Gaussian fluctuations for the wave equation under rough random perturbations

TL;DR

This paper studies the stochastic wave equation with fractional spatial noise, proving stationarity, ergodicity, and Gaussian fluctuation limits for spatial averages, including convergence rates and functional limit results.

Contribution

It establishes Gaussian fluctuation limits and convergence rates for the spatial averages of solutions to the wave equation with fractional noise, a novel result in this setting.

Findings

Solution is strictly stationary and ergodic in space

Spatial averages converge to normal distribution with rate estimates

Functional convergence of the spatial averages is proved

Abstract

In this article, we consider the stochastic wave equation in spatial dimension $d=1$, with linear term $\sigma(u)=u$ multiplying the noise. This equation is driven by a Gaussian noise which is white in time and fractional in space with Hurst index $H \in (\frac{1}{4},\frac{1}{2})$. First, we prove that the solution is strictly stationary and ergodic in the spatial variable. Then, we show that with proper normalization and centering, the spatial average of the solution converges to the standard normal distribution, and we estimate the rate of this convergence in the total variation distance. We also prove the corresponding functional convergence result.