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November 2020 A Central Limit Theorem for the stochastic wave equation with fractional noise
Francisco Delgado-Vences, David Nualart, Guangqu Zheng
Ann. Inst. H. Poincaré Probab. Statist. 56(4): 3020-3042 (November 2020). DOI: 10.1214/20-AIHP1069

Abstract

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.

Citation

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Francisco Delgado-Vences. David Nualart. Guangqu Zheng. "A Central Limit Theorem for the stochastic wave equation with fractional noise." Ann. Inst. H. Poincaré Probab. Statist. 56 (4) 3020 - 3042, November 2020. https://doi.org/10.1214/20-AIHP1069

Information

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

MathSciNet: MR4164864
Digital Object Identifier: 10.1214/20-AIHP1069

Subjects:
Primary: 60F05, 60G15, 60H07, 60H15

Rights: Copyright © 2020 Institut Henri Poincaré

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Vol.56 • No. 4 • November 2020
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