Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator

Haller-Dintelmann, Robert¹ ; Wiedl, Julian¹

¹ Technische Universität Darmstadt Fachbereich Mathematik Schlossgartenstr. 7 D-64289 Darmstadt, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 729-748

Résumé

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 $L^p\left(\Omega \right)$ for all $1\le p<\infty$ and for every domain $\Omega \subseteq {ℝ}^d$. For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given.

Détail

Classification : 47D06, 35K20