Weak convergence rates of splitting schemes for the stochastic Allen-Cahn equation

TL;DR

This paper establishes that explicit splitting schemes for the stochastic Allen-Cahn equation achieve a weak convergence rate of 1/2, despite the challenges posed by non-globally Lipschitz nonlinearities.

Contribution

It provides the first analysis of weak convergence rates for parabolic semilinear SPDEs with non-globally Lipschitz nonlinearities, using new estimates and regularity results.

Findings

Weak rate of convergence is 1/2 for the schemes.

New regularity results for solutions of Kolmogorov equations.

First analysis of weak convergence for such SPDEs with polynomial growth nonlinearities.

Abstract

This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the stochastic Allen-Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to $\frac12$, like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.