Convergence of densities of spatial averages of stochastic heat equation

Abstract

In this paper, we consider the one-dimensional stochastic heat equation driven by a space–time white noise. In two different scenarios, (i) initial condition u0=1 and general nonlinear coefficient σ, (ii) initial condition u0=δ0 and σ(x)=x (Parabolic Anderson Model), we establish rates of convergence with respect to the uniform distance between the density of spatial averages of (renormalized) solution and the density of the standard normal distribution. These results are based on a combination of Stein’s method for normal approximations and Malliavin calculus techniques. A key ingredient in Case (i) is a new estimate on the Lp-norm of the second Malliavin derivative.