Bi-Kolmogorov type operators and weighted Rellich’s inequalities

In this paper we consider the symmetric Kolmogorov operator L=Δ+∇μμ·∇ on L2(RN, dμ) , where μ is the density of a probability measure on RN. Under general conditions on μ we prove first weighted Rellich’s inequalities and deduce that the operators L and - L2 with domain H2(RN, dμ) and H4(RN, dμ) respectively, generate analytic semigroups of contractions on L2(RN, dμ). We observe that dμ is the unique invariant measure for the semigroup generated by - L2 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 L2(RN, dμ).