Central limit theorems for stochastic wave equations in dimensions one and two

Abstract

Fix $d\in \{1,2\}$, we consider a $d$-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang's condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise $L^p$-estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations.

Ver1: 19 pages

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Cited by in corpus (6)

• The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications
• Quantitative central limit theorems for the parabolic Anderson model driven by colored noises
• Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition
• Stochastic wave equation with Lévy white noise
• Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
• Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\ge 1$