Gaussian fluctuations for the parabolic Anderson model with L\'evy white noise

Jinxin Wang; Matis Le Gall; Raluca M. Balan

Gaussian fluctuations for the parabolic Anderson model with L\'evy white noise

T0 review · 0 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Spatial averages of the one-dimensional parabolic Anderson model with finite-variance Lévy white noise obey a quantitative CLT and a functional CLT in Skorohod space.

desk verdict Solid first QCLT and functional CLT for 1D pAm with finite-variance Lévy white noise; the heat-kernel estimates are the real work and they check out. read the letter →

arxiv 2607.02742 v1 pith:OCUT5PHI submitted 2026-07-02 math.PR

Raluca M. Balan , Matis Le Gall , Jinxin Wang This is my paper · ORCID

classification math.PR MSC 60H1560F0560G6060G51

keywords parabolicAndersonmodelLévywhitenoisequantitativeCLTMalliavincalculusPoissonspacespatialaveragefunctionalcentrallimittheoremSkorohod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies the parabolic Anderson model (the stochastic heat equation with multiplicative noise) driven by a finite-variance Lévy white noise in one space dimension. It shows that the spatial average of the solution, after centering and normalization by its standard deviation, converges in law to a standard normal random variable. The rate of this convergence is controlled in the Wasserstein, Kolmogorov and Fortet-Mourier distances and is of order R to a negative power that depends on the highest available moment of the Lévy measure. The same normalization also yields a functional central-limit theorem: the rescaled process converges in the Skorohod space of càdlàg paths to a continuous Gaussian process whose covariance is identified explicitly. The result is the first quantitative fluctuation theorem for this equation in an impulsive (jump) environment; the technical engine is a new moment bound on the first and second Malliavin derivatives of the solution that is strong enough to feed a second-order Poincaré inequality on the Poisson space.

What carries the argument

The key estimate is the moment bound on the Malliavin derivative: ||D_{r,y,z}u(t,x)||_p ≲ |z| (G_{t-r}(x-y)+G_{t-r}^{2/p}(x-y)), together with the analogous bound for the second derivative; these bounds close the second-order Poincaré inequality on the Poisson space and produce the quantitative rates.

What would settle it

Compute the spatial average for a concrete Lévy measure with only finite second moment (so that p cannot reach 3/2) and check whether the empirical Kolmogorov or Wasserstein distance to normality decays slower than any positive power of R; a slower observed rate would contradict the claimed bound.

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Extended reading notes

Core claim

For the one-dimensional parabolic Anderson model driven by finite-variance Lévy white noise, the normalized spatial average F_R(t)/σ_R(t) converges to N(0,1) at the explicit rate O(R^{-(1-1/p)}) in Wasserstein, Kolmogorov and Fortet-Mourier distances whenever there exists p∈(1,3/2) with both m_p and m_{2p} finite; moreover the process R^{-1/2} F_R converges in the Skorohod topology on D[0,T] to a continuous centered Gaussian process with covariance Σ_{t,s}.

Load-bearing premise

The Lévy measure must possess moments of order strictly less than three; without that integrability the fundamental heat-kernel powers are no longer integrable and the Malliavin-derivative estimates fail.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is the missing piece between the Gaussian heat-equation QCLT literature and Balan–Zheng’s Lévy wave results. They get a quantitative CLT for the spatial average of the 1D parabolic Anderson model driven by finite-variance Lévy white noise, with rate R^{-(1-1/p)} under the natural moment window p∈(1,3/2) with m_p and m_{2p} finite, plus the first functional CLT in D[0,T] for this class (continuous Gaussian limit, but the paths of F_R live in the Skorohod space because of the pure-jump part forced by the heat kernel).

What is actually new is the Malliavin bound (11) and its second-order cousin. The heat kernel does not satisfy the nice G_t^p ∼ G_t relation that made Rosenthal work cleanly for the wave equation, so they have to iterate carefully with the two-term Rosenthal estimate and track products of G^2 and G^p factors; the singularities cancel and you land on g_t^{(p)}=G_t+G_t^{2/p}. That argument is technical but self-contained and looks correct. Once those bounds are in hand, the second-order Poincaré estimates of Trauthwein close at the claimed rate, and the continuous-plus-jump decomposition (68) plus Kolmogorov–Chentsov + moment estimates for the martingale part give tightness in D[0,T].

