Gaussian fluctuation for spatial average of the space--time fractional stochastic heat equation

We study spatial averages of the mild solution to a one-dimensional space--time fractional stochastic heat equation driven by space--time white noise.

For fixed \(t>0\), we prove a quantitative central limit theorem for the normalized spatial average over \([-R,R]\): as \(R\to\infty\), its law converges to the standard normal law at rate \(R^{-1/2}\) in total variation distance.

The proof relies on the Malliavin--Stein method, combined with precise estimates for the space--time fractional heat kernel and for the Malliavin derivative of the mild solution.

We further establish a functional central limit theorem.