Almost sure central limit theorems for the parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions

Let u^(k)(t, x) be the solution to the parabolic Anderson model driven by a space-time white noise on the interval [0, k] with Neumann, Dirichlet, or periodic boundary conditions. In this paper, we investigate the almost sure central limit theorems for spatial averages of the form \({\int }_{0}^{k}{u}^{\left(k\right)}\left(t,x\right)\) ${\int }_0^k{u}^{\left(k\right)}\left(t,,,x\right)$ dx as k tends to infinity.