Definition 2.2.1.label Let $d \in \nat$, then a diffusion operator on $\real^{d}$ is a differential operator of the form
\[Lu = \dpn{A, D^2u}{\real^{d \times d}}+ \dpn{b, Du}{\real^d}\]
where $A \in C(\real^{d}; \real^{d \times d})$ is symmetric and non-negative, and $b \in C(\real^{d}; \real^{d})$.
It is assumed that there exists a Borel measure $\mu: \cb(\real) \to [0, \infty]$ equivalent to the Lebesgue measure such that
\[\int g Lf d\mu = \int f Lg d\mu\]
for all $f, g \in C_{c}^{\infty}(\real^{d}; \real)$.