Abstract We discrete the ergodic semilinear stochastic partial differential equations in space dimension d ≤ 3 with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is 1 − ϵ in space and 1 2 − ϵ in time for the space-time white noise case and 2 − ϵ in space and 1 − ϵ in time for the trace class noise case in space dimension d = 1 , with arbitrarily small ϵ > 0 . Numerical results are finally reported to confirm these theoretical findings.

中文翻译：

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抛物线随机偏微分方程不变测度的全离散指数欧拉近似

摘要 我们在空间维度 d ≤ 3 上用加性噪声离散遍历半线性随机偏微分方程，空间上通过谱伽辽金方法，时间上通过指数欧拉方案。结果表明，空间半离散化和时空全离散化都是遍历的。此外，数值不变测度的收敛阶数取决于噪声的规律性，基于简单的时间无关弱误差分析而不依赖于 Malliavin 演算来恢复。准确地说，空间维度 d 中的空间-时间白噪声情况的收敛阶数为 1 − ϵ 空间和时间 1 2 − ϵ，空间维度为 2 − ϵ，空间维度 d 中迹类噪声情况的收敛阶数为 1 − ϵ = 1 ，任意小 ϵ > 0 。最后报告了数值结果以证实这些理论发现。