Gaussian fluctuations for the parabolic Anderson model with Lévy white noise

Abstract: In this article, we consider the parabolic Anderson model driven by a Lévy white noise with finite variance in dimension 1, and we study the asymptotic behaviour of the spatial average of the solution. The main result shows that, with appropriate normalization and centering, the spatial integral converges in distribution to the standard normal distribution, and gives an estimate for the rate of this convergence in the Wasserstein distance, the Kolmogorov distance, and the Fortet-Mourier distance. We also prove the functional limit theorem corresponding to this result.

• The paper establishes rigorous quantitative central limit theorems for spatial averages of the parabolic Anderson model driven by finite-variance Lévy white noise.
• It develops advanced Malliavin calculus on Poisson space, deriving new moment bounds and derivative estimates essential for applying the Malliavin-Stein method.
• The study provides explicit polynomial convergence rates and a functional limit theorem in Skorohod space, bridging results between Gaussian and impulsive noise settings.

Gaussian Fluctuations in the Parabolic Anderson Model with Lévy White Noise

Introduction and Motivation

This work addresses the asymptotic behavior of spatial averages for the parabolic Anderson model (PAM) on the real line, when the system is driven by a finite-variance Lévy white noise rather than the classical Gaussian white noise. The study of spatial averages of SPDEs has been central to the quantitative central limit theorem (QCLT) program for intermittent systems, and understanding the extension to impulsive noise (pure-jump Lévy processes) is highly nontrivial due to the lack of martingale or comparison principles and the presence of discontinuities.

Main Results

The authors establish robust quantitative central limit theorems and functional limit theorems for the spatial average of the unique mild solution $u(t, x)$ to the PAM:

$$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$

where $L$ is a space-time Lévy white noise with finite variance in dimension one. Key technical contributions include new moment and Malliavin derivative bounds, which are essential for applying Malliavin-Stein's method and for obtaining rates of convergence in classical probability distances (Wasserstein, Kolmogorov, Fortet-Mourier).

Quantitative Central Limit Theorem (QCLT)

Let $F_R(t) = \int_{-R}^R (u(t, x) - 1) dx$ denote the spatial average of the fluctuation field. The authors prove that after normalization,

$\frac{F_R(t)}{\sqrt{\operatorname{Var}(F_R(t))}}$

converges in distribution to a standard normal as $R \to \infty$, with explicit polynomial rates of convergence in the cited probability metrics.

Strong numerical result: For Lévy measures with finite moments $m_p$ for all $p \in (1, 3)$, the convergence rate is $R^{-(p-1)}$ for any $p \in (1, 2)$, contrasting the classical rate $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$0 for Gaussian noise. If the Lévy noise is Gamma white noise, the result holds for all $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$1.

Functional Central Limit Theorem (FCLT)

For the process $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$2, after normalization, finite-dimensional distributions converge to a zero-mean Gaussian process with explicitly computed covariance structure. The process admits a càdlàg modification in the Skorohod space $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$3 (as opposed to the pathwise continuous limit in $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$4 for Gaussian cases), due to the jump nature of the Lévy noise.

Ergodicity and Covariance Structure

The solution $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$5 is strictly stationary and ergodic, meaning the spatial average converges almost surely and in $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$6 to zero as the region diverges. The limiting covariance between $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$7 and $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$8 is provided explicitly via integrals involving the Lévy noise variance and the heat kernel.

Methodological Advances

A major novelty lies in the derivation of high-moment bounds and Malliavin derivative estimates for the pure-jump (Poisson) case. The analysis relies on Malliavin calculus on Poisson space, notably using Rosenthal-type inequalities rather than Burkholder-Davis-Gundy. The paper presents sharp $$\frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}(t, x) + u(t, x) L(t, x),$$9 bounds for both first and second Malliavin derivatives with $L$0, crucial for the quantitative normal approximations.

The QCLT proof leverages the second-order Poincaré inequality [27], extended for the Poisson case, in conjunction with these new analytic bounds. Unlike established methods for wave kernel (hyperbolic Anderson model), the transition to the parabolic case required further innovations, especially due to the different moment structure and heat kernel properties.

The limiting functional results uncover that, due to the stochastic heat kernel's non-vanishing $L$1-limit at $L$2, the normalized average process cannot be shown to be continuous in the limit but is only càdlàg, with the jump part explicitly described.

Implications and Connections

The results provide, for the first time, rigorous CLTs for spatial averages in a SPDE with impulsive (non-Gaussian) noise exhibiting jumps, and a full functional limit theorem in the Skorohod topology. These findings bridge a significant gap between theoretical studies of spatial central limit theorems for Gaussian-driven SPDEs and their impulsive-noise counterparts.

The theoretical implications are especially pertinent for the KPZ universality class. Since the PAM driven by Gaussian white noise is related to the KPZ equation via the Hopf–Cole transformation, the extension to Lévy noise suggests possible directions for studying KPZ-type phenomena in impulsive environments. However, the nonnegativity of the solution, necessary for carrying out the Hopf-Cole transform, remains unresolved for Lévy noise due to the lack of a valid comparison principle.

On the practical side, the insights into the fluctuation behavior under pure-jump noise are relevant for models in physics and finance where Lévy processes more accurately capture observed heavy-tail or jump behavior than Gaussian approximations.

Future Directions

Natural extensions include:

• Removing the restriction $L$3 or analyzing higher-dimensional cases.
• Studying more general non-linearities or spatial operators (e.g., inclusion of drift, colored Lévy noise).
• Investigating positivity of the solution, which is essential for KPZ-type universality and for potential application of the Hopf-Cole transformation.
• Extending the methods to establish similar Gaussian fluctuation results for other nonlinear SPDEs with impulsive noise, possibly in higher spatial dimensions or for more singular kernels.

Conclusion

This work rigorously establishes Gaussian fluctuations and precise quantitative central limit theorems for spatial averages of the one-dimensional parabolic Anderson model driven by finite-variance Lévy white noise. It provides explicit convergence rates and functional limit theorems in the Skorohod space, via advanced Malliavin calculus techniques tailored to the Poisson framework. The contributions significantly advance the probabilistic theory of SPDEs in impulsive environments, delineating the nuanced differences between Gaussian and non-Gaussian (pure-jump) universality in fluctuation theory (2607.02742).