Density convergence of spatial average of solution to a one dimensional stochastic wave equation

In this paper we study the spatial averages of the solution of a one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise, which is white in time and has a homogeneous spatial covariance described by the Riesz kernel. We establish the rate of convergence for the uniform distance between the density of spatial averages and the standard normal density. The proof combines Malliavin calculus with Stein’s method for normal approximations. The key technical challenges lie in estimating the Lp-norm of the second Malliavin derivative and the existence of negative moments of Malliavin covariance matrix.