Definition 2.2.4.label Let $L$ be a diffusion operator. For each $f, g \in C_{c}^{\infty}(\real^{d}; \real)$,
\begin{align*}\dpn{f, g}{\ce}&= \dpn{Df, ADg}{L^2(\real^d, \mu; \real^d)}\\&= -\dpn{f, Lg}{L^2(\real^d, \mu)}= -\dpn{g, Lf}{L^2(\real^d, \mu)}\end{align*}
is the energy form of $L$.