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

TL;DR

This paper proves a central limit theorem for the spatial average of the parabolic Anderson model with various boundary conditions, showing the fluctuations converge to a universal Gaussian distribution as the domain size grows.

Contribution

It establishes the Gaussian fluctuation limit for the spatial integral of the parabolic Anderson model under different boundary conditions using the Malliavin-Stein method.

Findings

Gaussian fluctuation for spatial average established

Limit distribution is universal across boundary conditions

Convergence holds as domain size tends to infinity

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 $\mathbb{R}$.