Temporal approximation of stochastic evolution equations with irregular nonlinearities

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.