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

Abstract

Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to the linear (fractional) stochastic heat equation. We establish convergence rates with respect to the uniform distance between the density of spatial averages of the solution and the density of the standard normal distribution in some different scenarios. We first consider the case when $u_0\equiv1$ and the stochastic fractional heat equation is driven by a space-time white noise. When $\a=2$ (parabolic Anderson model, PAM for short), and the stochastic heat equation is driven by colored noise in space, we present the rates of convergence respectively for $u_0\equiv1$, $d\geq1$ and $u_0=\de_0$, $d=1$ under the additional condition $\hat f(\R^d)<\infty$. Our results are obtained by using a combination of the Malliavin calculus and Stein's method for normal approximations.