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 splitting/exponential Euler, implicit Euler, and Crank-Nicolson, 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). The logarithmic factor can be removed if the splitting scheme 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.

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的时间离散化方案的收敛率，其中主算子$A$是希尔伯特空间$X$上强连续半群$S$的生成元，重点研究非抛物型问题。主要结果为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，$U^j$通过时间离散化方案获得，$k$为步长，$N_k = T/k$。常见方案如分裂/指数欧拉法、隐式欧拉法、克兰克-尼科尔森法等均为特例。在非线性项和噪声的特定条件下，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，收缩$S$）；- $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$（非线性波方程，乘性噪声）。若使用（拟）收缩$S$的分裂方案，可去除对数因子。所得界与随机微分方程的最优界一致。现有文献多关注更简单的逐点强误差：$$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}.$$ 本文结果应用于麦克斯韦方程、薛定谔方程及波动方程，通过统一方法改进并重新证明了若干现有结论。