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Averaging 2D stochastic wave equation. (English) Zbl 1477.60095

The authors consider the following 2D stochastic wave equation \[ \frac{\partial^2u}{\partial t^2}=\triangle u+\sigma(u)\dot{W} \] with the boundary conditions \({u(0,x)=1}\) and \({\frac{\partial u}{\partial t}(0,x)=0}.\) Here \(\triangle\) is the Laplacian in the space variables, \(\dot{W}\) is a Gaussian centered noise with covariance \[ \mathbb{E}(\dot{W}(t,x)\dot{W}(s,y))=\delta_0(t-s)\|x-y\|^{-\beta} \] for some \({\beta\in(0,2)}\) and \({\sigma:\mathbb{R}\to\mathbb{R}}\) is a Lipschitz function with \({\sigma(1)\neq0}.\) The solution \(u(x,t)\) is viewed as functional over the homogeneous Gaussian random field \(W\) (where \({\dot{W}=\frac{\partial^3W}{\partial t\partial x_1\partial x_2}}\)). The main result asserts that the process \[ R^{\frac{\beta}{2}-2}\int\limits_{\|x\|\leq R}(u(t,x)-1)\,dx,\ \ t\in\mathbb{R}_+ \] converges in law to some centered Gaussian process as \({R\to\infty}.\) Also the rate of convergence is estimated using the total-variation distance. The proof is based on Stein’s method of normal approximation and Malliavin’s differential calculus.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
60G15 Gaussian processes
60H07 Stochastic calculus of variations and the Malliavin calculus
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[1] Breuer, P. and Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. Journal of Multivariate Analysis, 13, 1983, 425-441. · Zbl 0518.60023
[2] Campese, S., Nourdin, I. and Nualart, D.: Continuous Breuer-Major theorem: tightness and non-stationarity. Ann. Probab. 48(1), 2020, 147-177. · Zbl 1468.60043
[3] Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F.: Spatial ergodicity for SPDEs via Poincaré-type inequalities. (2019) 1907.11553
[4] Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F.: Poincaré inequality, and central limit theorems for parabolic stochastic partial differential equations. To appear in: Ann. Inst. Henri Poincaré Probab. Stat. (2021+)
[5] Dalang, R.C.: Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.’s. Electron. J. Probab. 4(6), 1999, 29pp. · Zbl 0922.60056
[6] Dalang, R.C.: The Stochastic wave equation. In: Khoshnevisan D., Rassoul-Agha F. (eds) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol 1962. Springer, Berlin, Heidelberg (2009) · Zbl 1165.60023
[7] Delgado-Vences, F., Nualart, D. and Zheng, G.: A Central Limit Theorem for the stochastic wave equation with fractional noise. Ann. Inst. Henri Poincaré Probab. Stat. 56(4), 2020, 3020-3042. · Zbl 1466.60127
[8] Gaveau, B. and Trauber, P.: L’intégrale stochastique comme opérateur de divergence dans l’espace founctionnel. J. Funct. Anal. 46, 1982, 230-238. · Zbl 0488.60068
[9] Huang, J., Nualart, D. and Viitasaari, L.: A central limit theorem for the stochastic heat equation. Stochastic Processes and Their Applications. 130(12), 2020, 7170-7184. · Zbl 1458.60072
[10] Huang, J., Nualart, D., Viitasaari, L. and Zheng, G.: Gaussian fluctuations for the stochastic heat equation with colored noise. Stoch. PDE: Anal. Comp 8(2), 2020, 402-421. · Zbl 1451.60066
[11] Kallenberg, O.: Foundations of Modern Probability. Second edition. Probability and Its Applications, Springer (2002). · Zbl 0996.60001
[12] Khoshnevisan, D.: Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics, 119. Published for the Conference Board of the Mathematical Sciences, Washington DC; by the American Mathematical Society, Providence, RI, 2014. viii+116 pp.
[13] Lebedev, N.N.: Special functions and their applications. (1972) Revised English edition, translated and edited by Richard A. Silverman. Dover Publications. · Zbl 0271.33001
[14] Millet, A. and Sanz-Solé, M.: A stochastic wave equation in two dimension: Smoothness of the law. Ann. Probab. 27(2), 1999, 803-844. · Zbl 0944.60067
[15] Nourdin, I. and Peccati, G.: Normal approximations with Malliavin calculus: From Stein’s method to universality. Cambridge Tracts in Mathematics, 192. Cambridge University Press, Cambridge, 2012. xiv+239 pp. · Zbl 1266.60001
[16] Nualart, D.: The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. · Zbl 1099.60003
[17] Nualart, D. and Nualart, E.: Introduction to Malliavin Calculus. IMS Textbooks, Cambridge University Press, 2018. · Zbl 1425.60002
[18] Nualart, D. and Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Re. Fields 78(4), 1988, 535-581. · Zbl 0629.60061
[19] Nualart, D. and Zheng, G.: Oscillatory Breuer-Major theorem with application to the random corrector problem. Asymptotic Analysis, 119, 2020, no. 3-4, 281-300. · Zbl 1472.60065
[20] Nualart, D. and Zheng, G.: Averaging Gaussian functionals. Electron. J. Probab. 25(1), 2020, 1-54. · Zbl 1441.60049
[21] Nualart, D. and Zhou, H.: Total variation estimates in the Breuer-Major theorem. Ann. Inst. Henri Poincaré Probab. Stat. 57(2), 2021, 740-777. · Zbl 1472.60049
[22] Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp. · Zbl 0207.13501
[23] Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, 265-439. Lecture Notes in Math. 1180, Springer, Berlin, 1986. · Zbl 0579.00013
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