---
title: Ornstein-Uhlenbeck Semigroup
url: https://www.emergentmind.com/topics/ornstein-uhlenbeck-semigroup-90d950f6-cf9a-43e0-b5ed-59ae6774dc0f
type: topic
---

# Ornstein-Uhlenbeck Semigroup

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 $\sigma(B) \subset \{\Re \lambda < 0\}$ and $\int_{|y|>1} \log|y|\,\nu(dy) < \infty$, there is a unique invariant probability measure $\mu$ on $\mathbb{R}^d$. This invariant law is infinitely divisible, with characteristic exponent
\[
\Psi(\xi) = - (Q\xi, \xi) + \psi(\xi),
\]
where $Q$ arises from the Gaussian part, and $\psi(\xi)$ is a function of a "re-weighted" Lévy measure. The invariant density $p$ is smooth ($C^\infty$), and its decay at infinity is governed by the behavior of $\nu$ [2502.15183].

## 2. Spectrum and Multiplicities on $L^p(\mu)$

Let $P_t$ be the semigroup generated by $\mathcal{L}$, with $A_p$ its generator in $L^p(\mu)$ for $1 < p < \infty$. The associated eigenvalue set is
\[
\mathcal{N}(B) = \left\{ n \cdot \lambda : n \in \mathbb{N}_0^d, \lambda_1, ..., \lambda_d\ \text{the eigenvalues of } B \right\},
\]
where $n \cdot \lambda = \sum_{k=1}^d n_k \lambda_k$. The central isospectrality result asserts:
\[
\sigma(P_t; L^p(\mu)) \setminus \{0\} = \{ e^{-t \alpha} : \alpha \in \mathcal{N}(B) \},
\]
and
\[
\sigma(A_p; L^p(\mu)) = -\mathcal{N}(B).
\]
The geometric and algebraic multiplicity of an eigenvalue $-\alpha$ equals the number of multi-indices $n$ for which $n \cdot \lambda = \alpha$, and these multiplicities are independent of both $\nu$ and $p$. The point spectrum and eigenfunction structure thus precisely mirror those of the classical diffusion-driven semigroup [2502.15183].

## 3. Explicit Hermite-Like Eigenfunctions and Biorthogonality

When $B$ is diagonalizable with real eigenvalues, $B = M^{-1} \Lambda M$ with $\Lambda = \mathrm{diag}(-\lambda_1, ..., -\lambda_d)$ and each $\lambda_k > 0$, and letting $\Sigma = Q^{-1}$, the following biorthogonal families span $L^2(\mu)$:
\[
\begin{aligned}
H_n(x) &= (-1)^{|n|} n!^{-1/2} \frac{(D^n p)(x)}{p(x)}, \\
G_n(x) &= H_n(x) p(x).
\end{aligned}
\]
These generalize Hermite polynomials and co-exist as eigenfunctions and co-eigenfunctions, with
\[
P_t H_n = e^{-t (n \cdot \lambda)} H_n, \quad (P_t)^* G_n = e^{-t (n \cdot \lambda)} G_n, \quad (H_n, G_m)_{L^2(\mu)} = \delta_{nm}.
\]
Span$\{H_n : n \in \mathbb{N}_0^d\}$ is dense in $L^2(\mu)$. Such explicit formulas for (co-)eigenfunctions in the Lévy-Ornstein-Uhlenbeck case with diagonalizable drift have not previously appeared in the literature [2502.15183].

## 4. Intertwining Relation and Spectral Transfer Mechanism

Spectral isospectrality arises from a Markov intertwining between Lévy-driven and Gaussian-driven OU semigroups. Define
\[
T f(x) = \int_{\mathbb{R}^d} f(x+y)\, \eta(dy),
\]
where $\eta$ is a response convolution measure with characteristic function satisfying $\widehat{\mu}(\xi) = \widehat{\mu^0}(\xi) \widehat{\eta}(\xi)$. Then, for every $t \ge 0$ and $1 < p < \infty$, the intertwining
\[
P_t T = T P_t^0 \quad \text{on } L^p(\mu)
\]
holds, where $P_t^0$ 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 [2502.15183].

## 5. Compactness, Non-Compactness, and Polynomials in $L^p(\mu)$

The compactness of $P_t$ in $L^p(\mu)$ is determined by the moment finiteness of the invariant measure:
- If $\mu$ has finite polynomial moments of all orders (i.e., $\int_{|y|>1} |y|^n \nu(dy) < \infty$ for all $n$), $P_t$ is compact for all $t > 0$.
- If $\nu$ is heavy-tailed (e.g., symmetric $\alpha$-stable, $\nu(dy) = c |y|^{-d-\alpha} dy$, $0 < \alpha < 2$), $P_t$ is not compact, $\mu$ has power-law polynomial density at infinity, and sufficiently high-degree polynomials are not in $L^p(\mu)$ [2502.15183].

Failure of compactness is thus directly linked to the inclusion of polynomial eigenfunctions in $L^p(\mu)$. When these fail, $P_t$ 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 $L^p(\mu)$ weighted by invariant law. The spectrum consists of discrete eigenvalues $-\mathcal{N}(B)$, 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 [2502.15183].

Source: https://www.emergentmind.com/topics/ornstein-uhlenbeck-semigroup-90d950f6-cf9a-43e0-b5ed-59ae6774dc0f