Bi-Kolmogorov type operators and weighted Rellich's inequalities

In this paper we consider the symmetric Kolmogorov operator $L=\Delta +\frac{\nabla \mu}{\mu}\cdot \nabla$on$L^2(\mathbb R^N,d\mu)$,where$\mu$ is the density of a probability measure on ${\mathbb{R}}^N$. Under general conditions on $\mu$ we prove first weighted Rellich’s inequalities with optimal constants and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb R^N,d\mu)$and$H^4(\mathbb R^N,d\mu)$ respectively, generate analytic semigroups of contractions on $L^2({\mathbb{R}}^N,d\mu )$. We observe that $d\mu$ is the unique invariant measure for the semigroup generated by $-L^2$ and as a consequence we describe the asymptotic behaviour of such semigroup and obtain some local positivity properties. As an application we study the bi-Ornstein-Uhlenbeck operator and its semigroup on $L^2({\mathbb{R}}^N,d\mu )$.