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).
Bi-Kolmogorov type operators and weighted Rellich’s inequalities / Addona, D.; Gregorio, F.; Rhandi, A.; Tacelli, C.. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 29:2(2022). [10.1007/s00030-021-00747-y]
Bi-Kolmogorov type operators and weighted Rellich’s inequalities
Addona D.
;Rhandi A.;
2022-01-01
Abstract
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).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
