Ornstein-Uhlenbeck Semigroup

• The Ornstein-Uhlenbeck semigroup is the evolution family of Markov processes with linear drift and additive noise, underpinning both classical diffusion and non-local Lévy operator studies.
• Spectral analysis reveals discrete eigenvalues and Hermite-like eigenfunctions with biorthogonality, providing a robust framework for understanding convergence rates and ergodic properties.
• Research advancements highlight intertwining relations that transfer spectral data from Gaussian-driven to Lévy-driven systems, emphasizing conditions for compactness and the role of invariant measures.

The Ornstein-Uhlenbeck semigroup is the evolution family associated with Ornstein-Uhlenbeck processes, which are Markov processes driven by additive noise and linear drift. The prototypical generator combines a first-order linear drift and a noise term, the latter possibly corresponding to either local (diffusive) or non-local (jump-driven) processes. Recent research has focused not only on the diffusion-driven case but also on semigroups generated by non-local Lévy-type operators, leading to extensive developments in spectral theory, functional inequalities, ergodic properties, and the structure of eigenfunctions.

1. Generators, Invariant Measures, and Structural Setting

Let $B$ denote a real $d \times d$ matrix with $\sigma(B) \subset \{\Re \lambda < 0\}$, and let $\nu$ be a Lévy measure on $\mathbb{R}^d$ such that $\nu(\{0\}) = 0$ and $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$. The Lévy-Ornstein-Uhlenbeck generator $\mathcal{L}$ is given by

$$\mathcal{L} f(x) = \langle B x, \nabla f(x) \rangle + \int_{\mathbb{R}^d} \Big( f(x+y) - f(x) - \langle \nabla f(x), y \rangle \mathbf{1}_{|y|<1} \Big) \nu(dy).$$

If $\nu \equiv 0$, this reduces to the classical diffusion generator. Under the conditions $d \times d$0 and $d \times d$1, there is a unique invariant probability measure $d \times d$2 on $d \times d$3. This invariant law is infinitely divisible, with characteristic exponent

$d \times d$4

where $d \times d$5 arises from the Gaussian part, and $d \times d$6 is a function of a "re-weighted" Lévy measure. The invariant density $d \times d$7 is smooth ($d \times d$8), and its decay at infinity is governed by the behavior of $d \times d$9 (Sarkar, 21 Feb 2025).

2. Spectrum and Multiplicities on $\sigma(B) \subset \{\Re \lambda < 0\}$0

Let $\sigma(B) \subset \{\Re \lambda < 0\}$1 be the semigroup generated by $\sigma(B) \subset \{\Re \lambda < 0\}$2, with $\sigma(B) \subset \{\Re \lambda < 0\}$3 its generator in $\sigma(B) \subset \{\Re \lambda < 0\}$4 for $\sigma(B) \subset \{\Re \lambda < 0\}$5. The associated eigenvalue set is

$\sigma(B) \subset \{\Re \lambda < 0\}$6

where $\sigma(B) \subset \{\Re \lambda < 0\}$7. The central isospectrality result asserts: $\sigma(B) \subset \{\Re \lambda < 0\}$8 and

$\sigma(B) \subset \{\Re \lambda < 0\}$9

The geometric and algebraic multiplicity of an eigenvalue $\nu$0 equals the number of multi-indices $\nu$1 for which $\nu$2, and these multiplicities are independent of both $\nu$3 and $\nu$4. The point spectrum and eigenfunction structure thus precisely mirror those of the classical diffusion-driven semigroup (Sarkar, 21 Feb 2025).

3. Explicit Hermite-Like Eigenfunctions and Biorthogonality

When $\nu$5 is diagonalizable with real eigenvalues, $\nu$6 with $\nu$7 and each $\nu$8, and letting $\nu$9, the following biorthogonal families span $\mathbb{R}^d$0: $\mathbb{R}^d$1 These generalize Hermite polynomials and co-exist as eigenfunctions and co-eigenfunctions, with

$\mathbb{R}^d$2

Span$\mathbb{R}^d$3 is dense in $\mathbb{R}^d$4. Such explicit formulas for (co-)eigenfunctions in the Lévy-Ornstein-Uhlenbeck case with diagonalizable drift have not previously appeared in the literature (Sarkar, 21 Feb 2025).

4. Intertwining Relation and Spectral Transfer Mechanism

Spectral isospectrality arises from a Markov intertwining between Lévy-driven and Gaussian-driven OU semigroups. Define

$\mathbb{R}^d$5

where $\mathbb{R}^d$6 is a response convolution measure with characteristic function satisfying $\mathbb{R}^d$7. Then, for every $\mathbb{R}^d$8 and $\mathbb{R}^d$9, the intertwining

$\nu(\{0\}) = 0$0

holds, where $\nu(\{0\}) = 0$1 is the semigroup for the classical (diffusion) OU process. Hence, all spectral data (point spectrum, multiplicities, discrete spectrum) transfer from the classical to the Lévy-Ornstein-Uhlenbeck semigroup (Sarkar, 21 Feb 2025).

5. Compactness, Non-Compactness, and Polynomials in $\nu(\{0\}) = 0$2

The compactness of $\nu(\{0\}) = 0$3 in $\nu(\{0\}) = 0$4 is determined by the moment finiteness of the invariant measure:

• If $\nu(\{0\}) = 0$5 has finite polynomial moments of all orders (i.e., $\nu(\{0\}) = 0$6 for all $\nu(\{0\}) = 0$7), $\nu(\{0\}) = 0$8 is compact for all $\nu(\{0\}) = 0$9.
• If $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$0 is heavy-tailed (e.g., symmetric $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$1-stable, $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$2, $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$3), $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$4 is not compact, $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$5 has power-law polynomial density at infinity, and sufficiently high-degree polynomials are not in $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$6 (Sarkar, 21 Feb 2025).

Failure of compactness is thus directly linked to the inclusion of polynomial eigenfunctions in $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$7. When these fail, $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$8 exhibits continuous spectrum accumulating at zero. Nonetheless, the discrete eigenvalues, corresponding multiplicities, Hermite-like eigenfunctions, and convergence rates remain preserved.

6. Conclusion and Further Implications

Lévy-Ornstein-Uhlenbeck semigroups, including those with non-local and non-self-adjoint generators, are isospectral with their classical diffusion analogues when considered as operators on $\int (1 \wedge |y|^2)\, \nu(dy) < \infty$9 weighted by invariant law. The spectrum consists of discrete eigenvalues $\mathcal{L}$0, with multiplicities and exponential convergence rates independent of the specific Lévy measure. The underlying intertwining operator provides a mechanism that universally identifies the spectrum, multiplicities, and spectral structure.

These results showcase a robust spectral rigidity: even in the presence of heavy-tailed noise and non-locality, the fundamental spectral properties and polynomial eigenfunction structure are preserved, up to possible non-compactness driven by moment growth. This interconnectedness underpins further developments in ergodic theory, spectral theory for non-local operators, and applications in non-Gaussian stochastic models (Sarkar, 21 Feb 2025).