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 $$E{k}^{\infty} := \Big(\mathbb{E} \sup{j\in {0, \ldots, Nk}} |U(tj) - U^{j|_Xp\Big){1/p}} \to 0\quad (k \to 0),$$ 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 $Nk = T/k$ 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 $$Ek := \bigg(\sup{j\in {0,\ldots,Nk}}\mathbb{E} |U(tj) - U^{j}|X^{p\bigg){1/p},$$} 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.