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Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator

Robert Haller-Dintelmann; Julian Wiedl — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Replacing the gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup on $ℝ^{d}$ , 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, (quasi-) contractive $C_{0}$ -semigroups on $L^{p} (Ω)$ for all $1 \leq p < \infty$ and for every domain $Ω \subseteq ℝ^{d}$ . For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given.

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