Functional central limit theorems for spatial averages of the parabolic Anderson model with delta initial condition in dimension $d\ge 1$

Let $\{u(t,x){\}}_{t>0,x\in {\mathbb{R}}^d}$ denote the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a multiplicative noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$. Let $S_{N,t}:=N^{-d}{\int }_{{[0,N]}^d}[U(t,x)-1]\mathrm{d}x$ denote the spatial average on ${\mathbb{R}}^d$. We obtain various functional central limit theorems (CLTs) for spatial averages based on the quantitative analysis of $f$ and spatial dimension $d$. In particular, when $f$ is given by Riesz kernel, that is, $f(\mathrm{x})={\mathrm{\parallel }x\mathrm{\parallel }}^{-\beta }\mathrm{d}x$, $\beta \in (0,2\wedge d)$, the functional CLT is also based on the index $\beta$.