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.
References
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- 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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- Jorge Clarke De la Cerda and Ciprian 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, DOI 10.1007/s13348-014-0108-9
- Francisco Delgado-Vences, David Nualart, and Guangqu 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 (English, with English and French summaries). MR 4164864, DOI 10.1214/20-AIHP1069
- 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.
- Jingyu Huang, David Nualart, and Lauri Viitasaari, A central limit theorem for the stochastic heat equation, Stochastic Process. Appl. 130 (2020), no. 12, 7170–7184. MR 4167203, DOI 10.1016/j.spa.2020.07.010
- Jingyu Huang, David Nualart, Lauri Viitasaari, and Guangqu 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, DOI 10.1007/s40072-019-00149-3
- Makoto Maejima and Ciprian 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, DOI 10.1080/07362990701540519
- Ivan Nourdin and Giovanni Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. MR 2962301, DOI 10.1017/CBO9781139084659
- David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
- D. Nualart and G. Zheng, 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.
- David Nualart and Guangqu Zheng, Spatial ergodicity of stochastic wave equations in dimensions 1, 2 and 3, Electron. Commun. Probab. 25 (2020), Paper No. 80, 11. MR 4187721, DOI 10.1214/20-ecp361
- Meryem Slaoui and C. A. Tudor, The linear stochastic heat equation with Hermite noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22 (2019), no. 3, 1950022, 23. MR 4064931, DOI 10.1142/S021902571950022X
- François Trèves, Basic linear partial differential equations, Pure and Applied Mathematics, Vol. 62, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0447753
- Ciprian A. Tudor, Analysis of variations for self-similar processes, Probability and its Applications (New York), Springer, Cham, 2013. A stochastic calculus approach. MR 3112799, DOI 10.1007/978-3-319-00936-0
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.
- L. Chen, D. Khoshnevisan, D. Nualart, F. Pu, 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
- D. Nualart, Malliavin Calculus and Related Topics., second edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
- D. Nualart and G. Zheng, 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
