In this paper we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly continuous semigroup $S$ on a Hilbert space $X$, and the focus is on non-parabolic problems. The main results are optimal bounds for the uniform strong error $$\mathrm{E}_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|^p\Big)^{1/p},$$ where $p \in [2,\infty)$, $U$ is the mild solution, $U^j$ is obtained from a time discretisation scheme, $k$ is the step size, and $N_k = T/k$. The usual schemes such as the exponential Euler, the implicit Euler, and the Crank-Nicolson method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show - $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$ (linear equation, additive noise, general $S$); - $\mathrm{E}_{k}^{\infty}\lesssim \sqrt{k} \log(T/k)$ (nonlinear equation, multiplicative noise, contractive $S$); - $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$ (nonlinear wave equation, multiplicative noise) for a large class of time discretisation schemes. The logarithmic factor can be removed if the exponential Euler method is used with a (quasi)-contractive $S$. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong error $$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}.$$ Applications to Maxwell equations, Schr\"odinger equations, and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the implicit Euler and the Crank-Nicolson method.

翻译：本文证明了带有加性或乘性高斯噪声的半线性随机发展方程时间离散格式的收敛速率，其中主导算子$A$是希尔伯特空间$X$上强连续半群$S$的生成元，研究重点为非抛物问题。主要结果给出了如下一致强误差的最优界：$$\mathrm{E}_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|^p\Big)^{1/p},$$ 其中$p \in [2,\infty)$，$U$为温和解，$U^j$由时间离散格式获得，$k$为步长，$N_k = T/k$。常见格式如指数欧拉法、隐式欧拉法和Crank-Nicolson方法等均作为特例包含在内。在非线性和噪声满足特定条件时，我们证明：- $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$（线性方程、加性噪声、一般$S$）；- $\mathrm{E}_{k}^{\infty}\lesssim \sqrt{k} \log(T/k)$（非线性方程、乘性噪声、压缩$S$）；- $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$（非线性波动方程、乘性噪声），该结论适用于一大类时间离散格式。若采用指数欧拉法且$S$具有（拟）压缩性，则可消除对数因子。所得界与随机微分方程的最优界一致。现有文献大多关注较简单的点态强误差界：$$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}.$$ 本文结果可应用于麦克斯韦方程、薛定谔方程及波动方程。对于这些方程，我们的研究通过统一方法改进并重新证明了若干现有结果，同时首次给出了隐式欧拉法和Crank-Nicolson方法的收敛性结论。