Generalized Ornstein–Uhlenbeck operators perturbed by multipolar inverse square potentials in $L^{2}$-spaces

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.