Lemma 2.2.2.label Let $L$ be a diffusion operator, then for any $f \in C^{\infty}(\real^{d}; \real)$ and $g \in C_{c}^{\infty}(\real^{d}; \real)$,
\[\int g Lf d\mu = \int f Lg d\mu\]
In particular,
\[\int Lg d\mu = 0\]
Proof. Let $\eta \in C_{c}^{\infty}(\real^{d}; \real)$ such that $\eta|_{\supp{g}}= 1$, then
\[\int g Lf d\mu = \int g L(\eta f)d\mu = \int (\eta f)Lgd\mu = \int f Lgd\mu\]
$\square$