Kolmogorov kernel estimates for the Ornstein–Uhlenbeck operator. (English) Zbl 1171.47302

Wiedl, Julian; Haller-Dintelmann, Robert

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 4, 729-748 (2005).

Summary: 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, (quasi-)contractive \(C_0\)-semigroups on \(L^p(\Omega)\) for all \(1\leq p<\infty\) 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.

MSC:

47D06	One-parameter semigroups and linear evolution equations
35K20	Initial-boundary value problems for second-order parabolic equations

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