Approximating the invariant measure and the expectation of the functionals for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients is an active research area and is far from being well understood. In this article, we study such problem in terms of a full discretization based on the spectral Galerkin method and the temporal implicit Euler scheme. By deriving the a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus, we establish the sharp weak convergence rate of the full discretization. When the SPDE admits a unique V -uniformly ergodic invariant measure, we prove that the invariant measure can be approximated by the full discretization. The key ingredients lie on the time-independent weak convergence analysis and time-independent regularity estimates of the corresponding Kolmogorov equation. Finally, numerical experiments confirm the theoretical findings.

中文翻译：

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具有非全局 Lipschitz 系数的抛物线 SPDE 的完全离散化的弱收敛和不变测度

对具有非全局 Lipschitz 系数的抛物线随机偏微分方程 (SPDE) 的函数的不变测度和期望进行逼近是一个活跃的研究领域，并且远未得到很好的理解。在本文中，我们在基于谱伽辽金方法和时间隐式欧拉方案的完全离散化方面研究了此类问题。通过通过变分方法和 Malliavin 演算导出数值解的先验估计和正则估计，我们建立了完全离散化的锐弱收敛率。当 SPDE 承认唯一的 V -均匀遍历不变测度时，我们证明了该不变测度可以通过完全离散化来近似。关键成分在于相应 Kolmogorov 方程的时间无关弱收敛分析和时间无关正则性估计。最后，数值实验证实了理论发现。