Let \({\left\{u\left(t,x\right)\right\}}_{t>0,x\in {\mathbb{R}}^{d}}\) 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 \({S}_{N,t} : ={N}^{-d}{\int }_{{\left[0,N\right]}^{d}}\text{d}x\left[U\left(t,x\right)-1\right]\) denote the spatial average on ℝd. We obtain various functional central limit theorems for spatial averages based on the quantitative analysis of f and spatial dimension d. In particular, when f is given by the Riesz kernel, that is, f(x) = ∥x∥−β dx, β ∈ (0, 2 ∧ d), the functional central limit theorem is also based on the index β.
