The generator an OU process is given by 

$$A = \operatorname{tr}(QD^2)+\langle Bx,D\rangle.$$

This one possesses an invariant measure given by
$$d\mu(x) = b(x) \ dx \text{ with } b(x) = \frac{1}{(4\pi)^{N/2} \vert Q_{\infty} \vert^{1/2}} e^{-\langle Q_{\infty}^{-1}x,x \rangle /4},$$
where $Q_{\infty}= \int_0^{\infty} e^{sB} Q e^{sB^*} \ ds.$

I read that $A$ is self-adjoint on $L^2(\mathbb R^n, d\mu)$ if and only if $QA^*=AQ,$ but I have never seen an argument for this. 

In particular, what I am mainly interested in, is there anywhere an explicit representation of the adjoint operator of $A$?