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：本文研究了带加性或乘性高斯噪声的半线性随机发展方程的时间离散格式的收敛速度。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，重点考虑非抛物型问题。主要结果给出了均匀强误差$$\mathrm{E}_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|^p\Big)^{1/p},$$ 的最优界，其中$p \in [2,\infty)$，$U$是温和解，$U^j$由时间离散格式得到，$k$为步长，$N_k = T/k$。常见格式如分裂/指数欧拉、隐式欧拉、克兰克-尼科尔森等作为特例包含在内。在非线性和噪声的适当条件下，我们证明了 - $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$（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$S$）； - $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$（非线性波动方程，乘性噪声）。若采用（拟）收缩$S$的分裂格式，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 $$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}.$$ 文中还包含了麦克斯韦方程、薛定谔方程和波动方程的应用。对于这些方程，我们的结果以统一方法改进并重新证明了若干现有结果。