标题： 具有非全局Lipschitz系数的SPDEs显式全离散化方案的一致时间弱误差估计 显示英文标题

标题： Uniform-in-time weak error estimates of explicit full-discretization schemes for SPDEs with non-globally Lipschitz coefficients

摘要： 本文致力于研究在有界域$\mathcal{D} \subset \mathbb{R}^d$,$d \leq 3$中演化的一类具有非全局Lipschitz且可能非收缩系数的随机偏微分方程（SPDEs）的长时间弱逼近。对于所考虑的SPDEs，分析了时空白噪声（$d=1$）和多维迹类噪声（$d=2,3$）。基于谱Galerkin空间半离散化，我们提出了一类新型的指数型全离散化格式，这些格式是显式的、易于实现的，并且保持了原耗散SPDEs的遍历性，即使系数可能不是收缩的。在低正则性和非收缩环境下，仔细分析了时间一致的弱逼近误差，并获得了时间一致的弱收敛速率。关键之处在于在超线性设置中建立所提出的全离散化格式的时一致矩界（在$L^{4q-2}$-范数，$q \geq 1$下）。这对于显式全离散化格式来说是非常不平凡的，通过充分利用半群在$L^{4q-2}$中的收缩性质、非线性的耗散性以及压制策略的特殊优势，提出了新的论证方法。最后报告了数值实验以验证理论结果。 显示英文摘要

摘要： This article is devoted to long-time weak approximations of stochastic partial differential equations (SPDEs) evolving in a bounded domain $\mathcal{D} \subset \mathbb{R}^d$, $d \leq 3$, with non-globally Lipschitz and possibly non-contractive coefficients. Both the space-time white noise ($d=1$) and the trace-class noise in multiple dimensions $d=2,3$ are examined for the considered SPDEs. Based on a spectral Galerkin spatial semi-discretization, we propose a class of novel full-discretization schemes of exponential type, which are explicit, easily implementable and preserve the ergodicity of the original dissipative SPDEs with possibly non-contractive coefficients. The uniform-in-time weak approximation errors are carefully analyzed in a low regularity and non-contractive setting, with uniform-in-time weak convergence rates obtained. A key ingredient is to establish the uniform-in-time moment bounds (in $L^{4q-2}$-norm, $q \geq 1$) for the proposed fully discrete schemes in a super-linear setting. This is highly non-trivial for the explicit full-discretization schemes and new arguments are elaborated by fully exploiting a contractive property of the semi-group in $L^{4q-2}$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy. Numerical experiments are finally reported to verify the theoretical findings.