Multivariate Lévy Noise

• Multivariate Lévy noise is a stochastic process defined as an extension of Gaussian white noise, characterized by infinite divisibility, heavy tails, and jumps in multiple dimensions.
• It is rigorously constructed via characteristic functionals and offers versatile representations such as spectral and chaos expansions, facilitating analysis in weighted function spaces.
• Its practical applications span SPDEs, stochastic differential equations, and data-driven model calibration, providing a robust framework for analyzing complex systems.

Multivariate Lévy noise, or multidimensional Lévy white noise, is a stochastic process generalizing classical Gaussian white noise to incorporate infinite divisibility, heavy tails, and jump behavior in $\mathbb{R}^d$. Its rigorous construction, pathwise regularity, spectral structure, and applications in stochastic analysis, PDEs, and data-driven modeling are foundational in modern probability theory and stochastic modeling.

1. Definition and Construction

A $d$-dimensional Lévy white noise $w$ is a generalized random distribution $w\in S'(\mathbb{R}^d)$, constructed via its characteristic functional. Given the Schwartz space $S(\mathbb{R}^d)$ of test functions, $w$ is specified by

$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$

for any $\phi \in S(\mathbb{R}^d)$, where $f$ is the characteristic exponent of an infinitely divisible law satisfying a finite moment (the "Schwartz condition") (Fageot et al., 2016). By the Minlos–Bochner theorem, this defines a unique law on tempered distributions. Equivalently, $w$ may be viewed as a field of independent, infinitely divisible random variables or as the $d$0-fold weak derivative of a Lévy sheet.

Several constructions exist for specific classes:

• General Lévy fields: Based on the Lévy–Khintchine formula with characteristic triplet $d$1, the noise is realized as the unique random field with

$d$2

where $d$3 is the Lévy symbol (Ernst et al., 2020).

• Multiparameter Lévy noise: Constructed via the compensated Poisson measure of an $d$4-parameter Lévy sheet, with the field

$d$5

where $d$6 is the compensated jump measure (Draouil et al., 29 Oct 2025).

2. Regularity and Function Spaces

The path properties of multivariate Lévy noise are most naturally described in weighted Besov spaces $d$7, characterized via wavelet expansions. The almost-sure membership of $d$8 in certain Besov spaces is governed by the Blumenthal–Getoor indices of the characteristic exponent $d$9:

• Lower index $w$0, controlling tail integrability.
• Upper index $w$1, controlling local singularity.

Key results (Fageot et al., 2016):

• For parameters $w$2, $w$3, $w$4 almost surely when

$w$5

• For local Besov regularity, the sharp condition is $w$6.
• Special classes:
  • Gaussian noise ($w$7): recovers standard $w$8 Sobolev results.
  • $w$9-stable noise: $w\in S'(\mathbb{R}^d)$0.
  • Compound Poisson or Laplace noise: more restrictive regularity, linked to small-jump and large-jump structure.

Smoothing multivariate Lévy noise with Matérn kernels or other mollifiers yields random fields with improved Sobolev or Hölder regularity (Ernst et al., 2020). The mapping $w\in S'(\mathbb{R}^d)$1 is continuous or even Hölder under explicit conditions on the kernel's smoothness and the Lévy measure.

3. Spectral and Chaos Expansions

A spectral representation for multivariate regularly varying Lévy noise of index $w\in S'(\mathbb{R}^d)$2 is constructed via a random content $w\in S'(\mathbb{R}^d)$3 on elementary subsets of $w\in S'(\mathbb{R}^d)$4 such that for $w\in S'(\mathbb{R}^d)$5 (Fuchs et al., 2011): $w\in S'(\mathbb{R}^d)$6 For $w\in S'(\mathbb{R}^d)$7, this yields a random orthogonal measure and familiar $w\in S'(\mathbb{R}^d)$8 spectral calculus; for $w\in S'(\mathbb{R}^d)$9, the representation converges only in probability (summability sense).

In infinite-dimensional settings, chaos expansions are available for Lévy noise via iterated stochastic integrals with respect to the compensated Poisson measure. The basis $S(\mathbb{R}^d)$0, indexed by multi-indices, provides a complete orthogonal representation of $S(\mathbb{R}^d)$1 functionals of the noise, leading to precise stochastic distribution and test function spaces ($S(\mathbb{R}^d)$2 and $S(\mathbb{R}^d)$3) (Draouil et al., 29 Oct 2025).

White noise calculus induces structure such as:

• S-transform and injection,
• Wick product and stochastic derivatives,
• Skorokhod integrals (divergence) and Malliavin differentiation, enabling the derivation of Itô formulas, stochastic integration, and analytic tools for SPDEs.

