pith. sign in

arxiv: 2504.18672 · v1 · submitted 2025-04-25 · 🧮 math.PR

Central limit theorem for stochastic nonlinear wave equation with pure-jump L\'evy white noise

Raluca M. Balan , Guangqu Zheng This is my paper

Pith reviewed 2026-05-22 18:21 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic nonlinear wave equationcentral limit theoremLevy white noiseMalliavin calculusWasserstein distancespatial ergodicityGaussian fluctuation
0
0 comments X

The pith

The spatial integrals of the solution to the stochastic nonlinear wave equation with pure-jump Lévy white noise satisfy a central limit theorem with explicit Wasserstein convergence rate.

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

This paper establishes Malliavin differentiability of the random field solution to the stochastic nonlinear wave equation with multiplicative Lipschitz noise driven by pure-jump Lévy white noise of finite variance. The differentiability supplies sharp moment bounds that enable spatial ergodicity, which immediately yields a law of large numbers for integrals of the solution over expanding intervals. The central result applies an adapted discrete Malliavin-Stein bound to obtain Gaussian fluctuations for the normalized integrals together with a quantitative rate in Wasserstein distance. A reader should care because the argument handles the nonlinear case without relying on explicit chaos expansions that were available only for the linear equation.

Core claim

We prove that the solution to the SNLW with constant initial conditions and multiplicative noise σ(u) driven by pure-jump Lévy white noise is Malliavin differentiable, with sharp moment bounds on the derivatives obtained via Itô and Malliavin calculus under the Lipschitz assumption on σ. These bounds allow direct application of an adapted discrete Malliavin-Stein bound, which produces a central limit theorem for the spatial integrals over [-R, R] as R tends to infinity, including a rate of convergence in Wasserstein distance. Functional and almost sure versions of the CLT are established, together with quantitative asymptotic independence of integrals from different spatial regions.

What carries the argument

Malliavin differentiability of the solution with sharp moment bounds on the derivatives, which permits application of the discrete Malliavin-Stein bound adapted to the Poisson setting.

If this is right

  • Spatial ergodicity of the solution implies a law of large numbers for the spatial integrals as the interval length tends to infinity.
  • A functional central limit theorem holds when the integrals are viewed as a process indexed by interval endpoints.
  • An almost sure version of the central limit theorem is valid for the spatial integrals.
  • Quantitative asymptotic independence holds between spatial integrals supported on disjoint intervals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Malliavin bounds may be tested on numerical realizations of the wave equation to confirm the predicted Wasserstein rates for moderate interval sizes.
  • The adaptation of the Malliavin-Stein method could be examined for other nonlinear SPDEs driven by jump noise where chaos expansions are unavailable.
  • The asymptotic independence result suggests possible links to spatial mixing rates in infinite-dimensional systems with discontinuous driving noise.

Load-bearing premise

The solution admits Malliavin differentiability with sharp moment bounds on the derivatives, which holds under the Lipschitz condition on σ and finite variance of the Lévy noise.

What would settle it

Numerical computation of the Wasserstein distance between the law of the normalized spatial integral over [-R, R] and a standard Gaussian, checking whether the distance decays at the precise rate stated by the adapted Malliavin-Stein bound as R increases.

