# A Central Limit Theorem for the stochastic wave equation with fractional noise

## TL;DR

This paper proves a central limit theorem for the spatial average of solutions to a one-dimensional stochastic wave equation driven by fractional noise, showing convergence to a normal distribution as the averaging domain grows.

## Contribution

It establishes a new CLT for the stochastic wave equation with fractional noise, including a total variation convergence and a functional CLT, extending prior results to fractional noise settings.

## Key findings

- Normalized spatial averages converge to a normal distribution
- Total variation distance tends to zero as domain size increases
- Functional central limit theorem is established

## Abstract

We study the one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in [1/2,1)$ in the spatial variable. We show that the normalized spacial average of the solution over $[-R,R]$ converges in total variation distance to a normal distribution, as $R$ tends to infinity. We also provide a functional central limit theorem.