Gaussian fluctuations for the wave equation under rough random perturbations

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.