read the original abstract

In this paper, we study the random field solution to the stochastic nonlinear wave equation (SNLW) with constant initial conditions and multiplicative noise $\sigma(u)\dot{L}$, where the nonlinearity is encoded in a Lipschitz function $\sigma: \mathbb{R}\to\mathbb{R}$ and $\dot{L}$ denotes a pure-jump L\'evy white noise on $\mathbb{R}_+\times\mathbb{R}$ with finite variance. Combining tools from It\^o calculus and Malliavin calculus, we are able to establish the Malliavin differentiability of the solution with sharp moment bounds for the Malliavin derivatives. As an easy consequence, we obtain the spatial ergodicity of the solution to SNLW that leads to a law of large number result for the spatial integrals of the solution over $[-R, R]$ as $R\to\infty$. One of the main results of this paper is the obtention of the corresponding Gaussian fluctuation with rate of convergence in Wasserstein distance. To achieve this goal, we adapt the discrete Malliavin-Stein bound from Peccati, Sol\'e, Taqqu, and Utzet ({\it Ann. Probab.}, 2010), and further combine it with the aforementioned moment bounds of Malliavin derivatives and It\^o tools. Our work substantially improves our previous results (\textit{Trans.~Amer.~Math.~Soc.}, 2024) on the linear equation that heavily relied on the explicit chaos expansion of the solution. In current work, we also establish a functional version, an almost sure version of the central limit theorems, and the (quantitative) asymptotic independence of spatial integrals from the solution. The asymptotic independence result is established based on an observation of L. Pimentel (\textit{Ann.~Probab.}, 2022) and a further adaptation of Tudor's generalization (\textit{Trans.~Amer.~Math.~Soc.}, 2025) to the Poisson setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes Malliavin differentiability of the mild solution to the stochastic nonlinear wave equation with multiplicative pure-jump Lévy white noise of finite variance, under a Lipschitz condition on σ. It derives sharp moment bounds on the Malliavin derivatives via Itô and Malliavin calculus, obtains spatial ergodicity and a law of large numbers for the spatial integrals I_R = ∫_{-R}^R u(t,x) dx as R→∞, and proves a quantitative central limit theorem for the centered and normalized integrals in Wasserstein distance by adapting the discrete Malliavin-Stein bound of Peccati et al. (2010). Additional results include functional and almost-sure versions of the CLT as well as asymptotic independence of spatial integrals, extending the authors' prior linear-case results that used explicit chaos expansions.

Significance. If the Malliavin moment bounds scale appropriately with the integration length R, the work provides a technically substantial extension of quantitative CLTs to nonlinear SPDEs driven by jump noise, avoiding reliance on explicit Wiener-Itô chaos expansions. The adaptation of Stein bounds to the Poisson setting for the wave equation, combined with finite-speed-of-propagation properties, is a clear strength and yields explicit rates in Wasserstein distance.

major comments (2)
  1. [§4] §4 (moment bounds): The Gronwall estimates control E[|Du(t,x)|^p] pointwise, but the manuscript does not explicitly compute or bound the integrated quantity E[‖D I_R‖_{L^2(μ)}^2] or show that it is O(R). Without this scaling, the Stein factor E[|1 − ⟨D I_R, −L^{-1} D I_R⟩|] may fail to be O(R^{-1/2}), blocking a vanishing Wasserstein rate.
  2. [§5] §5 (adapted Malliavin-Stein bound): The application of the discrete bound from Peccati-Solé-Taqqu-Utzet (2010) to the integrated field I_R requires verifying that the Malliavin covariance term vanishes at the claimed rate; the current argument invokes the pointwise bounds without an explicit integration-over-cones estimate that accounts for the union of backward light cones.
minor comments (2)
  1. [Abstract] The abstract states 'sharp moment bounds' without indicating their R-dependence; adding a brief scaling statement would improve readability.
  2. [§2] Notation for the Lévy measure and the difference operator realizing the Malliavin derivative could be introduced earlier to ease comparison with the Gaussian case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions regarding explicit scaling of integrated Malliavin quantities and cone estimates are helpful, and we address each major comment below. We will strengthen the presentation by adding the requested explicit calculations in the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (moment bounds): The Gronwall estimates control E[|Du(t,x)|^p] pointwise, but the manuscript does not explicitly compute or bound the integrated quantity E[‖D I_R‖_{L^2(μ)}^2] or show that it is O(R). Without this scaling, the Stein factor E[|1 − ⟨D I_R, −L^{-1} D I_R⟩|] may fail to be O(R^{-1/2}), blocking a vanishing Wasserstein rate.

