Abstract

Replacing the gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup on $\mathbb{R}^d$, we define the notion of Kolmogorov kernel estimates. This allows us to show that under Dirichlet boundary conditions Ornstein-Uhlenbeck operators are generators of consistent, positive, contractive $C_0$-semigroups on $L^p$ for all $1 \le p < \infty $ and for every domain $\Omega \subseteq \mathbb{R}^d$. For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given