In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild It\^{o} formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions.

翻译：本文针对具有乘性噪声的随机波动方程（特别是双曲安德森模型）的时间离散化，建立了弱收敛速率。对于该类随机偏微分方程，我们所获得的弱收敛速率确为已知强收敛速率的两倍。据我们所知，本研究是科学文献中首次为具有乘性噪声的随机波动方程的时间离散化提供本质上尖锐的弱收敛速率。证明的关键思想在于对误差进行精细分解，并应用近期引入的温和伊藤公式。我们通过Python数值模拟，针对四个不同测试函数下随机偏微分方程弱近似误差的衰减行为，对理论分析结果进行了补充验证。