    Authors: We thank the referee for this observation. The pointwise bounds on E[|D u(t,x)|^p] are obtained in Section 4 via Gronwall's inequality applied to the mild formulation. Because of the finite speed of propagation for the wave equation, D_{s,y} u(t,x) vanishes outside the backward light cone from (t,x). Integrating the pointwise bounds over the space-time support relevant to I_R = ∫_{-R}^R u(t,x) dx therefore yields E[‖D I_R‖_{L^2(μ)}^2] = O(R) by a direct Fubini argument whose measure is linear in R. We will insert an explicit lemma computing this integrated quantity and confirming the O(R) scaling, which guarantees that the Stein factor is O(R^{-1/2}) and the Wasserstein rate vanishes. This addition will be made in the revised manuscript. revision: yes

  2. Referee: [§5] §5 (adapted Malliavin-Stein bound): The application of the discrete bound from Peccati-Solé-Taqqu-Utzet (2010) to the integrated field I_R requires verifying that the Malliavin covariance term vanishes at the claimed rate; the current argument invokes the pointwise bounds without an explicit integration-over-cones estimate that accounts for the union of backward light cones.

    Authors: We agree that an explicit integration-over-cones estimate would make the argument in Section 5 fully transparent. The present proof combines the pointwise Malliavin moment bounds with the geometry of the wave equation's light cones. To address the referee's concern, we will add a detailed computation in the revision that evaluates the measure of the union of backward light cones emanating from points in [-R,R]. This calculation shows that the Malliavin covariance term is O(R^{-1/2}), as needed for the quantitative CLT via the discrete Malliavin-Stein bound of Peccati et al. (2010). The revised text will include this estimate explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity: new Malliavin bounds derived independently and fed into external Stein bound

full rationale

The paper derives Malliavin differentiability of the mild solution to the nonlinear SNLW together with sharp moment bounds on the derivatives via Itô and Malliavin calculus under the Lipschitz condition on σ and finite variance of the Lévy noise. These newly obtained bounds are then inserted into an adapted discrete Malliavin-Stein inequality taken from the external reference Peccati et al. (2010). The approach is explicitly contrasted with the authors' earlier 2024 work on the linear equation, which relied on chaos expansions; the present argument does not invoke that prior result as a load-bearing premise. Citations to Pimentel (2022) and Tudor (2025) are likewise external. No equation or claim reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain, so the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions of stochastic analysis for Lévy-driven SPDEs; no new entities or fitted parameters are introduced.

axioms (2)
  • domain assumption The pure-jump Lévy white noise has finite variance
    Invoked to obtain moment bounds and to apply the Malliavin-Stein method.
  • domain assumption The nonlinearity σ is Lipschitz continuous
    Required for existence of the random-field solution and for Malliavin differentiability with sharp moment bounds.

pith-pipeline@v0.9.0 · 5895 in / 1552 out tokens · 72966 ms · 2026-05-22T18:21:51.873668+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We adapt the discrete Malliavin-Stein bound from Peccati, Solé, Taqqu, and Utzet (Ann. Probab., 2010), and further combine it with the aforementioned moment bounds of Malliavin derivatives and Itô tools.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Applebaum, L´ evy Processes and Stochastic Calculus,Second edition, Cambridge Studies in Advanced Mathematics 116, Cambridge University Press, Cambridge, 2 009

    D. Applebaum, L´ evy Processes and Stochastic Calculus,Second edition, Cambridge Studies in Advanced Mathematics 116, Cambridge University Press, Cambridge, 2 009

    work page

  2. [2]

    Balan, Integration with respect to L´ evy colored noise, with appli cations to SPDEs, Stochastics 87 (2015), no

    R.M. Balan, Integration with respect to L´ evy colored noise, with appli cations to SPDEs, Stochastics 87 (2015), no. 3, 363–381

    work page 2015

  3. [3]

    Balan, C.B

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

    work page 2016

  4. [4]

    Balan, C.B

    R.M. Balan, C.B. Ndongo, Malliavin differentiability of solutions of SPDEs with L´ evy white noise, Int. J. Stoch. Anal. 2017, Art. ID 9693153, 9 pp

    work page 2017

  5. [5]

    Balan, D

    R.M. Balan, D. Nualart, L. Quer-Sardanyons, G. Zheng, The hyperbolic Anderson model: moment esti- mates of the Malliavin derivatives and applications , Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), no. 3, 757–827

    work page 2022

  6. [6]

