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 - $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$ (linear equation, additive noise, general $S$); - $\mathrm{E}_{k}^{\infty}\lesssim \sqrt{k} \sqrt{\log(T/k)}$ (nonlinear equation, multiplicative noise, contractive $S$); - $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$ (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：本文针对具有加性或乘性高斯噪声的半线性随机发展方程，证明了时间离散化格式的收敛速率。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，且重点研究非抛物型问题。主要结果为一致强误差$$\mathrm{E}_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|^p\Big)^{1/p}$$的最优界，其中$p \in [2,\infty)$，$U$为温和解，$U^j$来自时间离散格式，$k$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法等均作为特例包含在内。在非线性和噪声满足特定条件下，我们证明：- $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$（线性方程、加性噪声、一般$S$）；- $\mathrm{E}_{k}^{\infty}\lesssim \sqrt{k} \sqrt{\log(T/k)}$（非线性方程、乘性噪声、压缩$S$）；- $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$（非线性波动方程、乘性噪声），该结果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SDE的最优界一致。现有文献大多关注更简单的点态强误差$$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}$$的界。本文结果可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法的收敛性结论。