This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

翻译：本文研究非全局Lipschitz系数下半线性随机微分方程不变测度的弱逼近问题。为此，我们提出一种线性theta投影欧拉（LTPE）格式，该格式同样具有不变测度，以处理线性刚度项的潜在影响。在特定假设下，证明随机微分方程及其对应的LTPE方法分别指数收敛于各自的不变测度。此外，借助相应Kolmogorov方程与时间无关的正则性估计，可保证数值不变测度与原不变测度之间的弱误差具有一阶收敛性。在计算复杂度方面，所提出的保遍历性格式对非线性项进行显式处理，相较于文献中保遍历性隐式欧拉方法具有显著优势。数值实验验证了理论结果的正确性。