    Balan, L

    R.M. Balan, L. Quer-Sardanyons, J. Song, Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise , Electron. J. Probab. 24 (2019), no. 106, 1–43

    work page 2019

  7. [7]

    Balan, J

    R.M. Balan, J. Song, Hyperbolic Anderson model with space-time homogeneous Gau ssian noise, ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017), no. 2, 799–849

    work page 2017

  8. [8]

    Balan, P

    R.M. Balan, P. Xia, G. Zheng, Almost sure central limit theorem for the hyperbolic Anders on model with L´ evy white noise, arXiv:2310.10784 [math.PR]; to appear in: Proc. Amer. Mat h. Soc. 46 R.M. BALAN AND G. ZHENG

    work page arXiv

  9. [9]

    Balan, G

    R.M. Balan, G. Zheng, Hyperbolic Anderson model with L´ evy white noise: spatial e rgodicity and fluctua- tion, Trans. Amer. Math. Soc. 377 (2024), no. 6, 4171–4221

    work page 2024

  10. [10]

    Billingsley, Convergence of Probability Measures , John Wiley & Sons, Inc., New York-London-Sydney, 1968, xii+253 pp

    P. Billingsley, Convergence of Probability Measures , John Wiley & Sons, Inc., New York-London-Sydney, 1968, xii+253 pp

    work page 1968

  11. [11]

    Bola˜ nos-Guerrero, D

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

    work page 2021

  12. [12]

    Bourguin, G

    S. Bourguin, G. Peccati, The Malliavin-Stein method on the Poisson space, Stochastic analysis for Poisson point processes, 185–228. Bocconi Springer Ser., 7 Bocconi University Press, 2016

    work page 2016

  13. [13]

    L. Chen, D. Khoshnevisan, D. Nualart, F. Pu, Spatial ergodicity for SPDEs via Poincar´ e-type inequalities, Electron. J. Probab. 26 (2021), Paper No. 140, 37 pp

    work page 2021

  14. [14]

    L. Chen, D. Khoshnevisan, D. Nualart, F. Pu, Central limit theorems for parabolic stochastic partial differential equations, Ann. Inst. Henri Poincar´ e Probab. Stat. 58 (2022), no. 2, 10 52–1077

    work page 2022

  15. [15]

    L. Chen, D. Khoshnevisan, D. Nualart, F. Pu, Spatial ergodicity and central limit theorems for paraboli c Anderson model with delta initial condition, J. Funct. Anal. 282 (2022), no. 2, Paper No. 109290, 35 pp

    work page 2022

  16. [16]

    Chong, R.C

    C. Chong, R.C. Dalang, T. Humeau, Path properties of the solution to the stochastic heat equat ion with L´ evy noise,Stoch. Partial Differ. Equ. Anal. Comput. 7 (2019), no. 1, 123 –168

    work page 2019

  17. [17]

    Dalang, Extending the martingale measure stochastic integral with applications to spatially homoge- neous s.p.d.e.’s, Electron

    R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homoge- neous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29 pp

    work page 1999

  18. [18]

    Davydov, S.M

    Y.A. Davydov, S.M. Novikov, Remarks on asymptotic independence, Teor. Veroyatn. Primen. 66 (2021), no. 1, 55–72; translation in Theory Probab. Appl. 66 (2021), no. 1, 44–57; arXiv:1910.04243 [math.PR]

    work page arXiv 2021

  19. [19]

    Delgado-Vences, D

    F. Delgado-Vences, D. Nualart, G. Zheng, A central limit theorem for the stochastic wave equation wit h fractional noise, Ann. Inst. Henri Poincar´ e Probab. Stat. 56 (2020), no. 4, 30 20–3042

    work page 2020

  20. [20]

    Dudley, Real analysis and probability, Revised reprint of the 1989 original Cambridge Stud

    R.M. Dudley, Real analysis and probability, Revised reprint of the 1989 original Cambridge Stud. Adv. Math., 74 Cambridge University Press, Cambridge, 2002. x+5 55 pp

    work page 1989

  21. [21]

