Dispersive equations with invariant measures

In mathematical physics it is of interest to study Schrödinger equations with friction and possessing an invariant measure. The focus of this paper is the Cauchy problem for the Schrödinger equation ${\text{p}}_{t}f-i\mathcal{L}f=0$, where $\mathcal{L}=\mathrm{\Delta }-\text{sa}x,\mathrm{\nabla }\text{da}$ is the Ornstein-Uhlenbeck operator. We use this as a model to stimulate interest in a new class of possibly degenerate dispersive equations which cannot be treated by the existing theory.