The soft spot is real but transparent: white-noise integrability of G_t^p forces p<3, so the best rate you can extract is strictly worse than the Gaussian R^{-1/2}. They state the limitation clearly and do not overclaim. Existence of p-moments for p∈[2,3) is also new and needed. Citations are appropriate; the self-cites to earlier Balan–Zheng papers are background tools, not circular.

This is for people who already care about quantitative CLTs for SPDEs or Malliavin calculus on Poisson space. It is not a broad-audience paper, but the math is solid and the gap was genuine. I would send it to a serious referee without hesitation; it deserves to be in the literature.

Referee Report

0 major / 5 minor

Summary. The paper studies the parabolic Anderson model (stochastic heat equation with multiplicative noise) driven by a finite-variance Lévy white noise in dimension 1. It establishes a quantitative central limit theorem for the spatial averages F_R(t) = ∫_{|x|<R}(u(t,x)-1) dx: under the moment condition that there exists p∈(1,3/2) with m_p<∞ and m_{2p}<∞, F_R(t)/σ_R(t) converges to N(0,1) at rate O(R^{-(1-1/p)}) in Wasserstein, Kolmogorov and Fortet-Mourier distances (Theorem 1.2). The same hypotheses yield a functional CLT: R^{-1/2} F_R converges in the Skorohod space D[0,T] (J_1 and uniform topologies) to a continuous centered Gaussian process with explicit covariance Σ_{t,s} (Theorem 1.3). Supporting results include spatial ergodicity and the limiting covariance (Theorem 1.1), finite p-moments of the solution for p∈[2,3) (Theorem 2.7), and the key first- and second-order Malliavin derivative bounds (Theorems 3.1–3.2) obtained by iteration with Rosenthal’s inequality. The proofs rely on Malliavin calculus on the Poisson space and Trauthwein’s second-order Poincaré inequality.

Significance. This is the first quantitative CLT and functional CLT for spatial averages of the parabolic Anderson model driven by Lévy white noise. Earlier work treated the hyperbolic Anderson model (wave equation) with Lévy noise and the heat equation with Gaussian noise; the heat kernel lacks the algebraic identity G_t^p = c_p G_t that simplifies the wave case, so the Malliavin bounds require a more delicate iteration that produces the auxiliary kernels g^{(p)} and H^{(p)}. The functional limit in D[0,T] rather than C[0,T] is a genuine novelty forced by the heat kernel and is handled cleanly via the continuous-plus-pure-jump decomposition (68). The rate R^{-(1-1/p)} with p arbitrarily close to 3/2 is essentially optimal under the white-noise integrability restriction p<3. The arguments are self-contained, use standard tools (Rosenthal, second-order Poincaré, Kolmogorov–Chentsov), and supply the first rigorous asymptotics for spatial averages of pAm in an impulsive environment.

minor comments (5)

[Abstract / Theorem 1.2] In the abstract and introduction the rate is stated as R^{-(1-1/p)}; it would help the reader to note explicitly that the admissible range is p∈(1,3/2), so the best rate obtainable under the present hypotheses is any ε<1/3.
[Theorem 1.2] The constant C_t in Theorem 1.2 is said to depend on t; a brief remark that it also depends on the Lévy moments m_p, m_{2p} and on the constants appearing in Rosenthal’s inequality would make the dependence transparent.
[Section 2.3, proof of Theorem 2.7] In the proof of Theorem 2.7 the bound (32) for E|J_n|^p contains the factor (n!)^{-(3-p)/2}; the subsequent comparison with the Gamma function via Stirling is correct but could be referenced more cleanly (e.g., by citing a standard asymptotic for Γ(an+b)).
[Section 4.2, Lemma 4.2] Lemma 4.2 introduces the auxiliary kernels K^{(p)} and Φ^{(p)}; a one-line comparison with the earlier g^{(p)} and H^{(p)} would clarify that they differ only by the harmless rescaling of the time argument that appears in (20).
[Throughout / Eq. (39)] A few typographical slips: “Lévy” is occasionally written without the accent; “Fortet-Mourier” should be consistently hyphenated; in (39) the lower bound t-t_1/p ≤ γ^{(ℓ)} uses the same symbol γ for two different objects in nearby displays.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the QCLT and functional CLT are derived from the mild formulation, Rosenthal iteration, and an external second-order Poincaré inequality.

