Gaussian fluctuations for the stochastic wave equation with drift

Source abstract

In this article, we study Gaussian fluctuations of spatial averages of the solution to the stochastic wave equation with a nonlinear drift and multiplicative Gaussian noise in dimensions one and two. The noise is white in time, and its spatial covariance is either integrable or given by a Riesz kernel; space-time white noise in dimension one is also included. We establish spatial ergodicity and convergence of the rescaled covariance, and prove a quantitative central limit theorem for the centered spatial average over a ball of radius $R$. The bounds in total variation distance are of order $R^{-d/2}$ in the integrable case and $R^{-β/2}$ for a Riesz kernel of order $β$. We also obtain a functional central limit theorem in the space of continuous functions. Our approach combines a second-order Gaussian Poincaré inequality with spatially integrated estimates for the second Malliavin derivative, exploiting the compact support of the wave kernel. Further arguments based on the Clark-Ocone formula are used to control the drift contributions.