Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

TL;DR

This paper analyzes the weak convergence rate of spatial finite element approximations for the nonlinear stochastic heat equation with both white and colored noise, extending previous results to higher dimensions.

Contribution

It extends earlier work by deriving weak convergence rates for spatial discretizations of the nonlinear stochastic heat equation in higher dimensions with colored noise.

Findings

Weak convergence rate is approximately twice the strong convergence rate.

Results apply to both white noise in one dimension and colored noise in higher dimensions.

Uses Malliavin calculus for the proof.

Abstract

We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.