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.

翻译：针对一维、二维和三维空间中分别带有加性噪声和乘性色噪声的随机Allen-Cahn方程，我们证明了其结构保持时间离散格式（步长为τ）的强误差阶分别为${\mathcal O}(\tau)$和弱误差阶${\mathcal O}(\tau)$。在加性噪声情形下，直接变分论证利用三次非线性的单侧Lipschitz性质建立了强一阶收敛率；正是该性质使得相关Kolmogorov方程解的导数具有一致有界性，进而推导出乘性噪声情况下弱误差阶为${\mathcal O}(\tau)$。因此，我们获得的收敛率是已知乘性噪声下强误差收敛率的两倍。