Weak error analysis for the stochastic Allen-Cahn equation

TL;DR

This paper establishes strong and weak convergence rates of order one for a structure-preserving numerical scheme applied to the stochastic Allen-Cahn equation with additive and multiplicative noise across dimensions 1 to 3, leveraging variational techniques.

Contribution

It provides the first proof of strong and weak convergence rates of order one for a structure-preserving discretization of the stochastic Allen-Cahn equation with colored noise.

Findings

Strong and weak convergence rates of order one are proven.

The method exploits the one-sided Lipschitz property of the nonlinearity.

Results apply to dimensions 1, 2, and 3.

Abstract

We prove strong rate resp. weak rate ${\mathcal O}(\tau)$ for a structure preserving temporal discretization (with $\tau$ the step size) of the stochastic Allen-Cahn equation with additive resp. multiplicative colored noise in $d=1,2,3$ dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate ${\mathcal O}(\tau)$ in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.