Limit theorems for the one-dimensional parabolic Anderson model with white noise potential

We consider the parabolic Anderson model $\partial_t u=\partial_x^2 u+ξu$ on $\mathbb{R}_+\times\mathbb{R}$ with $u(0,\cdot)\equiv 1$, where $ξ$ is a spatial white noise. We study the long-time behavior of the spatial integral $U(t):=\int_{-L(t)/2}^{L(t)/2}u(t,x)\,dx$, where $L(t)=\exp(α^3 t^3/24)$