Gaussian fluctuation for spatial average of parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions
Fei Pu · Transactions of the American Mathematical Society · 2021
Consider the parabolic Anderson model ∂ t u = 1 2 ∂ x 2 u + u η \partial _tu=\frac {1}{2}\partial _x^2u+u\, \eta on the interval [ 0 , L ] [0, L] with Neumann, Dirichlet or periodic boundary conditions, driven by space-time white noise η \eta . Using Malliavin-Stein method, we establish the central limit theorem for the fluctuation of the spatial integral ∫ 0 L u ( t , x ) d x \int _0^Lu(t\,, x)\, \mathrm {d} x as L 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 R \mathbb {R} .