Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A Central Limit Theorem for the stochastic wave equation with fractional noise

Francisco Delgado-Vences, David Nualart, and Guangqu Zheng

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We study the one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise, which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in [1/2,1)$ in the spatial variable. We show that the normalized spatial average of the solution over $[-R,R]$ converges in total variation distance to a normal distribution, as $R$ tends to infinity. We also provide a functional Central Limit Theorem.


Nous étudions l’équation des ondes en une dimension, perturbée par un bruit gaussien multiplicatif, qui est blanc en temps et qui a la covariance d’un mouvement brownien fractionnaire avec paramètre de Hurst $H\in [1/2,1)$ dans la variable d’espace. Nous démontrons que la moyenne spatiale normalisée de la solution sur un intervalle $[-R,R]$ converge, en la distance de la variation totale, vers une loi normale, quand $R$ tend vers l’infini. Nous prouvons aussi un théorème central limite fonctionnel.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 4 (2020), 3020-3042.

Received: 28 December 2018
Revised: 2 March 2020
Accepted: 11 May 2020
First available in Project Euclid: 21 October 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H07: Stochastic calculus of variations and the Malliavin calculus 60G15: Gaussian processes 60F05: Central limit and other weak theorems

Stochastic wave equation Central Limit Theorem Malliavin calculus Stein’s method


Delgado-Vences, Francisco; Nualart, David; Zheng, Guangqu. A Central Limit Theorem for the stochastic wave equation with fractional noise. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 4, 3020--3042. doi:10.1214/20-AIHP1069.

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