Spectral theory for Lévy and Lévy-Ornstein-Uhlenbeck semigroups on step 2 Carnot groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911201744650240 |
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| author | Gordina, Maria Sarkar, Rohan |
| author_facet | Gordina, Maria Sarkar, Rohan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral theory for Lévy and Lévy-Ornstein-Uhlenbeck semigroups on step 2 Carnot groups Gordina, Maria Sarkar, Rohan Probability Analysis of PDEs Functional Analysis 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. |
| title | Spectral theory for Lévy and Lévy-Ornstein-Uhlenbeck semigroups on step 2 Carnot groups |
| topic | Probability Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2510.08866 |