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.

翻译：考虑利用全显式时空有限差分格式逼近随机Allen-Cahn型方程（即$1+1$维时空白噪声驱动的具多项式非线性项$F$且满足$F(\pm \infty)=\mp \infty$的随机偏微分方程）。文献中已有共识（并经严格下界支持）认为，在抛物线网格尺寸下，强收敛率$1/2$是最优预期。本文通过暂时“假定”该随机偏微分方程为奇异的，证明了在适当的负贝索夫范数下测量误差时，可达到几乎必然收敛率$1$（且无法更优）。