We show new estimates for the total variation and Wasserstein distances in the framework of the Breuer-Major theorem. The results are based on the combination of Stein's method for normal approximations and Malliavin calculus together with Wiener chaos expansions. Mathematics Subject Classifications (2010): 60H15, 60H07, 60G15, 60F05.

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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 and (ii): initial condition u0 = δ0 and σ(x) = x (Parabolic Anderson Model), we establish rates of convergence for the uniform distance between the density of (renormalized) spatial averages and the standard normal density. These results are based on the combination of Stein method for normal approximations and Malliavin calculus techniques. A key ingredient in Case (i) is a new estimate on the L p -norm of the second Malliavin derivative.

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