Abstract:This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[
E_h^\infty := \Big(\mathbb{E}\Big[\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big]\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Moreover, root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schrödinger equation. Further applications to Maxwell's and transport equations are included.
From: Katharina Klioba [view email]
[v1]
Mon, 22 Dec 2025 18:19:45 UTC (53 KB)
[v2]
Thu, 15 Jan 2026 18:09:30 UTC (54 KB)
[v3]
Mon, 2 Feb 2026 16:21:34 UTC (52 KB)
[v4]
Wed, 17 Jun 2026 13:15:33 UTC (58 KB)
[v5]
Wed, 26 Aug 2026 14:27:37 UTC (59 KB)
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