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 stochastic partial differential equation 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.

中文翻译：

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以随机纳维-斯托克斯方程为范式的 SPDE 数值近似的长期精度


这项工作介绍了一个通用框架，用于建立可分离巴拿赫空间上马尔可夫动力系统近似的长期精度。我们的结果阐明了近似动力学的 Wasserstein 收缩率的一定均匀性对长期精度估计的影响。特别是，我们的方法在 ${\mathbb{R}}^{+}$ 上产生弱一致性界限，同时提供了一种方法来回避常见情况，即某些高阶矩界限对于近似动力学不可用。此外，为了促进我们方法的分析核心，我们对某些“弱哈里斯定理”进行了改进。这一扩展将此类 Wasserstein 收缩估计的适用范围扩展到各种有趣的随机偏微分方程示例，这些示例涉及比现有文献所涵盖的更弱的耗散或更强的非线性。作为一个指导性和范例性的例子，我们将我们的形式应用于随机二维纳维-斯托克斯方程以及该系统的时间和谱伽辽金的半隐式空间数值逼近。在数值近似的情况下，我们对不变测度的近似建立定量估计，并证明 ${\mathbb{R}}^{+}$ 的弱一致性。为了开发这些数值分析结果，我们与现有文献相比，提供了 $L^{2}_{x}$ 精度范围的细化，这些文献是独立感兴趣的结果。