Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations

TL;DR

This paper demonstrates that, for stochastic Allen-Cahn-type equations, the strong convergence rate of a fully discrete explicit finite difference scheme can reach almost sure rate 1 in negative Besov norms, surpassing the expected rate 1/2.

Contribution

It shows that, by considering the SPDE as singular, one can achieve an almost sure convergence rate of nearly 1, improving upon the standard rate of 1/2.

Findings

Achieves almost sure convergence rate close to 1 in negative Besov norms.

Surpasses the traditionally expected rate of 1/2 for discretisations of these SPDEs.

Uses a novel perspective of treating the SPDE as singular to obtain improved convergence.

Abstract

Consider the approximation of stochastic Allen-Cahn-type equations (i.e. $1+1$-dimensional space-time white noise-driven stochastic PDEs with polynomial nonlinearities $F$ such that $F(\pm \infty)=\mp \infty$) by a fully discrete space-time explicit finite difference scheme. The consensus in literature, supported by rigorous lower bounds, is that strong convergence rate $1/2$ with respect to the parabolic grid meshsize is expected to be optimal. We show that one can reach almost sure convergence rate $1$ (and no better) when measuring the error in appropriate negative Besov norms, by temporarily `pretending' that the SPDE is singular.