Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise

TL;DR

This paper introduces a novel approach for analyzing weak convergence rates of a full discretization scheme for the stochastic Allen-Cahn equation with additive noise, achieving higher convergence rates than strong error estimates.

Contribution

It develops a new direct method for weak error analysis that does not rely on Kolmogorov equations, applicable to non-Markovian stochastic equations.

Findings

Weak convergence rates are roughly twice as high as strong convergence rates.

Weak convergence depends on the regularity of the noise.

Numerical experiments confirm theoretical predictions.

Abstract

We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the use of the associated Kolmogorov equation or It\^{o}'s formula and is therefore non-Markovian in nature. Such an approach thus has a potential to be applied to non-Markovian equations such as stochastic Volterra equations or other types of fractional SPDEs, which suffer from the lack of Kolmogorov equations. It turns out that the obtained weak convergence rates are, in both spatial and temporal direction, essentially twice as high as the strong convergence rates. Also, it is revealed how the weak convergence rates depend on the regularity of the noise. Numerical experiments are finally reported to confirm the theoretical conclusion.