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It is shown that the Ornstein–Uhlenbeck operator perturbed by a multipolar inverse square potential $$\begin{aligned} A_{\Phi ,G}+V=\Delta -\nabla \Phi \cdot \nabla +G\cdot \nabla +\sum \limits _{i=1}^{n}\frac{c}{|x-a_{i}|^{2}} \end{aligned}$$ with suitable domain… Click to show full abstract
It is shown that the Ornstein–Uhlenbeck operator perturbed by a multipolar inverse square potential $$\begin{aligned} A_{\Phi ,G}+V=\Delta -\nabla \Phi \cdot \nabla +G\cdot \nabla +\sum \limits _{i=1}^{n}\frac{c}{|x-a_{i}|^{2}} \end{aligned}$$ with suitable domain generates a quasi-contractive and positive analytic $$C_{0}$$ -semigroup on the weighted space $$L^{2}(\mathbb {R}^{N},d\mu )$$ , where $$d\mu =\exp (-\Phi (x))dx$$ , $$\Phi \in C^{2}(\mathbb {R}^{N}, \mathbb {R})$$ , $$G \in C^{1}(\mathbb {R}^{N},\mathbb {R}^{N})$$ , $$c>0$$ , and $$a_{1},\ldots , a_{n}\in \mathbb {R}^{N}$$ . The proofs are based on an $$L^{2}$$ -weighted Hardy inequality and bilinear form techniques.
Journal Title: Archiv der Mathematik
Year Published: 2021
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