Hyperbolic Anderson model with time-independent rough noise: Gaussian fluctuations

In this article, we study the hyperbolic Anderson model in dimension 1,
driven by a time-independent rough noise, i.e. the noise associated with the
fractional Brownian motion of Hurst index $H \in (1/4,1/2)$. We prove that,
with appropriate normalization and centering, the spatial integral of the
solution converges in distribution to the standard normal distribution, and we
estimate the speed of this convergence in the total variation distance. We also
prove the corresponding functional limit result. Our method is based on a
version of the second-order Gaussian Poincar\'e inequality developed recently
in [27], and relies on delicate moment estimates for the increments of the
first and second Malliavin derivatives of the solution. These estimates are
obtained using a connection with the wave equation with delta initial velocity,
a method which is different than the one used in [27] for the parabolic
Anderson model.