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 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) 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 implicit Euler and Crank-Nicolson.

翻译：本文研究加性或多乘性高斯噪声驱动的半线性随机演化方程时间离散格式的收敛速率，其中主导算子$A$为希尔伯特空间$X$上强连续半群$S$的生成元，重点聚焦于非抛物型问题。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$p \in [2,\infty)$，$U$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、隐式欧拉、克兰克-尼科尔森等）均作为特例包含在内。在非线性和噪声的适当条件下，我们证明： - $\mathrm{E}_{k}^{\infty}\lesssim k \log(T/k)$（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） 上述结果适用于一大类时间离散格式。若采用（拟）压缩$S$的指数欧拉方法，可去除对数因子。所得界与随机微分方程的最优界一致。现有文献大多关注更简单的逐点强误差$$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}$$的估计。本文结果还应用于麦克斯韦方程、薛定谔方程和波动方程。对于这些方程，我们以统一方法改进并重新证明了若干现有结果，并首次给出了隐式欧拉和克兰克-尼科尔森格式的已知结论。