4. Applications in Stochastic Differential Equations and SPDEs

Affine Term Structure Models

Multivariate Lévy noise serves as the driving source in affine term structure equations, generalizing the Cox–Ingersoll–Ross (CIR) model. Under independent or spherical classes of multivariate Lévy martingales, short-rate SDEs of the form

$S(\mathbb{R}^d)$4

admit strong solutions with explicit affine bond pricing representations (Barski et al., 2022). Under regular variation or radial symmetry conditions, the law of $S(\mathbb{R}^d)$5 is equivalent to that of a system driven by a finite number of independent stable processes or by a single stable process (in the spherical case).

Stochastic Partial Differential Equations

Multivariate Lévy noise is a natural input to parabolic SPDEs, e.g.,

$S(\mathbb{R}^d)$6

with $S(\mathbb{R}^d)$7 an infinite-dimensional Lévy process (Kirchner et al., 2015). The covariance structure of the solution is determined by deterministic space-time variational problems posed on projective and injective tensor product spaces, providing a deterministic equation for the covariance function and well-posedness under operator-norm conditions.

Fractional stochastic heat equations can be driven by multiparameter Lévy noise, where chaos expansions and Skorokhod integration yield explicit mild solutions in the stochastic distribution space $S(\mathbb{R}^d)$8 (Draouil et al., 29 Oct 2025).

Elliptic and Parabolic PDEs with Random Coefficients

Diffusion equations with coefficients given by transforms of smoothed Lévy fields,

$S(\mathbb{R}^d)$9

admit unique $w$0 pathwise solutions under mild growth and tail conditions. Kernel-based approximations yield explicit error rates and convergence in $w$1 (Ernst et al., 2020).

5. Data-Driven Identification and Calibration

Parameter estimation and model calibration for multivariate Lévy processes are addressed via both spectral/characteristic function methods and short-time statistics:

• Neural network-based nonparametric estimation of the Lévy density from low-frequency increments, matching empirical and model characteristic functions via quadrature and automatic differentiation (Xu et al., 2018). Empirical studies show that neural networks outperform piecewise-linear and radial basis functions in reconstructing both smooth and nonsmooth jump densities.
• The multivariate nonlocal Kramers–Moyal framework enables direct, data-driven identification of all coefficients—drift, diffusion, and locally state-dependent Lévy measure—in SDEs with multivariate Lévy noise using closed-form short-time moment formulas and sparse regression (Li et al., 27 Jan 2026). Convergence rates and error bounds are available, and the approach is validated in complex systems.

6. Theoretical Tools: Calculus, Regularity, and Itô Formula

White noise calculus and chaos expansions underpin much of the functional analysis for multivariate Lévy noise:

• The S-transform, Wick products, and Hida–Malliavin derivatives extend to the Poisson/infinitely divisible setting.
• The multiparameter Itô formula holds in the stochastic distribution sense for functionals $w$2 of Lévy sheets, decomposing into stochastic and compensator integrals (Draouil et al., 29 Oct 2025).
• Regularity in function or distribution spaces (e.g., weighted Besov or Sobolev) is critically determined by jump-indices and the spectral structure, with precise characterization available for a large class of noise (Fageot et al., 2016).

7. Summary Table: Regularity Thresholds (Selected Cases)

Noise Type	Blumenthal–Getoor Indices $w$3	Required $w$4, $w$5 for Besov Regularity
Gaussian	$w$6	$w$7, $w$8
$w$9-stable	$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$0	$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$1, $$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$2
Compound Poisson	$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$3	$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$4, no global tempering unless $$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$5
Laplace	$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$6	$$\Phi_w(\phi) = \exp\left( \int_{\mathbb{R}^d} f(\phi(x)) \, dx \right),$$7, local as for compound Poisson

The regularity and pathwise structure of multivariate Lévy noise is fully determined by the jump-indices and the associated function space embeddings (Fageot et al., 2016).

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References:

• "Multidimensional Lévy White Noises in Weighted Besov Spaces" (Fageot et al., 2016)
• "CIR equations with multivariate Lévy noise" (Barski et al., 2022)
• "Integrability and Approximability of Solutions to the Stationary Diffusion Equation with Lévy Coefficient" (Ernst et al., 2020)
• "Spectral Representation of Multivariate Regularly Varying Lévy and CARMA processes" (Fuchs et al., 2011)
• "Calibrating Multivariate Lévy Processes with Neural Networks" (Xu et al., 2018)
• "Multiparameter Lévy white noise theory and applications" (Draouil et al., 29 Oct 2025)
• "Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise" (Kirchner et al., 2015)
• "Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative Lévy noise" (Li et al., 27 Jan 2026)