Weak approximation of stochastic partial differential equations: the non linear case

TL;DR

This paper investigates the weak convergence rate of the Euler scheme for nonlinear stochastic partial differential equations, demonstrating that the weak order is twice the strong order, using Malliavin calculus for error analysis.

Contribution

It establishes the weak convergence rate for nonlinear SPDEs and applies Malliavin calculus to handle irregular error terms, extending previous results to the nonlinear case.

Findings

Weak order of Euler scheme is twice the strong order for nonlinear SPDEs

Malliavin calculus effectively handles irregular error terms

Application to semilinear stochastic heat equation with white noise

Abstract

We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise.