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

Abstract

We consider non-local perturbations $Δ^ψ_G$ 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 $Δ^ψ_G$ and some strongly continuous contraction semigroups on Euclidean spaces with purely continuous spectrum, and as a result we identify the spectrum of $Δ^ψ_G$. Further we introduce the Lévy-Ornstein-Uhlenbeck (OU) semigroup corresponding to $Δ^ψ_G$. We prove that these Markov semigroups are ergodic, though they are not normal operators on $L^2$ space with respect to the invariant distribution $\mathsf{p}_ψ$. 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.

Theorems & Definitions (97)

• Proposition 3.2: Plancherel's theorem
• Lemma 3.4
• proof
• Definition 3.5: Weyl transform
• Remark 3.6
• Proposition 3.7
• proof
• Lemma 3.9
• proof
• Proposition 3.10