Almost sure central limit theorems for the parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions (Q6622508)

The parabolic Anderson model driven by space-time white noise on the interval \([0, k]\) is considered, with Neumann, Dirichlet, or periodic boundary conditions. The spatial averages of the form \(\int_{0}^{k} u^{(k)}(t, x) \, dx\), where \(u^{(k)}(t, x)\) represents the solution to the model, are investigated. The almost sure central limit theorems for these spatial averages are established as \(k\) tends to infinity. The probabilistic properties and asymptotic behavior of the solutions are analyzed under varying boundary conditions. Insights into the statistical structure of the model are derived, contributing to a deeper understanding of its large-scale behavior.