Temporal approximation of stochastic evolution equations with irregular nonlinearities
Katharina Klioba, Mark Veraar
TL;DR
The paper addresses temporal discretisation of stochastic evolution equations $dU=(AU+F(U))dt+G(U)dW_H$ on a 2-smooth Banach space $X$, proving pathwise uniform convergence of contractive time-stepping schemes to the mild solution even when nonlinearities and noise are irregular. A regularisation strategy $F_m=mR(m,A)F$, $G_m=mR(m,A)G$, $u_{0m}=mR(m,A)u_0$ is used to obtain stability and convergence for the regularised problem, aided by a novel maximal inequality for discrete convolutions and a discrete Gronwall argument; convergence is then extended to the original problem as $m\to\infty$. The results generalise prior Hilbert-space findings to 2-smooth Banach spaces and apply to the stochastic Schrödinger equation, demonstrating convergence for common schemes such as exponential Euler, implicit Euler, and Crank–Nicolson, even with rough initial data. The work relies on maximal inequalities for stochastic convolutions and martingale-based stability analyses, offering a robust framework for temporal discretisation of SPDEs with irregular nonlinearities and noise, and lays groundwork for future locally Lipschitz analyses.
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 $$E_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|_X^p\Big)^{1/p} \to 0\quad (k \to 0),$$ 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 Schrödinger equation, for which previous convergence results were not applicable.