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Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to thelinear (fractional) stochastic heat equation. We establish rates of convergencewith respect to the uniform distance between the density of spatial averages ofsolution and the density of the standard normal distribution in some differentscenarios. We first consider the case that $u_0\equiv1$, and the stochasticfractional heat equation is driven by a space-time white noise. When $\alpha=2$(parabolic Anderson model, PAM for short) and the stochastic heat equation isdriven by colored noise in space, we present the rates of convergencerespectively in the case that $u_0\equiv1$, $d\geq1$ and $u_0=\delta_0$, $d=1$under an additional condition $\hat f(\mathbb{R}^d)<\infty$. Our results areobtained by using a combination of the Malliavin calculus and Stein's methodfor normal approximations.

Convergence of densities of spatial averages of the linear stochastic heat equation