I have the next exercice:
Let $dX_t=-aX_tdt+dW_t$, with $a>0$. Show that $\mathcal{N}(0,\frac{1}{2a})$ is an invariant measure of the process.
My idea:
We say that a measure $\mu$ is invariant for the process $X_t$ if $$\mu P_t = \mu,$$ where $P_tf(x)=\mathbb{E}[f(X_t)|X_0 =x]$ et $\mu = \frac{\sqrt{a}}{\sqrt{\pi}}\exp(-ay^2)$. Then
$$\mu P_t = \int_{\mathbb{R}}\frac{\sqrt{a}}{\sqrt{\pi}}\exp(-ay^2)f(x)\mathbb{P}(X_t|X_0=x)dx$$
and using an appropriate change of variables I could complet the prove.
Any suggestion is welcome.
Thanks!