    Ebina, Central limit theorems for nonlinear stochastic wave equat ions in dimension three, Stoch

    M. Ebina, Central limit theorems for nonlinear stochastic wave equat ions in dimension three, Stoch. Partial Differ. Equ. Anal. Comput. 12 (2024), no. 2, 1141–120 0

    work page 2024

  22. [22]

    Ebina, Central limit theorems for stochastic wave equations in hig h dimensions , Electron

    M. Ebina, Central limit theorems for stochastic wave equations in hig h dimensions , Electron. J. Probab. 30 (2025), 1–56

    work page 2025

  23. [23]

    Evans, R.F

    L.C. Evans, R.F. Gariepy, Measure theory and fine properties of functions, Revised edition, Textb. Math. CRC Press, Boca Raton, FL, 2015. xiv+299 pp

    work page 2015

  24. [24]

    Y. Hu, J. Huang, D. Nualart, S. Tindel, Stochastic heat equations with general multiplicative Gau ssian noises: H¨ older continuity and intermittency, Electron. J. Probab. 20 (2015), no. 55, 50 pp

    work page 2015

  25. [25]

    Huang, D

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

    work page 2020

  26. [26]

    Huang, D

    J. Huang, D. Nualart, L. Viitasaari, G. Zheng, Gaussian fluctuations for the stochastic heat equation with colored noise, Stoch. Partial Differ. Equ. Anal. Comput. 8 (2020), no. 2, 402–421

    work page 2020

  27. [27]

    Last, Stochastic analysis for Poisson processes, In G

    G. Last, Stochastic analysis for Poisson processes, In G. Peccati and M. Reitzner, editors, Stochastic analysis for Poisson point processes , Mathematics, Statistics, Finance and Economics, chapter 1, pages 1–36. Bocconi Springer Ser., 7, Bocconi Univ. Press, 2016

    work page 2016

  28. [28]

    G. Last, I. Molchanov, M. Schulte, normal approximation of Kabanov-Skorohod integrals on Poi sson spaces, J. Theoret. Probab. 37 (2024), no. 2, 1124–1167

    work page 2024

  29. [29]

    G. Last, G. Peccati, M. Schulte, Normal approximation on Poisson spaces: Mehler’s formula, second order Poincar´ e inequalities and stabilization,Probab. Theory Related Fields 165 (2016), no. 3-4, 667–723

    work page 2016

  30. [30]

    G. Last, M. Penrose, Lectures on the Poisson process, Institute of Mathematical Statistics Textbooks, 7. Cambridge University Press, Cambridge, 2018. xx+293 pp

    work page 2018

  31. [31]

    Nourdin and G

    I. Nourdin and G. Peccati, Stein ’s method on Wiener chaos , Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118

    work page 2009

  32. [32]

    Nourdin and G

    I. Nourdin and G. Peccati, Normal approximations with Malliavin calculus: from Stein ’s method to uni- versality, Cambridge Tracts in Mathematics 192. Cambridge Universit y Press, Cambridge, 2012, xiv+239

    work page 2012

  33. [33]

    Nualart, The Malliavin calculus and related topics, Second edition

    D. Nualart, The Malliavin calculus and related topics, Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp

    work page 2006

  34. [34]

    Nualart, E

    D. Nualart, E. Nualart, Introduction to Malliavin calculus, Institute of Mathematical Statistics Textbooks,

    work page

  35. [35]

    xii+236 pp

    Cambridge University Press, Cambridge, 2018. xii+236 pp

    work page 2018

  36. [36]

    Nualart, X

    D. Nualart, X. Song, G. Zheng, Spatial averages for the parabolic Anderson model driven by rough noise, ALEA Lat. Am. J. Probab. Math. Stat. 18 (2021), no. 1, 907–943 . CLT FOR SNL W WITH PURE-JUMP L ´EVY WHITE NOISE 47

    work page 2021

  37. [37]

    Nualart, P

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

    work page 2022

  38. [38]

    Nualart, G

    D. Nualart, G. Zheng, Averaging Gaussian functionals, Electron. J. Probab. 25 (2020), Paper No. 48, 54 pp

    work page 2020

  39. [39]

