标题： 时间一致的显式全离散方案对于具有非全局Lipschitz系数的SPDEs的弱误差估计 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$\mathcal{D} \subset \mathbb{R}^d$,$d \leq 3$中演化的随机偏微分方程（SPDEs）的长时间弱逼近，其系数具有非全局Lipschitz和可能非收缩的特性。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格式，这些格式是显式的、易于实现的，并且保留了原耗散SPDEs的遍历性，即使系数可能不是非收缩的。在低正则性和非收缩设置下，仔细分析了时间均匀的弱逼近误差，并获得了时间均匀的弱收敛速率。关键在于在超线性设置下建立所提出的全离散格式的时不变矩界（在$L^{4q-2}$-范数，$q \geq 1$）。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.