Bi-Kolmogorov type operators and weighted Rellich’s inequalities

Abstract

AbstractIn this paper we consider the symmetric Kolmogorov operator $$L=\Delta +\frac{\nabla \mu }{\mu }\cdot \nabla $$ L = Δ + ∇ μ μ · ∇ on $$L^2({\mathbb {R}}^N,d\mu )$$ L 2 ( R N , d μ ) , where $$\mu $$ μ is the density of a probability measure on $${\mathbb {R}}^N$$ R N . Under general conditions on $$\mu $$ μ we prove first weighted Rellich’s inequalities and deduce that the operators L and $$-L^2$$ - L 2 with domain $$H^2({\mathbb {R}}^N,d\mu )$$ H 2 ( R N , d μ ) and $$H^4({\mathbb {R}}^N,d\mu )$$ H 4 ( R N , d μ ) respectively, generate analytic semigroups of contractions on $$L^2({\mathbb {R}}^N,d\mu )$$ L 2 ( R N , d μ ) . We observe that $$d\mu $$ d μ is the unique invariant measure for the semigroup generated by $$-L^2$$ - 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 $$L^2({\mathbb {R}}^N,d\mu )$$ L 2 ( R N , d μ ) .