    Nualart, G

    D. Nualart, G. Zheng, Spatial ergodicity of stochastic wave equations in dimensi ons 1, 2 and 3 , Electron. Commun. Probab. 25 (2020), Paper No. 80, 11 pp

    work page 2020

  40. [40]

    Nualart, G

    D. Nualart, G. Zheng, Central limit theorems for stochastic wave equations in dim ensions one and two, Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), no. 2, 39 2–418

    work page 2022

  41. [41]

    Peccati, J.L

    G. Peccati, J.L. Sol´ e, M.S. Taqqu, F. Utzet, Stein ’s method and normal approximation of Poisson func- tionals, Ann. Probab. 38 (2010), no.2, 443–478

    work page 2010

  42. [42]

    Peccati, C

    G. Peccati, C. Th¨ ale, Gamma limits and U -statistics on the Poisson space, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 525–560

    work page 2013

  43. [43]

    Peszat, J

    S. Peszat, J. Zabczyk, Stochastic partial differential equations with L´ evy noise. An evolution equation ap- proach, Encyclopedia of Mathematics and its Applications, 113. Cam bridge University Press, Cambridge,

    work page

  44. [44]

    Pu, Gaussian fluctuation for spatial average of parabolic Ander son model with Neumann/Dirichlet/p- eriodic boundary conditions, Trans

    F. Pu, Gaussian fluctuation for spatial average of parabolic Ander son model with Neumann/Dirichlet/p- eriodic boundary conditions, Trans. Amer. Math. Soc. 375 (2022), no. 4, 2481–2509

    work page 2022

  45. [45]

    Pimentel, Integration by parts and the KPZ two-point function , Ann

    L. Pimentel, Integration by parts and the KPZ two-point function , Ann. Probab. 50, 2022, 1755–1780

    work page 2022

  46. [46]

    Ross, Fundamentals of Stein ’s method, Probab

    N. Ross, Fundamentals of Stein ’s method, Probab. Surv. 8 (2011), 210–293

    work page 2011

  47. [47]

    Trauthwein, Quantitative CLTs on the Poisson space via Skorohod estimat es and p-Poincar´ e inequal- ities, arXiv:2212.03782 [math.PR]; to appear in: Ann

    T. Trauthwein, Quantitative CLTs on the Poisson space via Skorohod estimat es and p-Poincar´ e inequal- ities, arXiv:2212.03782 [math.PR]; to appear in: Ann. Appl. Prob ab

    work page arXiv

  48. [48]

    Tudor, Multidimensional Stein method and quantitative asymptoti c independence, Trans

    C.A. Tudor, Multidimensional Stein method and quantitative asymptoti c independence, Trans. Amer. Math. Soc. 378 (2025), no. 2, 1127–1165

    work page 2025

  49. [49]

    Tudor, J

    C.A. Tudor, J. Zurcher, The spatial average of solutions to SPDEs is asymptotically independent of the solution, arxiv:2404.11147 [math.PR]

    work page arXiv

  50. [50]

    Walsh, An introduction to stochastic partial differential equation s, Ecole d’Et´ e de Probabilit´ es de Saint-Flour, XIV–1984, 265–439

    J.B. Walsh, An introduction to stochastic partial differential equation s, Ecole d’Et´ e de Probabilit´ es de Saint-Flour, XIV–1984, 265–439. Lecture Notes in Math., 11 80 Springer-Verlag, Berlin, 1986

    work page 1984

  51. [51]

    Zheng, Recent developments around the Malliavin-Stein approach (Fourth moment phenomena via exchangeable pairs), Ph.D dissertation, Universit´ e du Luxembourg, 2018

    G. Zheng, Recent developments around the Malliavin-Stein approach (Fourth moment phenomena via exchangeable pairs), Ph.D dissertation, Universit´ e du Luxembourg, 2018. Raluca M. Balan, Department of Mathematics and Statistics, University of Ottaw a, 150 Louis P asteur Private, Ottaw a, ON, K1N 6N5, Canada Email address : Raluca.Balan@uottawa.ca Guangqu Z...

    work page 2018