Central limit theorems for stochastic wave equations in high dimensions
Masahisa Ebina
TL;DR
This work advances the theory of stochastic wave equations in high dimensions by establishing central and functional central limit theorems for spatial averages driven by Gaussian noise with spatial correlation. The authors overcome the fundamental obstacle posed by the nonnegativity failure of the Green function in $d\ge 4$ by Fourier-analytic substitution with a nonnegative kernel, enabling a Malliavin-Stein strategy on an approximation sequence. They prove that, for $\gamma(x)=|x|^{-{\beta}}$ with $0<{\beta}<2$, the normalized spatial integral converges to a Gaussian law with explicitly characterized covariance $\tau_{\beta}\int_0^{t\land s}(t-u)(s-u)\mathbb{E}[\sigma(U(u,0))]^2du$, and they establish a functional CLT to a Gaussian process with the same structure. The approach combines careful regularization of the Green function, Malliavin calculus, and robust covariance-limit arguments to extend known results from low dimensions to the high-dimensional regime, with potential applicability to other SPDEs with rough Green kernels.
Abstract
We consider stochastic wave equations in spatial dimensions $d \geq 4$. We assume that the driving noise is given by a Gaussian noise that is white in time and has some spatial correlation. When the spatial correlation is given by the Riesz kernel, we also establish that the spatial integral of the solution with proper normalization converges to the standard normal distribution under the Wasserstein distance. The convergence is obtained by first constructing the approximation sequence to the solution and then applying Malliavin-Stein's method to the normalized spatial integral of the sequence. The corresponding functional central limit theorem is presented as well.