Space-Time Lévy White Noise Overview

• Space-Time Lévy White Noise is a pure-jump generalization of Gaussian white noise defined as an independently scattered, infinitely divisible random measure using compensated Poisson integrals.
• It incorporates intricate embedding conditions into tempered distributions and precise moment criteria to ensure meaningful application in stochastic partial differential equations.
• Its framework, covering examples like symmetric α-stable and variance gamma noises, underpins advanced stochastic integration, SPDE solution theory, and numerical discretizations.

Space-time Lévy white noise is the prototypical pure-jump generalization of Gaussian space-time white noise. It is a random generalized field on space-time, mathematically formalized as an independently scattered, infinitely divisible random measure or, equivalently, as a process on the space of tempered distributions. Canonical construction employs compensated Poisson random measures with Lévy intensity, encoding the jump structure and statistical laws of the noise. The precise embedding of such noise into functional-analytic frameworks and its intricate sample path regularity are foundational for the analysis of stochastic partial differential equations (SPDEs) with non-Gaussian driving terms.

1. Mathematical Definition and Lévy–Itô Structure

Let $d\ge1$ and $\nu$ a Lévy measure on $\mathbb{R}\setminus\{0\}$ with

$$\nu(\{0\})=0,\quad \int_{\mathbb{R}} (1 \wedge z^2) \,\nu(dz)<\infty.$$

Space-time Lévy white noise is the independently scattered, mean-zero random measure $L$ on $\mathbb{R}^d\times\mathbb{R}$ defined by the compensated Poisson integral

$$L(\phi) = \int_{\mathbb{R}^{d+1}} \int_{\mathbb{R}\setminus\{0\}} \phi(x,t) z\,\tilde{N}(dx,dt,dz)$$

for $\phi\in C_c^\infty(\mathbb{R}^{d+1})$, where $N$ is a Poisson random measure of intensity $dx\,dt\,\nu(dz)$ and $\tilde{N}=N-dx\,dt\,\nu(dz)$ is the compensated measure. The characteristic functional is

$$\mathbb{E}\left[\exp\left(iL(\phi)\right)\right] = \exp\left\{ \int_{\mathbb{R}^{d+1}}\int_{\mathbb{R}\setminus\{0\}} \left(e^{iz\phi(x,t)} - 1 - i z \phi(x,t) 1_{|z|<1} \right)\nu(dz)\,dx\,dt \right\}.$$

This structure yields independent increments on disjoint Borel sets, stationary increments, and allows a direct analogy to white Gaussian noise via the Lévy–Khintchine formula (Balan, 2021, Dalang et al., 2015, Griffiths et al., 2019).

2. Embedding in Tempered Distributions

A fundamental result identifies when a space-time Lévy white noise $W$ may be viewed as a random tempered distribution, i.e., as an element of $S'(\mathbb{R}^{d+1})$ with probability one (Dalang et al., 2015, Griffiths et al., 2019). The necessary and sufficient condition is the existence of a positive moment: $$\exists p>0: \int_{|z|>1} |z|^p\,\nu(dz)<\infty \iff \mathbb{P}(W\in S'(\mathbb{R}^{d+1}))=1.$$ If no positive moment exists, $W$ does not define a tempered distribution. This dichotomy holds regardless of dimension and remains valid when both space and time indices are included—no further restriction arises from their interplay. This embedding is critical for developing SPDE theory in the space of distributions and for establishing Fourier analytic frameworks (Dalang et al., 2015, Griffiths et al., 2019).

3. Regularity: Besov, Sobolev, and Path Properties

Space-time Lévy white noise almost surely takes values in negative smoothness function spaces. Sample paths almost surely belong to $H^r$ (local Sobolev) with $r<-\tfrac12$ (one-dimensional spatial section) or $r<-1$ (two-dimensional spatial section) (Balan, 2021, Fageot et al., 2015). In the periodic setting, every periodic Lévy white noise $w$ on $\mathbb{T}^d$ satisfies for every $\varepsilon>0$,

$$w \in H^{-d/2-\varepsilon}_2(\mathbb{T}^d) \quad \text{a.s.}$$

Besov (and weighted Besov) sample regularity is characterized via Blumenthal–Getoor indices $\beta_1,\beta_2$ of $\nu$, controlling, respectively, local and tail moment behavior. For weighted Besov spaces $B_{p,q}^{-\tau}(\mathbb{R}^{d+1};-\mu)$, almost sure inclusion holds under

$$\tau > (d+1)\big(1-1/\max(p,\beta_1)\big),\quad \mu > (d+1)/\min(p,\beta_2).$$

Wavelet-based decompositions provide explicit, nearly sharp criteria for almost sure path-space membership (Fageot et al., 2016, Fageot et al., 2015).

4. Stochastic Integration and Calculus

Stochastic calculus is built via multiple integral formulations—both with respect to $L$ (finite variance) and more general Lévy bases. With finite second moment $\int z^2\nu(dz)<\infty$, a canonical $L^2$ isometry for stochastic integrals obtains: $$\mathbb{E}\left|\int_0^T\int_{\mathbb{R}^d} X(s,x)\,L(ds,dx)\right|^2 = \left(\int z^2\nu(dz)\right) \mathbb{E}\int_0^T\int_{\mathbb{R}^d} |X(s,x)|^2\,dx\,ds$$ (Dalang et al., 2018, Balan et al., 2015, Balan et al., 2023). For $p>2$ and finite $p$th moment, a Rosenthal-type maximal inequality applies,

$$\mathbb{E}\bigl[\sup_{0\leq s\leq T}|Y(s)|^p\bigr] \leq C_{p,T} \bigl\{\mathbb{E}\int_0^T\int_{\mathbb{R}^d} |X(s,x)|^p dx ds + (\mathbb{E}\int_0^T\int_{\mathbb{R}^d}|X(s,x)|^2 dx ds)^{p/2}\bigr\}$$

(Balan et al., 2015). In the infinite-variance case, stochastic integration is defined via Daniell-approximable predictable processes, with only convergence in probability guaranteed (Balan et al., 28 Nov 2025).

