=AP_{t}f$

any help/hints would be appreciated!
OU process is defined as$\displaystyle X_t=e^{-\alpha t}(X_0+W_{\frac{\sigma ^2}{2\alpha}(e^{2\alpha t}-1)$