Gaussian fluctuation for spatial average of parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions

Abstract: Consider the parabolic Anderson model $$\partial_tu=\frac{1}{2}\partial_x^2u+u\, \eta$$ on the interval $[0, L]$ with Neumann, Dirichlet or periodic boundary conditions, driven by space-time white noise $\eta$. Using Malliavin-Stein method, we establish the central limit theorem for the fluctuation of the spatial integral $\int_0^Lu(t\,, x)\, \mathrm{d} x$ as $L$ tends to infinity, where the limiting Gaussian distribution is independent of the choice of the boundary conditions and coincides with the Gaussian fluctuation for the spatial average of parabolic Anderson model on the whole space $\mathbb{R}$.