full rationale

The paper's central claims (Theorems 1.2 and 1.3) are obtained by a self-contained analytic chain: existence and L^p moments of the mild solution via Picard iteration and Rosenthal's inequality (Theorem 2.7), key Malliavin bounds (11) and Theorem 3.2 proved by induction on the Picard sequence with careful control of the product of heat-kernel singularities (Section 3), and quantitative normal approximation via Trauthwein's external second-order Poincaré inequality (Proposition 4.1, cited as [27]) applied to F_R(t) with the resulting γ_i estimated by the new bounds (Section 4.2). The limiting covariance Σ_{t,s} is identified by equating second moments with the Gaussian parabolic Anderson model (18) and evaluating the series explicitly (Section 4.1); this is an independent calculation, not a fit. Self-citations to earlier Balan–Zheng works supply standard Poisson-space Malliavin tools and the wave-equation analogue; they are not load-bearing for the heat-kernel estimates or the rate R^{-(1-1/p)}. The functional CLT uses the natural continuous-plus-jump decomposition forced by the heat kernel (68) and standard tightness criteria. No step reduces by construction to its own input, and no uniqueness or ansatz is smuggled via self-citation. Score 0 is therefore the correct assessment.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The paper works entirely inside standard stochastic analysis on the Poisson space. The only free parameters are the Lévy measure moments that define the admissible noise class; no numerical fitting occurs. Invented entities are limited to the usual solution process and its Malliavin derivatives. All background tools (Rosenthal, second-order Poincaré, chaos expansions) are cited from the literature.

free parameters (1)

p in (1,3/2) with m_p and m_{2p} finite
The admissible range of p is chosen so that the second-order Poincaré quantities gamma_i remain finite and yield a positive rate; it is not fitted to data but is a free modeling parameter that determines the obtainable rate.

assumptions (4)

standard math Rosenthal's inequality for stochastic integrals with respect to compensated Poisson measures (Proposition 2.4)
Used repeatedly to bound p-moments of the mild solution and of its Malliavin derivatives; taken from the literature (e.g. Marinelli-Röckner).
standard math Second-order Poincaré inequality on the Poisson space (Proposition 4.1 / Trauthwein 2025)
The sole source of the quantitative bounds on Wasserstein, Kolmogorov and Fortet-Mourier distances; assumed as a black-box theorem.
domain assumption Existence and uniqueness of a mild solution with finite second moments for the Lévy pAm (Balan-Ndongo 2017)
Invoked at the outset so that the spatial averages F_R(t) are well-defined L^2 random variables.
domain assumption Finite p-moments of the Lévy measure for some p in [2,3) (condition m_p < infinity)
Necessary for the integrability of G_t^p and for the application of Rosenthal; cannot be removed for space-time white noise.

invented entities (1)

The auxiliary kernels g^{(p)}_t = G_t + G_t^{2/p} and H^{(p)}_t = G_t^2 + G_t^p
purpose: To capture the two terms produced by Rosenthal's inequality in the iterative bounds for Du and D^2 u.
These are not physical entities but convenient majorants; they have no independent existence outside the estimates.

