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Banach空间X上，对具有不规则Lipschitz非线性、初值以及加性或乘性高斯噪声的半线性随机演化方程，压缩时间离散格式的收敛性。主算子A被假设在X上生成强连续半群S，重点研究非抛物型问题。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，$U^j$由时间离散格式得到，k为步长，$N_k = T/k$，最终时间$T>0$。这推广了先前结果，将可允许非线性与噪声的更大类别以及粗糙初值从Hilbert空间情形推广到更一般的空间。我们基于正则化论证给出证明。在此范围内，我们将先前针对更正则非线性与噪声的量化收敛结果从Hilbert空间推广到2-光滑Banach空间。一致强误差无法通过更简单的逐点强误差 $$E_k := \bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|_X^p\bigg)^{1/p}$$ 来估计，而现有文献主要关注后者。我们的结果通过薛定谔方程的一个变体加以说明，先前的收敛结果并不适用于该变体。