$L^p(\mu)$ of more general symbols.

We will now define the Ornstein-Uhlenbeck operator in our setting. We consider the Dirichlet form $E(f,g) = \int_{\mathbb{R}^d} \nabla f \nabla \overline{g} d\mu$ with domain $\mathcal{C}_\phi$. It follows from the general theory of Dirichlet forms (see [@FukushimaOshimaTakeda]) that the operator on $L^2(\mu)$ associated with the form $E$ is positive and generates a positive contraction $C_0$-semigroup, which extends to a positive contraction $C_0$-semigroup on $L^p(\mu)$ for all $p \in (1, \infty)$. We call this semigroup the Ornstein-Uhlenbeck semigroup and denote it by $\exp(-tL)$, and its generator the Ornstein-Uhlenbeck operator which we will denote by $L$. It is a simple computation to find that on $f \in \mathcal{C}_\phi$, $L$ has action $$\label{equationAction}
Lf (x) = -\Delta f(x) + \nabla\phi(x) \cdot\nabla f(x).$$ Due to our set-up, we have the fact:

\[corSomeHinfty\] For any potential $\phi \in C^2$ and $p \in (1, \infty)$, the corresponding Ornstein-Uhlenbeck operator has a bounded $H^\infty$ functional calculus of some angle less than $\pi/2$.

This follows from Theorem 10.7.13 of [@AnalysisInBanachSpacesV2], as below.

\[thmPosConImpliesHInfty\] Suppose $(\Omega, m)$ is a measure space ($\sigma$-algebra omitted). If an unbounded operator $T$ on $L^p(\Omega, m)$, $p \in (1, \infty)$ generates a positive contraction semigroup, then $T$ has a 