Non-central limit theorem for the spatial average of the solution to the wave equation with Rosenblatt noise

Authors:
R. Dhoyer and C. A. Tudor

Journal:
Theor. Probability and Math. Statist. **106** (2022), 105-119

MSC (2020):
Primary 60H15, 60H07, 60G15, 60F05

DOI:
https://doi.org/10.1090/tpms/1167

Published electronically:
May 16, 2022

MathSciNet review:
4438446

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Additional Information

Abstract: We analyze the limit behavior in distribution of the spatial average of the solution to the wave equation driven by the two-parameter Rosenblatt process in spatial dimension $d=1$. We prove that this spatial average satisfies a non-central limit theorem, more precisely it converges in law to a Wiener integral with respect to the Rosenblatt process. We also give a functional version of this limit theorem.

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References
- O. Assaad, D. Nualart, C. A. Tudor and L. Viitasaari,
*Quantitative normal approximations for the stochastic fractional heat equation. * Preprint, to appear in Stoch. Partial Differ. Equ. Anal. Comput., https://doi-org.libproxy.viko.lt/10.1007/s40072-021-00198-7, 2022. MR **4385408**
- R. M. Balan, D. Nualart, L. Quer-Sardanyons and G. Zheng,
* The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications. * Preprint, to appear in Stoch. Partial Differ. Equ. Anal. Comput., https://doi-org.libproxy.viko.lt/10.1007/s40072-021-00227-5, 2022.
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* Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method. * Preprint, to appear in *Stoch. Partial Differ. Equ. Anal. Comput.*, https://doi.org/10.1007/s40072-021-00224-8, 2022. MR **3222416**
- J. Clarke De la Cerda and C. A. Tudor,
* Wiener integrals with respect to the Hermite random field and applications to the wave equation. * Collect. Math., **65** (2014),no. 3, 341–356. MR **3240998**
- F. Delgado-Vences, D. Nualart, and G. Zheng,
*A central limit theorem for the stochastic wave equation with fractional noise.* Ann. Inst. Henri Poincaré Probab. Stat. **56** (2020),no. 4, 3020–3042. MR **4164864**
- R. Dhoyer and C. A. Tudor,
*Spatial average for the solution to the heat equation with Rosenblatt noise. * Preprint, to appear in Stochastic Analysis and Applications, https://doi.org/10.1080/07362994.2021.1972008, 2022.
- J. Huang, D. Nualart and L. Viitasaari,
* A central limit theorem for the stochastic heat equation. * Stochastic Process. Appl. **130** (2020), no. 12, 7170-7184. MR **4167203**
- J. Huang, D. Nualart, L. Viitasaari and G. Zheng,
*Gaussian fluctuations for the stochastic heat equation with colored noise. * Stoch. Partial Differ. Equ. Anal. Comput. **8** (2020), no. 2, 402-421. MR **4098872**
- M. Maejima and C. A. Tudor,
*Wiener Integrals with respect to the Hermite process and a non-central limit theorem.* Stoch. Anal. Appl. **25** (2007), no. 5, 1043–1056. MR **2352951**
- I. Nourdin, G. Peccati,
* Normal Approximations with Malliavin Calculus From Stein’s Method to Universality*, Cambridge Tracts in Mathematics, 192, Cambridge University Press, Cambridge, 2012. MR **2962301**
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*Malliavin Calculus and Related Topics.*, second edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR **2200233**
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*Central limit theorems for stochastic wave equations in dimensions one and two. * Preprint, to appear in Stoch. Partial Differ. Equ. Anal. Comput., https://doi-org.libproxy.viko.lt/10.1007/s40072-021-00209-7, 2022.
- D. Nualart and G. Zheng,
*Spatial ergodicity of stochastic wave equations in dimensions 1, 2 and 3.* Electron. Commun. Probab. **25** (2020), Paper No. 80, 11 pp. MR **4187721**
- M. Slaoui and C. A. Tudor,
* On the linear stochastic heat equation with Hermite noise. * Infin. Dimens. Anal. Quantum Probab. Relat. Top. **22** (2019), no. 3, 1950022, 23 pp. MR **4064931**
- F. Treves,
*Basic Linear Partial Differential Equations*, Pure and Applied Mathematics, Vol. 62, Academic Press, New York, 1975. MR **0447753**
- C. A. Tudor (2013).
*Analysis of variations for self-similar processes. A stochastic calculus approach. * Probability and its Applications (New York). Springer, Cham, 2013. MR **3112799**

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Additional Information

**R. Dhoyer**

Affiliation:
SAMM, Université de Paris 1 Panthéon-Sorbonne, 75013 Paris, France

Email:
remi.dhoyer@gmail.com

**C. A. Tudor**

Affiliation:
Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d’Ascq, France

Email:
tudor@math.univ-lille1.fr

Keywords:
Stochastic wave equation,
Rosenblatt sheet,
cumulants,
multiple stochastic integrals,
second Wiener chaos

Received by editor(s):
July 20, 2021

Accepted for publication:
October 18, 2021

Published electronically:
May 16, 2022

Additional Notes:
C. A. Tudor acknowledges partial support from the projects MATHAMSUD (22- MATH-08) and ECOS SUD (C2107), Labex CEMPI(ANR-11-LABX-007-01) and Japan Science and Technology Agency CREST JPMJCR2115.

Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv