This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on $\mathbb{R}^+$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain `weak Harris theorems'. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting SPDE examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier-Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on $\mathbb{R}^+$. To develop these numerical analysis results, we provide a refinement of $L^2_x$ accuracy bounds in comparison to the existing literature which are results of independent interest.

翻译：本文建立了一个通用框架，用于分析可分离巴拿赫空间上马尔可夫动力系统逼近的长期准确性。我们的研究阐明了逼近动力学中Wasserstein收缩率的某种一致性对长期准确性估计的作用。特别地，该方法在$\mathbb{R}^+$上给出了弱相容性界，同时避免了逼近动力学中某些高阶矩界通常不可用的常见情况。此外，为支撑分析核心，我们改进了某些"弱哈里斯定理"。这一扩展使得Wasserstein收缩估计的适用范围得以涵盖现有文献未涉及的、涉及更弱耗散或更强非线性的多种有趣SPDE实例。作为指导性范例，我们将该形式体系应用于随机二维Navier-Stokes方程及其时间半隐式-空间谱伽辽金数值逼近。在数值逼近情形下，我们建立了不变测度逼近的定量估计，并证明了$\mathbb{R}^+$上的弱相容性。为发展这些数值分析结果，我们改进了现有文献中的$L^2_x$准确性界，这些结果本身具有独立研究价值。