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Cite this review

Pith. "Pith review of Gaussian fluctuations for the parabolic Anderson model with L\'evy white noise." pith.science (2026). https://pith.science/paper/OCUT5PHI

@misc{pith2026260702742,
  author       = {Pith},
  title        = {Pith review of: Gaussian fluctuations for the parabolic Anderson model with L\'evy white noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OCUT5PHI}},
  note         = {Machine review of arXiv:2607.02742}
}

read the original abstract

In this article, we consider the parabolic Anderson model driven by a L\'evy 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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 2 linked inside Pith

[1]

Alberts, T., Janjigian C., Rassoul-Agha, F., and Sepp¨ al¨ ainen, T. (2022). The Green’s function of the parabolic Anderson model and the continuum directed polymer. Preprint arXiv2208.11255

arXiv 2022
[2]

and Ndongo, C.B

Balan, R.M. and Ndongo, C.B. (2016) Intermittency for the wave equation with L´ evy white noise.Statist. Probab. Lett.109, 214-223. 32

2016
[3]

and Ndongo, C.B

Balan, R.M. and Ndongo, C.B. (2017). Malliavin differentiability of solutions of SPDEs with L´ evy white noise.International Journal of Stochastic Analysis.Arti- cle ID 9693153

2017
[4]

and Zheng, G

Balan, R.M., Nualart, D., Quer-Sardanyons, L. and Zheng, G. (2022). The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications. Stoch. PDEs: Anal. Comput.10, 757–827

2022
[5]

and Salins, M

Balan, R.M. and Salins, M. (2026). Gaussian fluctuations for the nonlinear stochastic heat equation with drift.Electr. J. Probab.To appear

2026
[6]

and Zheng, G

Balan, R.M. and Zheng, G. (2024). Hyperbolic Anderson model with L´ evy white noise: spatial ergodicity. and fluctuation.Trans. AMS377, 4171-4221

2024
[7]

and Zheng, G

Balan, R.M. and Zheng, G. (2026). Central limit theorem for stochastic nonlinear wave equation with pure-jump L´ evy white noise.Electr. J. Probab.31, no. 70, 1-46

2026
[8]

(1999).Convergence of Probability Measures

Billingsley, P. (1999).Convergence of Probability Measures. Second Edition. Wiley, New York

1999

Show all 28 references

[9]

and Zheng, G

Bola˜ nos-Guerrero, R., Nualart, D. and Zheng, G. (2021). Averaging 2d stochastic wave equation,Electr. J. Probab.26, Paper No. 102, 32

2021
[10]

and Pu, F

Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2022). Spatial ergodity and central limit theorems for parabolic Anderson model with delta initial condition.J. Funct. Anal.282, 109290

2022
[11]

and Pu, F

Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2022). Poincar´ e inequality, and central limit theorems for parabolic stochastic partial differential equations.Ann. Inst. Henri Poincar´ e: Prob. Stat.58, 1052-1077

2022
[12]

and Pu, F

Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2023). Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method. Stoch. PDEs: Anal. & Comput.11, 122–176

2023
[13]

and Zheng, G

Delgato-Vences, F., Nualart, D. and Zheng, G. (2020). A central limit theorem for the stochastic wave equation with fractional noise.Ann. Inst. Henri Poincar´ e: Probab. Stat.56, 3020-3042

2020
[14]

and Valkeila, E

Dzhaparidze, K. and Valkeila, E. (1990). On the Hellinger type distances for filtered experiments.Probab. Th. Rel. Fields85, 105-117

1990
[15]

Ebina, M. (2022). Central limit theorems for nonlinear stochastic wave equations in dimension three, Preprint arXiv:2206.12957

2022 arXiv
[16]

and Viitasaari, L

Huang, J., Nualart, D. and Viitasaari, L. (2020). A central limit theorem for the stochastic heat equation,Stoch. Proc. Appl.130, no. 12, 7170–7184. 33

2020
[17]

and Zheng, G

Huang, J., Nualart, D., Viitasaari L. and Zheng, G. (2020). Gaussian fluctuations for the stochastic heat equation with colored noise.Stoch. PDE: Anal. & Comp.8, 402-421

2020
[18]

(2021).Foundations of Modern Probability

Kallenberg, O. (2021).Foundations of Modern Probability. Third Edition. Springer, Cham

2021
[19]

