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Spectrum of Lévy-Ornstein-Uhlenbeck semigroups on $\mathbb{R}^d$

Rohan Sarkar

TL;DR

The paper resolves the spectral theory of Lévy-driven Ornstein–Uhlenbeck semigroups on $\mathbb{R}^d$ by intertwining them with diffusion OU semigroups. It shows that, under mild hypotheses, the $L^p(\mu)$ point spectrum coincides with that of the diffusion OU, and, when $B$ is diagonalizable with real eigenvalues, provides explicit polynomial eigenfunctions and biorthogonal co-eigenfunctions, yielding a Mehler-type expansion for the transition density. The authors also establish isospectrality with respect to the driving Lévy process, derive regularity estimates, and characterize when Lévy–OU semigroups are or are not compact in $L^p(\mu)$, including concrete non-compactness results for $\alpha$-stable noise and compactness criteria via the invariant density tail. The results bridge nonlocal, non-self-adjoint operators with classical diffusion theory, delivering practical spectral decompositions and hypocoercivity insights for Lévy-driven dynamics. Overall, the work significantly extends diffusion OU spectral theory to the nonlocal, non-reversible setting through intertwining techniques and explicit eigenfunction constructions.

Abstract

We investigate the spectral properties of Markov semigroups associated with Ornstein-Uhlenbeck (OU) processes driven by Lévy processes. These semigroups are generated by non-local, non-self-adjoint operators. In the special case where the driving Lévy process is Brownian motion, one recovers the classical diffusion OU semigroup, whose spectral properties have been extensively studied over past few decades. Our main results establish that, under suitable conditions on the Lévy process, the spectrum of the Lévy-OU semigroup in the $L^p$-space weighted with the invariant distribution coincides with that of the diffusion OU semigroup. Furthermore, when the drift matrix $B$ is diagonalizable with real eigenvalues, we derive explicit formulas for eigenfunctions and co-eigenfunctions--an observation that, to the best of our knowledge, has not appeared in the literature. We also show that the multiplicities of the eigenvalues remain independent of the choice of the Lévy process. A key ingredient in our approach is intertwining relationship: we prove that every Lévy-OU semigroup is intertwined with a diffusion OU semigroup. Additionally, we examine the compactness properties of these semigroups and provide examples of non-compact Lévy-OU semigroups.

Spectrum of Lévy-Ornstein-Uhlenbeck semigroups on $\mathbb{R}^d$

TL;DR

The paper resolves the spectral theory of Lévy-driven Ornstein–Uhlenbeck semigroups on by intertwining them with diffusion OU semigroups. It shows that, under mild hypotheses, the point spectrum coincides with that of the diffusion OU, and, when is diagonalizable with real eigenvalues, provides explicit polynomial eigenfunctions and biorthogonal co-eigenfunctions, yielding a Mehler-type expansion for the transition density. The authors also establish isospectrality with respect to the driving Lévy process, derive regularity estimates, and characterize when Lévy–OU semigroups are or are not compact in , including concrete non-compactness results for -stable noise and compactness criteria via the invariant density tail. The results bridge nonlocal, non-self-adjoint operators with classical diffusion theory, delivering practical spectral decompositions and hypocoercivity insights for Lévy-driven dynamics. Overall, the work significantly extends diffusion OU spectral theory to the nonlocal, non-reversible setting through intertwining techniques and explicit eigenfunction constructions.

Abstract

We investigate the spectral properties of Markov semigroups associated with Ornstein-Uhlenbeck (OU) processes driven by Lévy processes. These semigroups are generated by non-local, non-self-adjoint operators. In the special case where the driving Lévy process is Brownian motion, one recovers the classical diffusion OU semigroup, whose spectral properties have been extensively studied over past few decades. Our main results establish that, under suitable conditions on the Lévy process, the spectrum of the Lévy-OU semigroup in the -space weighted with the invariant distribution coincides with that of the diffusion OU semigroup. Furthermore, when the drift matrix is diagonalizable with real eigenvalues, we derive explicit formulas for eigenfunctions and co-eigenfunctions--an observation that, to the best of our knowledge, has not appeared in the literature. We also show that the multiplicities of the eigenvalues remain independent of the choice of the Lévy process. A key ingredient in our approach is intertwining relationship: we prove that every Lévy-OU semigroup is intertwined with a diffusion OU semigroup. Additionally, we examine the compactness properties of these semigroups and provide examples of non-compact Lévy-OU semigroups.

Paper Structure

This paper contains 12 sections, 25 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Results
  4. Regularity of $P_t$
  5. Spectrum, eigenfunctions, and co-eigenfunctions
  6. Compactness of $P_t$
  7. Multiplicities of eigenvalues and isospectrality
  8. Examples
  9. Intertwining relations
  10. Intertwining with a self-adjoint diffusion
  11. Proof of the main results
  12. Generalized eigenfunctions of $A_p$

Key Result

Theorem 3.1

For all $t>0$, $P_t$ has smooth transition density. The invariant distribution $\mu$ of $P_t$ is absolutely continuous and denoting the density by $\mu$, we have $\mu\in C^\infty_0(\mathbb{R}^d)$. Finally, for any $1<p<\infty$ and $t\in (0,1]$, $P_t$ maps $\mathrm{L}^p(\mu)$ to $\mathrm{W}^{k,p}(\mu where $|n|=k$ and $C=C(k,p)$ is a positive constant depending on $k,p$.

Theorems & Definitions (56)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.7
  • Corollary 3.8: Generalized Mehler's formula
  • Proposition 3.9
  • Theorem 3.10
  • ...and 46 more