On the 1-dimensional complex Ornstein-Uhlenbeck operator

Abstract

We show that for any fixed $\theta \in (-\frac{\pi }{2},0)\cup (0,\frac{\pi }{2})$, the 1-dimensional complex Ornstein-Uhlenbeck operator \begin{equation*} \tilde{\mathcal{L}}_θ= 4\cosθ\frac{\partial^2}{\partial z\partial \bar{z}}-e^{\miθ} z \frac{\partial}{\partial z}-e^{-\miθ}\bar{z} \frac{\partial}{\partial \bar{z}}, \end{equation*} is a normal (but nonsymmetric) diffusion operator.

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