A general form for precise asymptotics for the stochastic wave equation

Abstract

Consider the stochastic wave equation driven by a general Gaussian multiplicative noise, which is temporally white and colored in space including the cases of the spatial covariance given by a fractional noise, a Riesz kernel, and an integrable function that satisfies Dalang's condition. In this paper, we present the precise asymptotics for complete convergence and complete moment convergence for the spatial averages of the solution to the equation over a Euclidean ball, as the radius of the ball diverges to infinity. Some general results on precise asymptotics are obtained, which can describe the relations among the boundary function, weighted function, convergence rate and the limit value.