Spectral theory for Lévy and Lévy-Ornstein-Uhlenbeck semigroups on step 2 Carnot groups

Abstract

We consider non-local perturbations ${\Delta }_G^{\psi }$ of sub-Laplacians on a step $2$ Carnot group $G$. The perturbations are by translation-invariant non-local operators acting along the vertical directions in $G$. We use harmonic analysis on $G$ to obtain intertwining relationship between the semigroups generated by ${\Delta }_G^{\psi }$ and some strongly continuous contraction semigroups on Euclidean spaces with purely continuous spectrum, and as a result we identify the spectrum of ${\Delta }_G^{\psi }$. Further we introduce the Lévy-Ornstein-Uhlenbeck (OU) semigroup corresponding to ${\Delta }_G^{\psi }$. We prove that these Markov semigroups are ergodic, though they are not normal operators on $L^2$ space with respect to the invariant distribution ${\mathsf{p}}_{\psi }$. The intertwining relationships allow us to show that all Lévy-OU generators on $G$ are isospectral, that is, they have the same eigenvalues with the same multiplicities. As a byproduct, we obtain a precise description of the eigenspaces, and also derive explicit formula for the co-eigenfunctions corresponding to some eigenvalues.

50 pages