For square-integrable functionals, chaos expansions and an Itô representation theorem parallel the Gaussian case, providing a complete martingale and multiple integral calculus for functionals of Lévy white noise (Balan et al., 2015).

5. SPDEs Driven by Lévy White Noise: Existence, Regularity, and Structure

Space-time Lévy white noise constitutes the canonical driving signal for a broad family of SPDEs, including parabolic (heat), hyperbolic (wave), and fractional equations—with both additive and multiplicative noise (Balan, 2021, Shiozawa et al., 28 Sep 2025, Balan et al., 28 Nov 2025). Under appropriate moment, integrability, and smoothing conditions on the Lévy measure and equation coefficients, results include:

• Existence and uniqueness of (local or global) mild and generalized solutions in function spaces such as $L^p$, with explicit Lyapunov exponents and growth indices quantifying intermittency (Shiozawa et al., 28 Sep 2025, Dalang et al., 2018, Guo et al., 15 Jun 2025).
• The necessary and sufficient integrability threshold for solutions to be function-valued is typically a finiteness of certain $p$th moments or an explicit integrability condition on the fundamental solution convolved against the driving noise (Dalang et al., 2018, Balan et al., 28 Nov 2025, Guo et al., 15 Jun 2025).

For the stochastic heat equation (SHE) and wave equation, sample regularity is preserved under structure-preserving discretizations, with weak intermittency, negative Sobolev regularity, and convergence rates maintained in fully discrete schemes (Chen et al., 21 Sep 2024). For the fractional Laplacian and critical index regimes, solution growth and propagation of high peaks are tightly linked to the heavy-tailed structure of the noise and the associated heat kernel decay (Shiozawa et al., 28 Sep 2025).

6. Paradigmatic Examples and Applications

Finite-variance Lévy noise: Classic case with all moments up to order two finite, yielding a martingale measure and a direct generalization of Gaussian white noise (Balan et al., 28 Nov 2025, Dalang et al., 2018).

Symmetric $\alpha$-stable noise: Lévy measure $\nu(dz)=C|z|^{-1-\alpha}dz$ with $0<\alpha<2$, leading to infinite variance for $\alpha<2$ but admitting tempered distribution realization when $p<\alpha$ moments exist (Dalang et al., 2015, Griffiths et al., 2019). Generalized and mild solution theory for SPDEs in this case requires moment conditions adapted to the stable index.

Compound Poisson noise: Finite Lévy measure with bounded jumps. All positive moments finite, so mild and generalized solutions exist in the standard function spaces (Dalang et al., 2015, Fageot et al., 2015).

Tables organizing these cases in terms of Lévy measure and sample path regularity:

Lévy Measure Type	$\int_{|z|>1} |z|^p \nu(dz)$ finite	Regularity in Distributions
Compound Poisson	For all $p>0$	$W \in S'(\mathbb{R}^{d+1})$ a.s.
Symm. $\alpha$-stable	For $p<\alpha$	$W \in S'$ a.s. if $p<\alpha$
Cauchy ($\alpha=1$)	For $p<1$	$W \in S'$ a.s. if $p<1$
Variance Gamma	For all $p>0$	$W \in S'$ a.s.

(Dalang et al., 2015)

7. Function Space Perspectives and Nonlinear Approximation

Sample paths of Lévy white noise invariably occupy function spaces of highly negative regularity. Besov and weighted Besov space inclusion criteria provide a sparse, wavelet-based quantification of the irregularity of generalized paths. These align with optimal $n$-term nonlinear approximation rates in wavelet bases and furnish an almost sure description of the heavy-tailed sparsity structure in both deterministic and stochastic contexts (Fageot et al., 2015, Fageot et al., 2016).

8. Outlook and Open Directions

Space-time Lévy white noise underpins a large family of non-Gaussian stochastic models for SPDEs, enabling a fine classification of regularity, intermittency, and integrability phenomena. Precise understanding of the embedding into tempered distributions, weighted functional spaces, and the impact of jump tails on solution theory continues to drive advances, particularly for equations with arbitrary Lévy drivers, fractional operators, and in high-dimensional or anisotropic contexts. Structure-preserving discretizations and numerical methods for Lévy-driven SPDEs are an emerging area of research, ensuring the accurate capture of non-Gaussian path properties in computation (Chen et al., 21 Sep 2024, Guo et al., 15 Jun 2025).

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References:

(Dalang et al., 2015, Fageot et al., 2015, Fageot et al., 2016, Dalang et al., 2018, Griffiths et al., 2019, Chen et al., 21 Sep 2024, Guo et al., 15 Jun 2025, Balan, 2021, Balan et al., 2015, Balan et al., 2023, Balan et al., 28 Nov 2025, Shiozawa et al., 28 Sep 2025)