In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on $2$-smooth Banach spaces $X$. The leading operator $A$ is assumed to generate a strongly continuous semigroup $S$ on $X$, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error $$E_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|_X^p\Big)^{1/p} \to 0\quad (k \to 0),$$ 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 $N_k = T/k$ for final time $T>0$. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to $2$-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error $$E_k := \bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|_X^p\bigg)^{1/p},$$ which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schr\"odinger equation, for which previous convergence results were not applicable.

翻译：本文证明了在$2$-光滑巴拿赫空间$X$上，具有不规则Lipschitz非线性项、初始值以及加性或乘性高斯噪声的半线性随机发展方程收缩时间离散格式的收敛性。主算子$A$被假定在$X$上生成强连续半群$S$，重点关注非抛物型问题。主要结果涉及一致强误差的收敛性： $$E_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|_X^p\Big)^{1/p} \to 0\quad (k \to 0),$$ 其中$p \in [2,\infty)$，$U$是温和解，$U^j$通过时间离散格式获得，$k$为步长，$N_k = T/k$（最终时间$T>0$）。该结果将先前研究推广至更广泛的可行非线性项与噪声类别，并允许从希尔伯特空间情形拓展至更一般的空间中的粗糙初始数据。我们基于正则化论证给出了证明。在此框架下，将先前针对更规则非线性项与噪声的量化收敛结果从希尔伯特空间推广至$2$-光滑巴拿赫空间。需要指出，一致强误差无法通过更简单的逐点强误差 $$E_k := \bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|_X^p\bigg)^{1/p}$$ 进行估计，而现有文献多聚焦于后者。我们通过薛定谔方程的一个变体验证了所得结论，该方程此前无法应用已有的收敛性结果。