Convergence of Densities of Spatial Averages of Stochastic Heat Equation

2021/08/21 by Sefika Kuzgun, David Nualart, Kuzgun, Sefika +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2108.09531

Abstract

In this paper, we consider the one-dimensional stochastic heat equation driven by a space time white noise. In two different scenarios: \it (i) initial condition u₀=1 and general nonlinear coefficient σ and \it (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.

Cited by

• Optimal local central limit theorems on Wiener chaos
• Density convergence of spatial average of solution to a one dimensional stochastic wave equation

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