Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
Katharina Klioba, Mark Veraar
TL;DR
This work establishes pathwise uniform convergence rates for time discretisation schemes applied to SPDEs driven by Gaussian noise in non-parabolic settings. By developing a Kato-type framework with two spaces $X$ and $Y$, the authors prove optimal bounds for the uniform strong error $E_k^{\infty}$ for a broad class of contractive time-stepping schemes, including exponential Euler, implicit Euler, and Crank–Nicolson, under additive and multiplicative noise. The results quantify how the convergence rate $\alpha$ on the intermediate space $Y$ governs the grid-point and full-interval errors, with logarithmic factors removable in key cases (e.g., EE under quasi-contractive semigroups). The theory is applied to Schrödinger, Maxwell, and wave equations, yielding unified, improved, and in some cases first results for implicit methods in hyperbolic SPDEs, complemented by numerical experiments that confirm the predicted rates.
Abstract
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly continuous semigroup $S$ on a Hilbert space $X$, and the focus is on non-parabolic problems. The main results are optimal bounds for the uniform strong error $$\mathrm{E}_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|^p\Big)^{1/p},$$ 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 $N_k = T/k$. The usual schemes such as the exponential Euler, the implicit Euler, and the Crank-Nicolson method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show - $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$ (linear equation, additive noise, general $S$); - $\mathrm{E}_{k}^{\infty}\lesssim \sqrt{k} \sqrt{\log(T/k)}$ (nonlinear equation, multiplicative noise, contractive $S$); - $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$ (nonlinear wave equation, multiplicative noise) for a large class of time discretisation schemes. The logarithmic factor can be removed if the exponential Euler method is used with a (quasi)-contractive $S$. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong error $$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}.$$ Applications to Maxwell equations, Schrödinger equations, and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the implicit Euler and the Crank-Nicolson method.