S´ eminaire des probabilit´ es XLVI

Marinelli, C. and R¨ ockner, M. (2014).On maximal inequalities for purely discontin- uous martingales in infinite dimensions. In: “S´ eminaire des probabilit´ es XLVI”, eds. Donati-Martin, C., Lejay, A. and Rouault, A.Lect. Notes Math.2123, Springer, 293-315

2014
[20]

(2006).The Malliavin Calculus and Related Topics

Nualart D. (2006).The Malliavin Calculus and Related Topics. Second edition. Springer-Verlag, Berlin

2006
[21]

and Nualart, E

Nualart, D. and Nualart, E. (2018).Introduction to Malliavin Calculus. Cambridge University Press, Cambridge

2018
[22]

and Zheng, G

Nualart, D., Song, X. and Zheng, G. (2020). Spatial averages for the Parabolic An- derson model driven by rough noise,ALEA Latin Amer. J. Probab. Math. Stat.18, no. 1, 907–943

2020
[23]

and Zheng, G

Nualart, D., Xia, P. and Zheng, G. (2022). Quantitative central limit theorems for the parabolic Anderson model driven by colored noises,Electr. J. Probab.27, Paper No. 120, 43

2022
[24]

and Zheng, G

Nualart, D. and Zheng, G. (2020). Averaging Gaussian functionals.Electr. J. Probab. 25, paper no. 48, 54 pp

2020
[25]

and Zheng, G

Nualart, D. and Zheng, G. (2022). Central limit theorems for stochastic wave equa- tions in dimensions one and two,Stoch. PDE: Anal. Comput.10, no. 2, 392–418

2022
[26]

and Zabczyk, J

Peszat, S. and Zabczyk, J. (2007).Stochastic partial differential equations with L´ evy noise. Cambridge University Press

2007
[27]

Trauthwein, T. (2025). Quantitative CLTs on the Poisson space via Skorohod esti- mates andp-Poincar´ e inequalities.Ann. Appl. Probab.35, 1716-1754

2025
[28]

(1980) Some useful functions for functional limit theorems.Mathematics of Operations Research5, 67-85

Whitt, W. (1980) Some useful functions for functional limit theorems.Mathematics of Operations Research5, 67-85. 34

1980

Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.

[1] [1]

Alberts, T., Janjigian C., Rassoul-Agha, F., and Sepp¨ al¨ ainen, T. (2022). The Green’s function of the parabolic Anderson model and the continuum directed polymer. Preprint arXiv2208.11255

arXiv 2022

[2] [2]

and Ndongo, C.B

Balan, R.M. and Ndongo, C.B. (2016) Intermittency for the wave equation with L´ evy white noise.Statist. Probab. Lett.109, 214-223. 32

2016

[3] [3]

and Ndongo, C.B

Balan, R.M. and Ndongo, C.B. (2017). Malliavin differentiability of solutions of SPDEs with L´ evy white noise.International Journal of Stochastic Analysis.Arti- cle ID 9693153

2017

[4] [4]

and Zheng, G

Balan, R.M., Nualart, D., Quer-Sardanyons, L. and Zheng, G. (2022). The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications. Stoch. PDEs: Anal. Comput.10, 757–827

2022

[5] [5]

and Salins, M

Balan, R.M. and Salins, M. (2026). Gaussian fluctuations for the nonlinear stochastic heat equation with drift.Electr. J. Probab.To appear

2026

[6] [6]

and Zheng, G

Balan, R.M. and Zheng, G. (2024). Hyperbolic Anderson model with L´ evy white noise: spatial ergodicity. and fluctuation.Trans. AMS377, 4171-4221

2024

[7] [7]

and Zheng, G

Balan, R.M. and Zheng, G. (2026). Central limit theorem for stochastic nonlinear wave equation with pure-jump L´ evy white noise.Electr. J. Probab.31, no. 70, 1-46

2026

[8] [8]

(1999).Convergence of Probability Measures

Billingsley, P. (1999).Convergence of Probability Measures. Second Edition. Wiley, New York

1999