Temporal approximation of stochastic evolution equations with irregular nonlinearities

Abstract

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 $$E_k^{\infty }:=(\mathbb{E}\underset{j\in \{0,\dots ,N_k\}}{\sup }\Vert U(t_j)-U^j{\Vert }_X^p{)}^{1/p}\to 0(k\to 0),$$ 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 $$E_k:=(\underset{j\in \{0,\dots ,N_k\}}{\sup }\mathbb{E}\Vert U(t_j)-U^j{\Vert }_X^p{)}^{1/p},$$ which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

arXiv comment24 pages, this version: minor changes

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