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