Spectral theory for L\'evy and L\'evy-Ornstein-Uhlenbeck semigroups on step 2 Carnot groups

Abstract

We consider non-local perturbations $\Delta^\psi_G$ 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 $\Delta^\psi_G$ and some strongly continuous contraction semigroups on Euclidean spaces with purely continuous spectrum, and as a result we identify the spectrum of $\Delta^\psi_G$. Further we introduce the L\'evy-Ornstein-Uhlenbeck (OU) semigroup corresponding to $\Delta^\psi_G$. 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 L\'evy-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.